Robust Replication of Volatility Derivatives

7/28/2019 Robust Replication of Volatility Derivatives

1/35

Robust Replication of Volatility Derivatives

by Peter Carr

Courant Institute, NYU

and Roger Lee

Stanford University

April, 2003

We thank Peter Friz, Alireza Javaheri, Dilip Madan,

Jeremy Staum, and Liuren Wu for helpful comments

Postscript/PDF files of these overheads can be downloaded from:

www.petercarr.netorwww.math.nyu.edu\research\carrp\papers

Columbia University, Oxford University, Risk Europe, Mannheim University


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Introduction

It is widely recognized that delta-hedged options positions area vehicle for trading volatility.

In particular, under relatively weak conditions, a static po-sition in European options maturing at T can be combined

with dynamic trading in the underlying over [t0, T] to create acontract paying the realized variance over [t0, T]:

1

T

Tt0

2t dt limt01

n

ni=1

Fi Fi1Fi1

2

.

The main conditions are frictionless markets, continuous posi-tive price paths, continuous path monitoring, continuous trad-ing in the underlying, and a continuum of option strikes. Noassumptions are made regarding the dynamics of volatility.

The ability to create the realized variance without requiringa model for volatility dynamics has been one of the reasonsbehind the emergence of an active over-the-counter market in

variance swaps.

See Neuberger (1990), Dupire(1992), Carr and Madan (1998)and Derman et. al. (1999) for more information on varianceswaps.

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Example of a Variance Swap

2


4/35

Nonlinear Functions of Realized Variance

Recall that the floating part of the payoff on a variance swapis:

1

T

Tt0

2t dt limt01

n

ni=1

Fi Fi1Fi1

2

.

In addition to variance swaps, there is also an active market

for volatility swaps, i.e. swaps whose floating part is the squareroot of the realized variance.

However, in contrast to a variance swap, no one has yet devel-oped a hedge which bypasses the specification of a stochasticprocess for volatility.

In this talk, we assume all of the conditions that lead to therobust replication of variance swaps. We show that by further allowing dynamic trading in the op-

tions and by modelling the correlation between volatility andreturns, we can synthesize practically any function of final priceand the final realized variance defined above.

In particular, we can synthesize volatility swaps and Europeanoptions on realized variance or volatility.

Our work refutes the widely held notion that dynamic replica-tion requires process restrictions.

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Example of a Vol Swap

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Overview

A review of static hedging of path independent payoffs (Bree-den and Litzenberger (1978)).

A review of variance swaps and similar contracts.

Synthesizing volatility derivatives when returns and volatilityare independent.

Modelling correlation between returns and volatility.

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Static Hedging of Path Indep. Payoffs

Appendix 1 proves that for any generalized functionf(S), S > 0 and any expansion point 0:

f(S) = f() + f()(S ) +

f(K)(S K)+dK

+0

f(K)(K S)+dK.

This decomposition may be interpreted as a Taylor series ex-pansion with remainder of the final payoff f() about the ex-pansion point .

The first two terms give the tangent to the payoff at ; thelast two terms continuously bend this tangent so it conformsto the nonlinear payoff.

The payoff from an arbitrary path-independent claim has beendecomposed into the payoff from f() bonds, f() forward

contracts with delivery price , f

(K)dK calls of all strikesK > , and f(K)dK puts of all strikes K < .

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From Payoffs to Prices

Recall the decomposition of the payoff function f(S) into pay-offs from bonds, forwards, and options:

f(S) = f() + f()(S )+

0

f(K)(K

S)+dK+

f(K)(S

K)+dK.

Assume the existence of a pure discount bond and allEuropeanoptions of maturity T.

Assume no arbitrage and hence the existence of a martingalemeasure Q equivalent to the physical probability measure.

Then the initial value V0[f] of the continuous payoff f() canbe expressed in terms of the initial prices of the bond B0, callsC0(K), and puts P0(K) respectively:

V0[f] = f()B0 + f()[C0() P0()]

+0 f

(K)P0(K)dK + f

(K)C0(K)dK.

Note that we did not restrict the underlying price process inany way!

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Example 1: Power Plays

Consider the risk-neutral moment generating function ofXT:M0(p, T) EQ0 epXT, p .

Since XT ln(FT/F0), M0(p, T) is just the forward price ofa claim whose payoffepXT =

FTF0

p, i.e. a power ofFT.

