Complete Models with Stochastic Volatility

Complete Mo dels with Sto chastic Volatility

David G. Hobson

Scho ol of Mathematical Sciences, University of Bath

and

L.C.G. Rogers

Scho ol of Mathematical Sciences, UniversityofBath

Abstract

The pap er prop oses an original class of mo dels for the continuous time price pro cess

of a nancial security with non-constantvolatility. The idea is to de ne instantaneous

volatility in terms of exp onentially-weighted moments of historic log-price. The instanta-

neous volatility is therefore driven by the same sto chastic factors as the price pro cess, so

that unlike many other mo dels of non-constantvolatility, it is not necessary to intro duce

additional sources of randomness. Thus the market is complete and there are unique,

preference-indep endent options prices.

We nd a partial di erential equation for the price of a Europ ean Call Option.

Smiles and skews are found in the resulting plots of implied volatility.

Keywords: Option pricing, sto chastic volatility, complete markets, smiles.

Acknow ledgemen t. It is a pleasure to thank the referees of an earlier draft of this pap er

whose p erceptive comments have resulted in manyimprovements.

Research supp orted in part byRecordTreasury Management

Research supp orted in part by SERCgrant GR/K00448

1 Sto chastic Volatility

The work on option pricing of BlackandScholes (1973) represents one of the most

striking developments in nancial economics. In practice b oth the pricing and hedging

of derivative securities is to daygoverned by Black-Scholes, to the extent that prices are

often quoted in terms of the volatility parameters implied by the mo del.

The Black-Scholes mo del is based up on the common assumption that the prop or-

tional price changes of the asset form a Gaussian pro cess with stationary indep endent

increments. This assumption has b een the sub ject of much attention over the interven-

ing years. An insight of the Black-Scholes mo del is that the crucial parameter is the

volatility of the underlying price pro cess: consequently research has fo cussed on this pa-

rameter. Empirical analysis of sto ckvolatility has shown that it is not constant | see

Blattb erg and Gonedes (1974), Scott (1987) and the references therein. Moreover the

prices at which derivatives (and esp ecially call options) are traded are inconsistentwith

a constantvolatility assumption.

For this reason a numb er of authors have suggested variants of the Black-Scholes

mo del. These alternative theories separate into two broad genres. (Since we are interested

in mo dels of changing volatilitywe exclude mo dels with jumps such as the jump-di usion

mo del of Merton (1976).) The rst approach, as represented byCox and Ross (1976),

Geske (1979), Rubinstein (1983) and recently Bensoussan et al (1994) describ es the sto ck

price as a di usion with level dep endentvolatility. This may either b e a mo delling

assumption or follow from more fundamental prop erties, for example by relating sto ck

price to the value and the debt of a rm. The second approach, exempli ed by Johnson

and Shanno (1987), Scott (1987), Hull and White (1987, 1988) and Wiggins (1987)

de nes the volatility as an autonomous di usion driven by a second Brownian motion.

(The asset price pro cess is driven by the rst Brownian motion.) Further details of these

two approaches, whichwe lab el as level-dep endentvolatility and sto chastic volatility,

are given in Sections 2.1 and 2.2 resp ectively. Hull (1993) and Merton (1990) are also a

valuable source of reference.

In this pap er we suggest a new class of non-constantvolatility mo dels, which can

b e extended to include the rst of the ab ove classes, but also share manycharacteristics

with the second approach. The volatility is non-constant, but it is an endogenous factor

in the sense that it is de ned in terms of the past b ehaviour of the sto ck price. This

is done in suchaway that the price and volatilityforma multi-dimensional Markov 1

pro cess.

We b elieve that this class of mo dels has twin advantages over existing sto chastic

volatility mo dels. Firstly, unlike the sto chastic volatility mo dels, since no new sources of

randomness are intro duced, the market remains complete and there are unique preference-

indep endent prices for contingent claims. Secondly,incontrast with the level-dep endent

volatility mo dels, it is p ossible to sp ecify a single, simple mo del within the new class

whichover time will exhibit smiles and skews of di erent directions.

