World Academy of Science, Engineering and Technology

International Journal of Social, Behavioral, Educational, Economic, Business and Industrial Engineering Vol:8, No:4, 2014

Numerical Methods versus Bjerksund and Stensland

Approximations for American Options Pricing
Marasovic Branka, Aljinovic Zdravka, Poklepovic Tea

One of main issue about option is how to determine the

Abstract—Numerical methods like binomial and trinomial trees option price. The price of an option (like the price of a bond
and finite difference methods can be used to price a wide range of and the price of a stock) will depend on a number of factors.
options contracts for which there are no known analytical solutions. Some of these factors are the price of the underlying, the strike
American options are the most famous of that kind of options.
price, and the time left to maturity.
Besides numerical methods, American options can be valued with the
approximation formulas, like Bjerksund-Stensland formulas from Because the values of option contracts depend on a number
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1993 and 2002. When the value of American option is approximated of different factors they are complex to value. There are many
by Bjerksund-Stensland formulas, the computer time spent to carry pricing models in use today. First and the most popular model
out that calculation is very short. The computer time spent using for pricing European type of options is Black-Scholes-Merton
numerical methods can vary from less than one second to several model ([4], [11]). The American option can be exercised at
minutes or even hours. However to be able to conduct a comparative
any time up to its expiration date. This added freedom
analysis of numerical methods and Bjerksund-Stensland formulas, we
will limit computer calculation time of numerical method to less than complicates the valuation of American options relative to their
one second. Therefore, we ask the question: Which method will be European counterparts. With a few exceptions, it is not
most accurate at nearly the same computer calculation time? possible to find an exact formula for the value of American
options. Several researchers have, however, come up with
Keywords—Bjerksund and Stensland approximations, excellent closed-form approximations [1]-[3]. These
Computational analysis, Finance, Options pricing, Numerical approximations have become especially popular because they
methods.
execute more quickly on computers than the numerical
techniques.
I.INTRODUCTION
Numerical methods that can be used for evaluation of

O PTIONS are part of a larger class of financial instruments

known as derivative products, or simply, derivatives. A
derivative is an instrument whose value depends on values of
American options are binomial and trinomial trees and finite
difference methods. These methods are more flexible then
analytical solutions and can be used to price a wide range of
other more basic underlying variables. options contracts for which there are no known analytical
Option is a security that gives its owner the right, but not solutions including the American options.
the obligation, to buy or sell another, underlying security, The binomial method was first published by Cox, Ross and
simply called underlying, at or before a future predetermined Rubinstein [7] and Rendleman and Bartter [12]. Trinomial
date for a predetermined price. The option that provides its trees were introduced in option pricing by Boyle [5] and are
owner the right to buy is called a call option. The option that similar to binomial trees. The use of finite difference methods
provides its owner the right to sell is called a put option. If the in finance was first described by Brennan and Schwartz [6].
owner of the option can buy or sell on a given date only, the Finite difference methods, also called grid models, are simply
option is called a European option. If the option gives the right a numerical technique to solve partial differential equations.
to buy or sell up to (and including) a given date, it is called an The main objection to these methods is that the computing
American option. If the owner decides to buy or sell, we say time required for their algorithms is longer than for the
that the owner exercises the option. The date on which the analytical expressions. But with the development of computer
option can be exercised (or the last date on which it can be technology computers become faster and the computation time
exercised for American options) is called maturity or the is reduced significantly. The question arises of whether the
expiration date. The predetermined price at which the option price of American options obtained by numerical methods in a
can be exercised is called the strike price or the exercise price. short time (less than one second) is closer to the correct value
Simple puts and calls written on basic assets such as stocks of the option than the price obtained by an approximation
and bonds are common options, often called plain vanilla formula. This paper will try to give answers to this question by
options. There are many other types of options payoffs, and evaluating 280 American options by various numerical
they are usually referred to as exotic options. methods and Bjerksund and Stensland formulas for
approximation values of American options.
B. Marasović is with the Faculty of Economics, University of Split, 21000 The paper is organized as follows: following this
Split, Croatia (phone: 00385914430697; fax: 0038521430601; e-mail: introduction, in Section II, we describe the binomial and
branka.marasovic@ efst.hr).
Z. Aljinović and T. Poklepović are with the Faculty of Economics, trinomial model for valuing options. Section III presents the
University of Split, 21000 Split, Croatia (e-mail: zdravka.aljinovic@ efst.hr, applications of finite difference method in option pricing. In
tea.poklepovic@efst.hr).