Taylor expand the function f(F) FF0

p about F = F0:

FT

F0

p

= 1 + pF0

(FT F0) +F00

p(p 1)K

F0

p2

(K FT)+d

+F0

p(p 1)K

F0

p2

(FT K)+d

Hence the initial value is given by:

V0

FT

F0

p = B0 +

F00

p(p 1)K

F0

p2

P0(K)dK

+F0

p(p 1)K

F0

p2

C0(K)dK.

The RHS is the initial cost of a static position in bonds andOTM options maturing at T. Its forward price is the MGF:

M0(p, T) =V0FTF0

p

B0.

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Example 2: Complex Exponential

For p real, the MGF M0(p, T) of XT ln(FT/F0) might notexist. Now suppose that p is complex:

M0(p, T) EQ0 epXT, p C.

Let pr Re(p) and pi Im(p), so p = pr + ipi. From Euler:epXT = eprXT cos(piXT) + ie

prXT sin(piXT)

=FT

F0

pr

cos(pi ln(FT/F0)) + iFT

F0

pr

sin(pi ln(FT/F0))

We can separately create the real part

FTF0

prcos(pi ln(FT/F0))

and the imaginary partFTF0

pr sin(pi ln(FT/F0)) using bonds

and options.

We refer to the real part of the payoff as the cosine claim andwe refer to the imaginary part of the payoff as the sine claim.

The initial cost of creating the payoff on the cosine claim is the

real part of the complex initial value. Likewise, the initial costof creating the sine claim is the imaginary part of the complexinitial value.

We will later find it useful to allow complex payoffs and values.

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Eg. 3: Continuously Compounded Return

Finally, suppose the payoff of interest is just XT ln(FT/F0). Taylor expand the function f(F) ln(F/F0) about F = F0:

ln

FT

F0

=

1

F0 (FT F0) F0

0

1

K2(K FT)+

dK

F0

1

K2(FT K)+dK.

Hence the initial value is given by:

V0

ln

F

F0

=

F00

1

K2

P0(K)dK

F0

1

K2

C0(K)dK.

The RHS is just the initial cost of a static position in OTMoptions maturing at T.

A variance swap is a contract paying Tt0

2tdt v at T wherethe fixed payment v is chosen so that the swap has zero cost

to enter. The ability to create the log price relative using options is thekey component in synthesizing a variance swap, as we illustratenext.

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(Realized) Variance Swaps

Now assume deterministic interest rates, no jumps in the fu-tures price F, and continuous marking-to-market.

No arbitrage implies the existence of a risk-neutral probabilitymeasure Q under which the futures price dynamics are:

dFtFt =

tdWt, t [t0, T],where t is the pre-jump volatility level at time t.

Let Xt lnFtF0

be the return over [t0, t]. Itos lemma implies:

XT =Tt

0

1

FtdFt 1

2

Tt

0

2tdt.

Re-arranging implies that the realized variance is just the sumof the payoffs from a static options position and a dynamictrading strategy in futures:

Tt0

2tdt = 2XT +Tt0

2

FtdFt.

Since futures are costless, the fair fixed payment to charge ona variance swap is:

V0 [2XT]B0

=F00

2

K2P0(K)

B0dK+F0

2

K2C0(K)

B0dK.

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Localizing in Space

We replicated the quadratic variation of returns Xt ln FtF0:

XT Tt0

2tdt = 2ln(FT/F0) +Tt0

2

FtdFt.

One can add and subtract the linear function 2FTF0K

:

XT = 2 FT

F0

K lnFT

F0 + 2

Tt0

1Ft

1

K dFt.

If K = F0, the first term is just twice the difference betweenthe return compounded discretely versus continuously.

More generally, the quadratic variation of returns realized whenFt (K K, K+ K) can also be replicated:Tt0

2t1(Ft (K K, K+ K))dt

= 2

FT

F0K

+F0F0

FTFT

lnFT

K

+ 2

Tt0

1Ft

1K

dFt,

where Ft

max[K

K, min(Ft, K+

K)].

Dividing by the width of the corridor and letting it approachzero, one can localize on the underlying futures price:

Tt0

2t(Ft K)dt =2

K2

(FT K)+

Tt0

1(Ft > K)dFt

.

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Example of a Corridor Variance Swap

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Localizing in Time

Recall that the quadratic variation along a strike (local time)can be replicated:

Tt0

2t(Ft K)dt =2

K2

(FT K)+

Tt0

1(Ft > K)dFt

.