A natural e ect of the mo del is to makevolatility self-reinforcing. Since volatility

is de ned in terms of past b ehaviour of the asset price it will b e high precisely when

there have b een large movements in the recent past. This is designed to re ect real

world p erceptions of market volatility, particularly if practitioners are to compare historic

volatility with implied volatility.

An observed e ect in options markets is the presence of `smiles' and `skews' in

the implied volatilities across strikes. Rubinstein (1985) has demonstrated the presence

of this phenomena in options data and noted moreover that the direction of the skew

maychange over time. Similarly Fung and Hsieh (1991) discovered smiles in the prices

of options based on underlying securities in each of the sto ck, b ond and currency mar-

kets. Fung and Hsieh also ask whether implied volatility is a b etter predictor of realised

volatilityover the lifetime of an option than historic volatility.

A p otential explanation for the presence of smiles and skews is the concept of non-

constant or sto chastic volatility. Hence it is natural to investigate the implied volatility

implications of any mo del of non-constantvolatility.Such analysis has b een attempted

for the Constant ElasticityofVariance mo del of Cox and Ross byBeckers (1980) and

for the sto chastic volatility mo del of Hull and White by Stein and Stein (1991) and

Paxson (1994). In this pap er weshow that the prop osed class of volatility mo dels has

the desirable feature of explaining smiles and skews.

There are similarities b etween the prop osed class of sto chastic volatility mo dels and

the ARCH (Engle (1982)) and GARCH (Bollerslev (1986)) mo dels favoured for nancial

time series mo delling by econometricians. These mo dels are formulated in discrete time

and p ostulate a log-price pro cess for the sto ckwhich has a conditional variance dep ending

on a set of exogenous and lagged endogenous variables and past residuals. See Duan

(1995) for a discussion of GARCH mo dels and option pricing.

The remainder of this pap er is structured as follows. Section 2 describ es the set of

existing mo dels with level-dep endent or sto chastic volatility and outlines their implica- 2

tions for option pricing. The new mo del itself is intro duced in Section 3. The p enultimate

section describ es the implications of this mo del for option pricing and demonstrates the

existence of skews and smiles via numerical solution of the option pricing partial di er-

ential equation. The direction of the skew is seen to b e a function of the recent history of

the asset price pro cess. The nal section provides a summary of the theoretical ndings

and a comparison with comp eting mo dels. The task of comparing the predictions made

by the new class of volatility mo dels with market exp erience is left to a subsequent pap er.

Finally, in an app endix, we discuss some of the technical results that we require to ensure

that when we switch to a martingale measure, the new (pricing) measure is equivalent

to the original (real world) measure. Our measures arise as the laws of solutions of SDEs

and the conditions for equivalence are of interest in their own right.

2 Non-constantVolatility Mo dels

The ubiquitous Black-Scholes pricing formula for sto ck options assumes that the price

(P ) of a sto ck is the solution to a sto chastic di erential equation (SDE)

t tT

(1) dP = P (dB + dt)

t t t

where is a known and constantvolatility parameter and B is a Brownian motion.

Notwithstanding the widespread use of the Black-Scholes formula, concern has b een

expressed ab out some of the assumptions necessary for the derivation of the formula. The

most criticised of these assumptions are the requirement of a p erfect frictionless market

(and in particular the absence of transaction costs) and the imp osition of a constant

volatility. It is exclusively the second of these assumptions that we address here.

The basic nancial environmentinwhich our asset trades consists of the sto ckwith

price pro cess P and a b ond whichpays a xed and constantrateofinterest r . There is

a p erfect market with no transaction costs and no restrictions on short selling of sto ck

or b ond provided that the net wealth of the trader remains non-negative. In particular

a trader may sell sto ck or b onds that he do es not own provided that bytheendofthe

trading p erio d he has repurchased sucientquantities to cover his obligations. 3

2.1 Level Dep endentVolatility

There is a long history of alternative mo dels for the sto ck price pro cess in which P

satis es the SDE

(2) = (P )dB + dt

P t t

where is a function of P .For example in their Constant ElasticityofVariance (CEV)

(1 )

mo del Cox and Ross (1976) take (x) x : This is a di usion mo del for the price

pro cess and following Harrison and Pliska (1981) the mo del is complete with unique,

preference-indep endent option prices. In general these prices may only b e known as the

solution to a partial di erential equation.

