Time Charter Contracts in the Shipping Industry

A Fair Valuation Perspective

Renathe Elven
MSc Finance

Supervisor: Peter Lchte Jrgensen

Department of Economics and Business

Aarhus University
Business and Social Sciences
August 2013

Abstract
This thesis studies a specific type of asset lease namely time charter contracts with purchase
options. Time charter contracts are common in the shipping industry and give the charterer the
operational control of the vessel leased, whereas the option to purchase the vessel gives the
charterer the right, but not the obligation, to purchase the vessel at the options expiration. The
options embedded in such contracts are often complex in nature such that they are granted for free
rather than for their fair value. The intention of this thesis is to introduce fair valuation of the total
value of time charter contracts with embedded options by introducing two potential models for
valuation purposes. Both models are one-factor models that are assumed to model the main source
of risk in the shipping industry namely the freight rate. The two adopted models ensure the freight
rate to evolve in continuous time, and one of them allows for the derivation of analytic solutions for
some simple freight rate contingent claims. The other model values the vessel underlying the
contract, as well as an embedded European option to buy the vessel by implementing Monte Carlo
simulation.

Acknowledgements
I will like to thank my supervisor, Peter Lchte Jrgensen, for his help and valuable comments during
this period.

Table of Contents
Abstract .................................................................................................................................................... I
Acknowledgements .................................................................................................................................. I
List of Tables ............................................................................................................................................ V
List of Figures........................................................................................................................................... V
1

Introduction ..................................................................................................................................... 1
1.1

Motivation ............................................................................................................................... 1

1.2

Aim........................................................................................................................................... 2

1.3

Structure .................................................................................................................................. 2

Theoretical Framework ................................................................................................................... 3

2.1

Introduction to Important Terms in the Shipping Industry ..................................................... 3

2.1.1

Freight Rates .................................................................................................................... 3

2.1.2

Spot Freight Rates ........................................................................................................... 3

2.1.3

Time Charter Equivalent Spot Freight Rates ................................................................... 4

2.2

The Shipping Industry .............................................................................................................. 4

2.2.1

Agents in the Shipping Industry ...................................................................................... 4

2.2.2

The Different Shipping Segments .................................................................................... 5

2.2.3

Vessels in the Shipping Industry ...................................................................................... 7

2.2.4

The Shipping Market Model ............................................................................................ 7

2.3

Costs in the Shipping Industry ............................................................................................... 13

2.4

Business Risks in Shipping ..................................................................................................... 14

2.4.1

Price Risk........................................................................................................................ 14

2.4.2

Credit Risk ...................................................................................................................... 15

2.4.3

Pure Risk ........................................................................................................................ 15

2.4.4

Summing Up - Analyzing and Managing Freight Rate Risk ............................................ 15

2.5

The Four Shipping Markets ................................................................................................... 16

2.5.1

The Freight Market ........................................................................................................ 16

2.5.2

The Sale and Purchase Market ...................................................................................... 18

2.5.3

The Newbuilding Market ............................................................................................... 18

2.5.4

The Demolition Market ................................................................................................. 18

2.6

Time Charter Contracts with Embedded Options ................................................................. 19

2.7

Background for the Models Selected .................................................................................... 21

II

2.8

The Dynamics of Freight Rates .............................................................................................. 24

2.8.1

Shipping Market Cycles ................................................................................................. 24

2.8.2

Freight Rate Dynamics ................................................................................................... 24

2.9

The Ornstein-Uhlenbeck Process .......................................................................................... 25

2.9.1
2.10

The Geometric Mean Reversion Process .............................................................................. 31

2.10.1
3

The Solution to the Ornstein-Uhlenbeck Process ......................................................... 28

The Solution to the Geometric Mean Reversion Process .............................................. 32

Analysis Section ............................................................................................................................. 35

3.1

Two Models Different Characteristics: A Comparison ....................................................... 36

3.2

Applications of the Ornstein-Uhlenbeck Process: Introducing Valuation of Freight Rate

Contingent Claims ............................................................................................................................. 38

3.2.1
3.3

Derivation of the Fundamental Partial Differential Equation ....................................... 39

Valuation of some simple Freight Rate Contingent Claims ................................................... 44

3.3.1

Claim to Receive Spot Freight Rate Flow from Time to Time .................................. 44

3.3.2

Fixed for Floating Freight Rate Swap ............................................................................. 46

3.3.3

The Value of a Vessel ..................................................................................................... 49

3.4

European Option to Buy a Vessel .......................................................................................... 52

3.5

Applications of the Geometric Mean Reversion Process: Vessel and European Option

Valuation ........................................................................................................................................... 55
3.5.1

Monte Carlo Simulation ................................................................................................ 56

3.5.2

The Value of a Vessel ..................................................................................................... 56

3.5.3

European Option to Buy a Vessel .................................................................................. 59

3.6
4

The Valuation Results: Comparisons ..................................................................................... 61

Limitations ..................................................................................................................................... 62
4.1

Limitations Caused by the Models Selected.......................................................................... 62

4.1.1

The Parametric Property of the Models ........................................................................ 62

4.1.2

One-Factor ..................................................................................................................... 63

4.2

The Assumptions ................................................................................................................... 64

4.2.1

Constant Market Price of Freight Rate Risk................................................................... 64

4.2.2

Constant Risk-Free Interest Rate ................................................................................... 65

Summary and Conclusions ............................................................................................................ 66

List of References .......................................................................................................................... 68

III

Appendix........................................................................................................................................ 70
7.1

The Ornstein-Uhlenbeck Process Detailed Solution........................................................... 70

7.2

The Ornstein-Uhlenbeck process - Derivation of the Mean and the Variance ..................... 72

7.2.1

The Time Conditional Mean ........................................................................................ 72

7.2.2

The Time Conditional Variance ................................................................................... 73

7.3

The Geometric Mean Reversion Process Detailed Solution ............................................... 74

7.4

The Ornstein-Uhlenbeck Process Claim to Receive Spot Freight Rate Flow from Time to

Time 77
7.5

The Ornstein-Uhlenbeck Process - European Option to Buy the Vessel ............................... 80

7.6

The VBA Codes ...................................................................................................................... 84

IV

List of Tables
Table 1: The variables in the shipping market model ..............................................................................8
Table 2: Base case parameter values .................................................................................................... 35
Table 3: The dependence of fair time charter rates.............................................................................. 47
Table 4: The value of a 5-year time charter contract ............................................................................ 49
Table 5: The dependence of vessel values ............................................................................................ 51
Table 6: Value of European option to buy a vessel ............................................................................... 54
Table 7: The value of a 5-year time charter contract with European purchase option ........................ 55
Table 8: The dependence of vessel values ............................................................................................ 58
Table 9: Value of European option to buy a vessel ............................................................................... 60

List of Figures
Figure 1: Simulated spot freight rate from the Ornstein-Uhlenbeck process....................................... 36
Figure 2: Simulated spot freight rate from the Geometric Mean Reversion process ........................... 37
Figure 3: Simulated and expected value of a vessel.............................................................................. 52

1 Introduction
Through the last decade, the shipping industry has been subject to extreme volatility in freight rates.
Over the period from 2003 to mid-2008, freight rates increased by almost 300 per cent to
exceptional levels. This large increase in freight rates was followed by a corresponding drop of 95 per
cent over the last quarter of 2008. Such high volatility in the market also implies a shipping industry
that is extremely risky. Therefore, the last decades market fluctuations have changed the way the
shipping industry views and manages its risks, and accordingly the derivatives market for freight have
accelerated and a commoditization of the freight market is present. Today, agents that may not be
involved in the underlying physical market, such that investment banks, hedge funds and other
traders, can be seen participating in the shipping industry.

1.1 Motivation
Thus, the accelerated derivatives market for freight has opened up possibilities for hedging and
managing risk stemming from large volatility in freight rates. This leads to the topic of this thesis,
namely fair valuation of time charter contracts with embedded options. Time charter contracts are
common contractual agreements in the shipping industry which will be described in detail later.
Options embedded in time charter contracts are wildly used and can be considered as a tool used to
hedge against freight rate risk, this will also be described in detail later. More specific, the options
which will be examined and valued here are call options which enable the charterer to purchase the
vessel either during the contract period, or at the end of the contract period. The options serve as an
insurance against undesirable movements in freight rates.
Time charter contracts with purchase options are interesting from both academic and practical
business management perspectives as they can be very complex and of significant economic
importance. Jrgensen and Giovanni (2010) mention that they are aware of several shipping
companies having a total net asset value where more than half of it stems from an estimated value of
their portfolio of time charter contracts with purchase options. Thus, properly valuation of these
contracts is extremely important in order to both support the stock markets valuation of shipping
companies, and in order to assist managers of such companies in the general process of operation
and risk management of their companies. High volatility in freight rates may create either a
significant decrease in a shipping companys total reported net asset value or a significant increase in
its total net asset value.

The complex nature of time charter contracts with purchase options makes fair valuation a difficult
task. The need for development and analysis of good valuation models are therefore increasingly
important. According to Alizadeh and Nomikos (2009), embedded options in the shipping industry
are very often granted for free or for a nominal fee without being properly valued.

1.2 Aim
Therefore, this thesis aims to shed light on the importance of fair valuation of time charter contracts
with embedded options, as well as valuations of such contracts and a comparison between models.
Two models will be introduced for valuation purposes and their valuation abilities will be compared.
The model that will be used to obtain total values of a time charter contracts with embedded
purchase options is adopted from Jrgensen and Giovanni (2010). The other model is adopted from
Tvedt (1997) and will be introduced as an alternative model for valuations to the one from Jrgensen
and Giovanni (2010).

1.3 Structure
This thesis is divided into three main parts; a theoretical framework, an analysis section and a
discussion section where limitations are elucidated.
First, a theoretical framework of the shipping industry will be established. A fundamental
introduction of the comprehensiveness of the shipping industry will be given, where the shipping
market model will be emphasized. This simplified model describes the mechanisms that make freight
rates evolve in accordance with the market cycles. Further, the time charter contracts with
embedded options in the shipping industry will be described, as well as the background for the
model selection. Finally, the dynamics of freight rates will be described forming the basis for the two
models selected before the theoretical framework ends by a detailed description of these models,
and for that purpose also a derivation of the two processes solution will be done.
As an introduction to the analysis section the two models characteristics will be compared and
discussed. Evidences of one model being more appropriate in freight rate modeling will be
presented. Further, derivations will be done in order to value time charter contracts with purchase
options. To a greater or lesser extent, both models will be applied for valuations.
At the end, some of the limitations regarding the two models will be presented before the thesis will
be summarized and concluded.
2

2 Theoretical Framework
This section will first give a basic understanding of the shipping industry before it continues with a
description of time charter contracts with embedded options. Further, the background that clarifies
the reasons for the two models chosen will be presented, as well as a description of the freight rate
dynamics. Finally, the two models adopted for the valuations will be presented in detail in which the
solutions to both of them also will be derived.

2.1 Introduction to Important Terms in the Shipping Industry

To prevent confusions, some relevant terms in relation to the shipping industry will be described in
this first section of the theoretical framework. This will serve as a soft introduction to the
comprehensive shipping industry.

2.1.1

Freight Rates

Freight rates represent the cost of providing the service of seaborne transportation (Alizadeh and
Nomikos 2009). Hence, freight rates are not tangible assets and can therefore not be stored.
Kavussanos and Visvikis (2006) do also stress this special feature of freight services where they
describe the demand for freight services as a derived demand. This is due to the fact that the freight
service provided by the vessel is gone if it is not utilized at the time it is available. Again, the freight
service is non-storable and it cannot be carried forward in time.
Freight rates evolve through time according to market cycles that are prevalent in the shipping
industry. These will be described later.

2.1.2

Spot Freight Rates

The spot freight rate represents the cost of providing seaborne transportation today. It reflects the
continuous balance between supply and demand for shipping services (Alizadeh and Nomikos 2009).
Factors determining supply and demand will be described in Section 2.2.4 where the freight rate
mechanism will be examined.

2.1.3

Time Charter Equivalent Spot Freight Rates

The time charter equivalent spot freight rate represents the spot freight rate less the voyage costs
(Tvedt 1997). During this thesis, it is actually the time charter equivalent spot freight rate that is
applied in the derivations. For reasons explained later, applying the time charter equivalent spot
freight rate is in fact beneficial when valuing time charter contracts with purchase options.
What is already important to note is that market-quoted spot freight rates embed varying degree of
costs, and thus the time charter equivalent spot freight rates will not be directly comparable with the
market-quoted spot freight rates. This is important to have in mind when comparing with market
data.

2.2 The Shipping Industry

This section will give an introduction to the shipping industry that plays a central role in the global
economy and has also been at the forefront of global development through times. The main assets in
the shipping industry are the vessels that can transport cargo from one part of the world to another.
Already, it is worth mentioning that competition is an important key word in relation to this
intriguing industry, and as for the bulk shipping segment that this thesis aim to address, the condition
of perfect competition is present. When nothing else is stated, the whole section will be based on
Stopford (2009).

2.2.1

Agents in the Shipping Industry

In the following, a description of the most important agents participating in the shipping industry will
be given. Hence, some agents will be left outside this explanation since they are less important
according to the aim of this thesis. The groups that will be explained here are the shipowners, the
charterers and the shipbrokers.
The shipowner has vessels for hire and enters the market with a vessel available free of cargo. The
vessel has particular characterizations and the vessels availability will be described in the contractual
agreements. The shipowner may be looking for a short charter for the vessel or a long charter, in
which will be dependent on the shipowners strategy.
The charterer can be either an individual or an organization; common to both of them is that they
have a volume of cargo they need to transport from one location to another. The vessel type needed
will be determined by the physical characteristics of the cargo.
4

The shipbroker operates as an intermediary between shipowners and charterers. According to who
have a need, the shipbroker will be contacted and his task is to discover what cargoes or vessels are
available, what expectations the shipowners/charterers have about what they will be paid or pay,
and what is reasonable given the state of the market. The deal for their client is negotiated, and
often in tense competition with other brokers.

