A Commentary by Harley Bassman:

The Convexity Maven

Not a Product of Credit Suisse Research Value Concepts from the Credit Suisse Trading Desk
For Distribution to Institutional Clients Only August 27, 2013



“The FVA: Forward Volatility Agreement”
Explained in full replicative detail by Danielle N. Mohney

Summary:

A Forward Volatility Agreement (FVA) is a forward on a vanilla swaption straddle. The

buyer agrees to purchase a straddle on a specified date (the strike date) for a price that
is determined today. The option becomes a standard option on strike date and is struck
at the usual ISDA fixing time at the then current ATM forward rate.

Keeping the option “unstruck” until strike date gives the buyer concentrated and relatively
constant exposure to Implied Volatility while mitigating exposure to the usually tricky
delta, gamma, and theta inherent in a standard option. Thus, the FVA demands reduced
management for a user wishing to hedge, or take a view on, changes in implied volatility.
The FVA uniquely offers the risk management tool of “Volatility as an Asset Class”.

1
A Case Study Example: An investor believes that implied volatility looks cheap but
does not have a strong view on the near-term actual volatility of interest rates. Moreover,
he does not care to manage the “greek risk” inherent in owning options. Via an FVA, the
investor can buy a six-month forward option on 1y10y (aka 6m1y10y) at XYZbps.

Today ( ): Customer commits to buy a 1y10y straddle in 6m for a price of XYZbps.

Strike date ( . ): At the standard ISDA fixing time, the straddle is struck at the ATM
forward rate. The Customer then has a few choices. He can be delivered a vanilla
1y10y straddle by paying the XYZbps premium as a “regular way” purchase. He can
then hold the position or he can sell it at any time to Credit Suisse (or to another counter-
party via an assignment). In the alternative, he can “cash settle” the contract with Credit
Suisse versus the bid-side of the market. If this bid-side price is above his contracted
price, the Customer is paid the difference, if it is lower, the Customer pays Credit Suisse.
In this latter format, the FVA is a reduced management financial tool that can be
employed to buy pure “Volatility as an Asset Class”.

Expiry date ( . ): If the Customer elects to take delivery and hold the option until the
final expiry date, he will exercise the payer or receiver side of the straddle into
a 10y swap on this date. The swap will settle “regular way”, t+2 from the expiry date.

Maturity date ( . ): The swap matures.

A Challenge: The FVA Replication Process

As a forward rate can be (mostly) replicated with spot rates, an FVA can be (partially)
replicated using vanilla options. As such, it would be helpful to review the replication of
forward rates.

The 1y10y rate, for example, can be replicated with 1y and 11y rates. One can gain a
short exposure to the forward rate by paying the 11y rate and receiving the 1y rate,
DV01-weighted using the following property:

01 , ∗ , 01 ∗ 01 ∗

Thus, one could replicate the exposure of paying 100k/bp of the 1y10y rate by doing the
following trades:

- Pay DV01 /DV01 , * 100k/bp of the 11y rate

- Rec DV01 /DV01 , * 100k/bp of the 1y rate

2
To replicate the FVA, we need to buy the variance from to . and sell back the
variance from to . (we are locking in the expected diffusion of the rate
from . , when the option is struck, to . , when it expires).

To do this, we buy a 1.5y10y straddle and sell a 0.5y expiry option on the same rate.
This second option is an option on a forward rate – a 0.5y option on the 1y10y rate.
An option on a forward rate is known as an (almost) vanilla “mid-curve” option.

A graphical explanation might be helpful. We buy 1.5y into10y:

, 1.5 , 10

. . .

And sell 0.5y into the 1y10y forward rate:

, 0.5 , 10

. . .

To gain exposure to 1y10y swaption vol, 6m forward:

,1 , 10

. . .

As such, one can gain exposure to 100k/nv of 6m forward 1y10y by doing the following:

-Buy 160k/nv 18m10y

-Sell 60k/nv 6m expiry options on the 1y10y rate (6m1y10y mid-curve)

3
There is slippage here due to stochastic volatility and skew effects since these options
are struck at a fixed-strike determined at , while the FVA has no fixed-strike before the
Strike Date. This is the most complete replication we can achieve without more exotic
options. (The calculations for the ratios above are discussed in the Appendix of this
primer.) NOTE: The critical “value add” for the FVA product is that it can never
be “fully replicated” via a limited series of fixed-strike options.