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Section IV we describe Bjerksund and Stensland formulas for At each final node of the tree i.e. at expiration of the option
approximation values of American options. In Section V we the option value is simply its intrinsic, or exercise, value
conduct a comparative analysis of specified numerical Max [ ( S n − K ), 0 ], for a call option
methods and approximation formulas. Section VI summarizes Max [ ( K − S n ), 0 ], for a put option,
the paper and indicates the possible directions for further
where K is the strike price and S n is the spot price of the
research.
underlying asset at the n th period.
II. BINOMIAL AND TRINOMIAL MODEL FOR VALUING OPTIONS Once the above step is complete, the option value is then
found for each node, starting at the penultimate time step, and
A. Binomial Model
working back to the first node of the tree (the valuation date)
The procedure followed by binomial model is to assume where the calculated result is the value of the option.
that the stock price follows a discrete time process. The life of Under the risk neutrality assumption, today's fair price of a
the option T – t is decomposed into n equal time steps of derivative is equal to the expected value of its future payoff
length (∆t = (T – t)/n). At each time interval (tj = j·∆t, j = 0, 1, discounted by the risk free rate. Therefore, expected value is
..., n), it is assumed that the underlying instrument will move calculated using the option values from the later two nodes
International Science Index, Economics and Management Engineering Vol:8, No:4, 2014 waset.org/Publication/9997945

up or down by a specific factor ( u or d where, by definition (Option up and Option down) weighted by their respective
u ≥ 1 , and 0 < d ≤ 1 ) per step of the tree with probability p, 1- probabilities (probability p of an up move in the underlying,
p respectively . So, if S is the current price, then in the next and probability 1-p of a down move). The expected value is
period the price will either be Sup = S ⋅ u or S down = S ⋅ d . The then discounted at r, the risk free rate corresponding to the life
of the option.
binomial tree of stock’s price is best illustrated in a Fig. 1. The following formula to compute the expectation value is
applied at each node:

Ct −Δt ,i = e− r Δt ( pCt ,i +1 + (1 − p )Ct ,i )

where Ct ,i is the option's value for the i th node at time t.

This result is the “Binomial Value”. It represents the fair
price of the derivative at a particular point in time (i.e. at each
node), given the evolution in the price of the underlying asset
to that point. It is the value of the option if it were to be held—
as opposed to exercised at that point.
For an American option, since the option may either be held
or exercised prior to expiry, the value at each node is: Max
(Binomial Value, Exercise Value). The value of the initial
Fig. 1 Binomial tree
node presents the required fair price of the option.
B. Trinomial Model
The up and down jump factors and corresponding Under the trinomial model, in each period, the prices can go
probabilities are chosen to match the first two moments of the up, down or remain unchanged. The term "lattice" implies two
stock price distribution (mean and variance). There are, or more branches protruding from the node of a tree. In the
however, more unknowns than there are equations in this set case of a binomial lattice there are two branches, three in the
of restrictions, implying that there are many ways of choosing case of a trinomial, and so on. Where there are more than two
the parameters and still satisfy the moment restrictions. Cox, branches, the lattice can be called a multinomial lattice.
Ross and Rubinstein [7] set the up and down parameters to A trinomial lattice works on the same principles as the
binomial lattice, but assumes that the prices may also remain
u = eσ Δt
, d = e −σ Δt
, constant. So in the first step, the prices may go up, down or
remain unchanged. For each of the three outcomes, there will
where σ is volatility of the relative price change of the be three outcomes each in the second time step, but the second
underlying stock price. The probability of the stock price outcome of the first node in the second step will be the same
increasing at the next time step is: as the first outcome of the second node in the second step and
so on.
e r Δt − d The expected results are attained much faster, as the
p= , branches become intractable at a much earlier period of time.
u−d
Trinomial trees can be used as an alternative to binomial trees,
where r is risk-free interest rate. where there are numerous time steps. It is to be noted that the

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International Journal of Social, Behavioral, Educational, Economic, Business and Industrial Engineering Vol:8, No:4, 2014

trinomial tree computation procedure is exactly the same as chosen finite difference techniques, as well as derivative’s
for the binomial model. contractual details. Here we will give an overview of the three
As the name suggests, trinomial model uses a similar most common finite difference techniques in option pricing:
approach to be binomial one. But the hedging and replication explicit finite difference method, implicit finite difference
arguments do not take place in constructing trinomial trees. method and Crank-Nicolson finite difference. The explicit
For a non - dividend paying stock, parameter values that finite difference method is more or less a generalization of the
match the mean and standard deviation of price changes are trinomial tree. The method approximates the PDE in (1) by
given below: ∂f
using numerical differentiation (see [10]). The is
∂t
u = eσ 3Δt , d = 1/ u , approximated by using the forward difference (naturally,
because time can only move forward):
Δt ⎛ 1 ⎞ 1
pu = 2 ⎜
r − σ2 ⎟+
12σ ⎝ 2 ⎠ 6, ∂f f j +1,i − f j ,i
≈
Δt ⎛ 1 2⎞ 1 ∂t Δt
pd = − ⎜r − 2σ ⎟+ 6
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12σ 2 ⎝ ⎠ , where f j ,i is the value of the derivative instrument at time