2.2.2

The Different Shipping Segments

Three different segments constitute the shipping industry; these are the liner shipping segment, the
bulk shipping segment and the specialized shipping segment. This clearly division of the industry is
necessary in order to meet specific needs of different customers; anything from grain to cars can be
transported by sea. Shipping companies characteristics differ depending on which segment they
operate in, this is due to differences in both the transported cargo, and in the dynamics of how the
agreement between the shipowner and the charterer is conducted. The characteristics of each
shipping segment will be described below. Despite the differences in shipping companies
characteristics across these three segments, one shipping company can often operate in more than
one segment and thus contribute to intense competition for the same cargo. Therefore, it is not
convenient to treat the shipping industry as a series of isolated segments, but rather as a single
market. Investors in the shipping industry move their investments from one segment to another, and
if there is a supply-demand imbalance in one of the segments, this will also move on to the other
segments.
As this thesis aims to address the bulk shipping segment, liner shipping and specialized shipping will
only shortly be described, whereas the bulk shipping segment will be examined in more detail.
The liner shipping segment offers transport for cargoes that are too small to fill a single vessel and
thus need to be grouped with others for transportation. Such cargoes are often highly valued and can
be delicate in nature; the shipper thus often requires a special shipping service with a fixed tariff
rather than a fluctuating market rate. Cargoes transported in the liner shipping segment are called
general cargo and include loose cargo, containers and pallets. This creates complex administrative
tasks and makes this segment very different from the bulk shipping segment. The specialized shipping
segment is recognized by properties lying in between the liner shipping segment and the bulk
shipping segment. This leads to a somewhat indefinite distinction between the specialized shipping
segment and the two other segments. Cargoes that are considered special include cars, forest
products, chemicals and refrigerated products.

The bulk shipping segment supplies transport for cargoes that need to be transported in large
homogeneous shiploads. Generally a whole vessel is hired for transportation of one type of cargo,
but it is also possible to carry different bulk cargoes in a single vessel. If so, each cargo occupies a
separate hold or possibly even part of a hold. Commodities that often need to be transported in
bulks are raw materials and bulky semi-manufactures. The bulk shipping segment is divided into dry
bulk and liquid bulk. Dry bulk cargo transported in shiploads is mainly raw materials such as iron ore,
coal and grain. Liquid bulk cargo includes crude oil, oil products, and liquid chemicals. Cost
minimization of providing safe transport, as well as effective management of vessel investments is
the main focus within the bulk shipping business. Costs can be minimized due to the characteristics
of the bulk segment in which few transactions are handled. Typically, a vessel completes about six
voyages with a single cargo each year.
As already mentioned, the bulk shipping segment is subject to conditions of perfect competition. This
creates rather volatile freight rates and prices (Kavussanos and Visvikis 2006). Perfect competition
arises due to the many buyers and sellers of freight services. They negotiate on a relatively
homogenous product, the freight service, and have no barriers to entry or exit the market. In
addition, the freight markets are well organized markets. The product can be assumed to be almost
perfectly homogenous since, in a particular route-trade, there are no significant differences in the
quality of the freight service offered. The fact that the product is almost perfectly homogenous
relates to some minor differences in vessel characteristics and customer relations that may exist
between shipping companies.
Another contribution to the condition of perfect competition is the availability of information in the
freight markets. The Baltic Exchange, among others, brings together participants wishing to buy or
sell the freight service. Relevant information on fixtures1, prices, and cargoes/vessels available are
collected and disseminated to the market. Such information is used by shipowners and charterers to
make informed decisions in the freight markets (Kavussanos and Visvikis 2006). The feature of
perfect competition arising in the bulk shipping segment, resulting in volatile freight rates, makes it
especially interesting to address this particular segment; investors and charterers are to a great
extent exposed to risk, and proper risk management becomes extremely important.
In summary, these three shipping segments face different tasks depending on the value and volume
of cargo, the number of transactions handled, and the commercial systems employed.

Stopford (2009) explains a fixture as an agreement where a vessel is chartered or a freight rate is agreed on.
Further, he explains that the arrangement happens in much the same way as any major international hiring or
subcontracting operation.

2.2.3

Vessels in the Shipping Industry

In order to give a comprehensive picture of the shipping industry a description of different vessel
types is, at this point, appropriate. Due to the differences in cargoes transported by sea, both within
each segment and also across the segments, vessels are built and adjusted to fit the cargoes they
transport. The result is several different vessel types and sizes in the world fleet2. However, this
section will only give a presentation of the different vessel types in the bulk shipping segment.
There exist four different types of vessels in the dry bulk sector where each of them are classified by
their size measured in dead weight tons (dwt). Handy bulk carriers are the smallest ones, those
vessels are of 10 000 to 40 000 dwt. Handymax bulk carriers are of 40 000 to 60 000 dwt, Panamax
are of 60 000 to 100 000 dwt, and Capesize are the largest vessels of over 100 000 dwt. As any other
physical asset vessels do also have a limited lifetime. The shipping industry is in continuous
technological progress and the vessels operating in the industry therefore suffer from obsoleteness
rather fast. Logically, as a vessel grows old or gets obsolete its value will also decrease, this reduction
in value continues until the vessels age lies between 20 and 30 years which is when it is normally
scrapped.
According to Kavussanos and Visvikis (2006), a useful tool to understand why different vessel sizes
are necessary is the Parcel Size Distribution (PSD) of each commodity. A parcel is an individual
consignment of cargo for shipment, each commodity can be transported in different parcel sizes, and
these different parcel sizes constitute the Parcel Size Distribution. Based on the observation that
some commodities are typically moved in larger sizes than others, the PSD function describes the
range of parcel sizes in which that commodity is transported. In addition to the parcel size
distribution, port and seaway restrictions have created differences in types and sizes of vessels.

2.2.4

The Shipping Market Model

To understand the freight market and how the freight market cycles are generated the mechanisms
determining spot freight rates will be explained next. These are the economic mechanisms that the
shipping industry uses to regulate supply and demand.
The ten most important economic variables determining supply and demand are collected in a
simplified model called The Shipping Market Model. Ten variables which have an influence on the
shipping market are listed and the purpose is to leave out less relevant details in order to create a

All existing vessels.

picture of how the spot freight rates are determined. Five of the variables influence demand in the
shipping market, and the other five influences supply. The variables are listed in Table 1 below.

Demand

Supply

1. The World Economy

1. The World Fleet

2. Seaborne Commodity Trades

2. Fleet Productivity

3. Average Haul

3. Shipbuilding Production

4. Random Shocks

4. Scrapping and Losses

5. Transport Costs

5. Freight Revenue

Table 1. Ten variables in the shipping market model. Source: Stopford (2009), page 136.

First, the demand for sea transport will be examined. According to Kavussanos and Visvikis (2006),
demand for freight services is a derived demand; the charterers demand is not for the vessel, but for
the service the vessel provides.

2.2.4.1 The World Economy

The world economy is considered as the most important single influence on vessel demand. This is
due to the world economy generating most of the demand for sea transport. Events in the world
economy that generate demand for sea transport is import of raw materials for the manufacturing
industry, and the trade in manufactured products. The relationship between the world industry and
the demand for sea transport is complex and consists of two aspects of the world economy; the
business cycle and the trade development cycle. These two aspects may create changes in the
demand for sea transport.
The business cycle has an important influence on the demand for sea transport in the short-term.
Fluctuations in the world economy are directly transferred to the shipping market. In this way, the
foundation for the cyclical behavior of freight rates is set.
The trade development cycle is related to the long-term relationship between sea transport and the
world economy. It says something about the speed of industrial growth relative to the speed of sea
trade growth. The sea trade growth of individual regions will change as time goes by due to both the

change in a countrys economic structure, and due to the ability of local resources of food and raw
materials to meet local demand.

2.2.4.2 Seaborne Commodity Trades

This variable explains the relationship between sea trade and the industrial economy both in the
short-term and in the long-term.
Short-term volatility arises due to seasonality in some trades. An example of such a trade is grain,
which is subject to seasonal variations caused by harvests. Due to the difficulties in planning
transportation of seasonal agricultural commodities, shippers rely heavily on the spot charter market
when demand for sea transport arises. Thus, fluctuations in the grain market have larger impact on
the spot charter market than other trades, like for example iron ore. Tonnage requirements in the
transportation of iron ore are almost always met through long-term contracts.
Demand affected by long-term trends in commodity trade is best identified by studying the economic
characteristics of the industries that produce and consume the traded commodities. Overall, there
are four types of changes that affect the demand for seaborne transport in the long-run; changes in
the demand for that particular commodity, changes in the source from which supplies of the
commodity are obtained, changes due to a relocation of processing plant changing the trade pattern,
and changes in the shippers transport policy.
Changes in demand for that particular commodity may have an effect on the tonnage requirements if
this particular commodity is imported. If, for example, the country decides to replace the imported
commodity by a domestic commodity, this will naturally affect the demand for seaborne transport.
Changes in the source from which supplies of the commodity are obtained happen when new sources
are discovered. These new sources may happen to be located near countries that earlier imported
the same commodity, this may result in imports of this commodity being redundant. Thus, the
demand for sea transport is changed.
Changes due to relocation applies to industrial raw materials, and may affect both the volume of
cargo transported by sea, and the type of vessel used to transport this cargo. Raw materials are often
transformed several times before the final product is made. If the transformation is done before
it is shipped rather than after, the volume and characteristics of the vessel may be changed.
Changes in the shippers transport policy relates to, for example, switching between using long-term
contracts and using the spot charter market. This will again affect the demand for sea transport.
9

2.2.4.3 Average Haul

Cargo shipped over larger distances generates more demand for sea transport than cargo shipped
over shorter distances. Therefore, demand is affected by the length of where the cargo is shipped.
The demand of sea transport is measured in ton miles making sure that the distance effect is taken
account for. Ton miles are defined by the tonnage of cargo shipped, multiplied by the average
distance over which it is transported.

2.2.4.4 Random Shocks

Random shocks can have major impact on the economic system, and in turn affect the cyclical
process. Random shocks of more or less severity include weather changes, wars, new resources and
commodity price changes. Economic shocks do often have the most important influence on the
shipping market, the reason why is that the timing is usually unpredictable and they bring about a
sudden and unexpected change in vessel demand.
Political events often have an indirect effect on vessel demand. Examples of such events are a
localized war, a revolution or strikes.

2.2.4.5 Transport Costs

Raw materials will only be transported from other destinations around the world if the
transportation costs are at a relatively low level, or if the quality of a product can be increased to a
level that gives major benefits.
Improved efficiency, bigger vessels and more effective organization of the shipping operation result
in reduced transport costs and higher quality of service. Thus, also the amount of seaborne transport
increases.

Even though these five factors influencing demand for sea transport are a simplified picture of
reality, they give an indication of the complex nature of seaborne demand. On the other hand, the
supply for sea transport is quite different in nature as also will be seen in the following.
The supply for seaborne transport is characterized as being slow in its adaptation to changes in
demand. This is due to the time-lag created in response to an increase in demand; several years are
needed for the completion of a new vessel. Responding to a decrease in demand is also a slow
10

process, once a vessel is built it is estimated to have a lifetime of 15-30 years (Stopford 2009).
Therefore, if a large surplus is to be removed, that process can take several years resulting in a timelag between the decreased demand and the surplus reduction.

2.2.4.6 The World Fleet

This variable measures the total amount of vessels existing all over the world. Scrapping old vessels
and deliveries of new vessels determine the rate of fleet growth, which can be positive or negative.
The world fleet contains all the different vessel types and sizes. Since a new vessel is estimated to
have a lifetime of 25 years on average only a few vessels are scrapped each year. The pace of
adjustments to changes in the market is therefore measured in years.
When demand for seaborne transport does not turn out as expected, supply is adjusted. This is the
key feature of the shipping market model (Stopford 2009).

2.2.4.7 Fleet Productivity

The productivity of the vessels that constitute the world fleet can vary. This creates a flexibility
element since a vessel often has several days where it does not transport cargo. Such ineffective
activities include ballast time, cargo handling, incidents, repair, lay-up, waiting, short-term storage
and long-term storage.
The fleet productivity is measured in ton miles per dead weight ton and depends upon four main
factors; speed, port time, deadweight utilization and loaded days at sea.
Speed is measured by the time a vessel uses on a voyage. New vessels are often designed to go
faster, but this reduces the transport capacity of the vessel. Also, older vessels are often subject to
hull fouling which will reduce the maximum operating speed. Port time relates to the time a vessel is
at port. Factors that determine the efficiency at port include the physical performance of the vessels
and terminals, and the organization of the transport operation. The deadweight utilization measures
how much of the total cargo capacity that is lost due to bunkers, stores, etc. Loaded days at sea are
the time where the vessel actually transports cargo at sea. It is desirable to increase loaded days at
sea to improve the efficiency of the world fleet.

11

2.2.4.8 Shipbuilding Production

Shipbuilding is an important adjustment factor to the world fleet. In times of increased demand,
shipbuilding can increase the world fleet to meet the demand required. But building new vessels is a
lengthy process, therefore, when deciding to build a new vessel it is important that the need for this
vessel in the future is identified. It can be difficult to predict future demand, and if the prediction
turns out to be wrong, this can lead to excess of vessels in relation to required demand.

2.2.4.9 Scrapping and Losses

The balance between delivery of new vessels and scrapping of old ones (or losses) determines the
growth rate of the world fleet. A new vessel is estimated to have a lifetime between 15-30 years,
indicating the difficulty in estimating exactly when the vessel is to be scrapped. The reason why is
that scrapping depends on the balance of a number of factors; age, technical obsolescence, scrap
prices, current earnings and market expectations. These factors create some flexibility to the
shipowner in deciding when a vessel is to be scrapped.

