001.on the pricing and hedging of volatility derivatives_howison

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On the pricing and hedgingof volatility derivatives

SAM HOWISONMathematical Institute, University of Oxford, 24 29 St. Giles,

OX1 3LB Oxford, UKEmail: [email protected]

AVRAAM RAFAILIDISDepartment of Mathematics, Kings College London,

Strand, London WC2R 2LS, UKEmail: [email protected]

HENRIK RASMUSSENMathematical Institute, University of Oxford, 24 29 St. Giles,

OX1 3LB Oxford, UKEmail: [email protected]

Abstract

We consider the pricing of a range of volatility derivatives, including volatilityand variance swaps and swaptions. Under risk-neutral valuation we provideclosed-form formulae for volatility-average and variance swaps for a varietyof diffusion and jump-diffusion models for volatility. We describe a generalpartial differential equation framework for derivatives that have an extra de-

pendence on an average of the volatility. We give approximate solutions ofthis equation for volatility products written on assets for which the volatilityprocess fluctuates on a time-scale that is fast compared with the lifetime ofthe contracts, analysing both the outer region and, by matched asymptoticexpansions, the inner boundary layer near expiry.

1 Introduction

In this paper we discuss derivative products that provide exposure to the realisedvolatilities or variances of asset returns, while avoiding direct exposure to the under-

lying assets themselves. These products are attractive to investors who either wishto hedge volatility risk or who wish to take a view on future realised volatilities.Indeed, much of the investor interest in volatility products seems to have been pro-vided by the LTCM collapse in 1998, which was accompanied by a dramatic increasein volatilities. As a result, a number of recent papers [4, 8, 9, 12, 15] address theevaluation of volatility products; see also Chapter 13 of [17].

Like several of these authors, we take a stochastic volatility model as our startingpoint; we also provide formulae for the case that the volatility follows a jump-diffusion process of the type described in [18]. The fact that stochastic volatilitymodels are able to fit skews and smiles, while simultaneously providing sensible

Greeks, have made these models a popular choice in the pricing of exotic options.Under this framework, we present a number of formulae for the fair delivery

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price for volatility and variance swaps, and show how other related contracts can bepriced. In addition to providing formulae for a range of volatility and variance swaps,we consider an asymptotic analysis under which we derive approximate solutionsfor volatility and variance swaptions (options on the realised average volatility or

variance). The main motivation for this approximation is the empirical evidencethat volatility is mean-reverting over a time-scale which is fast compared with thetypical lifetime of options and other contracts. That is, when considering the time-scale of months, stock and index volatility is observed to fluctuate rapidly, see forexample the discussion in [10] or [19] and references therein. In this way, we showhow to calculate accurate approximations to the price both at O(1) times beforeexpiry and in a temporal boundary layer near expiry.

In what follows, the asset S is assumed to follow the usual log-normal process

dStSt

= t (t, ) dt + t (t, ) dWt, (1)

where Wt is Brownian motion. For the rest of the paper we fix notation as follows:the conditional expectation at time t is denoted by Et = E[|Ft] where Ft is thefiltration up to time t and E0 is thus the initial value of the expectation. All expec-tations are considered with respect to the risk-neutral probability measure. Withina stochastic volatility framework, the market is typically incomplete, admitting aninfinite number of equivalent martingale measures. This is to say, the market priceof volatility risk is not unique, and it is an open question how one chooses in anoptimal way the appropriate measure to price derivatives. The view we take in thepresent investigation, which is the standard machinery for many practical purposes,

is that the risk-neutral probability measure is chosen by the market and this hasa number of immediate implications for calibration issues, mainly that parameterestimation is not possible from stock data but may be possible using derivatives.Except from a brief discussion in 4.6 we do not further address these issues here.

The rest of the paper is organized as follows: we begin section 2 by brieflydiscussing the contracts, while the stochastic volatility framework is introduced in3, and the general pricing equation is given in 3.3. In 4 we present in depththe asymptotic analysis which concludes with second order approximations for thevolatility derivatives of interest. In 4.5 we concentrate on pure volatility products,in which case the analysis greatly simplifies. Then we give examples in 5. Weconclude in

6.

2 Variance and volatility swaps

The variance swap is a forward contract in which the investor who is long pays a fixedamount Kvar/$1 nominal value at expiry and receives the floating amount vR/$1nominal value, where Kvar is the strike and vR = (

2)R, where is the volatility, isthe realized variance. The entering price must be zero, that is, it costs nothing toenter the contract; we use this condition to find the fair value Kvar. The measureof realized variance to be used is defined at the beginning of the contract; a typical

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formula for it is

1

T

M

i=1 Si Si1

Si1 2

,

which in continuous time we approximate by

vR =1

T

T0

2t dt.

The corresponding payoff is then

vR Kvar. (2)

It is also possible to construct contracts on the realised volatility. One measure ofthis is

1

T

Mi=1

Si Si1Si1

2 12derived from the standard deviation of the asset price random walk, and the corre-sponding continuous-time payoff for a volatility swap is

(vR)1/2 Ks/d =

1T

T0

2t dt 1

2 Ks/d; (3)

we term this contract a standard deviation swap. However, this is not the onlypossible measure of realised volatility. As discussed in [1], a more robust measure is

2MT

Mi=1

Si Si1Si1

, (4)and the associated continuous-time contract has payoff

R Kvol-ave = 1T

T0

t dt Kvol-ave, (5)

which involves the average of realised volatility, rather than the square root of theaverage realised variance as in (3). We term this contract a volatility-average swap.In addition, we shall consider products based on an average of a suitable impliedvolatility, for example the implied volatility it of the at-the-money call optionswith the same expiry as the volatility derivative; this implied volatility swap hascontinuous-time payoff

iR Ki-vol =1

T

T0

it dt Ki-vol. (6)

We could also use a single option throughout the life of the contract, for example

the option that is initially at-the-money; we could further construct implied varianceswaps, and so on.

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Generalizing further, we consider variance and volatility swaptions [17], a typicalpayoff being

max(R K, 0), max(K R, 0)

for volatility call and put swaptions, or

max(vR K, 0), max(K vR, 0)for variance swaptions. We can also contemplate contracts whose payoff depends onboth a realised volatility or variance and the asset; for example, the payoff

max(STe(0R)

T K, 0)

is a call option which pays more if the asset rises steadily without much volatilitythan if it rises in a volatile way; here 0 is a reference volatility and T is the contractslifetime.