pm = 1 − pu − pd ,
step j and price level i .
The delta and the gamma approximated by central
where u, d and r have the same meaning as in binomial differences (the asset price can naturally move in both
model, σ is stock volatility, while pu , pd and pm denote directions):
probabilities of the price going up, down or remaining
unchanged, respectively. ∂f f j +1,i +1 − f j +1,i −1
Once the tree of prices has been calculated, the option price ≈
∂S 2ΔS
is found at each node largely as for the binomial model, by
∂2 f f j +1,i +1 − 2 f j +1,i + f j +1,i −1
working backwards from the final nodes to today. The ≈ .
difference being that the option value at each non-final node is ∂S 2 ΔS 2
determined based on the three (as opposed to two) later nodes
and their corresponding probabilities. The implicit finite difference method is closely related to
the explicit finite difference method. The main difference is
III.FINITE DIFFERENCE METHOD IN OPTION PRICING ∂f ∂2 f
that we approximate and in PDE (1) by central
The finite difference method is basically a numerical ∂S ∂S 2
approximation of the partial difference equation. Here we will differentiation at time step j instead at j + 1 as in the explicit
give overview of the three most common finite difference finite difference method.
techniques in option pricing: explicit finite difference, implicit In Crank-Nicolson method the approximation of the PDE is
finite difference, Crank-Nicolson finite difference. In all the 1
finite difference models first we built a grid with time along done by central differences at time step j + instead of at
2
one dimension/axis and price along the other dimension/axis.
j + 1 as in explicit finite difference method, or at point j as
Time increases in increments of Δt , while the asset changes in
amount of ΔS . These increments are then used to construct a in the implicit finite difference method.
As we can see, the Crank-Nicolson method is a combination
grid of possible combinations of time and asset price levels.
of the explicit and implicit methods. It is more efficient than
The finite difference technique is then used to approximately
the others. In combination with the same boundary conditions
solve the relevant PDE on this grid. Just as in a tree model one
as in the implicit finite difference method, the Crank-Nicolson
starts at the end of the grid at time T, and rolls back through
method will make up a tridiagonal system of equations. For an
the grid. The finite difference models can be used to solve a
in-depth discussion of the Crank-Nicolson method applied to
large class of options. If we assume that the underlying asset
derivatives valuation, see [13].
follows a geometric Brownian motion, we get the following
Black-Scholes-Merton PDE ([4], [11]) for any single asset
IV. THE BJERKSUND AND STENSLAND (1993) AND (2002)
derivatives:
APPROXIMATION
∂f 1 2 2 ∂ 2 f ∂f The Bjerksund and Stensland, 1993 approximation can be
+ σ S + (r − q) S − rf = 0 (1) used to price American options on stocks, futures and
∂t 2 ∂S 2 ∂S
currencies. Bjerksund and Stensland's approximation is based
on an exercise strategy corresponding to a flat boundary I
where f is the value of a derivative security. We want to solve (trigger price).
this PDE along the grid for the particular derivative instrument Given this feasible but non-optimal strategy, the American
under consideration. How this is done will depend on the call boils down to: (i) a European up-and-out call with knock-

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out barrier I, strike K, and maturity date T; and (ii) a rebate I- that is valid from date t to date T, where 0 < t < T . Their
K that is received at the knock-out date if the option is American call approximation is:
knocked out prior to the maturity date.
Their American call approximation is c = α 2 S β − α 2φ ( S , t1 , β , I 2 , I 2 ) + φ ( S , t1 ,1, I 2 , I 2 ) − φ ( S , t1 ,1, I1 , I 2 ) −
− K φ ( S , t1 , 0, I 2 , I 2 ) + K φ ( S , t1 , 0, I1 , I 2 ) + α1φ ( S , t1 , β , I1 , I 2 ) −
c = α S β − αφ ( S , T , β , I , I ) + φ ( S , T ,1, I , I ) − φ ( S , T ,1, K , I ) −
− α1Ψ ( S , T , β , I1 , I 2 , I1 , t1 ) + Ψ ( S , T ,1, I1 , I 2 , I1 , t1 ) −
− Kφ ( S , T , 0, I , I ) + Kφ ( S , T , 0, K , I ) ,
− Ψ ( S , T ,1, K , I 2 , I1 , t1 ) − K Ψ ( S , T , 0, I1 , I 2 , I1 , t1 ) +
where + Ψ ( S , T , 0, K , I 2 , I1 , t1 ) ,
α = (I − K ) I −β
,
2
where:
⎛1 b ⎞ ⎛ b 1⎞ r
β =⎜ − 2 ⎟+ ⎜ 2 − ⎟ +2 2 . ⎛1 b ⎞ ⎛ b 1⎞ r
2

⎝2 σ ⎠ ⎝σ 2⎠ σ α1 = ( I1 − K ) I1− β , α 2 = ( I 2 − K ) I 2 − β , β = ⎜ − + − +2 2 .
⎝2 σ 2 ⎟⎠ ⎜⎝ σ 2 2 ⎟⎠ σ
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The function φ ( S , T , γ , H , I ) is given by:

The function φ ( S , T , γ , H , I ) is given by:

⎡ ⎛ ⎛I ⎞ ⎞⎤
⎢ ⎜ 2 ln ⎜ ⎟ ⎟⎥ ⎡ ⎛I⎞
κ
⎤
φ ( S , T , γ , H , I ) = eλ S γ ⎢ N ( − d ) − ⎜ N ( −d 2 )⎥ ,
κ
⎛I ⎞ ⎝S ⎠ ⎟⎥ , ⎟
φ ( S,T ,γ , H , I ) = e S N (d ) − ⎜ ⎟ N ⎜ d −
λ γ ⎢
⎢⎣ ⎝S⎠ ⎥⎦
⎢ ⎝ ⎠
S ⎜ σ T ⎟⎥
⎢ ⎜ ⎟⎥ 1
⎣⎢ ⎝ ⎠ ⎦⎥ λ = −r + γ b + γ ( γ − 1) σ 2 ,
2
⎡ 1 ⎤
λ = ⎢ −r + γ b + γ ( γ − 1) σ 2 ⎥ T , ⎛S ⎞ ⎡ ⎛ 1⎞ ⎤
⎣ 2 ⎦ ln ⎜ ⎟ + ⎢b + ⎜ γ − ⎟ σ 2 ⎥ T
⎝H⎠ ⎣ ⎝ 2⎠ ⎦
⎛S ⎞ ⎡ ⎛ 1⎞ 2⎤ d= ,
ln ⎜ ⎟ + ⎢b + ⎜ γ − 2 ⎟ σ ⎥ T σ T
⎝H ⎠ ⎣ ⎝ ⎠ ⎦ 2b
d= , κ = 2 + ( 2γ − 1) , ⎛ I2 ⎞ ⎡ ⎛ 1⎞ 2⎤
σ T σ ln ⎜ ⎟ + ⎢b + ⎜ γ − ⎟ σ ⎥ T
SH ⎠ ⎣ ⎝ 2⎠ ⎦ 2b
d2 = ⎝ , κ = 2 + ( 2γ − 1) ,
and the trigger price I is defined as σ T σ

(
I = B0 + ( B∞ − B0 ) 1 − e
h (T )
), The trigger price I is defined as:

(
⎛ B0
h (T ) = − bT + 2σ T ⎜ )
⎞
⎟ , B0 =
β
K, ( )
I1 = B0 + ( B∞ − B0 ) 1 − eh1 ,
⎝ B∞ − B0 β −1
⎠ I 2 = B0 + ( B − B ) (1 − e ) ,
∞ 0
h2

⎧ r ⎫
B∞ = max ⎨ K , K⎬. ⎛ ⎞
⎩ r −b ⎭ (
h1 = − bt1 + 2σ t1 ⎜
K2
)
⎜ ( B∞ − B0 ) B0 ⎟⎟
,
⎝ ⎠
If S>I, it is optimal to exercise the option immediately and ⎛ ⎞
the value must be equal to the intrinsic value of S-X. On the (
h2 = − bT + 2σ T ⎜
K2
)
⎜ ( B∞ − B0 ) B0 ⎟⎟
,
other hand, if b ≥ r , it will never be optimal to exercise the ⎝ ⎠
American call option before expiration, and the value can be
found using Black-Scholes formula [4]. The value of the
t1 =
1
2
5 −1 T ,( )
American put is given by Bjerksund and Stensland put-call β
transformation: B0 = K,
β −1
p ( S , K , T , r , b, σ ) = c ( S , K , T , r − b, − b , σ ) ⎧ r ⎫
B∞ = max ⎨ K , K⎬.
⎩ r − b ⎭
The Bjerksund and Stensland, 2002 approximation divides
the time to maturity into two parts, each with a separate flat Moreover, the function Ψ ( S , T , γ , H , I 2 , I1 , t1 , r , b, σ ) is
exercise boundary. They extend the flat boundary given by:
approximation above by allowing for one flat boundary I1
that is valid from date 0 to date t, and another flat boundary I 2