2.2.4.10 Freight Revenue

The freight rate is the most important regulator of the supply of sea transport. Freight rates are used
by the market to motivate decision-makers to adjust capacity in the short-term, and to find ways of
reducing their costs in the long-term.
The shipping industry consists of two main pricing regimes; the freight market and the liner market.
As explained above, the liner market can be thought of as a retail shipping business; transport of
cargo in small quantities is offered to many customers. The freight market (bulk shipping) is totally
different, this can be thought of as a wholesale operation; transport of cargo in shiploads is offered
to few customers at individually negotiated prices.
In the short-term, supply is adjusted in response to prices by changing the vessels operation speed
and move to and from lay up. In the long-term, investment decisions such as ordering new vessels
and scrapping old ones, are heavily influenced by the freight rate. The supply and demand
adjustment mechanism will be explained in more detail in Section 2.8 where freight rate dynamics
are examined.

12

2.2.4.11 Summing Up
According to Tvedt (1997) it is usual to assume that demand is quite inelastic to freight rates. The
shipping industry is characterized by large scale operations, and the cost of transportation at sea is a
minor share of the total oil price. Therefore, only to a very small extent, demand is supposed to
depend on freight rates. Further, he points out that the supply can be quite inelastic to freight rates
in the short run when there are no vessels available. The reason is that speed and efficiency in
loading and discharging only to a limited degree can be increased. On the other hand, when freight
rates have been low for a while many vessels may have been laid up. This makes it possible to
increase short run supply by re-entering vessels that are laid up.

2.3 Costs in the Shipping Industry

In this section, costs in the shipping industry will shortly be described. They are divided into four
categories; capital costs, operation costs, voyage costs, and cargo-handling costs. The whole section
will be based on Alizadeh and Nomikos (2009).
Capital costs are related to interest payments and capital repayments. These costs depend on how
the shipowner or the shipping company has financed their vessel purchases, and on the interest rate
level. Fleet financing can take several forms, some of which include full equity, bank loans, bonds,
public offerings and private placements. Shipping companies with high operational and financial
capabilities may enjoy better financial agreements than shipping companies with relatively lower
levels of credit and collateral. Operating costs are costs arising from the day-to-day running of the
vessel. These costs are generally in the responsibility of the shipowner, and incur whether the vessel
is active or idle. Operating costs include, among others, crew wages, stores and provisions,
maintenance, and insurance. Such costs do not vary over time, but they grow at a constant rate
normally in line with inflation. Voyage costs are costs related to a specific voyage. These costs include
fuel costs, port charges, pilotage and canal dues. The specific voyage undertaken, and the type and
size of the vessel are factors that decide the level of costs. Cargo-handling costs arise from the
loading, stowage, lightering and discharging of the cargo.

13

2.4 Business Risks in Shipping

The shipping industry is considered as one of the most volatile industries where participants in the
markets are exposed to substantial financial and business risks. Fluctuations in freight rates, bunker
prices, vessel prices, and even from fluctuations in the level of interest rates and exchange rates are
all reasons why this is an extremely risky industry (Alizadeh and Nomikos 2009). All these factors
have an impact on the cash flows of shipping investment and operations, thus they also influence the
profitability of shipping companies as well as their business viability. Alizadeh and Nomikos (2009)
divide business risk in shipping into three categories; price risk, credit risk and pure risk. In the
following, these will be described and risk stemming from fluctuations in freight rates will be relied
most weight.

2.4.1

Price Risk

Price risk refer to a shipping companys costs and earnings which is uncertain and outside of direct
control of the shipping company. The first source of price risk is freight rate risk which refers to the
variability in earnings of a shipping company due to changes in freight rates. Alizadeh and Nomikos
(2009) argue that this may be the most important source of risk for a shipping company due to the
direct impact volatility in freight rates have on the profitability of the company. The management of
risk arising from freight rates will be described after the other types of business risk in shipping have
been presented.
The second source of price risk is the operating-costs risk which refer to volatility in a shipping
companys costs. Sharp and unanticipated changes in, for example, bunker prices will have a major
impact on the operating profitability of shipping companies and vessel operators. The third source of
price risk is the risk arising from exposure to changes in interest-rates. Most vessels in the shipping
industry are financed through term loans priced on a floating rate basis, thus unanticipated changes
in interest rates may create cash flow and liquidity problems for companies which may no longer be
able to service their debt obligations. The fourth and last source of price risk is asset-price risk arising
from fluctuations in the price of the assets of the companies. In the shipping industry, vessels are the
most important assets, they are often used as collateral in vessel-finance transactions and a
reduction in vessel value may therefore affect the creditworthiness of a shipowner and its ability to
service debt obligations. Volatility in vessel values will also affect a shipping companys balance
sheet.

14

2.4.2

Credit Risk

Credit risk refers to counter-parties to transactions and their ability to perform their financial
obligations in full and on time. Therefore, credit risk is also known as counter-party risk. In the
shipping industry most of the deals, trades and contracts are negotiated directly between the
counterparties, their trust and commitment to honor the agreement therefore becomes extremely
important.

2.4.3

Pure Risk

Pure risk relates to a decrease in the value of the shipping companys assets due to physical damage,
accidents and losses. Also, risk of loss due to physical risks, technical failure and human error in the
operation of the assets of a company are covered. In addition, the risk of legal liability for damages as
a result of actions of the company is covered.

2.4.4

Summing Up - Analyzing and Managing Freight Rate Risk

For the shipowner to be able to analyze the freight rate risks which he is exposed to, it is convenient
to consider the vessels as investments as assets in portfolios (Kavussanos and Visvikis 2006).
Through the freight services that vessels offer to charterers a stream of income is generated and the
level of this income is dependent on the freight rate level at each point in time. Also, capital
gains/losses created by selling the vessels at a price higher/lower than what they were bought at is
part of the shipowners investment strategy. To manage the risks arising from freight rates
Kavussanos and Visvikis (2006) point out the use of financial derivatives. They explain that financial
derivatives have been used in the shipping industry since 1985, but also that the popularity of these
derivatives are far less popular than those available in other sectors of the economy. The time
charter contracts with purchase options, which this thesis aims to analyze and valuate, is an
instrument used to protect shipowners and charterers from risk arising from fluctuations in freight
rates.

15

2.5 The Four Shipping Markets

Within the shipping industry, markets play an extremely important role in the operation of the
international sea transport. Stopford (2009) mentions the nineteenth-century economist, Jevons,
who provided a definition of a market where the basic principles is still very suitable to the shipping
industry. The definition is quoted below.
Originally a market was a public place in a town where provisions and other objects were
exposed for sale; but the world has been generalized, so as to mean any body of persons who
are in intimate business relations and carry on extensive transactions in any commodity. A
great city may contain as many markets as there are important branches of trade, and these
markets may or may not be localized. The central point of a market is the central exchange,
mart or auction rooms where traders agree to meet and transact business But this
distinction of locality is not necessary. The traders may be spread over a whole town, or
region or country and yet make a market if they are in close communication with each
other. (Jevons, 1871, Ch. IV)
Within the shipping industry there exist four different markets trading in different commodities. The
freight market trades in sea transport, the sale and purchase market trades second-hand vessels, the
newbuilding market trades new vessels, and the demolition market deals in vessels for scrapping.
Since this thesis aims to model the valuation of contracts on sea transport, it is the freight market
that is considered, and thus the freight market will be relied most weight in the following. However,
for completeness the main characteristics of the three other markets will also be described. The
whole section is based on Stopford (2009).

2.5.1

The Freight Market

The freight market is the marketplace where sea transport is bought and sold. Within the freight
market different sectors are developed in order to support the different vessel types. The freight
rates within each sector often behave quite differently from each other in the short term, but since it
is the same broad group of agents participating in the shipping industry, what happens in one sector
eventually ripples through into the others. There exist two different types of transactions in the
freight market; the freight contract and the time charter. The freight contract is a fixed type of
contract where the shipper buys transport from the shipowner at a fixed price per ton of cargo, and
is used by shippers who prefer to pay an agreed sum and leave the management of the transport to
the shipowner. On the other hand, the time charter contract is based on the spot freight market, and

16

the vessel is hired by the day. Experienced vessel operators who prefer to manage the transport
themselves are the users of time charter contracts.
Four different contractual agreements are used in the freight market; the voyage charter, the
contract of affreightment, the time charter, and the bare boat charter. A voyage charter agrees on a
fixed cargo price, measured in price per ton. In a contract of affreightment, the shipowner agrees to
transport a series of cargo parcels for a fixed price per ton. These series of cargo parcels are agreed
to be transported within a fixed time interval, for example within two months. The details of each
voyage are in the concern of the shipowner. The vessels are then used in an efficient manner by,
among others, switching cargo between vessels and arrange backhaul cargoes.
The time charter contract takes a step further and gives the charterer the operational control of the
vessels that carry the cargo. When the charterer has the operational control he instructs the master
where to go and what cargo to load and discharge. Responsibilities left for the shipowner are
ownership and management of the vessel. The length of the charter can vary from the time taken to
complete a single voyage, to a period of months or years. When a vessel is chartered, the shipowner
continues to pay the operating costs of the vessel. Operating costs include crew, maintenance and
repair. Commercial operations, voyage expenses3 and cargo handling costs are left to the charterer.
Since time charters hand over the voyage costs to the charterer it is convenient to apply the time
charter equivalent spot freight rate when agreeing upon a time charter contract. By subtracting the
voyage costs from the spot freight rate time charter rates will reflect the net freight earnings through
shipping operations (Alizadeh and Nomikos 2009). Therefore, when valuing time charter contracts
with purchase options later on, the time charter equivalent spot freight rate will be applied.
A bare boat charter can be arranged if the charterer wishes to have full operational control of the
vessel without owning it. The owner of the vessel does not need to be a professional shipowner, it
can also be an investor buying a vessel, and then entering into a bare boat charter. The charter
period usually spans from ten to twenty years. Management of the vessel and operating and voyage
costs is in the charterers responsibility.

Voyage expenses include bunkers, port charges and canal dues (Stopford 2009).

17

2.5.2

The Sale and Purchase Market

In the sale and purchase market second-hand vessels are traded with high intensity. Generally, the
sale and purchase transactions are carried out through shipbrokers who have been instructed by the
shipowner to find a buyer for the vessel. Most commonly, competition is created by offering the
vessel through several broking companies. The sale and purchase market generates price volatility,
and asset play4 can result in high profits being an important source of income for shipping
investors.

2.5.3

The Newbuilding Market

As its name indicates, the newbuilding market trades vessels that are not yet built. For the building
process, specifications of the vessel must be decided. The shipyard generally has their own standard
designs, and it is therefore desirable that the buyer choose one of those. This will ease the
negotiation process, the pressure on design and estimating resources will be reduced, and the
shipyards standard designs are normally cheaper to build than a customized design.

2.5.4

The Demolition Market

The demolition market has similarities to the second-hand market, but now it is the scrap yards that
are the customers instead of the shipowners. When a shipowner is not able to sell his vessel in the
second-hand market, he offers it on the demolition market. Also here, a broker generally handles the
sale. Scrap values are determined by negotiation and depend on the availability of vessels for scrap
and the demand for scrap metal.

With the four shipping markets described, and with the different contractual agreements in the
freight market in hand, it is time to move on to the primary theme of this thesis; namely time charter
contracts and their embedded options.

Described by Stopford (2009) as well-timed buying and selling the vessels.

18

2.6 Time Charter Contracts with Embedded Options

This section will give an understanding of the various options that often are embedded in time
charter contracts in the shipping industry. Some of them are quite complex in nature, and for
valuation purposes advanced numerical methods are necessary. Therefore, this section serves as an
introduction to both simple options and more complex options, whereas simple European options
will be valued later on.
Time charter contracts with embedded options are common in the shipping industry. When agreeing
upon a time charter, options to extend the lease period and options to buy the vessel are often
embedded in the lease contract. The option to extend the lease period makes it possible for the
charterer to lengthen the life of the contract period (Hull 2012). The option to buy the vessel is a so
called purchase option (call-option) and gives the charterer the opportunity to buy the vessel at a
predetermined price. These options serve as an insurance for the charterer against undesirable
movements in the freight rate level over the contract period as he can terminate the contract by
purchasing the vessel.
Options embedded in contracts in the shipping industry are real options. This is options on physical
assets (Hull 2012), such as the vessels in the shipping industry. According to Alizadeh and Nomikos
(2009) the underlying assets of real options are cash flows affected by managerial decisions. In the
shipping industry, owning a vessel results in cash flows when operating it in the freight market; by
offering seaborne transport the shipowner captures the freight earnings.
Jrgensen and Giovanni (2010) point out that these embedded options can have more or less
complex properties, in addition to being of significant economic value. This makes such contracts
interesting from both academic and practical business management perspectives. Due to the
economic significance of such contracts, the need for development and analysis of good valuation
models is increasing. Good developed valuation models will support the stock markets valuation of
shipping companies and assist managers in the general process of operation and risk management of
their companies.
Embedded options can have different styles, the most common types of options in the shipping
industry is European options, American options and Bermudan options.
European options can only be exercised at a predetermined date (the expiration date) in the future
(Hull 2012). If the embedded options in a time charter contract are of European style, the lease
period can only be extended at one specific predetermined date. If the option to extend is exercised,
the option to purchase the vessel normally also is extended until the end of the lease period. When
19

the option to purchase the vessel is of European style, the vessel can only be bought at one
predetermined date in the future. If the option is not exercised at that point in time, the purchase
option ceases to exist. Normally, the expiration date of the purchase option is at the end of the
contract period.
American options can be exercised at any point in time until the end of the contract period (Hull
2012). If the embedded options in the time charter contract are of American style, the charterer can
extend the lease period or buy the vessel at any point in time in the contract period. Also here, if the
option to extend is exercised, the purchase option is normally also extended. If the purchase option
is exercised, the time charter contract ceases to exist.
Bermudan options are American options with non-standard features. The option holder have the
possibility to exercise the option at several predetermined dates in the contract period (Hull 2012). If
the embedded options in the time charter contract are of Bermudan style, the charterer can choose
to extend the lease period or purchase the vessel at several dates in the lease period. These dates
are often set to once a year. As before, if the option to extend is used, the option to purchase the
vessel is normally also extended.
Since options are financial derivatives, the availability of reliable price information on the underlying
asset is a necessary condition. Alizadeh and Nomikos (2009) emphasizes that available price
information on the underlying freight market is necessary in order to trade derivatives on freight.
Further, they explain that the price information on the underlying freight market needs to be
continuous, measurable and fully transparent.
The Baltic Exchange is the leading provider of freight market information (Alizadeh and Nomikos
2009). When derivative transactions are priced and settled in the freight market, they generally rely
on freight indices provided by the Baltic Exchange. Several different freight indices exist due to both
the differences between segments in the shipping industry, but also due to different vessel types and
sizes. Since this thesis focus on the dry bulk segment, the Baltic Dry Index (BDI) is the index of
interest. The BDI is an index calculated as the equally weighted average of the indices related to the
different vessel sizes in the dry bulk segment (Alizadeh and Nomikos 2009). This index is used as a
general market indicator reflecting the movements in the dry bulk segment.
In addition to being of European, American or Bermudan style, options embedded in time charter
contracts are written on the underlying average spot freight rate over the defined contract period;