3 Risk-neutral pricing techniques

Working within the standard risk-neutral pricing framework, we now outline threeapproaches to the valuation of volatility derivatives. In the case of volatility orvariance swaps, this shows that we need to choose

Kvar = E[vR] = E0

1T

T0

2t dt

,

and for a standard-deviation swap we need Ks/d = E[v1/2R ] (which may be less easyto calculate explicitly; an approximation in terms of higher moments of vt is givenin [4, 15]). For the volatility-average swap, we have

Kvolave = E[R] = E0

1T

T0

t dt

and similarly, for the more complicated implied volatility average swap. Volatilityswaptions and so forth are priced in the usual manner; for example, the price of avolatility swaption is

E0max 1T T0 t dt K, 0.There are, however, various approaches to the calculation of these prices, which wenow consider.

3.1 Pricing independently of the volatility model

As observed by [8], it is possible to derive risk-neutral prices for average-varianceproducts without making any assumption on the evolution of the volatility, althoughthe asset is still considered to follow the risk-neutral geometric Brownian motion

dStSt

= r dt + t dWt.

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Since

d(log St) = (r 12

2t ) dt + tdWt,

we have that

log ST log S0 =T0

(r 12

2t ) dt +

T0

tdWt.

Taking expectations to get risk-neutral prices, we have

1

2E0

T0

2t dt

= rT + log S0 E0 [log ST] ,

and the final term on the right-hand side is the value of a log-contract, which canbe decomposed into a continuously parametrised strip of call and put options in a

standard way. Hence variance swaps can be easily priced in terms of vanilla options.However, this method does not allow us to compute the Greeks, nor does it givestraightforward explicit formulae.

3.2 Pricing by expectations in a stochastic volatility frame-work

It is possible to derive values for the quantities

E[vR] = E0 1

T T

0

2t dt =1

T T

0

E0[2t ] dt

and

E[R] = E0

1

T

T0

t dt

=

1

T

T0

E0[t] dt

when either the volatility t or the variance vt follows a quite general random walk.In this way, we can immediately give risk-neutral prices for variance and volatility-average swaps, although standard-deviation swaps are less straightforward in thisframework.

In Appendix 1 we show how to calculate E[vR] and E[R] when t follows the

(risk-adjusted) process

dt = (a1 + a2t)dt + (a3 + a4t)dWt + (a5 + a6t)dNt, (7)

where Nt is a standard compound Poisson process with constant intensity andzero correlation with Wt, and a1, , a6 are constants.1 We also calculate E0[vR](but not E0[R]) when vt, instead oft, follows a similar process; lastly we calculateE0[vR] and E0[R] for the process

dt = (b1 + b2t)dt + b31/2t dWt (8)

1We expect a1 > 0, a2 < 0, to model mean-reversion, and we note that for certain choices of

the parameters this model allows negative values oft. Models of volatility with jumps have beenconsidered by [18].

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and, as above, E0[vR] if vt follows a similar process. Specific results for the mean-reverting log-normal process

dt = ( t)dt + tdWt, (9)

for constants , and , are reported in [13]. These, and the more general formulaeof the Appendix, serve to check the asymptotic approach developed in the followingsections.

This approach can also be used to calculate prices and hedge ratios at interme-diate times. For example, if we have a volatility-average swap, with payoff

1

T

T0

2sds Kvolave,

the value at earlier times t (0 t < T) is

er(Tt)Et

1T

T0

sds

Kvolave

=er(Tt)

T

t0

sds + Et

Tt

sds

T Kvolave

since the contribution

t0

sds to the final average of the volatility is known at timet. Using the formulae of Appendix 1, the conditional expectation

Et T

t

sds = F(t, t),say, is readily evaluated, and then the Vega of the contract is

1

Ter(Tt)

F

(10)

which is also readily evaluated once t is known.

3.2.1 Examples

Here we briefly give some examples for the price of volatility products, the derivation

of which is based on the results of the Appendix. The mean-reverting log-normalmodel

dt = ( t)dt + tdWt,is a special case of (7), and following the calculation of A1, under this model, wehave

Kvolave =1 eT

T(0 ) + . (11)

Similarly, the value at any time t T is

Vt = er(Tt)

Tt

0

(s )ds + 1

(e(Tt) 1)(0 t) . (12)6


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In addition, for the fair strike of the variance swap we have

Kvar =22

(2 2)T

T 1 e(22)T

2 2

+ 2(0 )( 2)T

1 eT

1 e(22

)T

2 2

+20

(2 2)T

1 e(22)T

. (13)

The formula for the price of the contract at time t is

Vt = er(Tt)

1T

t0

2sds +22

(2 2)T

(T t) 1 e(22)(Tt)

2 2

+

2(t

)

( 2)T 1 e(Tt)

1

e(2

2)(Tt)

2 2 +

2t2 2

1 e(22)(Tt)

Kvar

, (14)

with Kvar given in (13).Note that if we were to consider the variance vt as the underlying process sat-

isfying the mean reverting log-normal model, then the relevant expressions for thevariance swap would be different. We can illustrate the contrast between volatilityand variance driven models by considering the two cases

dt = k1(t )dt + k2dWt,and

dvt = k3 (vt v) dt + k4vtdWt.

Then, following the calculations in A2, we have for the volatility-average swap

Kvolave =1

T

0k1

ek1T 1 + k1T ek1T + 1 ,

in the first case, and we cannot price this contract explicitly in the second framework.However, the price of the fair variance strike is

Kvar =1

T

1T

2b2 2

b22

eb2T 1+ 1

2b2

20 +

2b2

+1

2b2

e2b2T 1 ,

for the first model and

Kvar =1

T

20

k3

ek3T 1+ vt k3T ek3T + 1

for the second; the various constants are as defined in the Appendix. It is apparent

that the second model, for vt rather than t, leads to considerably simpler formulae.

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3.3 Pricing via partial differential equations

Explicit formulae are in general only available for pure volatility products, and un-der the assumption that the coefficients in the process for volatility are independentof St. For more general cases, we must use either Monte-Carlo methods or nu-merical/asymptotic solutions of the pricing differential equation. In this section weconsider the latter.