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Ψ ( S , T , γ , H , I 2 , I1 , t1 , r , b, σ ) = eλT S γ ⋅ takes values of the interval [ 0.05,1] , and the current price of
⎡ ⎛ the stock values are taken from the interval [50,180] .
κ
t ⎞ ⎛I ⎞ ⎛ t ⎞
⋅ ⎢ M ⎜ −e1 , − f1 , 1 ⎟ − ⎜ 2 ⎟ M ⎜ −e2 , − f 2 , 1 ⎟ −
⎢⎣ ⎝⎜ ⎟
T ⎠ ⎝S ⎠ ⎜ T ⎟⎠
⎝ The option values obtained by the analysis are given in
κ
⎛ t ⎞ ⎛I ⎞
κ
⎛ t ⎞⎤
Tables I-VI. In applying the binomial and trinominal model, as
⎛I ⎞
− ⎜ 1 ⎟ M ⎜ −e3 , − f3 , 1 ⎟ + ⎜ 1 ⎟ M ⎜ −e4 , − f 4 , 1 ⎟ ⎥ , well as the Crank-Nicolson finite difference method, the
⎝S⎠ ⎜ ⎟
T ⎠ ⎝ I2 ⎠ ⎜ T ⎟⎠ ⎦⎥
⎝ ⎝ biggest number (rounded to the tens) was taken for the number
of periods, for which computer computation is less than one
where M (⋅, ⋅, ⋅) cumulative bivariate normal distribution and second.
The main aim is to find out whether the errors in the
⎛S⎞ ⎡ ⎛ 1 ⎞ 2⎤ ⎛ I 22 ⎞ ⎡ ⎛ 1 ⎞ 2⎤ observed methods differ significantly.
ln ⎜ ⎟ + ⎢b + ⎜ γ − ⎟ σ ⎥ t1 ln ⎜ ⎟ + ⎢b + ⎜ γ − ⎟ σ ⎥ t1
⎝ 2⎠ ⎦ ⎝ 2⎠ ⎦ For this purpose, we will apply the Friedman non-
e1 = ⎝ 1 ⎠ ⎣ , e2 = ⎝ 1 ⎠ ⎣
I SI
, parametric test.
σ t1 σ t1
This test is used for more than two dependent variable
⎛S⎞ ⎡ ⎛ 1⎞ ⎤ ⎛ I2 ⎞ ⎡ ⎛ 1⎞ ⎤
ln ⎜ ⎟ − ⎢b + ⎜ γ − ⎟ σ 2 ⎥ t1 ln ⎜ 2 ⎟ − ⎢b + ⎜ γ − ⎟ σ 2 ⎥ t1 samples measured using the sequence scale. The following
International Science Index, Economics and Management Engineering Vol:8, No:4, 2014 waset.org/Publication/9997945

⎝ 2⎠ ⎦ ⎝ 2⎠ ⎦
e3 = ⎝ 1 ⎠ ⎣ , e4 = ⎝ 1 ⎠ ⎣
I SI hypotheses are set:
,
σ t1 σ t1 H0. There is no difference in the rank of model errors,
H1. There is a difference in the rank of model errors.
⎛ S ⎞ ⎡ ⎛ 1 ⎞ 2⎤ ⎛ I 22 ⎞ ⎡ ⎛ 1 ⎞ 2⎤ Fig. 2 indicates the results of the conducted Friedman test.
ln ⎜ ⎟ + ⎢b + ⎜ γ − ⎟ σ ⎥ T ln ⎜ ⎟ + ⎢b + ⎜ γ − ⎟ σ ⎥ T
⎝H⎠ ⎣ ⎝ 2⎠ ⎦ ⎝ SH ⎠ ⎣ ⎝ 2⎠ ⎦ Friedman test was used to test the differences in the error
f1 = , f2 = ,
σ T σ T ranks for all five models based on the results obtained for the
option offer. The obtained results show that in both cases there
⎛ I ⎞ ⎡ ⎛
2
1⎞ ⎤ ⎛ SI ⎞ ⎡ ⎛
2
1⎞ ⎤ is a difference in ranks of error for the observed models, i.e.
ln ⎜ 1 ⎟ + ⎢b + ⎜ γ − ⎟ σ 2 ⎥ T ln ⎜ 12 ⎟ + ⎢b + ⎜ γ − ⎟ σ 2 ⎥ T
SH ⎠ ⎣ ⎝ 2⎠ ⎦ ⎝ 2⎠ ⎦ the initial hypothesis H 0 is rejected.
f3 = ⎝ , f4 = ⎝ 2 ⎠ ⎣
HI
.
σ T σ T The binomial model has shown to be the best, followed by
the trinomial, Crank-Nicolson finite difference method, and
The computer code for Bjerksund and Stensland, 1993 and Bjerksund-Stensland model 2002, with the Bjerksund-
2002 American option approximation is taken from [8]. Stensland model 1993 taking the last position.

V.COMPARISON OF NUMERICAL METHODS AND BJERKSUND -

STENSLAND APPROXIMATIONS
Comparative analysis of observed models will be carried
out by their application to pricing American put options on
nondividend-paying stocks. We will compare the Bjerksund
and Stensland 1993 and 2002 approximation with binomial
model, trinomial model and Crank-Nicolson finite difference
method. We will limit computer calculation time of numerical
method to less than one second, which nearly corresponds to
the calculation time of the Bjerksund and Stensland
approximation.
Since there is no formula that can calculate the exact value
of American options offer, for the calculation of reference
value, we will use trinomial model with a very large number
of steps (5000 steps) that achieves high precision and the
resulting value can be considered accurate. The calculation of
the reference value using trinomial model in this analysis
required over 100 hours of computer processing. The values
obtained by the observed models are compared with the
reference values. Errors of each particular model will be
represented by the absolute value of the difference between
the values obtained by the observed model and the reference
value (on the same way as in [9]).
The survey is conducted by evaluating 280 American
options with the exercise price of 150, and the volatility of
25%, with a risk-free interest rate of 6%. Time to maturity