20

they are path dependent5 and therefore of Asian style (Koekebakker, Adland et al. 2007). As
mentioned before, freight rates cannot be delivered in their physical form, they represent a cost of a
freight service that cannot be stored or carried forward in time (Kavussanos and Visvikis 2006). Their
non-storable feature is one reason why it is convenient to write freight rate claims on an average of
spot freight rates over a defined period of time. Further, Koekebakker, Adland et al. (2007) explain
that a charterer operating in the spot freight market is exposed to freight rate fluctuations during
some period of time. In order to capture freight rate fluctuations over a defined period of time, it is
convenient to treat freight rate contingent claims as path-dependent derivatives.
Another aspect worth mentioning is the freight revenue process when operating a vessel in the spot
freight market. The duration of a voyage can differ from a few weeks to several months; the
expected freight revenue from each voyage is given by an estimated average of the forecasted
fluctuations of the freight rates over the voyages duration. Therefore, the spot freight rate is itself
implicitly average based since it refers to fluctuations over a specific time period (Koekebakker,
Adland et al. 2007).

2.7 Background for the Models Selected

This section will give a presentation of previous empirical findings that support the two models
adopted for valuations. Different arguments are presented that together form an empirical
foundation for adopting both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion
process for valuation purposes. Due to the similarities between the two models, the following
arguments will apply to both processes. But first, clarifying arguments for the two models adopted
will be presented.
The main intention of this thesis is to introduce the concept of fair valuation of time charter
contracts with purchase options. For this purpose, the Ornstein-Uhlenbeck process is adopted and
analytic solutions for the valuations will be derived. The Ornstein-Uhlenbeck process is a one-factor
model with mean reversion properties which, as will be argumented for in the following, serves as
well suited when modeling freight rates.
However, two models will be introduced for valuation purposes and the second one is the Geometric
Mean Reversion process. As its name indicates, mean reversion is also a property of this process and
one factor is present also here. The Geometric Mean Reversion process is considered as more
5

According to Hull (2012) path-dependent options are options where the payoff depends on the path followed
by the price of the underlying asset, not just its final value.

21

realistic in freight rate modeling compared to the Ornstein-Uhlenbeck process, arguments pointing in
that direction will be examined later. Nevertheless, the Ornstein-Uhlenbeck process will be applied
for the valuation of time charter contracts with purchase options due to the nice analytic solutions
from Jrgensen and Giovanni (2010).
Due to the lack of analytic solutions from the Geometric Mean Reversion process, valuation of time
charter contracts with embedded options requires numerical routines such as the Finite Difference
method6. Such routines are complex and time intensive to implement, but Monte Carlo simulation7
can be used to value vessels and European options from the Geometric Mean Reversion process. This
will be done later on as an introduction to further studies of the use of this model.
Now that the model choices have been clarified it is time to move on to empirical arguments for why
these models are specially suited for freight rate modeling. First, both models are driven by one
factor which is assumed to be the spot freight rate. Stopford (2009) points out four factors that are
influential on the vessel value; freight rates, age, inflation and shipowners expectations for the
future. Further he says that freight rates are the one factor that primarily influences vessel prices. As
the freight market goes up and down, so will this continue to the sale and purchase market.
Also, Jrgensen and Giovanni (2010) argue that the spot freight rate is the major source of
uncertainty in the shipping industry, which results in the spot freight rate representing the main
source of business risk in shipping. By adopting the Ornstein-Uhlenbeck process and choose the spot
freight rate as the stochastic factor evolving through time, simple and more complex freight rate
contingent claims in the shipping industry can be fairly valued, in addition to valuation of the vessels
involved in these contingent claims. The Ornstein-Uhlenbeck process ensures the spot freight rate to
evolve through time according to a mean reverting stochastic process. Empirical arguments
according to the mean reversion property will be presented below, whereas an economic reasoning
for mean reversion in freight rate movements will be presented in the next section where freight rate
dynamics are examined.
Fair valuation of time charter contracts with purchase options include valuation of the vessels
underlying the contract. Therefore, another important justification for the models adopted and the
spot freight rate as the uncertain factor in the models is the evidence of high correlation between
spot freight rates and vessel prices. Adland and Koekebakker (2007) introduce their research by
mentioning that freight rates are relatively highly correlated with vessel values since peaks and
6

Finite Difference methods value a derivative by solving the differential equation that the derivative satisfies
(Hull 2012).
7
A procedure for randomly sampling changes in market variables in order to value a derivative (Hull 2012). This
procedure will be described in detail in Section 3.5.1.

22

troughs in the freight market have a tendency to quickly work their way into the sales and purchase
market. They propose that the most important factor determining the price of a vessel is the vessels
age; this is due to the depreciation of the vessels value during its lifetime. Further, they expect the
one-year time charter rate to be the second important factor to the price of a vessel. In their
research, they find strong evidence of correlation between freight rates and vessel prices. Also, they
find that the vessel value is an increasing function of the freight rate level (Adland and Koekebakker
2007).
Another argument for adopting both the Ornstein-Uhlenbeck process and the Geometric Mean
Reversion process for valuation purposes is the evidence of mean reversion in spot freight rates. In
his work, Adland (2000) finds only very slight mean reversion for low and medium values of the spot
freight rate. But when the freight rate increases beyond 35 000 per day, the mean reversion is
stronger. This is the same as saying that the drift decreases since stronger mean reversion imply
stronger attraction towards the long-term mean reversion level. Further, he explains that the
decline in drift at high freight rates prevents the freight rate from exploding towards infinity.
The evidence of mean reversion in freight rates is also acknowledged by Koekebakker, Adland et al.
(2006) who state that the freight rate is expected to be mean reverting in one sense or another. They
apply basic maritime theory to show why the spot freight rate process must be mean reverting
(Koekebakker, Adland et al. 2006). The same theory is also to be found in Stopford (2009). In Section
2.8 below, where the freight rate dynamics are described, the economic reasoning will be presented.
To sum up, the presented arguments point in the direction of a model that takes account of one
variable evolving in a stochastic manner through time (the spot freight rate), and which also have
mean reversion properties. Both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion
process have these properties, and it is therefore reasonable to assume these models to be
appropriate models for the valuations.

23

2.8 The Dynamics of Freight Rates

The following section will describe the dynamics of spot freight rates and how they evolve in a
stochastic manner through time. To understand the evolvements of spot freight rates a short
description of the predominant market cycles in the shipping industry is essential. Thereafter, the
freight rate dynamics will be described by first introducing the elasticity of freight rates in different
market conditions, and then the behavior of freight rates will be described.

2.8.1

Shipping Market Cycles

The shipping industry is pervaded by market cycles. Stopford (2009) identifies three components of a
typical cyclical time series. The first is the long-term cycles driven by technical, economic or regional
change. A long-term cycle moving upwards is good for business, whereas a long-term cycle moving
downwards is bad for business. The second is the short-term cycles characterized by short-lived
movements and are important drivers of the shipping market cycle. Short-term cycles are often
referred to as business cycles, they fluctuate up and down and a complete cycle can last anything
from 3 to 12 years from peak to peak. The third and last component is the seasonal cycles which are
regular fluctuations within the year. As for the dry bulk shipping segment, weak markets appear
often during July and August due to relatively little grain being shipped. Thus, seasonal cycles occur in
response to seasonal patterns of demand for sea transport.
Now that the shipping market cycles are shortly described it is time to move on to the dynamics of
freight rates, which in fact is the factor that fluctuates according to the market cycles.

2.8.2

Freight Rate Dynamics

According to Alizadeh and Nomikos (2009), the balance between supply and demand for shipping
services is at any point in time reflected by the spot freight rates. The crucial factors for demand and
supply are the ones listed in Table 1. When the shipping market is in recession and spot freight rates
are at very low levels, an overcapacity is developed and many vessels cannot find employment, are
laid up, slow steam or even carry part cargo. When the market is in such a recession, changes in
demand due to external factors8 can be absorbed by the extra available capacity. Thus, the impact on
spot freight rates would be relatively small and they are considered as being inelastic to changes in
demand. Gradually, market conditions will improve and freight rates will increase. Available vessels
8

Nomikos and Alizadeh (2009) propose seasonal changes in trade and random shocks (events) as external
factors.

24

will be employed again until the world fleet is fully utilized and any increase in supply is only possible
by increasing productivity through increasing speed and shortening port stays and ballast legs. When
market conditions are good and spot freight rates are at high levels, any changes in demand due to
external factors would create a relatively large movement in spot freight rates; spot freight rates are
considered elastic in relation to changes in demand.
Good market conditions imply high spot freight rates. Tvedt (1997) explains that very high spot
freight rates may appear in shorter periods of time due to the demand for seaborne transport being
inelastic to changes in spot freight rates, and that there is a short-term upper limit to supply. Further,
he explains that good market conditions with high levels of spot freight rates will not be a persistent
situation over time. The potential for supply adjustment, both newbuilding and demolition, will
guarantee that very high or very low spot freight rates will not be a persistent situation over time
(Adland and Cullinane 2006). High spot freight rates will tempt shipowners to order new vessels in
order to capture high freight earnings. Due to the time lag from ordering a new vessel until delivery,
a gambling position is created for the shipowner. Often, the market clears at a rate that is not high
enough to cover the investment costs of a new vessel. If the shipowner manages to order a new
vessel in time such that the rates are still high when the vessel is delivered, he may catch a high
reward. But ordering a vessel when the spot freight rates are high is often too late because the rates
will probably be back to normal low levels when the vessel is ready to be delivered.
Summing up, the shipping market cycles and the continuous stochastic evolution of spot freight rates
go hand in hand; spot freight rates will evolve in accordance with the current state of the shipping
market and its cycles. This serves as an economic reasoning for the mean reverting behavior of spot
freight rates which reinforces the empirical findings described in the Section 2.7 above.

2.9 The Ornstein-Uhlenbeck Process

In the following, the model adopted to value time charter contracts with purchase options, namely
the Ornstein-Uhlenbeck process, will be described in detail, along with the derivation of the solution
to the process. The Ornstein-Uhlenbeck process has been applied to the shipping industry by, among
others, Bjerksund and Ekern (1995), Tvedt (1997) and Jrgensen and Giovanni (2010).
The basic underlying assumption is that the stochastic component of the instantaneous cash flow
from owning a vessel, the spot freight rate, is characterized by an Ornstein-Uhlenbeck process
(Bjerksund and Ekern 1995). When the spot freight rate is said to be stochastic, its value will change
over time in an uncertain way (Hull 2012). Spot freight rates evolve in a stochastic manner and their
25

values can change at any point in time, they are therefore defined to be continuous-time stochastic
variables. When spot freight rates are assumed to follow an Ornstein-Uhlenbeck process, they are
ensured to behave like continuous-time stochastic variables due to the special feature of the model
being an It process. An It process consist of two terms; a drift term which is a function of the value
of the underlying variable9 and time, and a variance term which also is a function of the value of the
underlying variable and time. The variance term contains a standard Wiener process which is a
particular type of a stochastic process with mean of zero and variance of one per year, it is denoted
(Hull 2012). Both the drift term and the variance term are liable to change over time.
Bjerksund and Ekern (1995) assume that the instantaneous cash flow generated by an operating
vessel,

, may be described as follows:

(1)

where

represents the size of the cargo,

represents the operating cost-flow rate, and

represent

the uncertain spot freight rate (annualized) at time per unit of cargo. For ease of notification, it is
assumed that the spot freight rate is prevailing for the vessel as a whole and net of all costs. Thus,
and

(Jrgensen and Giovanni 2010), and

is left as the instantaneous cash flow from

operating a vessel. As described in Section 2.1.3 where time charter equivalent spot freight rates are
introduced, this assumption causes lack of the ability for direct market comparisons as marketquoted freight rates embed varying degree of costs.

The freight rate in this case.

26

As argumented for in Section 2.7 ,where the background for the model selected is examined, and in
Section 2.8, where the dynamics of freight rates are examined, in addition to following Bjerksund and
Ekern (1995), it is assumed that the spot freight rate can be modeled by the following stochastic
differential equation:

(2)

where

is the speed of mean reversion,

volatility of spot freight rates, and

space

is the constant long-term mean,

is the instantaneous

is a standard Wiener process defined on some probability

(Jrgensen and Giovanni 2010). The model is an It process where the first term is

considered as the drift term, whereas the last term is considered as the variability term. This process
is identical to the one Vasicek (1977) proposed for modeling interest rate dynamics.
In line with Bjerksund and Ekern (1995), technical descriptions of the model, the drift term, ensures
that the process always are pushed back to its long-term mean. When

, the drift term is

negative and the process will be pushed up to its long-term mean. On the other hand, when

the drift term is positive and the process will be pushed down to its long-term mean. Further, they
explain that higher values of

will create a stronger tendency of the stochastic process to move back

towards its long-term mean. Thus, the higher the , the higher the degree of mean reversion.
Further, they explain that the second term characterizes the volatility of the process with

being

an increment of a standard Wiener process with characteristics as explained above.