Cox and Ross suggest (2) in direct comp etition to the exp onential Brownian motion

mo del, suggesting that the form of the equation may representleverage e ects. Geske

(1979), Rubinstein (1985) and Bensoussan et al (1994) derive explicit forms for (2) by

reference to the value and debt of the rm.

Supp ose the aim is to price a Europ ean Call option with exercise date T and strike

price K . By considering a riskless p ortfolio consisting of the sto ck call option, asset and

cash, standard arguments show that the value H = H (P ;T t) of the call is the solution

@H @H @ H

0=rP rH + (P ) (3)

@P @t @P

with b oundary condition

H (P; 0) = (P K ) :

Beckers (1980) investigates the solution to (3) for the Constant ElasticityofVari-

ance mo del of Cox and Ross. Compared with standard Black-Scholes prices, options

prices for the CEV mo del are higher for in-the-money options and lower for out-of-the-

money options. Equivalently implied volatilities decrease as the strike of the option

increases.

Geske (1979) suggests mo delling sto ck prices as an option on the value of a rm

with debt. If the value V of the rm is given by an exp onential Brownian motion, (so

that it has constantvolatility ), then the sto ck-price P is given bytheBlack-Scholes

V t

formula with the strike set equal to the value of the debt D , and the exercise date set

equal to the maturity date of the debt T :

)(T t) log (V =D )+(r +

D t

P = V

t t

T t

V D 4

1 2

log (V =D )+(r )(T t)

t D

r (T t) V

De ;

T t

V D

where (:) denotes the cumulative normal distribution. In practice we infer V from the

observables P , D and T .Thevolatility of the sto ck price is then

@P V

t t

P V

P @V

log (V =D )+(r + V )(T t)

t t D

T t

V D

> :

The value of a call option in this mo del is given by the solution of (3) with this new

sp eci cation of . If D =0 or T = 1 then the option price collapses to the usual

P D

Black-Scholes result. Otherwise the e ect is to mo dify the Black-Scholes sto ckoption

price: if for example the sto ck price falls then sto ckvolatility rises and the Geskeoption

price rises relativetotheBlack-Scholes price, although it still falls in absolute terms.

Figure 1 shows the implied volatility curves for the CEV mo del and the Geske

mo del. The impled volatilityisthatvalue for the volatility, which, when substituted

into the Black-Scholes formula gives the options price as calculated from the alternative

mo del. For each alternative mo del it is seen that the implied volatility falls as the strike

price rises. Formul for the prices of call options are taken from Schro der (1989) and

Geske (1979). As Schro der p oints out =2=3 is a sp ecial case of the CEV mo del for

which the prices of call options can b e expressed in terms of normal distributions.

Rubinstein (1985) analysed options price data to see if there was any systematic

variation of implied volatility with strike. For one of the p erio ds under consideration he

found that implied volatility did indeed decrease as the option moved out-of-the-money.

However in a later p erio d the implied volatility increased as the strike increased. The

level-dep endentvolatility mo dels of Cox and Ross and Geske are unable to explain this

change over time in the direction of the implied volatilityskew.

In a pair of innovative articles Dupire (1993, 1994) incorp orates options price data

into the sp eci cation of a level dep endentvolatility mo del. Rather than p ostulate the

dynamics of the asset price pro cess he uses options price data to infer the form of the

volatility so that the mo del is guaranteed to price Europ ean call options consistently with

the market. In spirit this mo del is similar to term structure mo dels of interest rates.

Dupire assumes that the asset price is a di usion (and thus that the volatilityof

the asset dep ends on the price level and time alone), and that the price of an option is the 5 implied volatility

0.21

0.205

0.2

0.195

strike

0.8 0.9 1.1 1.2

Figure 1: Implied Volatilities for the CEV and Geske mo dels. In each case the

option is a Europ ean Call with time to exercise T =0:25, and the initial price is

unity.For the CEV mo del (less steep curve) =0:2and =2=3; for the Geske

mo del T =0:25, D = 1 and the volatilityofthevalue of the rm is 0.1.