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International Journal of Social, Behavioral, Educational, Economic, Business and Industrial Engineering Vol:8, No:4, 2014

TABLE I
EVALUATING THE AMERICAN PUT OPTIONS FROM THE SAMPLE USING THE TRINOMINAL MODEL (N=5000)
Asset price
120 130 140 150 160 170 180
0,05 30,00000 20,00000 10,19847 3,15076 0,47987 0,03346 0,00110
0,1 30,00000 20,00000 10,73557 4,34832 1,25706 0,25606 0,03739
Time to
maturity 0,15 30,00000 20,00815 11,26361 5,22684 1,95552 0,58906 0,14462
0,2 30,00000 20,07938 11,74576 5,94098 2,57269 0,95582 0,30776
0,25 30,00000 20,19474 12,18433 6,55056 3,12455 1,32659 0,50549
0,3 30,00000 20,33329 12,58522 7,08619 3,62410 1,68963 0,72317
0,35 30,00000 20,48377 12,95450 7,56603 4,08074 2,04012 0,95110
0,4 30,00000 20,63969 13,29662 8,00188 4,50215 2,37668 1,18346
0,45 30,00000 20,79738 13,61572 8,40195 4,89336 2,69902 1,41655
0,5 30,00000 20,95455 13,91454 8,77218 5,25892 3,00771 1,64802
0,55 30,00252 21,10980 14,19563 9,11706 5,60216 3,30326 1,87626
International Science Index, Economics and Management Engineering Vol:8, No:4, 2014 waset.org/Publication/9997945

0,6 30,01404 21,26219 14,46096 9,44008 5,92563 3,58663 2,10051

0,65 30,03380 21,41166 14,71236 9,74400 6,23132 3,85824 2,32038
0,7 30,05967 21,55743 14,95144 10,03107 6,52212 4,11939 2,53533
0,75 30,09046 21,69955 15,17871 10,30311 6,79823 4,37049 2,74496
0,8 30,12525 21,83806 15,39616 10,56167 7,06211 4,61209 2,94982
0,85 30,16341 21,97301 15,60362 10,80805 7,31371 4,84520 3,14959
0,9 30,20373 22,10444 15,80301 11,04336 7,55549 5,07012 3,34429
0,95 30,24636 22,23236 15,99396 11,26855 7,78698 5,28714 3,53410
1 30,29034 22,35679 16,17769 11,48444 8,00939 5,49662 3,71918

TABLE II
EVALUATING THE AMERICAN PUT OPTIONS FROM THE SAMPLE USING THE BJERKSUND-STENDSLAND (1993) MODEL
Asset price
120 130 140 150 160 170 180
0,05 30,00000 20,00000 10,17537 3,13141 0,47725 0,03334 0,00110
0,1 30,00000 20,00000 10,68140 4,30999 1,24605 0,25417 0,03717
0,15 30,00000 20,00110 11,18646 5,17112 1,93365 0,58305 0,14337
0,2 30,00000 20,04965 11,65041 5,86954 2,53929 0,94409 0,30431
0,25 30,00000 20,14402 12,07379 6,46493 3,07945 1,30789 0,49888
0,3 30,00000 20,26491 12,46203 6,98779 3,56752 1,66314 0,71235
0,35 30,00000 20,40066 12,82041 7,45614 4,01341 2,00560 0,93546
0,4 30,00000 20,54434 13,15332 7,88169 4,42440 2,33399 1,16250
0,45 30,00000 20,69174 13,46431 8,27250 4,80604 2,64827 1,38990
Time to 0,5 30,00000 20,84025 13,75623 8,63444 5,16257 2,94899 1,61546
maturity 0,55 30,00000 20,98821 14,03140 8,97191 5,49736 3,23691 1,83781
0,6 30,00276 21,13457 14,29174 9,28830 5,81309 3,51284 2,05612
0,65 30,01390 21,27867 14,53882 9,58632 6,11196 3,77760 2,26989
0,7 30,03210 21,42009 14,77398 9,86813 6,39579 4,03198 2,47887
0,75 30,05606 21,55859 14,99835 10,13552 6,66610 4,27668 2,68295
0,8 30,08470 21,69403 15,21289 10,38996 6,92418 4,51238 2,88210
0,85 30,11716 21,82635 15,41844 10,63273 7,17113 4,73967 3,07639
0,9 30,15273 21,95554 15,61573 10,86486 7,40789 4,95910 3,26589
0,95 30,19080 22,08162 15,80538 11,08730 7,63529 5,17117 3,45073
1 30,23091 22,20465 15,98797 11,30081 7,85406 5,37634 3,63104