Important to remember is that the model measures time in years, whereas actual spot freight rates
are normally quoted on a daily basis. This is important to have in mind when comparing with market
data where

have to be considered instead of

is one day measured in years, where one

year is assumed to contain 360 days (Jrgensen and Giovanni 2010).

Due to the properties of the Wiener process being standard normal distributed, Equation (2) implies
that future spot freight rates are normally distributed (Jrgensen and Giovanni 2010). The density
function10 will therefore have the classic bell shaped curve. In order to derive the mean and the
variance of the distribution of future spot rates the solution to Equation (2) have to be derived, this
will be done in the following section.
10

Plots the shape of the distribution curve of the random variable that is considered (Skovmand 2012), which
in this case is the spot freight rate.

27

2.9.1

The Solution to the Ornstein-Uhlenbeck Process

The solution to the process can be derived from the stochastic differential equation in Equation (2).
This is desirable both in order to calculate the freight rate at time

, and in order to derive the

mean and the variance of the distribution of future spot freight rates. From the mean of the
Ornstein-Uhlenbeck process the expected return from operating a vessel in the spot freight market
over a defined period, can be calculated. The variance of the Ornstein-Uhlenbeck process calculates
the volatility in spot freight rates over the same period. Both measures are useful for agents in the
shipping industry as they give an indication of how the market will develop in the future.
First, consider how to obtain the freight rate at time

which is given by todays freight rate plus

the sum of the dynamics of the spot freight rate evolving from time to time :

(3)

Equation (3) has to be solved explicitly in order to ensure the availability of analytic solutions. In
order to do this, a temporary variable have to be introduced (Hammer, Hafsaas et al. 2011):

(4)

It is desirable to obtain the dynamics of this temporary variable since the dynamics of the spot
freight rate are considered in Equation (2). The dynamics of Equation (4) are given by Itos lemma11:

(5)

(6)

11

Describe the behavior of functions of stochastic variables. Such a function can be the price of a derivative
which is dependent on the underlying stochastic variable and time (Hull 2012).

28

Now, the next step is to insert Equation (2) for

in the equation above:

(7)

Manipulations12 of Equation (7) give:

(8)

Continuing by integrating from time zero to time :

(9)

Calculating

Continuing with Equation (9):

(10)

12

A detailed derivation is attached in the Appendix.

29

(11)

(12)

Calculating

Finally, the solution to Equation (2) becomes:

(13)

From Equation (13) the time conditional mean and variance of the normal-distributed future spot
freight rate

can be stated as follows:

(14)

(15)

where the detailed derivations of Equations (14) and (15) are attached to the Appendix.

30

2.10 The Geometric Mean Reversion Process

The Geometric Mean Reversion process will now be presented and the solution will be derived. As
mentioned earlier, this process is more realistic in modeling freight rates, but to the authors
knowledge analytic solutions do not exist. Therefore, when calculating vessel values and European
option values numerical procedures need to be implemented. Those will be explained in more detail
below. Following Tvedt (1997), the increment of the process is given by the following stochastic
differential equation:

(16)

where

is the speed of mean reversion,

(annualized) spot freight rate at time ,

is the long-term mean reversion level,

is the

is the instantaneous volatility of spot freight rates, and

is the increment of a standard Wiener process.

As Tvedt (1997) describes, the Geometric Mean Reversion process is mean reverting and is also
downwards restricted since it is not possible to take the natural logarithm of spot freight rates that
equals zero or are negative - zero is an absorbing level.
Another important feature of the Geometric Mean Reversion process is that it ensures the volatility
to be progressively increasing in the freight rate level. This occurs in the last term of Equation (16)
where the volatility parameter is multiplied by the time spot freight rate level. In his research,
Adland (2000) finds evidence that the volatility is increasing in the freight rate level. More specific, he
finds that the diffusion function,

, is close to linear for low and medium freight rates, while it is

increasing progressively for very high freight rates (Adland 2000). Assuming that the spot freight rate
follows the Geometric Mean Reversion process is thus appropriate in relation to his findings.
In the following, the derivation of the solution to the Geometric Mean Reversion process in Equation
(16) will be done, whereas the time conditional mean and variance only will be presented. The
same applications for the solution, the mean and the variance as for the Ornstein-Uhlenbeck process
are applicable also here.

31

2.10.1 The Solution to the Geometric Mean Reversion Process

The solution to Equation (16) is derived following Tvedt (1997). In order to simplify later calculations
he suggests starting by dividing the whole process by

and multiplying it with the integrating factor

(17)

Further, he continues by defining a function

. Again, the increments of

this temporary function are desirable. They are given by Itos lemma:

(18)

(19)

Rearranging:

(20)

In order to proceed towards the solution two things have to be done. First, it is desirable to
substitute the Geometric Mean Reversion process for
and

are defined such that

. In order to ease the notification,

. This gives:

32

where the first and the second term tend to zero, and where the last term tends to

such that

.
Then, by rearranging Equation (17) to
process for

and substituting the

Equation (20) becomes:

(21)

The next step is to rearrange this equation and integrate it from time zero to time :

(22)

(23)

Finally, the solution is obtained by rearranging Equation (23) such that

is alone, the

rearrangement step by step is attached to the Appendix. The freight rate level at time ,

, can be

expressed as:

(24)

with mean and variance:

(25)

33

(26)

where:

34

3 Analysis Section
This section starts by an examination of the differences between the two considered models; the
Ornstein-Uhlenbeck process and the Geometric Mean Reversion process. Evidences of the Geometric
Mean Reversion process being more appropriate in freight rate modeling will be presented and
discussed. Further, the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process will
be examined separately. When investigating the Ornstein-Uhlenbeck process, valuation of simple
freight rate contingent claims will be introduced and for that purpose the partial differential equation
will be established. Analytic solutions will be derived before they are used to the valuation of simple
freight rate contingent claims. Finally, European option values to buy the vessel will be calculated,
which in turn is used to obtain the total value of time charter contracts with purchase options. Next,
the Geometric Mean Reversion process will be investigated and the purpose is to obtain vessel
values and European option values to buy the vessel by Monte Carlo simulation. Finally, the section
ends by a comparison of vessel values and option values obtained by these two models.
As for the calculations throughout this section base case parameter values will be used. They are
adopted from Tvedt (1997) where he has estimated parameter values in relation to both processes.
The base case parameter values are presented in Table 2 below. However, when tables are
presented for different volatilities and different spot freight rates, those specific varying parameter
values will be adopted from Jrgensen and Giovanni (2010).

Parameter

Ornstein-Uhlenbeck Process

Geometric Mean Reversion

Process

14 371

10,45

1 094

0,1184

0,00247

0,0033

1,5%

1,5%

Table 2. Base case parameter values for both processes.

35

3.1 Two Models Different Characteristics: A Comparison

This section will give an examination of the differences between the Ornstein-Uhlenbeck process and
the Geometric Mean Reversion process. At a first glance they look quite similar, and they do in fact
have some properties in common. But due to some important dissimilarities they behave quite
differently from each other.
In real life, negative values of the spot freight rate are impossible. This is due to the shipowners
option to lay up his vessel if the operational costs are not covered. In fact, due to the option to lay
up, the spot freight rate will generally never fall below the given lay up level (Tvedt 1997). The
Ornstein-Uhlenbeck process fails to take account of the impossibility of spot freight rates becoming
negative since the process is not downwards restricted. Large volatility will often give negative spot
freight rates because the process is normally distributed around a given mean. This is demonstrated
in Figure 1 below where the process is simulated over a period of almost six years with time steps
equal to one observation of the spot freight rate each day. The base case parameter values in Table 2
are applied and presented in the figure description. However, the volatility is adopted from
Jrgensen and Giovanni (2010) in order to capture the high volatility effect. From Figure 1 it is clearly
that the spot freight rates over the horizon can take on both negative and positive values. This is
somewhat unrealistic compared to a real life situation.

Simulated Freigt Rate Following an Ornstein-Uhlenbeck

Process
20 000,00

Freight Rate Value

15 000,00

10 000,00

5 000,00

Freight Rate

-5 000,00

-10 000,00

Time

Figure 1. Simulated spot freight rate from the Ornstein-Uhlenbeck process. =0,00247, =14 371,
=14 500,

=1/360 and =7 000.

36

In relation to negative freight rates, what actually can happen in reality during short intervals is that
the spot freight rates may become so low that the estimated time charter equivalent spot rate will
become negative. This happens when the voyage income is less than the total of fuel consumption
and harbor and channel costs, but in these cases the shipowner will probably rather lay up his vessel
than keeping it in operation. Therefore, the market almost always clears at a positive time charter
equivalent spot freight rate and the Geometric Mean Reversion process, which is downwards
restricted, may give a more appropriate and realistic description of the spot freight rate.
As Figure 2 below clearly indicate, the simulated Geometric Mean Reversion process only give
positive values of the spot freight rate and is therefore also more realistic in relation to a real life
situation. Again, the base case parameter values from Table 2 are applied and given in the figure
description. However, in order to capture the high volatility effect the volatility is set equal to one13.

Simulated Freight Rate Following a Geometric Mean

Reversion Process
35 000
30 000

Freight Rate Value

25 000

20 000
15 000

Freight Rate

10 000
5 000
0

Time

Figure 2. Simulated spot freight rate from the Geometric Mean Reversion Process. =0,0033, =10,45,
=14 500,

13

=1/360 and =1,0000.

Inspired by Tvedt (1997) where he at one point has set the volatility parameter equal to 0,93.

37

3.2 Applications of the Ornstein-Uhlenbeck Process: Introducing Valuation

of Freight Rate Contingent Claims
In the following, some simple freight rate contingent claims whose value,
the spot freight rate factor process,

, depend only on

, and time will be valued. Before this can be done, the

fundamental partial differential equation that

must satisfy in order to give the correct value

of such contingent claims will be established. According to Hull (2012), a key property of the
fundamental partial differential equation is that no variables affected by the risk preferences of
investors are involved. This opens for risk-neutral valuation of claims dependent on only the spot
freight rate factor process and time; the risk-neutral valuation result14 can be applied. When no risk
preferences enter the partial differential equation a simple assumption that all investors are riskneutral can be made. Thus, the risk-free rate of interest, , is assumed to be the expected rate of
return on all investments. Also, over the contract period the risk-free rate of interest is assumed to
be constant. The assumption of a risk-neutral world does therefore simplify the computations of the
claims which are dependent on only the spot freight rate factor process and time.
The spot freight rate essential in the partial differential equation is representing the price of a
service, and is therefore not a physical asset. In order to price freight rate contingent claims by the
partial differential equation, an assumption of the existence of a physical asset whose price is
perfectly correlated with the spot freight rate need to be done (Jrgensen and Giovanni 2010). They
explain that this is necessary to the application of standard no arbitrage arguments (Hull 2012).
Further they explain that the vessels underlying the option contracts could be used, but hedging
strategies based on trading vessels are very unpractical. Instead, they rely on the assumption of the
existence of a liquid market for the relevant futures. Such markets do in fact exist in the real world;
the Oslo-based International Maritime Exchange (IMAREX)15 is an example.
When both the assumption of a risk-neutral world and the assumption of the existence of a liquid
market for the relevant futures are done, Itos lemma and a risk-neutralizing hedge argument16 can
be used to derive the fundamental partial differential equation that

must satisfy in order to

give the correct value of the option contract.

14

The present value of any cash flow in a risk-neutral world can be obtained by discounting its expected value
at the risk-free rate (Hull 2012). The mathematical expression will be presented later.
15
Writes freight rate derivatives upon the freight indices obtained from the Baltic Exchange and from Platts.
16
Transformation of the true probability measure to the risk-neutral probability measure which will be
derived later.

38

3.2.1

Derivation of the Fundamental Partial Differential Equation

From Equation (2), the single factor spot freight rate process is given as:

where

is the increment of a standard Wiener process under the true probability measure, . It

is argumented that derivatives prices,

, influenced by the factor process,

, and time must

have dynamics described by (Vasicek 1977).

(27)

where:

and where

is the market price of freight rate risk. In order to discount by the risk-free interest

rate, , and thus apply the risk-neutral valuation result, the true probability measure have to be
changed to the risk-free probability measure, . In short, the risk-neutral valuation principle states
that a derivative can be valued by (a) calculating the expected payoff on the assumption that the
expected return from the underlying asset equals the risk-free interest rate and (b) discounting the
expected payoff at the risk-free rate (Hull, 2012, page 630).The transformation from the true
probability measure to the risk-free probability measure can be done by applying Girsanovs
Theorem. This procedure will be done in the following. Define:

(28)

39

where

is now the increment of a standard Wiener process under the risk-free probability

measure, .

is again the market price of freight rate risk. Then:

(29)

(30)

(31)

In the following, an assumption of a constant market price of freight rate risk is done such that
reduces to . Therefore, the factor process under

is given as:

(32)

(33)

(34)

(35)

40

Define

. Then:

(36)

This is the dynamics of

under the risk neutral probability measure, . The mean and the variance

are given as in Equations (14) and (15). The only difference is that

is replaced by

. Finally, Itos

lemma can be used to characterize the dynamics of the -process under :

(37)

The next step is to substitute the It process for

and

in Equation (37). The It process

under the risk-neutral probability measure is given in Equation (36):

To ease the calculations,

Calculating

and

are defined such that

41

In the limit,

tends to zero,

tends to

, and

tends to

(Campbell, Lo et al. 1997).