From now on we assume that it is t that drives the volatility; if instead theunderlying process is written in terms of vt =

2t the appropriate modifications are

easily made. We assume that St and t follow the process

dStSt

= t dt + t dWt,

dt = Mtdt + tdWt (15)where Mt and t may depend on St and t, and where the correlation coefficient ofWt and Wt is t. As with Asian options, we need to introduce a variable to measurethe average volatility to date. The payoff of the derivatives under considerationinvolves an average of the form

IT =

T0

F(s)ds;

for example,

IvarT =T0

2s ds

for a variance swap, for which the payoff (2) is

1

TIvarT Kvar,

and the same average is sufficient for the standard-deviation swap payoff (3), namely

1TIvarT 1/2

Ks/d

.

Likewise the volatility-average swap and the implied-volatility swap have payoffs (5)and (6), namely

1

TIvolaveT Kvolave,

1

TIivolT Kivol

respectively, where IvolaveT =T0

s ds, Iivolt =

T0

is ds, and so forth. For timesbefore expiry, we use the running average (denoted generally by It regardless of F)

It = t0

F(s) ds,

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and then it is a standard combination of stochastic volatility and Asian optionsanalysis [10, 21] to show that the value V(S,,I,t) of a derivative satisfies thepartial differential equation

V

t +1

2 2

S2

2V

S2 + S2V

S +1

2 2

2V

2 + F()V

I

+rSV

S+ (M ) V

rV = 0, (16)

where is the market price of volatility risk. We write the general payoff conditionin the form

V(S,,I,T) = P(S, I). (17)

It is straightforward to show that the formulae derived in 3 and the Appendixsatisfy this equation when = 0. For example for the mean-reverting log-normal

model (9) we have M = ( t), = t.and expressions (12), (14) satisfy equation (16).

4 Asymptotic analysis for fast mean-reversion

We now present an asymptotic analysis for the case of fast mean-reversion for t,in the spirit of the Fouque et al. (FPS) analysis [10] for equity and fixed-incomederivatives. The novelty here is first in the application to volatility products, andsecondly in that we provide a fairly complete description of the solution both at

O(1)

times before expiry which FPS do, and in the short boundary layer immediatelybefore expiry, which they do not. We initially make the simplifying assumptionsthat Mt and t, the coefficients in the process for t, are independent of St, andwe let t = 0, t = 0. The assumption of zero correlation is not as dramatic as itwould be for equity derivatives, and indeed if we consider payoffs depending only onvolatility averages it is irrelevant. We indicate briefly at the end of this section howthe analysis should be extended for nonzero correlation.

We introduce a small parameter , where 0 < 1, to measure the ratio of themean-reversion time-scale to the lifetime of an option, and we assume that Mt andt have the forms

Mt =mt

, t =t

1/2,

the relative sizes of these coefficients being chosen to ensure that t has a nontrivialinvariant distribution

p() = limt

p(, t|0, 0)

where p(, t|0, 0) is the transition density function for t starting from 0 at timezero.

With these assumptions the pricing equation becomes

V

t+

1

22S2

2V

S2+

1

22

2V

2+ rS

V

S+

m

V

+ F()

V

I rV = 0,

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which we write in the form 1L0 + L1

V = 0

where

L0 = 12

22

2+ m

,

L1 = t

+1

22S2

2

S2+ rS

S+ F()

I r.

We note immediately that p() satisfies

L0p =

2 1

22p

(mp) = 0

where L0 is the adjoint of L0, and we assume that 2, m are such that p exists; itis then proportional to e2

Rm(s)/2(s) ds/2(). We introduce the notation , for

the usual inner product over 0 < < , and note the identityL0u, v = u, L0v (18)

for suitable functions u and v, which we will use repeatedly.We now expand

V(S,,I,t)

V0(S,,I,t) + V1(S,,I,t) +

2V2(S,,I,t) +

,

so that

1

L0V0 + (L1V0 + L0V1) + (L1V1 + L0V2) + = 0.

At lowest order, O(1/), we haveL0V0 = 0

and so

V0 = V0(S,t,I)

since the operator L0 consists of derivatives with respect to only (the particularsolutions that depend on are in general ruled out by the conditions at large and/orsmall S); however V0 is as yet undetermined.

At O(1) we findL0V1 + L1V0 = 0, (19)

a Poisson equation for V1 considering V0 as known. Multiplying by p, integratingand using (18), we find that the solvability (Fredholm Alternative) condition for thisequation can be expressed as

L1V0, p = 0.

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That is, L1V0 is orthogonal to p, which, being a solution of the stationary forwardKolmogorov equation for , is an eigenfunction of L0. Thus,

V0

t

+1

2

S22V0

S2

0

p

()2d + rSV0

S

+V0

I

0

p

()F()d

rV0 = 0,

where we have used the result that V0 is independent of . Denoting the integralsby 2 and F = F() respectively to represent the fact that they are averages of 2

and F(), we write this as

L1V0 = V0t

+1

22S2

2V0S2

+ rSV0S

+ FV0I

rV0 = 0, (20)

(note that L1 = L1, p). Making the transformation

I = I + (T

t)F

and writing V0(S,t,I) = V0(S, t; I), reduces this further, to

V0t

+1

22S2

2V0S2

+ rSV0S

rV0 = 0, (21)

which is the BlackScholes equation with volatility

21/2

. The dependence on I

is retained parametrically via the payoff, which takes the form

V0(S, T; I) = V0(S , T, I ) = P(S, I),

and so we have V0(S,t,I) = V0(S,t,I + (T t)F). Of course, this formula isconsiderably simpler for pure volatility products, for which P/S = 0 so thatV/S 0.

The next step is to calculate V1. First, observe that L1V0 can be written as

L1V0 = 12

(2 2)S22V0

S2+ (F() F) V0

I.

Hence, (19) can be written as

L0V1 =1

2(2

2

)S2

2V0S2 + (F F())

V0I .