By differentiating w.r.t. T, one can localize the payoff in time:

2T(FT K) =2

K2

(FT K)+

T 1(FT > K)dFT

.

Wed like to further localize on quadratic variation withouthaving to specify a stochastic process for volatility:

2T(FT K)(XT L) =? The ability to create and price the LHS allows one to create

and price integrals of the formTt0

f(Xt, Xt)dXt.

These integrals are needed to create and price payoffs such asthe floating part of a volatility swap XT or an option onvariance (XT k)+.

We turn to this problem shortly.

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Literature on Volatility Derivatives

Grunbuchler and Longstaff (1993) assume the Heston (1993)SV model to value options on the instantaneous variance.

Brockhaus & Long (1999) use Heston to value volatility swaps.

Heston and Nandi (2000) value both volatility options andvolatility swaps using a discrete time GARCH process.

Javaheri, Wilmott, and Haug (2002) value volatility swaps us-ing Nelsons continuous time limit of the GARCH (1,1) process.

Brenner, Ou, and Zhang (2001) use Stein and Stein (1991). Howison, Rafailidis, and Rasmussen (2002) value both volatil-

ity average and variance swaps in an SV model where the in-stantanous volatility follows a mean-reverting lognormal proces

Matytsin (2000) values volatility swaps and options on variance

using a jump diffusion model with stochastic volatility.

Finally, Detemple and Osakwe (1999) value volatility optionsin a general equilibrium framework.

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Pros and Cons of SV Models

All of the literature on volatility derivatives employs some kindof stochastic volatility (SV) model.

Furthermore, the models typically (but not always) assumethat the price process and the volatility process are both con-tinuous over time.

To the extent that these assumptions are correct and that theparticular process specification is correct, one can use dynamictrading in one option and its underlying to hedge.

Most of the SV models used further assume that prices andinstantaneous volatility both diffuse, leading to a simple andparsimonious world view.

However, simple SV models cannot simultaneously fit optionprices at long and short maturities. Furthermore, since in-stantaneous volatility is not directly observable, the assump-tion that the diffusion coefficients of the volatility process are

known is debatable.

Indeed, the SV diffusion parameters implied from time seriestypically differ from the risk-neutral parameters, contradictingthe implications of Girsanovs theorem.

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Pros and Cons of SV Models

The great tragedy of science - the slaying of a beautifulhypothesis by an ugly fact Thomas Henry Huxley.

Complicating the pricing and hedging of volatility derivativesis the ugly fact that prices jump.

Furthermore, when prices jump by a large amount, it is widelybelieved that expectations of future realized volatility jump aswell. The high levels of mean reversion and vol vol impliedfrom option prices further suggests that volatility jumps.

When price and/or volatility can jump from one level to anyother, then in a model with two or more sources of uncer-tainty, perfect replication typically requires dynamic tradingin a portfolio of options.

While option bid/ask spreads typically render this strategy asprohibitively expensive at the individual contract level, knowl-edge of desired option hedges can be enacted at the aggregate

portfolio level.

Can we use dynamic trading in a portfolio of options to developa theory which does not require a complete specification of thestochastic process for volatility?

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Space Time Harmonic Payoffs

For the rest of my talk, assume zero interest rates for simplicity.We now temporarily drop the assumption that options trade.

Consider some C2,1 function u(x, q). Note that by Itos lemma:u(XT, XT) = u(0, 0) +

Tt0

ux(Xt, Xt)dXt+Tt0

uxx(Xt, Xt)2 + uq(Xt, Xt) dXt,

where recall Xt ln(Ft/F0) and hence dXt = dFtFt 12dXt. Substituting into the top equation implies:

u(XT,

X

T) = u(0, 0) +

T

t0

ux(Xt,

X

t)

dFt

Ft

+Tt0

uxx(Xt, Xt)

2 ux(Xt, Xt)

2+ uq(Xt, Xt)

dXt.

Now suppose that the function u(x, q) solves the PDE:1

2uxx(x, q) 1

2ux(x, q) + uq(x, q) = 0.

Then u(XT, XT) is just the sum of the payoffs from a staticbond position and a dynamic futures position:

u(XT, XT) = u(0, 0) +Tt0

ux(Xt, Xt)Ft

dFt.