Hence, the first and the second term tend to zero, and the last term tends to
equation to

. Inserting both

and

. This reduces

into Equation (37):

(38)

This is equal to:

(39)

(40)

(41)

42

According to Jrgensen and Giovanni (2010), absence of arbitrage requires that:

(42)

(43)

where

is the dividend rate received by the claim. By introducing

as the

absolute cash flow from the claim the following partial differential equation is the one

must

satisfy:

(44)

Or, equivalently:

(45)

This is the partial differential equation that all claims depending on the spot freight rate process,
and time need to satisfy in order to be priced arbitrage free. Solving for

will give the price of a

particular freight rate contingent claim. Another way of solving Equation (45) is to manipulate the
probabilistic Feynman-Kac representation of the solution to the partial differential equation. A partial
differential equation can then be solved by an expectation of the discounted payoff of the derivative,
, modified replacing the true probability measure, , by a risk-free probability measure,
(Duffie 2001). By doing this, the risk-free interest rate, , can be used as the expected rate of return
when discounting backwards in time. The probabilistic Feynman-Kac representation of the solution
to the partial differential equation takes the following form (Jrgensen and Giovanni 2010):
43

(46)

where

would typically be the expiration date at which the value of the claim would be given as a

known function of

. Equation (46) is quite intuitive, it indicates that the time value of the claim is

the expected value of the continuous flow of net dividends plus the value of the payoff at the
maturity date, both discounted back to time .
Finally, when the partial differential equation is established, with its solution, the valuation of some
simple freight rate contingent claims can be done.

3.3 Valuation of some simple Freight Rate Contingent Claims

The following section will present derivations of analytic solutions for evaluating simple freight rate
contingent claims, as well as valuation results for each simple contingent claim.

3.3.1

Claim to Receive Spot Freight Rate Flow from Time to Time

The first claim that will be valued is the claim to receive the spot freight rate on a continuous basis
from time , when the current spot freight rate is

, to time . To do this, the probabilistic Feynman-

Kac representation from Equation (46) is used. In the following, the main steps of how Equation (46)
is used to derive an analytic solution of the value of such a claim are shown, whereas a detailed step
by step representation is attached to the Appendix. Defining the current value of receiving the spot
freight rate on a continuous basis as:

(47)

44

The freight rate process,

(48)

Inserting

in Equation (47) gives:

(47)

Since the expectation of the increment of a standard Wiener process equals zero (Hull 2012), the last
term becomes equal to zero and only the two first terms are left. Further:

(49)

(50)

(51)

(52)

where

is an annuity factor of the present value using the discount rate

of

receiving a unitized continuous cash flow for years (Jrgensen and Giovanni 2010).

45

3.3.2

Fixed for Floating Freight Rate Swap

When a shipowner and a charterer agrees on a time charter contract, the shipowner receives a fixed
daily freight rate over the lifetime of the contract, and leaves the operation of the vessel in the
charterers responsibility. If the shipowner had not chartered his vessel, he could have operated it in
the spot freight market himself and received the floating spot freight rate over the same period of
time. Therefore, when a shipowner enters into a time charter contract, he agrees to receive a fixed
daily freight rate in exchange for a floating one. This is the same mechanism as an interest rate swap;
an exchange of a fixed rate of interest on a certain notional principal for a floating rate of interest on
the same notional principal (Hull 2012).
It is in the shipowners interest to determine the constant freight rate,

, fixed at time that will

be equivalent to receiving the variable spot freight rate over the period from time to time . In this
way, the value of the freight rate swap will be equal to zero at initiation. This is also a property equal
to an interest rate swap agreement (Hull 2012).
To determine the fixed freight rate that will make the shipowner indifferent in his choice of whether
to operate the vessel on his own, or charter it, Jrgensen and Giovanni (2010) apply Equation (52)
and solve for

(53)

Equation (53) indicate that the present value of the fixed freight rate have to be equal to the present
value of receiving the spot freight rate on a continuous basis from time to . When ensuring this
equality, the shipowner will be indifferent from receiving the fixed freight rate over the horizon, or
receiving the floating spot freight rate over the same horizon. Solving for

(54)

where

will be the fair continuously paid time charter rate (fixed freight rate) during the life of

the contract.

46

In the following, some fair valued time charter rates for different values of the speed of mean
reversion and current spot freight rates will be presented. The calculation of these fair valued time
charter rates is done by applying Equation (54) above. The base case parameter values from Table 2
are applied and given in the table description. As for this example, both the spot freight rates and the
fair time charter rates are presented in daily values. It is worth mentioning that when the analytic
solutions are used for valuations the parameter values have to be scaled up into yearly values so that
the time units are proportionate to each other. This is the case for all remaining calculations during
the next sections.

Spot Freight Rate,

5 000

10 000

15 000

20 000

25 000

30 000

35 000

40 000

0,10

6 974

10 921

14 868

18 814

22 761

26 708

30 655

34 602

0,25

8 981

11 857

14 733

17 609

20 484

23 360

26 236

29 112

0,50

10 881

12 743

14 605

16 467

18 329

20 191

22 053

23 915

1,00

12 466

13 483

14 499

15 515

16 531

17 548

18 564

19 580

2,00

13 406

13 921

14 436

14 951

15 466

15 981

16 496

17 011

5,00

13 983

14 190

14 397

14 604

14 811

15 018

15 225

15 432

10,00

14 177

14 280

14 384

14 488

14 591

14 695

14 799

14 902

Table 3. The dependence of fair time charter rates (daily) on speed of mean reversion,
spot freight rate,

. =1,5% and

and current

=14 371 (daily).

The results in Table 3 indicate that when the spot freight rate is 15 000 per day, which is close to the
long-term mean of 14 371 per day, the fair time charter rate will also lie close to the long-term mean
for all values of the speed of mean-reversion parameter. This is in line with intuition; when the spot
freight rate is close to the long-term mean it is not expected to increase or decrease wildly the next
five years, thus also the fair time charter rates should lay close to the spot freight rate.
The story is different when the current spot freight rate differs from the long-term mean. For current
spot freight rates that are below the long-term mean, the fair time charter rates are far from the
long-term mean when the speed of mean reversion-parameter takes on low values, but are closer to
the long-term mean when the speed of mean reversion-parameter takes on high values. This makes
sense since when the speed of mean reversion is high, it is expected that the spot freight rate will
47

quickly return to the long-term mean. Therefore, the fair time charter rate should be close to the
long-term mean. Opposite, when the speed of mean reversion is low, it is expected that the spot
freight rate will return slowly to its long-term mean. The fair time charter rate should in that case be
set closer to the current spot freight rate to secure as low difference between the two rates as
possible during the contract period.
For current spot freight rates lying above the long-term mean the opposite is present; when the
speed of mean reversion is high the fair time charter rate is closer to the long-term mean than is the
case when the speed of mean-reversion is low. Also this makes perfectly sense; low speed of mean
reversion creates expectations of the spot rate to slowly return to the long-term mean. And again,
high speed of mean reversion creates expectations of the spot freight rate to quickly return to the
long-term mean, implying a fair time charter rate closer to the long term mean to reduce the
possibility of large differences between these two rates.

At the time a time charter contract is agreed on and the fair time charter rate is fixed, the value of
the swap is, as mentioned before, equal to zero. However, Equation (47) clearly indicates that the
value of receiving the current spot freight rate on a continuous basis during the contract period is
based on an expectation. This implies uncertainty that the spot freight rate actually will evolve as
expected. If the spot freight rate moves in other directions than expected, the fair time charter rate
will also be different than the one that is fixed in the contract. This difference between the new fair
time charter rate and the one that was set at initiation gives the swap contract a value - positive or
negative. It is possible to calculate the value of a time charter contract entered into at an earlier
point in time. Jrgensen and Giovanni (2010) present this valuation formula for a contract that
receives the floating spot rate and pays a fixed time charter rate:

(55)

where time

is the prevailing fair time charter rate and

is the contracted fixed time

charter rate. This equation indicate that if the prevailing fair rate is higher than the contracted fixed
time charter rate the swap has a positive value, and otherwise if the contracted fixed time charter
rate is higher than the prevailing fair time charter rate. The annuity factor,

, ensures that

the value of the swap equals the discounted value of the continuous flow of the spread difference.
48

Valuation of a swap for different spot freight rates, fixed contracted time charter rates, and fair time
charter rates will be done next. The base case parameter values from Table 2 are applied and
presented in the table description. Table 4 below show the value of a 5-year time charter contract
which is dependent on the current fair time charter rate, and the previously contracted time charter
rate. In the table, spot freight rates, current fair time charter rates and previously contracted rates
are presented in daily values, whereas the values for the 5-year contract are yearly17 values.

The Value of a 5-Year Time Charter Contract

Spot and Current Time

, Previously Contracted Rate

Charter Rate
5 000

10 000

20 000

30 000

5 000

5 057

98 690

-8 572 091

-25 913 655

-43 255 218

10 000

10 027

8 716 815

46 033

-17 295 530

-34 637 094

20 000

19 966

25 953 063

17 282 282

-59 282

-17 400 845

30 000

29 905

43 189 312

34 518 531

17 176 967

-164 596

Table 4. The value of a 5-year time charter contract. r = 1,5%,

= 14 371 (daily),

= 0,00247.

As expected, when the previously contracted time charter rate differ from the current fair time
charter rate, the swap contract does either have a positive value or a negative value. A current fair
time charter rate higher than the previously contracted time charter rate gives the swap contract a
positive value. Otherwise, a current fair time charter rate lower than the previously contracted time
charter rate results in a negative value of the swap contract. Also, when the difference between the
two rates is large, the value of the swap is also large - positive or negative.

3.3.3

The Value of a Vessel

According to Jrgensen and Giovanni (2010), a vessel is a physical asset that earns rents to its owner
through the flow of net spot freight rates during the vessels lifetime. Therefore, they recommend
using Equation (47) to value a vessel. But since the vessel will have a scrap value18 when it reaches
the end of its useful economic life, an extension of this formula has to be applied. Assuming that the

17
18

One year is assumed to contain 360 days.

When the vessel is scrapped the remaining steel can be sold to the steel industry (Stopford 2009).

49

final service date, , of a vessel as well as its scrap value, , are known with certainty, the vessel
value can be calculated using this extension of Equation (52):

(56)

Equation (56) signify that the valuation formula for a vessel follows a Gaussian process with
deterministically time-varying drift and diffusion coefficients (Jrgensen and Giovanni 2010).
Therefore, future vessel values at any time

are normally distributed under both the true

probability measure and the risk-neutral probability measure. The time conditional mean and
variance under both probability measures are given below (Jrgensen and Giovanni 2010):

(57)

(58)

(59)

In Table 5 below, vessel values as a function of different spot freight rates and varying remaining
vessel lifetimes is calculated using Equation (56). The parameter values are again the ones from Table
2 and are given in the table description. The spot freight rates are given in daily values whereas the
vessel values are given in yearly values.

50

Remaining Vessel Life (

5

10

15

30

5 000

13 408 189

21 392 759

28 968 933

36 152 125

42 957 810

10 000

22 026 314

37 908 156

52 721 046

66 535 648

79 418 068

15 000

30 644 438

54 423 553

76 473 159

96 919 171

115 878 325

20 000

36 262 562

70 938 950

100 225 272

127 302 694

152 338 583

25 000

47 880 687

87 454 347

123 977 385

157 686 217

188 798 841

30 000

56 498 811

103 969 744

147 729 498

188 069 740

225 259 098

35 000

65 116 936

120 485 141

171 481 610

218 453 263

261 719 356

40 000

73 735 060

137 000 539

195 233 723

248 836 786

298 179 613

Table 5. The dependence of vessel values on remaining vessel value (

(

20

). =0,00247,

) and the spot freight rate

=14 371 per day, =1,5% and =5 000 000.

In line with intuition, Table 5 clearly indicates that vessel values are an increasing function of both
the spot freight rate and the remaining vessel life. Higher spot freight rate gives higher cash inflow to
the shipowner and therefore also the vessel value is increased. Longer remaining lifetime of a vessel
also imply higher vessel value compared to a vessel with shorter remaining lifetime.
In Figure 3 below, Equation (56) is used to calculate the evolution of a vessels value over a lifetime of
25 years. This is the jagged line where the simulated vessel value is calculated over its lifetime. The
expected vessel value is also calculated under the risk-neutral probability measure using Equation
(57) above. For this purpose, all parameter values are adopted from Jrgensen and Giovanni (2010)
and are presented in the figure description. In line with intuition, we can see that the value of a
vessel is expected to decrease as it ages.

51

120000000

Simulated and Expected Vessel Value

100000000

Value

80000000

60000000

Simulated Vessel Value

Expected Vessel Value

40000000

20000000

0
0

10

15

20

25

30

Time

Figure 3. Simulated and expected value of a vessel. =0,25, =0,05, =25, =5 000 000,

=20 000,

=5 000 and t=1/360.

3.4 European Option to Buy a Vessel

As described earlier, a European option to buy a vessel gives the charterer the right, but not the
obligation, to buy the vessel at expiration. If the option is not exercised, the contract with the
purchase option will expire and the vessel is handed over to the shipowner. Purchase options in
shipping contracts are common practice, but simple European options are generally not used. As
mentioned earlier, complex options happen to be used more often in shipping contracts. Although
European purchase options are somewhat unrealistic in relation to real life, such options will be
valued in order to demonstrate the interpretations of the nice analytic solutions derived through the
sections above.
Following Jrgensen and Giovanni (2010), the current date is denoted by and the expiration date of
a European option to buy a vessel is denoted by . Also, the vessel must be scrapped at date
value of . Further,

for a

will denote the exercise price19 of the purchase option. The payoff function at

expiry is then given as:

(60)

19

The price at which the vessel may be bought at in the time charter contract (Hull 2012).