We seek a solution of the form

V1(S,t,,I) = f2()S2

2V0S2

+ f1()V0I

+ H(S,t,I), (22)

where H is independent of . The functions f1 and f2 are then the appropriatesolutions of the equations

1

22()

d2f2d2

+ m()df2d

=1

2(2 2),

1

22()

d2f1d2

+ m()df1d

= F F(). (23)

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and can readily be found in integral form; one of the complementary solutionsis a constant and can be absorbed into H, and the other is unbounded at infinity(because p exists). However, the function H(S,t,I) can only be determined byproceeding to next order and applying the solvability condition to the equation

L0V2 + L1V1 = 0. (24)We further see that the solution (22), which depends on , cannot satisfy the pay-off condition V1(S,,I,T) = 0. This discrepancy is resolved by a boundary layeranalysis in which T t = O() (if the payoff has discontinuities, as for a volatilityoption, further local analysis near these points is also necessary). We point out thatthe lowest-order analysis is quite general, and not specific to the random walk (15).Only at higher order do we need to know more about these details, and even thenonly certain moments need be calculated.

Now, the solvability condition for (24) takes the form:

L1V1, p = 0or

L1

f2()S2

2V0S2

, p + L1

f1()

V0I

, p + L1H, p = 0,

that is,

L1

f2()S2

2V0S2

, p + L1

f1()

V0I

, p + L1H = 0, (25)

where L1 is as in (20). Before solving (25), we note that since L1V0 = 0, we alsohave

L1

SnnV0Sn

= 0, n 1, L1

V0I

= 0. (26)

We can therefore write the first term in (25) as

L1

f2()S2

2V0S2

, p = 1

2S2

2

S2

S2

2V0S2

2f2() 2 f2()

+

IS22V0

S2 F()f2() F() f2() . (27)

The second term, after some algebra, takes the form

L1

f1()V0I

, p = 1

2S2

3V0IS2

2f1() 2 f1()

+

2V0I2

F()f1() F() f1()

.

As a result we can rewrite equation (25) as

L1H = A1S

2 3V0

IS2+ A2

2V0

I2+ A3S

2 2

S2 S22V0

S2 = W(S,t,I), (28)

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say, where we have defined

A1 =

F() f2() F()f2()

+1

2

2 f1() 2f1()

,

A2 = F() f1() F()f1(),

A3 =1

2

2 f2() 2f2()

. (29)

However, from (26), L1W = 0, which implies that L1 ((T t)W) = W. Thisallows us to find a particular solution of the form (T t)W, and we can thenexpress the general solution of (28) as

H(S,t,I) = (T t)A1S2 3V0

IS2 + A2

2V0

I2

+ A3S2

2

S2

S2

2V0S2

+ H1(S,t,I), (30)

where H1 solves the equation

L1H1 = 0. (31)The solution of this last equation can be obtained directly via the transformationI = I + F(T t), which transforms (31) into the Black-Scholes equation with

volatility 21/2.4.1 Boundary Layer Analysis

We have already satisfied the payoff condition with V0, and so we expect that

V1(S,T,,I) = 0.

But this is not possible, since V1 is a function of whereas the payoff is not. Theremedy is to introduce a boundary layer in t near t = T, of thickness ofO(). Wedefine the scaled inner variable via

t = T + , < 0.

Our goal is now to find the expansion in the boundary layer (the inner solution),which we subsequently match with the solution outside the boundary layer (theouter solution) using Van Dykes matching principle [20]. We introduce the followingoperators:

L0 =

+ L0, L1 = 12

2S22

S2+ rS

S+ F()

I r,

and we have

L = 1L0 + L1,

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but we note that L0, unlike L0 above, contains the time derivative /. We writeV(S,,,I) for the value of the derivative in the boundary layer, and expandV

V0 +

V1 + .As a result equation (16) becomes

1

L0V0 + L1V0 + L0V1 + = 0. (32)

At lowest order, L0V0 = 0with the condition

V0(S, 0, , I ) = P(S, I).

The solution is easily seen to beV0(S,,,I) = P(S, I). (33)Note that this matches automatically with our outer solution V0(S,t,I) as t T, .

At the next order we have

L0V1 = L1 V0 = L1P ,which can be written as

V1 + L0V1 = 12 2 2S22PS2 + F() F() PI L1P (34)with the final condition V1(S, 0, , I ) = 0(note that P/ = 0). We now need the solution of (34) as in order tomatch with the outer solution. A particular solution of (34) is

V1 = f2()S22PS2

+ f1()P

I L1P + H(S, I) (35)

where H(S, I) is arbitrary, and we conjecture that this is the correct form for theasymptotic behaviour of V1(S,,,I) as . Now L0 V1, p = V1

+ L0V1, p = V1

, p

since L0V1, p = 0. We further note that from (34),

V1, p = V1

, p = L1P, p = L1P

and therefore

V1, p = L1P, (36)14


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where we have used the condition V1(S, 0, , I ) = 0. Now from (35) we have

V1 , p = L1P + f2()S2

2P

S2+ f1()

P

I+

H(S, I). (37)

We now compare (36) with (37) and we immediately have

H(S, I) = S22PS2

f2() PI

f1().

Thus, as ,

V1 V1 = S2 f2() f2() 2PS2 + f1() f1() PI L1P, (38)since subtracting of the particular solution leaves a complementary function

V1

V1 whose inner product with p vanishes, which satisfies the homogeneous versionof the parabolic equation (34), and which therefore vanishes as .4.2 Matching

We now consider the matching. Expanding the outer solution to O() in the innervariable , we have

V0(S,t,I) = V0S, T + ,I+ F()(T t)= V0S, T + ,I F V0(S , T, I ) + V0

t

S , T, I

FV0I

(S,t,I)

= P(S , T, I ) + V0

t(S , T, I ) FV0

I(S,t,I)

(note that for t = T we have I = I). Using (20) this reduces to

V0(S,t,I) P(S, I) L1P,

and so, combining with (22), the two-term inner expansion of V0 + V1 is

P(S, I) +

LP + f1()PI

+ f2()S2

2P

S2+ H(S , T, I )

. (39)

Similarly, from (33) and (38), the two-term expansion of the inner solution in termsof the outer variable t is

P(S, I) +

f2() f2()

S2

2P

S2+

f1() f1() P

I L1P

. (40)

Matching (40) and (39), we see that the final condition for H(S,t,I) is

H(S , T, I ) = f2()S22P

S2 f1()P

I,

15


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and therefore (30) becomes

H1(S , T, I ) = f2()S22P

S2 f1() P

I,

and the boundary layer analysis thus leads to the missing final condition for H1.As stated above, we can reduce LH1 = 0 to the Black-Scholes equation using thesubstitution I = I+ (T t)F, and hence we find H1(S, t; I) = H1(S,t,I) by solving

LBSH1 = 0,

with

H1(S, T; I) = H1(S , T, I ) = f2()S22P

S2 f1() P

I.