We have now reduced the trade to two struck options; a vanilla option and an option on a
forward rate (mid-curve option).

We can (approximately) replicate the mid-curve with vanilla options as well.

Let’s consider the profile of the mid-curve option again. Replication of this profile is
analogous to the replication of forward swaps previously detailed. To mimic the exposure
of the 6m1y10y mid-curve option:

We buy 6m11y:

, 0.5 , 11

. . .

And sell 6m1y:

, 0.5 ,1

. . .

Leaving us with exposure to the 6m1y10y mid-curve:

, 0.5 , 10

. . .

4
Thus, one can approximately replicate a -60k/nv exposure on the 6m1y10y mid-curve
by:

-Sell 66k/nv 6m11y

-Buy 6k/nv 6m1y

This vanilla replication is an approximation as it fails to incorporate the correlation

exposure from the mid-curve. (This too will be addressed in the Appendix.)

In summary, we have developed an intuitive framework for the pricing of a Forward

Volatility Agreement. As per our example of buying a 6m1y10y FVA, we have effectively
created a long position in 18m10y (our predominant exposure), a short position in
6m11y, and a long position in 6m1y. While this is not a closed-form solution, it does
capture the main risk vectors and identifies the general risk structure. The Appendix will
now further detail the mathematics behind this replication process.

Appendix

For readers with backgrounds in other asset classes who are not accustomed to seeing
normal volatility, we include a brief note on the topic. We measure the diffusion process
in terms of absolute changes (basis point moves in the forward rate), signifying
~N , . Thus, is notation for the standard deviation of absolute changes in the
forward rate, in basis points per year.

Calculating the Implied Forward Volatility of the FVA

We generalize our framework below.

We define an FVA as a straddle that is struck at time and expires on into a swap
that matures on . The mid-curve in our replication is struck on , expiring on into a
swap starting on and maturing on .

As variance is proportional to time, it follows that:

5
The implied volatility of the mid-curve can be calculated using the DV01-weighted spread
option formula. If 01 , is the DV01 of a swap that starts on T1 and matures on T2:

01 , , 01 , , 2 , , , 01 , , 01 , ,

01 ,

The only variable that is not readily transparent in the market is the implied correlation
between the , and , rates. One could calculate this level using historical
delivered correlation, or back out the market implied correlation from a expiry option on
the , spread. Practically, the forward volatility of the mid-curve trades
above the level implied using a correlation of one (partially because this replication
approach ignores skew effects that give value to the forward volatility.)

Using the above formulas, we can calculate the implied forward volatility using only vanilla
options and the DV01s of the forwards:

01 , , 01 , , 2 , , , 01 , , 01 , ,
, 01 ,

Vega Calculations:

The price of the FVA is equal to:

2 ,
01 , 1
2

Thus, the vega is:

2 ,
01 , 1
2

6
Using the chain rule, we can calculate the sensitivity of the normal volatility to a change
in implied normal volatility of the vanilla and the mid-curve:

1 2 ,

, 2 ,

1 2
2 ,

Thus, the normal vega is:

1 2 , 2
∗
, 2 ,
,
∗ 01 , ∗ 1
2

1 2 2
∗
2 ,
,
∗ 01 , ∗ 1
2

We apply the same logic to generate vega sensitivities of the vanillas to replicate the mid-
curve option:

,
1 1
2
01 , , 01 , , 2 , , , 01 , , 01 , ,

2 01 , , 2 , , , 01 , 01 , ,
01 ,

,
1 1
2
01 , , 01 , , 2 , , , 01 , , 01 , ,

2 01 , , 2 , , , 01 , 01 , ,
01 ,

7
 ,
1 1
2
01 , , 01 , , 2 , , , 01 , , 01 , ,

, , , , , , , ,
∗ ∗ 01 ,
,

,
1 1
2
01 , , 01 , , 2 , , , 01 , , 01 , ,

, , , , , , , ,
∗ ∗ 01 ,
,

An Example: 6m1y10y

We return to our initial example: A Customer would like to purchase 100k/nv exposure
to a 6m1y10y FVA.

At this rate level, the DV01 of the 18m10y rate is 8.6, that’s equivalent to approximately
146mm notional of the FVA.

Let’s make the following volatility surface assumptions: The normal volatility for the
6m1y is 37bp, the 6m11y is 99bp, the 1y10y is 100bp, and the 18m10y is 99bp.
Additionally, the DV01 of the 6m1y is 1.0 and the DV01 of 6m11y is 9.6. For simplicity,
we use a 6-month discount factor of one. Finally, we assume a correlation of one for the
mid-curve.