52

Due to the analytic solution for the vessel value in Equation (56), an analytic solution for the time
value of this European call option can be derived (Jrgensen and Giovanni 2010). In the following,
only the results will be presented, whereas a detailed step by step derivation will be attached to the
Appendix. Thus, from the Appendix the time value of the European call option in Equation (60) is
given as:

(61)

where:

(62)

(63)

(64)

(65)

and where

and

denote the standard normal cumulative probability and density functions,

respectively.
Now, when the framework for valuing a European option to buy a vessel is established, purchase
options for varying spot freight rates and freight rate volatilities will be valued using Equation (61).
The parameter values are again taken from Table 2 and are given in the table description. Also, the
spot freight rates and the freight rate volatilities are daily values, whereas the option values are
annualized values.
53

Freight Rate Volatility,

1 000

3 000

5 000

7 000

9 000

5 000

44

1 421 456

7 120 274

14 938 207

23 654 772

10 000

126 219

5 886 695

14 715 677

24 191 691

33 895 366

15 000

6 998 523

16 859 312

26 828 385

36 813 079

46 802 986

20 000

31 459 010

35 670 469

43 790 420

52 928 505

62 438 019

25 000

59 276 730

60 198 581

65 052 295

72 326 786

80 698 860

30 000

87 118 860

87 249 455

89 463 264

94 518 454

101 339 629

Spot Freight
Rate,

Table 6. Value of European option to buy a vessel. =1,5%, =0,00247,

=5 000 000, =0, =5,

=14 371 (daily),

=25 and =93 000 000.

The option values in Table 6 confirm well-known option theory; as the freight rate volatility increase,
so does the option value. Higher volatility imply higher probability of upside gains, thus the option
value also increases. In addition, increased spot freight rate will also increase the option value since
the cash flow from owning the vessel is increased.
Since the option is European, the total value of the time charter contract with a purchase option can
safely be decomposed into its leasing contract component and its option contract component
(Jrgensen and Giovanni 2010). The total value of the 5-year time charter contract with an
embedded European purchase option can be found by adding the value of the 5-year contract to the
value of the embedded purchase option. The value of the 5-year contract is calculated by using
Equation (55). Table 7 below show total fair contract value for varying spot freight rates and fixed
time charter rates. Again, the base case parameter values from Table 2 are applied and given in the
table description.

54

Fixed Time Charter Rate

5 000

10 000

15 000

20 000

25 000

30 000

5 000

7 218 964

-1 451 817

-10 122 599

-18 793 381

-27 464 162

-36 134 944

10 000

23 432 491

14 761 710

6 090 928

-2 579 854

-11 250 635

-19 921 417

15 000

44 163 324

35 492 542

26 821 760

18 150 979

9 480 197

809 416

20 000

69 743 483

61 072 702

52 401 920

43 731 138

35 060 357

26 389 575

25 000

99 623 483

90 952 701

82 281 920

73 611 138

64 940 356

56 269 575

30 000

132 652 577

123 981 795

115 311 013

106 640 232

97 969 450

89 298 668

Spot Freight
Rate,

Table 7. The value of a 5-year time charter contract with European purchase option. r=1,5%,
=0,00247,

=14 371 (daily), =5 000 (daily), =5 000 000, =0, =5, =25, =93 000 000.

Table 7 indicates that the 5-year time charter contract with an embedded European option to buy
the vessel can take on a negative value. Generally, this is the case when the spot freight rate is
sufficiently low compared to the fixed time charter rate. This is a logical result since the spot freight
rate represents the cash inflow to the charterer, and the fixed time charter rate has to be paid; cash
inflow that is lower than the cash outflow creates a loss to the charterer, and the contract value
therefore becomes negative and unfavorable. This is also the reason why the contract value
decreases as the fixed time charter rate increases; higher rate which have to be paid will reduce the
contract value.

3.5 Applications of the Geometric Mean Reversion Process: Vessel and

European Option Valuation
As described in Section 2.6 where time charter contracts with embedded options were examined, the
options embedded in time charter contracts are path-dependent which imply that the payoff
depends on the path followed by the price of the underlying asset, not just its final value. In this case,
the payoff will depend on the path in which the freight rate follows over the contract period; the
freight rate represents the price of operating a vessel in the spot freight market. To the authors
knowledge there exist no analytic solutions to the Geometric Mean Reversion process. Thus, in order
to value both the vessel and the option to buy the vessel, numerical procedures must be applied. For
this purpose Monte Carlo simulation is implemented.
55

3.5.1

Monte Carlo Simulation

Monte Carlo simulation is a numerical procedure which is applied when analytic results do not exist
(Hull 2012). Especially, when the derivatives are path-dependent, as is the case for the options
embedded in the time charter contracts, Monte Carlo simulation is a highly popular tool. The idea
underlying Monte Carlo simulation is the feature of random sampling. When Monte Carlo simulation
is applied to the valuation of an option the risk-neutral valuation result is used; several paths are
sampled to obtain the expected payoff in a risk-neutral world and then the average payoff is
discounted at the risk-free rate in order to achieve the option value. An assumption of a constant
risk-free interest rate is done when applying Monte Carlo simulation, which fit perfectly in this case
since the risk-free interest rate was assumed to be constant in Section 3.2 where valuation of simple
freight rate contingent claims was introduced. The main drawback with Monte Carlo simulation is
that it is extremely time consuming in the achievement of the required level of accuracy. This will be
exemplified later.

3.5.2

The Value of a Vessel

This section will introduce the use of Monte Carlo simulation when valuing a vessel applying the
Geometric Mean Reversion process. The model adopted to the valuation is obtained from Tvedt
(1997). However, some simplifying assumptions will be done for consistency compared to the ones
done when investigating the Ornstein-Uhlenbeck process. The simplifying assumptions done will be
explained when proceeding. According to Tvedt (1997) the instantaneous cash flow from operation
and lay up until the vessel is scrapped, is given by:

(66)

where

is the time charter equivalent spot freight rate,

related costs20,

is the operation costs except for voyage

is the costs of keeping the vessel mothballed and

event , where

is an indicator function of the

. Keeping the vessel in operation is the optimal policy for the

shipowner when the spot freight rate plus lay up costs are above the operation costs. When this is
the case,

is equal to one and

. Otherwise,

is equal to zero and

. Again, for

ease of notation it is assumed that the spot freight rate is quoted for the entire vessel and net of all
20

Those are subtracted from the spot freight rate in order to obtain the time charter equivalent spot freight
rate.

56

costs. This results in the spot freight rate being the instantaneous net profits from an operating
vessel:

(67)

Further, Tvedt (1997) explains that when the vessel reaches its maximum age at time , its value
must be equal to the value of the vessel as scrap, . But if the value of the vessel as a going concern
is less than the demolition value, the vessel may be scrapped before the estimated end of its lifetime.
Formally, the termination date is equal to the stopping time given by:

(68)

where

is as defined below. Equation (68) indicates that the termination date is reached the first

time that the value of the vessel as a going concern is equal to or below the scrap value.
Finally, the value of a vessel at time can be presented. It is represented by the market value of the
cash flow generated from time to , and is given by:

(69)

where

is the risk-neutral probability measure. Equation (69) indicates that the vessel value equal

the discounted sum of the spot freight rates from time to , plus the discounted scrap value.
Again, to the authors knowledge, no analytic solutions exist to Equation (69). To calculate the vessel
value Monte Carlo simulation is therefore implemented in a Visual Basic for Applications (VBA)
function, the codes will be attached in the Appendix. In the following, the procedure will shortly be
explained.

57

First, vessel values for several points in time have to be calculated. Random numbers are therefore
generated from the standard normal distribution for the increment of the standard Wiener process,
, in the Geometric Mean Reversion process. The Geometric Mean Reversion process is then
applied to obtain thousand different values for the freight rate evolving from time to time

by

Monte Carlo simulation. Each simulated freight rate is in turn used in Equation (68) to calculate
several different vessel values in time . Finally, to obtain the vessel value in time , the vessel values
are averaged over the thousand simulated paths. The VBA code will be attached to the Appendix.
Vessel values as a function of different spot freight rates and varying remaining vessel life are
presented in the table below. Also here, the base case parameter values given in Table 2 are applied
and presented in the table description below. The number of yearly subdivisions21 is set to 200, and
the number of simulated paths is set to 1000.

Remaining Vessel Life (T-t)

5

10

15

20

25

X(t)
5 000

12 957 020

20 024 764

26 148 386

31 171 237

35 707 173

10 000

21 412 215

35 540 770

47 441 184

57 675 263

66 562 843

15 000

29 623 056

50 647 868

68 102 516

84 102 145

96 709 234

20 000

37 703 051

65 869 093

89 329 727

110 266 254

127 514 082

25 000

46 113 328

81 182 557

110 679 791

134 738 187

156 181 838

30 000

54 495 751

95 208 781

130 570 383

160 705 809

185 371 697

35 000

62 634 579

110 674 635

152 576 992

185 889 464

213 720 792

40 000

71 269 729

125 249 842

171 712 908

211 957 959

244 816 516

Table 8. The dependence of vessel values for different freight rates and different remaining vessel
lifetimes. =0,0033,

=10,45, =1,5%, =0,1184 and =5 000 000.

Table 8 clearly indicates that vessel values are an increasing function of both the spot freight rate and
the remaining vessel life. As is in line with intuition, the longer the remaining vessel lifetime, the
higher the vessel value. Also, higher spot freight rates indicate higher vessel value which is also in line

21

One year is divided into 200 equally time steps.

58

with intuition; higher spot freight rates generate higher cash inflow from owning a vessel, which in
turn will increase the value of owning this vessel.

3.5.3

European Option to Buy a Vessel

When an analytic solution does not exist, a concept called Monte Carlo on Monte Carlo simulation
can be used to calculate the value of a European option to buy a vessel. This is done by a new
simulation of freight rates using the Geometric Mean Reversion process. For each simulated spot
freight rate from time

to time vessel values are calculated using Equation (69) which is done by

the same procedure as the calculation of vessel values above. In time , the payoff function is given
by:

(70)

where

is the exercise price of the purchase option. For each vessel value in time the payoff

function is calculated. To find the option value in time , the discounted average of the payoff
functions obtained in time is calculated using the risk-neutral valuation result:

(71)

where is the risk-free interest rate which is assumed to be constant over the defined period. Also
this is done in VBA and the codes will be attached in the Appendix. Option values for different spot
freight rates and different volatilities are presented in Table 9 below. The number of yearly
subdivisions is again sat to 200, and the number of simulated paths has been sat to 500. The
parameter values are again obtained from Table 2 and are presented in the table description. The
exercise price is obtained from Jrgensen and Giovanni (2010).

59

Freight Rate Volatility,

0,1

0,3

0,5

0,7

0,9

Spot Freight
Rate, X(0)
5 000

183 775

702 012

1 097 792

260 950

10 000

3 028 103

5 536 125

8 553 248

5 832 592

15 000

6 379

6 169 871

11 481 198

1 921 091

855 705

20 000

216 678

8 900 324

33 333 346

14 663 615

11 260 378

25 000

1 415 484

15 739 370

38 285 950

17 309 911

26 778 091

30 000

2 970 948

20 728 850

16 371 935

86 278 677

27 623 172

Table 9. Value of European purchase option. Dependence on spot freight rate and freight rate
volatility. =0,0033,

=10,45, =1,5%, Maturity Option=15 years, Maturity Vessel=25 years,

=93 000 000, Scrap Value=5 000 000.

The European option value is expected to be an increasing function of both the spot freight rate and
the freight rate volatility. As the spot freight rate increases, the vessel will become more valuable due
to higher cash inflow, and thus the option to buy such a valuable vessel will also increase. When the
freight rate volatility increases the probability of large upside gains will increase, thus, the option
value will also increase. Most of the values in Table 9 are acting in line with what is expected, but
some values are somewhat strange. There can be several reasons for these strange results, but an
obvious reason is that extremely few paths are Monte Carlo simulated. A modest number of 500
simulations are done due to the limited abilities of Microsoft Office Excel22.
The efficiency of Monte Carlo simulation can be increased by implementing variance-reduction
techniques. The antithetic variates method is an example of such a technique; correlation is created
across simulated paths in order to reduce the variance of the sum of random variables (Campbell, Lo
et al. 1997). This technique ensures a doubled amount of simulated paths by utilizing the symmetry
present in the normal distribution. Each simulated path can be reflected through its mean to produce
a mirror-image with the same statistical properties (Campbell, Lo et al. 1997); a mirrored random
variable is created with exactly the same statistical properties, but with the opposite sign.

22

A simulation of 500 paths ran a whole night before it was done.

60

3.6 The Valuation Results: Comparisons

This section will shortly comment the vessel values and the European option values obtained from
both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process.
Important to note is that estimation procedures lie outside the scope of this thesis. This creates
difficulties when comparing the valuation results. For correct comparison the parameter values to
both models should have been estimated from the exact same dataset in order to ensure that they
matches relatively to each other. This may be the reason why the vessel valuations give ambiguous
results; it is expected that the Geometric Mean Reversion process will give higher vessel values as
negative values are impossible. Also comparisons of the option values becomes difficult.
By comparing the vessel values from the Ornstein-Uhlenbeck process and the Geometric Mean
Reversion process in Table 5 and Table 8 respectively, they clearly indicate quite similar results.
Generally, the vessel values obtained from the Ornstein-Uhlenbeck process lie above the values
obtained from the Geometric Mean Reversion process, which is ambiguous as the Geometric Mean
Reversion process is expected to give higher values. Also, as both the spot freight rate increases and
the remaining lifetime increases, so does the difference between the values also increase.
As for the European option values, comparison becomes difficult due to both the estimation issue,
and due to the reduced accuracy when obtaining values from the Geometric Mean Reversion
process. A modest amount of 500 paths are Monte Carlo simulated, which in fact is too few when
accurate results are desirable. However, the valuations serve as an introduction to further studies
where Finite Difference methods could have been implemented for more accurate results.