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TABLE III
EVALUATING THE AMERICAN PUT OPTIONS FROM THE SAMPLE USING THE BJERKSUND-STENSLAND (2002) MODEL
Asset price
120 130 140 150 160 170 180
0,05 30,00000 20,00000 10,18504 3,13842 0,47776 0,03335 0,00110
0,1 30,00000 20,00000 10,70275 4,32352 1,24882 0,25446 0,03719
0,15 30,00000 20,00192 11,21543 5,19049 1,93969 0,58423 0,14353
0,2 30,00000 20,05988 11,68517 5,89411 2,54901 0,94675 0,30488
0,25 30,00000 20,16173 12,11320 6,49413 3,09293 1,31248 0,50015
0,3 30,00000 20,28857 12,50527 7,02110 3,58470 1,66997 0,71462
0,35 30,00000 20,42911 12,86686 7,49311 4,03413 2,01486 0,93899
0,4 30,00000 20,57670 13,20248 7,92190 4,44849 2,34576 1,16748
0,45 30,00000 20,72732 13,51576 8,31560 4,83327 2,66257 1,39649
Time to 0,5 30,00000 20,87850 13,80963 8,68009 5,19272 2,96580 1,62377
maturity 0,55 30,00000 21,02868 14,08646 9,01982 5,53021 3,25616 1,84790
International Science Index, Economics and Management Engineering Vol:8, No:4, 2014 waset.org/Publication/9997945

0,6 30,00535 21,17689 14,34819 9,33821 5,84842 3,53445 2,06803

0,65 30,01939 21,32254 14,59644 9,63798 6,14958 3,80147 2,28363
0,7 30,04016 21,46525 14,83258 9,92134 6,43550 4,05800 2,49442
0,75 30,06638 21,60482 15,05775 10,19008 6,70771 4,30474 2,70028
0,8 30,09703 21,74114 15,27295 10,44570 6,96752 4,54235 2,90117
0,85 30,13127 21,87418 15,47901 10,68948 7,21604 4,77144 3,09714
0,9 30,16841 22,00394 15,67670 10,92249 7,45421 4,99255 3,28826
0,95 30,20787 22,13047 15,86664 11,14566 7,68289 5,20618 3,47466
1 30,24920 22,25384 16,04942 11,35980 7,90280 5,41280 3,65646

TABLE IV
EVALUATING THE AMERICAN PUT OPTIONS FROM THE SAMPLE USING THE BINOMINAL MODEL (N=350)
Asset price
120 130 140 150 160 170 180
0,05 30,00000 20,00000 10,19905 3,14915 0,47988 0,03310 0,00108
0,1 30,00000 20,00000 10,73589 4,34628 1,25886 0,25574 0,03726
0,15 30,00000 20,00773 11,26556 5,22452 1,95659 0,58774 0,14459
0,2 30,00000 20,07851 11,74720 5,93846 2,57603 0,95751 0,30741
0,25 30,00000 20,19513 12,18542 6,54787 3,12583 1,32843 0,50610
0,3 30,00000 20,33354 12,58790 7,08335 3,62837 1,69249 0,72422
0,35 30,00000 20,48266 12,95861 7,56305 4,07923 2,03889 0,95270
0,4 30,00000 20,64152 13,29634 7,99880 4,50425 2,38003 1,18397
0,45 30,00000 20,79536 13,61625 8,39877 4,89911 2,69582 1,41589
Time to 0,5 30,00000 20,95667 13,91870 8,76892 5,26463 3,01202 1,65048
maturity 0,55 30,00000 21,11104 14,20110 9,11371 5,60535 3,30652 1,87639
0,6 30,01409 21,26105 14,46596 9,43666 5,92498 3,58333 2,10171
0,65 30,03241 21,41378 14,71539 9,74051 6,22770 3,86069 2,32419
0,7 30,05792 21,56030 14,95113 10,02750 6,52274 4,12527 2,53458
0,75 30,09084 21,70058 15,17562 10,29946 6,80229 4,37606 2,74487
0,8 30,12435 21,83530 15,39544 10,55796 7,06798 4,61475 2,95430
0,85 30,16074 21,97351 15,60558 10,80426 7,32124 4,84284 3,15380
0,9 30,20302 22,10717 15,80654 11,03948 7,56311 5,06833 3,34467
0,95 30,24708 22,23600 15,99896 11,26460 7,79462 5,29013 3,52932
1 30,29083 22,36017 16,18358 11,48044 8,01666 5,50280 3,72087