Recalling that V0(S, t; I) satisfies LBSV0 = 0 with V0(S, T; I) = P(S, I), and (26),we see that for t T,

H1(S, t; I) = f2()S22V0

S2 f1() V0

I.

4.3 Summary

In summary, the outer expansion takes the form

V(S,t,,I) V0(S,t,I) + V1(S,t,,I)

whereV0(S,t,I) = V0(S, t; I + (T t)F)

and V0(S, t; I) satisfies the Black-Scholes problem

LBSV0 = 0, V0(S, T; I) = P(S, I)

with volatility (2)1/2. The correction V1(S,t,,I) takes the form

V1(S,t,,I) = S2

2V0S2

f2() f2()

+

V0I

f1() f1()

(T t)A1S2

3V0

IS2 + A2

2V0

I2 + A3S

2 2

S2 S22V0

S2 ,(41)where the functions f1(), f2() are defined in (23) and the constants A1, A2, A3 in(29). In the boundary layer, the solution takes the form

V(S,,,I) P(S, I) + V1,where V1 satisfies the parabolic problem (34) with final data V1(S, 0, , I ) = 0; thissolution cannot be found explicitly but its large-time behaviour is sufficient to allowmatching with the outer solution to this order.

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4.4 Extensions

4.4.1 Payoff singularities

It is implicit in our analysis that the payoff is smooth enough for the expansion to be

valid. For example, in (30) we have terms such as 2V0/I2 which is a delta functionat expiry if the contract is a volatility call swaption with payoff max(I/T K, 0).Roughly speaking, such a discontinuity will propagate along the line I+ (T t)F =KT, where K is the strike, and a separate analysis is necessary in a small regionaround this line; the result is an interesting mathematical problem which we brieflydiscuss in 5.5. We also note that we could (but do not) also consider the implicationsof discontinuities in the S-dependence of the payoff, which (because of the secondS-derivative in the pricing equation) may be expected to induce small square rootof time to expiry region in which the expansion is not valid.

4.4.2 Correlation and the market price of riskWe also briefly consider the effects of correlation and the market price of volatilityrisk. Ift or t is of O(1/2) then their effect is to modify the operator L1 (we donot consider this special case here), but if they are ofO(1) then our pricing equationtakes the form 1

L0 + 1

1/2L 1

2

+ L1

V = 0

where L0 and L1 are as before, and

L 12

= S 2

S

.

The expansion for V is now in the form

V V0 + 1/2V12

+ V1 + 3/2V3

2

+ ,

and the first four equations are, in order,

L0V0 = 0,

L0V12

+ L 12

V0 = 0,

L0V1 + L 12

V12

+ L1V0 = 0, (42)

L0V32

+ L 12

V1 + L1V12

= 0. (43)

We still have that V0 is a function of (S , I, t) alone, hence L 12

V0 = 0 and so V12

is

also a function of (S , I, t). Thus L 12

V12

= 0 and the solvability condition for (42)

gives L1V0, p = 0

17


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just as before, so also as before V0 = V0(S, t; I+ (T t)F). We then have

V1 = f2()S2

2V0S2

+ f1()V0I

+ H(S,t,I),

also as before, and we note that L 12 V1 = 0 but L 12 H = 0. Proceeding to thesolvability condition for (43), we have L 1

2

V1 + L1V12

, p = 0, which becomes

LV12

= f2S

S

S2

2V0S2

f1S

2V0SI

+ f2S2

2V0S2

+ f1V0I

, (44)

where we denote by f1 and f2 the corresponding derivatives with respect to . The

solution for V12

that vanishes at expiry is readily found to be

V12

(S , I, t) = (T t)

f2S3

3V0S3

+ 2S22V0S2

+ f1S2V0SI

f2S22V0S2

f1V0I

(45)

where V0 is already known. The calculation of V1, from a solvability condition atO(2), is straightforward but even more cumbersome and we do not give details here;in the next section we give them for a pure volatility product. In the boundary layer,we have 1

L0 + 1

1/2L 1

2

+

L1V = 0,

V

V0 +

1/2

V1

2

+

V1 +

where L0 and L1 are as before. Again V0 = P(S, I), matching automatically withV0, and the problem for V1

2

is

L0V12

= L 12

V0 = 0;the solution is simply V1

2

= 0, and this is consistent with matching with the two-

term outer expansion V0 + 1/2V1

2

, since in inner variables the T t in V12

means

that this term only contributes O(3/2) to the inner expansion of the outer solution.Again,

V1 can be calculated and matched although we do not give details here.

In summary, with nonzero

O(1) correlation, the outer solution is given by

V(S,t,,I) V0(S,t,I) + 1/2V12

(S,t,I)

where V0 is the solution of a Black-Scholes problem and V12

is given in (45).

4.5 Pure volatility products

The analysis above is greatly simplified for pure volatility products for which all S-derivatives vanish. Again using t as the underlying volatility variable, the pricingequation is simply

Vt

+ 12

2

2V2

+ m

1/2

V

+ F()VI

rV = 0

18


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with payoffV( , I, T) = P(I).

For certain models and products it is possible to write down explicit solutions notonly for swapsas in

3.2but also for swaptions, since corresponding results for

Asian options can simply be transferred. In particular, for a Hull-White type modelin which = 0 and M = M0, so that t follows Geometric BrownianMotion, there is a series solution for a volatility-average swaption and, if one wereto construct a swaption based on the continuously sampled geometric mean of t,that too would have an explicit solution because the running geometric average ofa Geometric Brownian Motion is log-normally distributed.