D 6

discounted exp ected value of its payo under some equivalent martingale measure. He

shows that if there is a continuum of market prices for Europ ean calls of every p otential

strike, and for every p otential exercise date then this uniquely sp eci es the volatility.

However this mo del cannot explain smiles or skews, since they are taken as inputs.

Platen and Schweizer (1995) havedevelop ed a further mo del in this category. They

p ostulate a mo del in which a non-constantlevel-dep endentvolatility arises endogenously.

They assume that market traders have p ortfolios consisting of b oth the underlying asset

and option liabilities which they must dynamically hedge. The e ect of the hedging

requirements is to a ect the asset volatility. As in this pap er the authors derive a partial

di erential equation for an option price which they solvenumerically.

2.2 Sto chastic Volatility via an SDE

Several authors have prop osed mo dels of volatilityinwhich the volatility is de ned

via a sto chastic equation. The following mo del is due to Hull and White (1987). For

related pap ers see Johnson and Shanno (1987), Scott (1987), Wiggins (1987) and Hull

and White (1988). Hofmann et al (1993) consider a more general Markovian mo del which

contains the mo del of Hull and White as a sp ecial case.

Let the sto ck price P and the volatility b e de ned via the pair of sto chastic

t t

di erential equations

(4) dP = P ( dB + (P ; ;t)dt)

t t t t t t

(5) dV = V ( ( ;t)dW + ( ;t)dt)

t t t t t

where V and B and W are Brownian motions with covariance dB dW = %dt,for

some correlation 1 <%<1.

If the volatility is a traded asset then there are two risky securities and the market

is complete. Otherwise the intro duction of the second Brownian motion W makes this

mo del incomplete. In particular there are no unique prices for sto ck options.

It is feasible to consider the ab ove mo del with j%j = 1. In this case completeness

is regained, but the volatility b ecomes a complicated function of the history of the price

pro cess.

Now consider the option pricing implications of these sto chastic volatilitymodels.

Our analysis follows Scott (1987). Supp ose that the aim is to price a Europ ean call option

with exercise date T and strike price K .Thenby considering a p ortfolio consisting of 7

the riskless b ond, the asset and two call options with di erent maturities, Scott shows

that the option pricing function H H (P ;V ;T t)must satisfy

t t

2 3=2 2 2

1 1

vp H + % v pH + v H rH + rpH = bH ; H +

pp pv vv p v t

2 2

sub ject to the b oundary condition

H (P; V ; 0) = (P K ) :

Here b b(P ;V ;t) is indep endent of the exercise date T .

t t

The function b cannot b e deduced from arbitrage considerations alone. Conse-

quently there is no unique option pricing function.

Although there is no unique price for the option some authors have suggested

particular choices. When % = 0 the choice b = V corresp onds to setting the market

price of risk to b e zero (see for example Stein and Stein (1991); Wiggins (1987) builds on

the equilibrium approachofCox, Ingersoll and Ross (1985) to provide some justi cation).

It is also equivalent to pricing options under the minimal martingale measure of Follmer

and Schweizer (1990). Suppp ort for this idea is to b e found in Hofmann et al (1993).

However other authors prop ose di erent criteria for determining b.For a utility based

approach see for example Karatzas et al (1991) and Due and Skiadas (1994).

The implications for options pricing of a sto chastic volatilitymodelhavebeen

considered bymany authors. Stein and Stein (1991), who assume no correlation b etween

the pair of Brownian motions driving the asset price and the volatility, nd implied

volatility smiles. By allowing a negative correlation b etween the asset and the volatility

Wiggins (1987) shows that implied volatilitymay b e higher for in-the-money options than

out-of-the-money options. Both he and Scott (1987) nd that, when considered across a

range of strikes, options prices from mo dels with sto chastic volatilityprovide a sup erior

t to market prices when compared with a constantvolatilitymodel.Paxson (1994) is

able to conclude that, in most but not all cases, sto chastic volatility can account for b oth

smiles and skews. On a theoretical note, Renault and Touzi (1992) haveshown that

under a zero correlation (% = 0) assumption a sto chastic volatilitymodelmust exhibit

volatility smiles in the option price.