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TABLE V
EVALUATING THE AMERICAN PUT OPTIONS FROM THE SAMPLE USING THE TRINOMINAL MODEL (N=100)
Asset price
120 130 140 150 160 170 180
0,05 30,00000 20,00000 10,19825 3,14746 0,47911 0,03304 0,00103
0,1 30,00000 20,00000 10,73506 4,34383 1,25827 0,25513 0,03699
0,15 30,00000 20,00346 11,25985 5,22140 1,95855 0,58726 0,14413
0,2 30,00000 20,07530 11,74857 5,93480 2,56940 0,95335 0,30810
0,25 30,00000 20,19105 12,18049 6,54370 3,13005 1,32292 0,50548
0,3 30,00000 20,33235 12,58898 7,07865 3,62876 1,69367 0,72438
0,35 30,00000 20,47753 12,95696 7,55781 4,07649 2,03364 0,95214
0,4 30,00000 20,63658 13,29196 7,99310 4,50051 2,38195 1,18100
0,45 30,00000 20,78916 13,60944 8,39265 4,89876 2,70119 1,41891
Time to 0,5 30,00000 20,95150 13,91357 8,76237 5,26711 2,99895 1,64913
maturity 0,55 30,00000 21,10956 14,19801 9,10670 5,61026 3,30600 1,87051
International Science Index, Economics and Management Engineering Vol:8, No:4, 2014 waset.org/Publication/9997945

0,6 30,00000 21,25993 14,46479 9,42915 5,93174 3,59367 2,10467

0,65 30,01188 21,40273 14,71600 9,73249 6,23434 3,86434 2,32448
0,7 30,04478 21,55168 14,95357 10,01905 6,52075 4,12035 2,53260
0,75 30,08104 21,69705 15,17882 10,29062 6,79247 4,36384 2,73814
0,8 30,11786 21,83681 15,39303 10,54880 7,05161 4,60438 2,95212
0,85 30,15865 21,97092 15,59722 10,79485 7,30321 4,84506 3,15566
0,9 30,20119 22,09973 15,79234 11,02985 7,54903 5,07466 3,34992
0,95 30,24326 22,22363 15,97915 11,25474 7,78412 5,29440 3,53604
1 30,28451 22,34286 16,16452 11,47026 8,00937 5,50504 3,71516

TABLE VI
EVALUATING THE AMERICAN PUT OPTIONS FROM THE SAMPLE USING THE CRANK-NICOLSON MODEL (N=150; M=150)
Asset price
120 130 140 150 160 170 180
0,05 30,00000 20,00000 10,19673 3,14138 0,47926 0,03422 0,00120
0,1 30,00000 20,00000 10,72998 4,33476 1,25531 0,25698 0,03847
0,15 30,00000 20,00122 11,25180 5,21001 1,95298 0,58814 0,14649
0,2 30,00000 20,07238 11,73842 5,92137 2,56189 0,95233 0,31046
0,25 30,00000 20,18465 12,16851 6,52851 3,12039 1,32056 0,50739
0,3 30,00000 20,32522 12,57519 7,06189 3,61778 1,68956 0,72539
0,35 30,00000 20,46933 12,94124 7,53969 4,06389 2,02739 0,95191
0,4 30,00000 20,62565 13,27441 7,97367 4,48572 2,37461 1,17998
0,45 30,00000 20,77601 13,59137 8,37199 4,88287 2,69297 1,41632
Time to 0,5 30,00000 20,93751 13,89462 8,74057 5,25039 2,98835 1,64589
maturity 0,55 30,00000 21,09411 14,17784 9,08385 5,59274 3,29428 1,86526
0,6 30,00000 21,24302 14,44345 9,40527 5,91339 3,58133 2,09867
0,65 30,00188 21,38453 14,69358 9,70765 6,21515 3,85124 2,31784
0,7 30,02923 21,53359 14,92995 9,99327 6,50055 4,10633 2,52499
0,75 30,06405 21,67833 15,15414 10,26399 6,77126 4,34870 2,72824
0,8 30,10029 21,81724 15,36731 10,52135 7,02897 4,58666 2,94170
0,85 30,14155 21,95049 15,57050 10,76664 7,27844 4,82683 3,14474
0,9 30,18461 22,07841 15,76461 11,00088 7,52367 5,05602 3,33846
0,95 30,22723 22,20131 15,95042 11,22495 7,75824 5,27525 3,52395
1 30,26906 22,31957 16,13626 11,43970 7,98308 5,48541 3,70209

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Fig. 2 Results of the Friedman test for the error sample obtained by the Bjerksund-Stensland 1993 model, Bjerksund-Stensland 2002 model,
binominal, trinominal and Crank-Nicolson model in evaluating American options
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VI.CONCLUSION [13] Wilmott, P., “Paul Wilmott on Quantitative Finance,” John Wiley &
Sons, New York, 2000.
Taking into account the development of computer
technology, i.e. architecture improvements and the increased
speed of the new computer models, it is clear that the
calculation accuracy of numerical methods in the same time
period will be significantly higher on the modern computers
than it was at the time when Bjerksund-Stensland models were
published. The results of this study confirmed our assumptions
and proved that the numerical methods provide a greater
precision of calculations when compared to the Bjerksund-
Stensland model if the computation time is limited to one
second. Out of the set of numerical methods presented for the
evaluation of plain vanilla American options, it was the
binomial model that proved to be the most precise, followed
by the trinomial model and the Crank-Nicolson finite
difference method.

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