61

4 Limitations
When choosing a mathematical model for valuation purposes there will always be a trade-off
between analytical tractability and goodness of fit to observations. This is important to have in mind
when considering the valuation results during this thesis. This section will examine some of the
simplifications done when modeling freight rates which create a gap between the valuation results
and reality.

4.1 Limitations Caused by the Models Selected

By choosing the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process for
valuation purposes, important considerations going beyond the properties of the model will
automatically not be taken into account. Some of the considerations left behind when choosing
these two models will be discussed in this section.

4.1.1

The Parametric Property of the Models

Both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process are parametric in
nature; their distributions are assumed to belong to a parametric23 family (Jorion 2007), which in this
case is the normal distribution. In his article, Adland (2000) proposes a non-parametric model to
model the time charter equivalent spot freight rate by using monthly data over a period of ten years.
By doing this, misspecification of the density functions shape of the time charter equivalent spot
freight rates is avoided, and estimations may appear more accurate compared to estimations based
on a misspecified model. By estimating the marginal density of the time charter equivalent spot
freight rate he finds that the distribution is slightly skewed to the right. This result does not
correspond to the assumption of the time charter equivalent spot freight rates being normally
distributed in both the Ornstein-Uhlenbeck process and in the Geometric Mean Reversion process.
Adland (2000)s findings imply misspecification when assuming normally distributed time charter
equivalent spot freight rates. Subsequent to this, the results appearing in this thesis will need to be
considered with this possible misspecification in mind. But also Adland (2000)s results have
shortcomings; a non-parametric model opens for greater estimation error than parametric models
do, in addition he stresses that too few observations are obtained in order to get a statistically
confident estimate.

23

To obtain parameter values in parametric models estimation procedures are necessary (Jorion 2007).

62

4.1.2

One-Factor

The model adopted to the valuation of time charter contracts with purchase options is a one-factor
model with an assumption of the spot freight rate being the factor influencing the value of a vessel.
This assumption will disregard all information that is not embedded in the current spot freight rate
level and the dynamics of the spot freight rate process observed in the past. Important fundamental
market information that is expected to influence future freight rate dynamics is thus not taken into
account at all (Adland and Strandenes 2007). A gap is then created between theory and reality due to
the existence of several factors influencing vessel values.
According to Alizadeh and Nomikos (2009), factors influencing vessel values can be classified into two
groups; vessel-specific factors and market-specific factors. Vessel-specific factors generate a more
intuitive understanding of why vessel values vary as those are related to the particulars and
condition of each vessel. Examples of such factors are size, type, age and general condition. Marketspecific factors relates to the general state of the freight market. Those factors are more complex
than the vessel-specific factors. Current and expected freight rate levels and market conditions are
among the most important market-specific factors.
Stopford (2009) argues that the vessels age is the second most important factor in relation to vessel
values. As also the vessel valuation results from both models indicate, a newer vessel is worth more
than an older vessel. As a vessel gains age, it may lose performance and also higher maintenance
costs may appear. Thus, this will lead to a lower vessel value in total, and when its market value falls
below the scrap value the vessel is likely to be sold for scrapping. Due to the importance of the
vessels age, a multifactor model, where both the freight rate and the vessels age are taken into
account, could have been considered when valuing vessels. This would have included more realism
to the model.

63

4.2 The Assumptions

This section will address some of the simplifying assumptions done through this thesis, and describe
how more reality would have been included if they were not taken.

4.2.1

Constant Market Price of Freight Rate Risk

Since the risk-free interest rate have been applied as the discount factor the true probability
measure, , had to be transformed to the risk-neutral probability measure, . To do this
transformation Girsanovs Theorem was used, and the market price of freight rate risk, , was
introduced and integrated in the Ornstein-Uhlenbeck process. Further, the market price of interest
rate risk was assumed to be constant.
The market price of freight rate risk can be thought of as the difference between implied forward
freight rate24 and the expected spot freight rate (Adland 2003). Therefore, the value of the market
price of risk is closely related to the expectations of the term structure of freight rates in the future.
The assumption of this being constant has been disproved by Adland (2003). He presents qualitative
arguments and suggests that the market price of risk in the freight market in bulk shipping must be
time varying and depend on the state of the spot freight market and the duration of the time charter
in a systematic fashion. He argues for the existence of other factors than volatility in freight rates
that support the hypothesis of a non-constant market price of risk. Finally, he finds it reasonable to
consider the market price of risk as an increasing function of the spot freight rate level.
An earlier research also done by Adland (2000) supports his conclusions discussed above. This is a
quantitative based research which contribute with two separate results; that shipowners are not
compensated for the risk associated with trading in the spot freight market when freight rates are
low, and that the market price of risk is an increasing function of the freight rate level (Adland 2000).
The first result may appear due to the lack of considering other risk factors than the volatility of
freight rates when determining the market price of risk; when freight rates are low, the volatility of
freight rates is also low, and the risk associated with trading in the spot market may be considered as
small. The second result support his qualitative arguments discussed above that the market price of
risk is increasing in the freight rate level. This indicate that shipowners are compensated for bearing
freight rate risk when freight rates are at high levels; operating in the spot freight rate market is
more risky in this case since high freight rates implies high volatilities.

24

The freight rate evolvements in the future.

64

4.2.2

Constant Risk-Free Interest Rate

Throughout this thesis the risk-free interest rate is assumed to be constant. The risk-free interest rate
do in fact vary through time, thus this is another assumption made to simplify the calculations and
for the ability to apply the risk-neutral valuation result. To accommodate the shortcoming of
constant risk-free interest rate, one approach could be to model the time structure of the risk-free
interest rate outside the models. This could be done by using the Vasicek (1977) model which is
identical to the Ornstein-Uhlenbeck process. The estimated time structure could then be
implemented in the model as a simple variable.

65

5 Summary and Conclusions

The aim of this thesis was to shed light on the importance of fair valuation of embedded options in
time charter contracts, with options to purchase the vessel underlying the contract as the primary
area of research. As these options can be very complex in nature, they are often granted for free
rather than for their fair value. This creates misleading information of a shipping companys total net
asset value as the embedded options are of highly economic significance to the company. Also, large
volatility in freight rates can lead to large decreases or increases in the option values, which in turn
can affect the shipping companys viability. Thus, important to stress is also the proper management
of the risks a shipping company faces. Also, this thesis aimed to introduce two possible models for
the valuations where each of them was examined and their properties were compared. This gave an
indication of one model being more appropriate in freight rate modeling than the other.
First, the comprehensive shipping industry was introduced. The shipping market model that
describes the mechanisms controlling the shipping market cycles shortly described in Section 2.8.1,
was examined. Further, various costs and risks occurring in this industry was described, as well as a
description of the four existing markets. With the theories of the shipping industry in hand,
embedded options in time charter contracts were examined before a comprehensive description of
the two models could be established.
The first model introduced for valuation purposes was the Ornstein-Uhlenbeck process which has
been applied for valuation of time charter contracts with embedded purchase options. The solution
to this process was derived before the derivation of the partial differential equation that all freight
rate dependent claims must satisfy in order to be priced arbitrage free. Further, the partial
differential equation enabled for derivations of analytic solutions to simple freight rate contingent
claims - equal to the approach seen in Jrgensen and Giovanni (2010). The analytic solutions were
then applied to valuations of simple freight rate contingent claims, as well as vessel valuation,
European options to buy the vessel and finally, the total fair value of the time charter contract with
purchase options.
The second model introduced for valuation purposes was the Geometric Mean Reversion process. To
the authors knowledge no analytic solutions exist and numerical routines were therefore needed for
valuations. Vessel values and European options to purchase a vessel were valued by the application
of Monte Carlo simulation. The Geometric Mean Reversion process was introduced as an alternative
and more realistic valuation model. However, for valuation of time charter contracts with purchase
options, implementation of Finite Difference methods would have been necessary. This may be an
interesting area of further research.
66

The Ornstein-Uhlenbeck process gave analytic solutions such that time charter contracts with
purchase options for different fixed time charter rates and spot freight rates could be valued. The
results imply that the value of the time charter contract increase in the spot freight rate level, but
decrease in the fixed time charter rate level. This was also expected as higher spot freight rates imply
higher cash inflow and thus an increase in the value of the option to purchase the vessel. Whereas
higher fixed time charter rates creates a larger gap from the long-term mean of 14 371, and thus also
increasing the gap between the fixed time charter rate that would have been fair from the agreed
fixed time charter rate.
The Geometric Mean Reversion process facilitated for the use of Monte Carlo simulation. Vessel
values and European option values for purchasing the vessel were obtained. However, accuracy in
the results was reduced at the expense of an illustration of how Monte Carlo on Monte Carlo
simulation can be applied to obtain option values when analytic solutions do not exist. A larger
amount25 of simulated paths would have been beneficial in relation to more accurate results, but this
would have been very time consuming due to the shortcomings of Microsoft Office Excel.
The intention behind the introduction of both models was the opportunity for comparisons both
the model characteristics and the valuation results. The comparison of the valuation results
highlighted the weakness when lack of estimation is present. In order to obtain correct comparisons
between the valuation results obtained from both models, the parameter values need to match
relatively to each other. This could have been ensured by using the exact same dataset when
conducting the estimation procedure to both models, and at least the ambiguous vessel values
obtained could have been avoided.
The introduction of two models opened for an examination of model characteristics and a thoroughly
review of which model that would have been best fitted to freight rate modeling. The discussion
points in the direction of the Geometric Mean Reversion process being most appropriate, which is
due to the fact that the process is downwards restricted with zero as an absorbing level. With the
impossibility of freight rates being negative, this property becomes valuable compared to the
Ornstein-Uhlenbeck process which is not downwards restricted. Thus, this thesis ends by concluding
that the Geometric Mean Reversion process is more realistic in freight rate modeling compared to
the Ornstein-Uhlenbeck process.

25

A suggestion of 20 000 to 30 000 simulated paths would have been beneficial.

67

6 List of References
Adland, R. and K. Cullinane (2006). "The Non-Linear Dynamics of Spot Freight Rates in Tanker
Markets." Transportation Research: Part E 42(3): 211-224.
Adland, R. and S. Koekebakker (2007). "Ship Valuation Using Cross-Sectional Sales Data: A
Multivariate Non-Parametric Approach." Palgrave Macmillan Journals: 105-118.
Adland, R. and S. P. Strandenes (2007). "A Discrete-Time Stochastic Partial Equilibrium Model of the
Spot Freight Market." Journal of Transport Economics & Policy 41(2): 189-218.
Adland, R. O. (2000). "A Non-Parametric Model of the Timecharter-Equivalent Spot Freight Rate in
the Very Large Crude Oil Carrier Market." Foundation for Research in Economics and Business
Administration.
Adland, R. O. (2003). The Stochastic Behavior of Spot Freight Rates and the Risk Premium in Bulk
Shipping. The Department of Ocean Engineering, Massachusetts Institute of Technology. Ph.D in
Ocean Systems Management
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7 Appendix
The Excel spreadsheets where the tables and figures have been obtained are saved in a USB stick
enclosed to this paper.

7.1 The Ornstein-Uhlenbeck Process Detailed Solution

The freight rate at time

is given by todays freight rate plus the sum of the dynamics of the

spot freight rate evolving from time to time :

A temporary variable is introduced in order to solve the equation above explicitly:

The dynamics of this temporary variable are given by Itos lemma:

The Ornstein-Uhlenbeck process is substituted for

above:

70

Manipulations of the equation above give:

Continuing by integrating from time zero to time :

Calculating

Using this result in the continuation:

71

Calculating

Finally, the solution to the Ornstein-Uhlenbeck process becomes:

7.2 The Ornstein-Uhlenbeck process - Derivation of the Mean and the

Variance

7.2.1

The Time Conditional Mean

Taking the expectation of each term in the solution to the Ornstein-Uhlenbeck process:

Since the expectation of a Wiener process equals zero the mean of the process becomes:

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7.2.2

The Time Conditional Variance

Taking the variance of each term in the solution to the Ornstein-Uhlenbeck process:

Since the variance of the constants in the two first terms equals zero we are left with the following
expression:

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7.3 The Geometric Mean Reversion Process Detailed Solution

To simplify later calculations:

A temporary variable is introduced in order to solve the equation above explicitly:

The increments of this temporary variable are given by Itos lemma:

Rearranging:

74

Now, it is desirable to substitute the process for

and

are defined such that

. In order to ease the notifications

. This gives:

where the first and the second term tend to zero, and where the last term tends to

such that:

Then, by rearranging the manipulated Geometric Mean Reversion process to

and substituting the process for

The next step is to rearrange this equation and integrating it from time zero to time :

75

What is left now is to rearrange this equation such that

is alone:

76

Finally, the freight rate level at time ,

, can be expressed as:

7.4 The Ornstein-Uhlenbeck Process Claim to Receive Spot Freight Rate

Flow from Time to Time
Defining the current value of receiving the spot freight rate on a continuous basis as:

The freight rate process,

Inserting

in the expression for

77

Since the expectation of a standard Wiener process equals zero (Hull 2012), the last term equals zero
and only the two first terms are left. Further:

78

where

is an annuity factor.

79

7.5 The Ornstein-Uhlenbeck Process - European Option to Buy the Vessel

Define

and the value of a vessel is then given by:

The next step in the derivation of an analytic formula for the option value is to evaluate

+. Again, by following Jrgensen and Giovanni (2010) consider:

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where

is the density function that describes freight rates (Skovmand 2012). The density

function plots the shape of the distribution curve of the random variable that is considered, which is
the freight rate in this case. Since the freight rates are assumed to be normally distributed by their
density function plots the classic Bell Curve which takes the form (Skovmand 2012):

which in this case will look like this (Jrgensen and Giovanni 2010):

Continuing by inserting the expression for the density function gives:

Further, define

. Then:

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Finally, the time t value of the European call option is given by premultiplying the above result with
times the discount factor:

82

where:

and where

and

denote the standard normal cumulative probability and density functions,

respectively.

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7.6 The VBA Codes

84

85