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Example: Exponential Payoffs

Recall that if u(x, q) solves the PDE:1

2uxx(x, q) 1

2ux(x, q) + uq(x, q) = 0,

then the payoffu(XT, XT) is just the sum of the payoffs froma static bond position and a dynamic futures position:

u(XT, XT) = u(0, 0) +Tt0

ux(Xt, Xt)Ft

dFt.

For example, G(x, q) ep()xq solves the PDE, where:

p()

1

2

1

4

+ 2.

Hence ep()XTXT = 1 + Tt0 p()GtFt dFt as X0 = X0 = 0.

Thus, the process Gt ep()XtXt is a Q martingale startedat one. For > 18, GT > 0, so it is also a likelihood ratio.

If < 18, then G is a complex-valued martingale, but one

need only take real or imaginary parts to obtain real results.

If we can break the link between p and , then we can deter-mine the joint MGF ofXT and XT, and we are done.

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MGF of Quadratic Variation

Recall spanning an exponential payoff with bonds and futures:

GT ep()XTXT = 1 +Tt0

p()GtFt

dFt, t [t0, T].

Despite the dependence of the payoff on

X

T, increments in

G depend only on increments in F:dGtGt

= p()dFtFt

.

Let Mt EQt eXT. Since it is a conditional expectation of aterminal random variable, it is a martingale under Q.

For now, we assume that the martingale F and the martingaleM are orthogonal:

F, Mt = 0, t [t0, T]. We will relax this arguable assumption later, but some mod-

elling of the co-movement ofF and M will always be required.

Since increments in G depend only on increments in F, themartingale Gt ep()XtXt and the martingale M are alsoorthogonal:

G, Mt = 0, t [t0, T].

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Aside on Conditional Independence

Our orthogonality assumption implies that dMt = dEQt [eXT]and dFt are conditionally independent for each t [0, T].

To illustrate, consider the following bivariate diffusion:dFtFt

=

vtdW1t (1)

dvt = (vt, t)dt + (vt, t)dW2t,where W1 & W2 are independent standard Brownian motions.

We assume that Ft and vt are adapted to the filtration F. (1) dXt = vtdt so XT = T0 vtdt and eXT = e

T0 vtdt.

In this model, the pair (Xt, vt) is Markov in itself, and henceEQ

[eX

T|Ft] = m(Xt, vt, t) for some function m(q , v, t). By Itos lemma and the fact that EQt [eXT] is a martingale:

dEQt [eXT] =

vm(Xt, vt, t)(vt, t)dW2t. (2)

Since W1 and W2 are independent, increments in EQt [eXT]and in F are conditionally independent. While dEQt [e

XT]and dFt both depend on vt, this variable is part of the infor-mation set that we are conditioning on, and hence they areconditionally independent.

This ends the example.21


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Determining the MGF

Recall that Mt EQt eXT and Gt ep()XtXt are bothQ martingales.

Now, G, Mt is defined as the predictable process which onesubtracts from MtGt to get a Q martingale.

Since G, Mt = 0, MtGt is a Q martingale and hence:EQt [MTGT] = MtGt, t [t0, T].

Since MT eXT and GT ep()XTXT:MtGt = E

Qt [e

p()XT] Pt, t [t0, T].

Since the payoffep()XT depends only on the futures price FT,we re-introduce the existence of options of all strikes maturingat T. Pt is the observable price at time t of the static portfolioof options that spans the payoff ep()XT at T.

We allow the payoff, ep()XT, and hence its price, Pt, to haveimaginary parts. Hence, Mt, Gt, and Pt can all be complex.

The desired MGF is just the ratio of two observables:Mt =

PtGt

, t [t0, T].

In particular, M0 = P0 = EQ0 ep()XT, since G0 = 1.22


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Replicating the Exponential of QV

Recall our pricing formula for Mt EQt eXT:Mt =

PtGt

, t [t0, T],

where Gt ep()XtXt, Pt EQt [ep()XT], t [t0, T].

Taking the total derivative: dMt =1

Gt dPt Pt

G2t dGt,where higher order terms vanish because M is a martingale.

Recall that G is driven by just F, so dGtGt

= p()dFtFt

.

Hence, dMt can be represented in terms of dPt and dFt:dMt =

1

Gt

dPt

Ptp()

GtFt

dFt.

Integrating over time:eXT = EQ0 e

p()XT +Tt0

1

GtdPt Tt0

Ptp()

FtGtdFt.