4.5.1 Asymptotics for pure volatility products

As mentioned above, the approximate analysis is rather simpler for a pure volatility

product, since then all S-derivatives vanish. We therefore give more details of theanalysis of4.4, in Appendix 2; in summary, the outer expansion takes the formV V0 + 1/2V1/2 + V1 +

where

V0(I, t) = er(Tt)P(I + (T t)F), (46)

V12

(I, t) = (T t) f1V0I

=

(T

t) er(Tt)f1P

(I + (T

t)F), (47)

and

V1(I, t) =

f1() f1() (T t) f3 V0

I

+

f1

2 (T t)22

A2 (T t)

2V0I2

= er(Tt)

f1() f1() (T t) f3

P(I+ (T t)F)

+ er(Tt)f1

2 (T t)22

A2 (T t)P(I+ (T t)F). (48)

The function f3() is defined in Appendix 2. The boundary-layer solution is asbefore.

Notice that the leading order term in the outer solution has a natural financialinterpretation. If, say, the contract is a call swaption, the outer solution is zero for

I+ (T t) F < KT;since I is the contribution to the payoff already in the bank and (T t)F is theapproximate expected remaining contribution (as in a fast mean-reverting model thefluctuations in average out at this order), the option is unlikely to pay out if the

sum of these is an O(1) amount less than KT. The discontinuity at I+ (T t)F =KT must be resolved by an inner expansion which we do not deal with here.

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4.6 Calibration

We note briefly that, like the original FPS [10] model, a great deal of calibration ofthe current framework can be accomplished without reference to a specific model.The first two terms in (45) can be calibrated to pure equitie option volatility smiles asin the FPS scheme. It is more likely that calibration is neseccary for pure volatilityproducts, and gives, say, a series of volatility swaptions of different prices, (46)and (47) can be used to calculate F and f1. The calibration to O() is morecomplicated and we do not discuss it here, although it is in principle possible.

5 Examples

In this section we illustrate the theory with several different products. The first threeare the pure volatility swaps described in 2 where explicit formulae are available (see3.2.1); the asymptotic results can be shown to be correct. For the implied-volatilityswap, there is some S-dependence via i, although the payoff is still independent ofI. Finally we look briefly at volatility-average swaptions.

5.1 The variance swap

For this contract, we have F() = 2 and so F = 2, the average variance to be usedin the Black-Scholes equation (21). The payoff is Ivar/T Kvar = Ivar/T Kvarand the first three terms in the expression for the variance swap value are

V(t,,I)

er(Tt)

Ivart + 2(T t)

T Kvar

1/2er(Tt)(T t) f1

T

+ er(Tt)

f1() f1() (T t)f3

T

.

For the random walk (9) for which 2 = 22/(2 2) = 2a2/(2a b2), it iseasily confirmed that we recover the O(1) term of the exact result (14).

5.2 The standard-deviation swap

For the standard-deviation swap payoff (3), we still have F() = 2 but now thepayoff is (Ivar/T)1/2

Ks/d. Hence, the standard-deviation swap price to order

O()

is

V(t,,I) er(Tt)Ivart + 2(T t)

T

1/2 Ks/d

1/2 e

r(Tt)

2T(T t) f1

Ivart +

2(T t)T

1/2

+ er(Tt)

4T2

2T

f1() f1() (T t) f3

Ivart + 2(T t)T

1/2

+ (T t)A2 f12 (T t)2 Ivart + 2(T t)T 3/2

.20


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5.3 The volatility-average swap

For the payoff (5) we have F() = and we find that

V(t,,I) er(T

t)Ivolavet + (T t)T Kvol-ave

1/2 er(Tt)

T(T t) f1

+ er(Tt)

f1() f1() + (T t) f3

T

Again here, we recover the O(1) term of the exact solution given by (11) and (12)for large mean reversion coefficient.

5.4 The implied volatility swapIn this case F() is the implied volatility of an at the money option, say a call, withprice C(S,t,). We can of course apply the same procedure to pure equity options(this is the FPS analysis) to give

C(S,t,) CBS

S,t,

21/2

+ 1/2 (T t)

f2

S3

3CBSS3

+ 2S22CBS

S2

f2S2

2CBSS2

+

O(). (49)

The implied volatility of this option can also be expanded in the form

i i0 + 1/2i1 +

and since by definition

C(S,t,) = CBS(S,t,i),

we have

C(S,t,) CBS(S,t,i0) + 1/2i1CBS

|i0 + O(). (50)Comparing (49) with (50) we clearly have

i0 =

21/2

and, setting the strike of C equal to S and substituting from the Black-Scholesformulae,

i1 =

f2 12

2

r f

2

2

23/2 .

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Hence we see an additional complication in that the averaging function F() itselfhas an expansion

F()

2

1/2

+ 1/2i1 + O().

Thus the price operator takes the form1

L0 + 1/2L 1

2

+ L1 + 3/2L 32

+

where

L 32

= i1

I.

This means in turn that the right-hand side of (44) has an extra term

L 32

V0, p = i1V0I

and so there is an extra term (T t)i1V0/I on the right hand side of (45), thatis an extra O(1/2) correction to V; it has the obvious financial interpretation asthe vega with respect to the average I. The O() correction, however, is morecomplicated because of the S-dependence in the implied volatility and we do notgive details here.

5.5 Volatility-average swaptions

For the volatility-average swaption we have F() = so F = , and

V(t,,I) er(Tt) maxI+ (T t)T K, 0 1/2 e

r(Tt)

T(T t) f1H (I+ (T t) T K)

+ er(Tt)

T

f1() f1() (T t) f3

H(I+ (T t) T K)

+

f1

2 (T t)22

A2 (T t)

(I+ (T t) T K)

,

where H is the Heaviside function and (x) is the delta function. As discussed above,the O() contribution is singular on the line I+ (T t)F and the expansion is notvalid in this region.

The singularity near I = KT (T t) is resolved by the introduction of aninner layer coordinate defined by

I = KT (T t) + ,leading to the problem

( ) V

+ L0V + O() = 0

where the O() term contains the remaining derivatives and undifferentiated terms(if the averaging is with respect to F(), the coefficient of V/ is F() F()).