Using these assumptions, the normal volatility of the 6m1y10y mid-curve is 107bp;
where is 1.6 and is 0.6. So for a 100k/nv exposure FVA, we need to buy

160k 18m10y and sell 60k of the 6m1y10y mid-curve. Since is 8.4bp, we

need to buy 186mm 18m10y. We calculate that is 4.9bp, so we need to sell

116mm of the mid-curve.

8
This leads to the ultimate value added proposition – when the Spot Implied Volatility of a
1y10y is trading at 100nv, a client can purchase it six months forward at 95nv, a 5%
discount to the spot level.

On a practical note, we should highlight that most clients trade notionally neutral in the
vanilla and the mid-curve fixed-strike options when attempting a replication. While this is
fine for a terminal replication, it does not match the actual vega exposure of the FVA at
inception; as such, there can be considerable mark-to-market variation.

If you have fully absorbed this appendix and sample application, we look forward to
helping you duplicate our modeling process. If instead you have just paged to the end
looking to see if we have any new colors to offer, we will just close by reminding you that
the FVA is a truly special product in that it allows an investor to gain exposure to Volatility
as an Asset Class (executed on your existing ISDA) without the need to engage in a
costly and complicated “replication process”.

In a nutshell, the FVA is just plain old-fashioned Financial Engineering in the classic
manner of traditional Wall Street.

Danielle N. Mohney
Harley S. Bassman
Credit Suisse US Rates Trading
August 27, 2013

Clear Notice: This Commentary reflects the view of its author and is independent of CS Strategy.

This material has been prepared by individual sales and/or trading personnel of Credit Suisse AG or its subsidiaries or affiliates (collectively "Credit
Suisse") and not by Credit Suisse's research department. It is not investment research or a research recommendation for the purposes of FSA rules as it
does not constitute substantive research. All Credit Suisse research recommendations can be accessed through the following hyperlink: https://s.research-
and-analytics.csfb.com/login.asp subject to the use of approved login arrangements. This material is provided for information purposes, is intended for
your use only and does not constitute an invitation or offer to subscribe for or purchase any of the products or services mentioned. Any pricing
information provided is indicative only and does not represent a level at which an actual trade could be executed. The information provided is not
intended to provide a sufficient basis on which to make an investment decision. Credit Suisse may trade as principal or have proprietary positions in
securities or other financial instruments that are the subject of this material. It is intended only to provide observations and views of the said individual
sales and/or trading personnel, which may be different from, or inconsistent with, the observations and views of Credit Suisse analysts or other Credit
Suisse sales and/or trading personnel, or the proprietary positions of Credit Suisse. Observations and views of the salesperson or trader may change at any
time without notice. Information and opinions presented in this material have been obtained or derived from sources believed by Credit Suisse to be
reliable, but Credit Suisse makes no representation as to their accuracy or completeness. Credit Suisse accepts no liability for loss arising from the use of
this material. Nothing in this material constitutes investment, legal, accounting or tax advice, or a representation that any investment or strategy is suitable
or appropriate to your individual circumstances. Any discussions of past performance should not be taken as an indication of future results, and no
representation, expressed or implied, is made regarding future results. Trade report information is preliminary and subject to our formal written
confirmation.

CS may provide various services to municipal entities or obligated persons ("municipalities"), including suggesting individual transactions or trades and
entering into such transactions. Any services CS provides to municipalities are not viewed as “advice” within the meaning of Section 975 of the Dodd-
Frank Wall Street Reform and Consumer Protection Act. CS is providing any such services and related information solely on an arm’s length basis and
not as an advisor or fiduciary to the municipality. In connection with the provision of the any such services, there is no agreement, direct or indirect,
between any municipality (including the officials, management, employees or agents thereof) and CS for CS to provide advice to the municipality.
Municipalities should consult with their financial, accounting and legal advisors regarding any such services provided by CS. In addition, CS is not acting
for direct or indirect compensation to solicit the municipality on behalf of an unaffiliated broker, dealer, municipal securities dealer, municipal advisor, or
investment adviser for the purpose of obtaining or retaining an engagement by the municipality for or in connection with Municipal Financial Products,
the issuance of municipal securities, or of an investment adviser to provide investment advisory services to or on behalf of the municipality.