If everything is real-valued, then the payoff eXT is con-structed by charging a premium of EQ0 e

p()XT dollars, and

holding 1Gt plays and Ptp()FtGt futures at each t [t0, T]. Thepremium is used to finance the initial purchase of 1

G0= 1 play.

Thus, the exhibited dynamic strategy in power plays and fu-tures is self-financing, non-anticipating, and replicating.

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Complex Replication

Recall that the exponential function ofXT is replicated as:eXT = EQ0 e

p()XT +Tt0

1

GtdPt Tt0

Ptp()

FtGtdFt.

If M,G, and P have imaginary parts, then one must takethe real part of both sides to determine the trading strategy

required to create the real part ofM.

Likewise, one takes the imaginary part of both sides to deter-mine the trading strategy which creates the imaginary part ofM.

If P has an imaginary part, then the real part of P is givenby cosine claims, while the imaginary part is given by a sineclaim.

Our results bear out Hadamards famous dictum that:The shortest path between two results in the real domainpasses through the complex domain.

If is real, then the Post Widder algorithm can be used toinvert the Laplace transform. If is complex, then efficientLaplace Transform inversion algorithms such as Abate Whittcan be used. In either case, a wide class of functions of XTcan be created including

XT

and (XT k)+.24


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Volatility Swaps

The floating part of a volatility swap has payoffXT. The following identity is proved in Appendix 2:

q =

1

2

0

1 esqs

32

ds for all q 0. (3)

Evaluating at q = XT and taking risk-neutral expectations,the fixed part of a volatility swap is:

EQ0XT = 1

2

0

1 EQ0 esXTs

32

ds

=1

2

0

1

EQ0 e

p(s)XT

s32ds

where recall p(s) 12 14 2s.

For s (0, 18), p is real, while for s > 18, p is complex:

EQ0

X

T =

1

2

180

1 EQ0 e

12

142sXT

s3

2

ds

+1

2

18

1 EQ0 e12XT cos(

2s 14XT)s

32

ds.

We need to numerically check the robustness of this inversion.25


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A Recap

Under our assumptions, claims on quadratic variation XTmaturing at T can be priced relative to the initial futures priceand the initial prices of all European options maturing at T.

This information is the same as for a variance swap. However,robust replication of a nonlinear function ofXT requires dy-namic trading in futures andall options of maturity T.

Notice that we have not assumed a Markov process for thestate variables in the problem. It may well be for examplethat the whole surface of implied volatilities is following someunknown stochastic process.

However, if we do assume some Markov process eg. that time,price, and the instantaneous variance rate vt are Markov inthemselves, then note that we have not assumed that vt iscontinuous over time, i.e. vt is just an arbitrary process withunknown characteristics.

Even if we do assume a bivariate diffusion for returns Xand itsvariance rate vt, we do not need to specify the drift or diffusioncoefficients of the SDE governing vt.

In contrast to the variance swap, we assumed that returns andthe MGF of XT are orthogonal. We relax this assumptionnext.

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Modelling Correlation

Recall that we assumed that X, Mt = 0, whereXt ln(Ft/F0) and Mt EQt eXT.

We now redefine M as Mt(, ) EQt eXT+XT, , C. The joint MGF Mt(, ) is still a Q martingale and it reverts

to its former definition if = 0. To relax the orthogonality assumption, suppose instead:dMtMt

= (Xt, , )dXt + dNt,

where X and the noise N are orthogonal, i.e. X, Nt = 0.

The function (x,,) is known and we allow (x, 0, )

= 0.

Under our new assumption, dX,MtMt

= (Xt, , )dXt. Extend Gt C(Xt)eXt, where C(x) solves the ODE:

1

2Cxx(x) +(x,,) 1

2

Cx(x) C(x) = 0.

If is independent ofx, then C(x) = ep(,)x

where:

p(, ) 12

(, ) 1

2 (, )2

+ 2.

If (x,,) = 0, then G reverts to its previous definition.27


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Multiplication Makes Martingale

Recall Gt C(Xt)eXt where C(x) solves:1

2Cxx(x) +(x,,) 1

2

Cx(x) C(x) = 0.

From Itos lemma:dGt = GxdFtFt

+12Cx + 12Cxx C

eXtdXt

= GxdFtFt

(x,,)CxeXtdXt

= GxdFtFt

Gx(x,,)dXt.