22


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This is to be solved for all ; that is, the leading order inner value V0 satisfies theforward-backward parabolic equation

1

2

22V0

2

+ mV0

= (

)

V0

for < < , 0 < < , with matching conditions

V() 0, , and V() , +

for a call swaption and these conditions interchanged for a put swaption. This ratherunconventional problem has been considered in [2] in the context of inner layers formodels of neutron transport. It is crucial that the coefficient of the time derivativeV0/ satisfies , p = 0, and given this condition it can be shown that thereis a unique solution connecting the behaviour as to that as +, eventhough the equation itself is not standard. Similar remarks apply to derivatives withother payoff singularities such as digital options.

6 Conclusion

We have described a range of approaches to the pricing and hedging problem for avariety of products depending on realised volatility. Some of these, especially thosebased on realised variance, are already traded; but as we point out in 2, from astatistical point of view the realised first variation (4) is a more robust estimatorand hence we have described the application of the theory to volatility-averaged

options as well as to implied volatility averages, which are the type measured bythe VIX index [7]. We have presented an asymptotic analysis which leads to adescription of the derivative price even in the time leading up to expiry, and thisshould be an accurate approximation whenever volatility is fast mean-reverting, aswell as straightforward to calibrate from implied volatility smiles.

A Appendix 1: Derivation of certain expectations

A.1 The random walk (7)

Let yt satisfy the stochastic differential equation

dyt = (a1 + a2yt)dt + (a3 + a4yt)dWt + (a5 + a6yt)dNt, (51)

with zero correlation between Wt and Nt. Define

E1 = E[y|y0], E1 =0

E1t dt,

and

E2 = E[y2

|y0], E2 =

0

E2t dt.

23


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Recalling that E[dNt] = dt, we have

dE1t = (a1 + a2E1t + (a5 + a6E1t)) dt

= (0 + 1E1t)dt,

where0 = a1 + 5, 1 = a2 + a6,

from which, since E10 = y0,

E1 = y0e1 0

1(1 e1), (52)

and

E1 =y01

(e1 1) 021

(1 e1 + 1). (53)

For a mean-reverting process 1 < 0, so the unconditional expectation of yt is

y = lim

E1 = 01

,

and the long-term average of yt is the same:

lim

1E1 = 01

;

note the relatively slow (algebraic) decay of the contribution to E1 from the initial

value y0.Now define zt = y2t . We easily find that

dzt =

2yt(a1 + a2yt) + (a3 + a4yt)2

dt

+ 2yt(a3 + a4yt)dWt + (a5 + a6yt) (a5 + (a6 + 2) yt) dNt. (54)

Thus E2 = E[z|z0 = y20] satisfiesdE2t = (0 + 1E1t + 2E2t) dt

where

0 = a23 + a25,

1 = 2(a1 + a5) + 2(a5a6 + a3a4) = 2(0 + 1),

2 = 2(a2 + a6) + a26 + a

24 = 2(1 + 2), (55)

with 0, 1 as before, and

1 = a5a6 + a3a4, 2 =a26 + a

24

2.

Using (53), we find that

E2 = 01 1012

1 e2 1(1y0 + 0)1(1 2)

e2 e1 + y20e2,24


25/32

and we note that E2 grows exponentially in unless 2 < 0, a condition analogousto 1 < 0 for E1. In this case, the unconditional expectation of E2, namely

z = lim

E2,

is (01 10)/12.Lastly, we calculate the averaged expectation of zt,

E2 =

01 10

12

+

01 10

12+

1(1y0 + 0)

1(1 2) y20

1 e2

2

1(1y0 + 0)

21(1 2)(1 e1) . (56)

A.1.1 Special cases

We note some special cases:

(i) The case 1 = 0, = 0 corresponds to the condition = 2/6, and then wehave

E1 = y0 + 0, E1 = y0 +1

20

2,

E2 =0 + 1y0 + 2y

2

02 e2 + 1 + 2y

2

0 10

2 ,

and

E2 =0 + 1y0 + 2y

20

22(e2 1)

2(0 + 1y0) 1

220

2.

(ii) The case 1 = 2 = 0 corresponds to 1 + a26 + a24 = 0, and then we have E1,E1 as in the general case, whereas

E2 =

01 10

21 (1 e1) + 1(1y0 + 0)

1e1 + y20e

1,

and

E2 =

01 10

21

+

10 10 21y2031

(1 e1) .

(iii) The case 1 = 2 = 0 corresponds to a4 = a6 = 0, and we have

E2 = y20 + (0 + 20y0) +

20

2,

E2 = y20 + (0 + 20y0)

2

2+

20

3

3.

25


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(iv) Finally, consider the case 1 = 0, 2 = 0. Then we have that 2a2 + 2a6 +a26 + a

24 = 0 and we have E1 and E1 as in the general case, and

E2 = (10 01)1 1 (1y0 + 0)21 (1 e1) + y20

and

E2 =(10 01) 2

21 1 (1y0 + 0)

31(1 e1 + 1) + y20.

A.1.2 Derivatives pricing

Using the general expressions (52), (53), (56) and (56), or any of the particular casesdescribed above, either volatility-average or variance swaps can be priced by taking

yt = t (in which case zt = 2t = vt) or by taking yt = vt directly, depending onwhether the volatility model is for t or vt. For example, the strike for a volatility-average swap is

Kvolave =1

TE0

T0

t dt

=1

TE1T|y0=0,

where it is understood that the process for yt is the required model for volatility. Ina similar way, the expectation needed to calculate the vega (10), namely,

Et

Tt

s ds

,

is equal toE1(Tt)|y0=t

that is, y0 is replaced by t and by T t.

A.2 The process (8)

Suppose now that

dyt = (b1 + b2yt)dt + b3y1/2t dWt (57)

and define E1, E1, E2, E2 as before. Also define zt = y2t so that

dzt = (2yt(b1 + b2yt) + b23yt)dt + 2b3y

3/2t dWt

= ((2b1 + b23)yt + 2b2zt)dt + 2b3y

3/2t dWt.

Then, proceeding as above, we find linear ordinary differential equations first forE1, then E1, E2 and E2, yielding

dE1t = (b1 + b2E1t)dt,

26


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so that

E1 = y0eb2 b1

b2(1 eb2),

E1 =y0b2 (e

b2 1) b1b22 (b

2 eb2 + 1), (58)

and then

dE2tdt

= (2b1 + b23)E1t + 2b2E2t

= (2b1 + b23)

y0eb2t b1

b2(1 eb2t)

+ 2b2E2t

= 1 + 2eb2t + 2b2E2t,

where

1 = b1(2b1 + b23)

b2, 2 = y0(2b1 + b

23) 1.