Note that G is no longer a Q martingale. Furthermore, thecovariation of M with X induces covariation ofM with G:

dG, MtMt

= GxdX, Mt

Mt= Gx(x,,)dXt.

From integration by parts:

d(GM)t = GtdMt + Mt

dGt +dG, M

t

Mt

= GtdMt + MtGxdFtFt

,

and hence MG is still a Q martingale.

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Determining the MGF

Recall Gt C(Xt)eXt where C(x) solves an ODE andMt EQt eXT+XT.

Also recall that MG is still a Q martingale and hence:EQt [MTGT] = MtGt, t [t0, T].

Since MT eX

T+X

T and GT C(XT)eX

T:MtGt = E

Qt [C(XT)e

XT] Pt, t [t0, T]. Pt is now the observable price at time t of the static portfolio

of options that spans the payoff C(XT)eXT at T.

If the payoff has an imaginary part, so does Pt. The desired MGF is again just the ratio of two observables:

Mt =PtGt

.

Replication proceeds as before. Hence we can model somecovariation of returns with volatility and still obtain robustreplication.

If the desired option maturities are not available in the mar-ket initially, then one can let price and quadratic variation beMarkov state variables and use our results to roll over shortermaturity options (introducing more model risk).

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A Problem

Recall that for the joint MGF Mt EQt eXT+XT and thereturn Xt ln(Ft/F0), we assumed that:

dMtMt

= (Xt, , )dXt + dNt,

where X, Nt = 0 and (x,,) is known. If = 0, then as time increases, the MGF M ofXT should

vary less as more of its payoff gets determined.

In fact, as t T, the covariation of M with X should tendtowards 0.

However, is independent of time, so we cant capture thiseffect.

Future research needs to address this issue, perhaps by tradingin options of all maturities as well as all strikes.

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Summary and Extensions

Just as a linear payoff on price can be robustly synthesized bya static position in stock and bond, a linear payoff on quadraticvariation such as a variance swap can be robustly synthesizedusing a static position in options on price.

However, just as a nonlinear payoff on price usually requiresdynamic trading in the stock, robust replication of a nonlinearpayoff on quadratic variation seems to require dynamic tradingin options on price.

The claims we considered mature at a fixed time. For continu-ous processes, it is actually easier to have the maturity occur at

the first time that quadratic variation or price crosses a level,since one random variable is determined at the payoff time.

We have extended this work to X = g(F) and to jumps. It would be interesting to extend this work to the multivariate

setting.

It would also be interesting to try to replicate claims on othernotions of volatility, such as local volatility or exponentiallyweighted quadratic variation.

In the interests of brevity, these extensions are best left forfuture research.

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App. 1: Replicating with Bonds & Options

For any fixed , the fundamental theorem of calculus implies:f(S) = f() + 1S>

S f

(u)du 1SS

f() +u f

(v)dv

du

1SS

Sv

f(v)dudv

+1S

S

f(v)(S v)dv+1S


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App. 2: The Square Root Function

Let () 0

t1etdt be the gamma function with a positive real. Then it is well known that:

1

2

= . (4)i.e.

=0

ett

dt. (5)

Consider the change of variables s = tq for q > 0. Then t = sq, dt = qds and hence:

=0

esqsq

qds =

q

0

esqs

ds.

Solving for

q gives one representation:

q =

0

esqs

ds. (6)

Integrating (5) by parts, let:

u =1

tdv = etdt

du = 12t3/2

dt v = 1 et (7)

Hence: =

1 ett

t=t=0

+1

2

0

1 ett3/2

dt (8)

or

2

=0

1 ett3/2

dt. (9)

Again consider the change of variables s = tq

for q > 0:

2

=0

1 esq(sq)3/2

qds =1

q

0

1 esqs3/2

ds. (10)

Solving for q gives a second representation:

q =

1

2

0

1 esqs3

2

ds. (11)

33


35/35

References

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Bond and Currency Options, Review of Financial Studies, 6, 327343.

[11] Heston S., and S. Nandi, 2000, Derivatives on Volatility: Some Simple Solutions Based on Observ-ables, Federal Reserve Bank of Atlanta Working paper.

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[14] Matytsin, A., 2000, Modelling Volatility and Volatility Derivatives, presentation available at

www.math.columbia.edu/ smirnov/Matytsin.pdf

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Robust Replication of Volatility Derivatives