Integrating, we have

E2 = 12b2

2b2

eb2 +

y20 +2b2

+1

2b2

e2b2

and so

E2 = 1

2b2 2b22 (e

b2

1) +1

2b2y20 + 2b2 + 12b2(e2b2 1).A.3 Popular Stochastic Volatility Models

For completeness we give the results of these calculations for some popular stochasticvolatility models:(i) The Hull-White model [14]. This is a geometric Brownian motion for the variance2t :

d2t = 2t dt +

2t dWt.

We use (51) with yt = 2t , so that

0 = 0, 1 = .

We obtain

E1 = 20e

, E1 =20

(e 1) . (59)

(ii) Analogous to (i), we consider a geometric random walk for the volatility,

dt = tdt + tdWt.

27


28/32

Then in (51) we have 0 = 0, 1 = , 0 = 0, 1 = 0, 2 = 2 + 2, and we have

E1 = 0e, E1 =

0

(e 1) ,

and

E2 = 20e

(2+2), E2 = 20

2 + 2

1 e(2+2)

.

(iii) The mean reverting-version of the Ornstein-Uhlenbeck model for the volatilityis

dt = ( t) dt + dWt.We take yt = t, 0 = , 1 = , 0 = 2, 1 = 2 and 2 = 2 in (51). Thenit is straightforward to show that

E1 = (0 )e + , E1 = 0

1 e + ,

and that

E2 =22 + 2

2

1 e2 + 2( 0) e2 e + 20e2,

E2 =

22 + 2

2

+

20 2(0 )

22 + 2

2

1 e2

2

2( 0)

1 e .

(iv) The Heston model [11]. This is

d2t =

2t

dt + tdWt.

In this case we use (57) with

yt = 2t , b1 = , b2 = , and b3 = .

Then from (58) we have:

E1 = 20e +

1 e , E1 = 20

1 e + .

(v) Finally we consider the mean reverting log-normal model [13]

dt = ( t)dt + ttdWt.We use (51) with yt = t, 0 = , 1 = , 0 = 0, 1 = 2, 2 = 2 + 2. Asexpected, we obtain the following expressions:

E1 = 0e + 1 e ,

E1 =0

1 e + ,

28


29/32

E2 =22

2 2

1 e(22)

+ 2 (0 ) 2 e e(22) + 20e(22),E2 =

22

2 2 +

22

2 2 +2( 0)

2 20

e(2

2) 12 2

2( 0) 2

1 e .

B Appendix 2: Analysis for pure volatility prod-ucts

Following 4.4, we have L0V0 = 0, so V0 = V0(S, t); similarly L0V12

= L 12

V0 = 0,

so V12

= V12

(I, t). From the solvability condition for (42), we have

L1V0 = V0t

+ FV0I

rV0 = 0

whose solution with V0(I, T) = P(I) is

V0(I, t) = er(Tt)P

I+ (T t)F .

Then, solving (42),

V1 = f1()V0I

+ H(I, t).

where H is as yet undetermined.Now the solvability condition for (43) gives

LV12 = f1V

0I

so that the solution satisfying V12

(I, T) = 0 is

V12

(I, t) = (T t) f1V0I

.

Thus far we have paralleled the analysis of 4.4. Now we continue by finding thesolution of (43): it is

L0V32

= L 12

V1 L1V12

=

f1 f1 V0I (T t) f1 F F() 2V0I229


30/32

so that

V32

= f3()V0I

(T t) f1f1()2V0I2

+ H32

(I, t)

where f3() satisfies

1

22

d2f3d2

+ mdf3d

= f1() f1

and H32

is arbitrary. Now at O(),L0V2 + L 1

2

V32

+ L1V1 = 0;

our final application of the solvability condition, in the form L 12

V32

+L1V1, p = 0,gives

L1f1() V0I + H , p = L 12 V32 , p,that is,

L1H =

A2 (T t)

f12 2V0

I2+ f3

V0I

,

where A2 is defined in (29). The solution is

H(I, t) =

A2 (T t) +

f1

2 (T t)2

2

2V0I2

f3 (T t) V0I + H1(I, t),

where L1H1 = 0, so that H1 = H1

I + (T t)F; this last unknown function isdetermined by matching with the boundary layer.

In the boundary layer, we have1

L0 + 1

1/2L 1

2

+ L1V0 + 1/2V12

+ V1 + = 0,as before, and it is easy to see that V0(I, t) = P(I), V12 (I, t) = 0, and that, as before,

V1

+ L0V1 = F() F() dPdI

L1P

so that, as above,

V1(I, ) f1() f1() dPdI

L1P

as . Hence the matching condition is unaffected at this order by the marketprice of risk term, and the required final condition for H1(I, t) is

H1(I, T) = f1() dPdI

.

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Hence H1(I, t) = er(Tt)f1()P(I+ (T t)F), where = d/dI.

Acknowledgments SDH wishes to thank the Universita degli Studi di Siena forhospitality during the completion of this work; AR wishes to thank the participants

of the second world congress of the Bachelier Finance Society, Crete, June 2002, forstimulating discussions. AR acknowledges financial support from the School of Phys-ical Sciences and Engineering, Kings College London. We are grateful to GeorgePapanicolaou for helpful comments and especially for drawing the reference [2] toour attention.

References

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[5] Buraschi, A. and Jackwerth J. C. The Price of a Smile: Hedging and Spanning

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[10] Fouque J. P., Papanicolaou G. and Sircar, K. R. Derivatives in FinancialMarkets with Stochastic Volatility, Cambridge University Press, 2000.

[11] Heston, S. L. A closed-form Solution for Options with Stochastic Volatility withApplications to Bond and Currency Options. Review of Financial Studies, 6(2):237-343, 1993.

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[13] Howison, S. D., Rafailidis, A. and Rasmussen, H. O. A Note on the Pricingand Hedging of Volatility Derivatives. Bachelier Finance Society Second Worldcongress proceedings, Crete, June 2002.

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001.on the pricing and hedging of volatility derivatives_howison

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