2.3 GARCH Mo dels

The acronym GARCH stands for generalised autoregressive conditional heteroskedastic

and describ es a p opular class of discrete time mo dels used to mo del time series with 8

non-constantvolatility. In discrete time the GARCH(1,1) mo del for the log-price pro cess

Z and conditional variance is of the form

t t

Z = Z + +

t t1 t t t

(6)

2 2 2 2

= ! + +

t t1 t1 t1

Here is an i.i.d. sequence of zero mean unit variance random variables. If is a

t t

function of then the mo del is GARCH(1,1)-M. More generally GARCH mo dels allow

for to b e an arbitrary function of past conditional variances and past residuals.

GARCH mo dels generate data with fatter tails than those from a mo del where

is constant in (6). This is consistent with many studies on observed sto ck prices.

Whilst GARCH mo dels may capture essential qualitative prop erties of an asset

price pro cess, they are an unsuitable class of mo dels for a prosp ective option replicator

since, save in the simplest binomial cases, exact replication is infeasible in discrete time.

In general there is no natural continuous time analogue of the discrete time GARCH

pro cess. With judicious choice of parameter values Nelson (1990) has demonstrated

convergence in the Skorokho d top ology to a continuous-time pro cess similar to the mo del

describ ed in Section 2.2; however if this limit pro cess is sampled at equally spaced discrete

time p oints then the resulting pro cess is not GARCH. Recently however a new class of

weak ARCH mo dels has b een prop osed by Drost and Nijman (1993). These mo dels have

the emb eddibility prop erty thatifaweak ARCH pro cess is sampled at regular intervals

then the resulting pro cess is again weak ARCH.

Kind et al (1991) proved a convergence result for a di erent sto chastic volatility

mo del. In their GARCH-typ e mo del the quadratic variation of the sto ck has an interpre-

tation as the `historic' volatility de ned over a nite time window. Unfortunately in the

continuous time limit the volatility pro cess is deterministic. In either case the GARCH

mo del fails to yield preference indep endent option pricing in the presence of sto chastic

volatility.However under assumptions on the utilityoftheinvestor Duan (1995) is able

to derive a unique price for an option.

3 A Complete Mo del with Sto chastic Volatility

In this section we de ne a new class of sto ck-price mo dels. The new feature is the

sp eci cation of instantaneous volatility in terms of exp onentially weighted moments of

the historic log-price. This intro duces a feedback e ect into the volatility pro cess: present

sho cks in the asset price result in higher future uncertainty. 9

The rst step is to intro duce some convenient notation. De ne the discounted log-

price pro cess Z by Z =log(P e ). De ne also the o set function of order m, denoted

t t t

(m)

S ,by

(m)

u m

(7) S = e (Z Z ) du

t tu

The constant is a parameter of the mo del which describ es the rate at which past

information is discounted. Then, for some value n,

Assumption 3.1 Z solves the SDE

(n) (1) (n) (1)

(8) )dt ;:::;S )dB + (S ;:::;S dZ = (S

t t

t t t t

where (:) and (:) are Lipschitz functions, and (:) is strictly positive.

Remark 3.1 More generally it is p ossible to allow (:) to b e a function of the price level

P also. In this sense this mo del can b e extended to include the class of level-dep endent

volatility pro cesses as a sp ecial case.

Remark 3.2 The key feature to note at this early stage is that no new Brownian motions

(or other sources of uncertainty) have b een intro duced in the sp eci cation of the price

pro cess. This will imply that the mo del yields unique option prices without the need to

sp ecify market prices for risk.

Remark 3.3 The mo del is designed so that movements in the price of an asset may

result in changes in the volatility of that asset. If the volatility of sto ck prices is related

to the amount of trading in a sto ck then this corresp onds to the scenario that price

changes encourage further interest and activity in the market.

(m)

The reason for our de nition of the pro cesses S is seen in the following lemma.

(1) (n)

(m)

Lemma 3.1 (Z ;S ;:::;S ) forms a Markov process. The o set processes S sat-

t t

isfy the coupled SDEs

m(m 1)

(m1) (m) (m) (m2)

dZ + = mS dS (9) dt dhZ i S S

t t

t t t t

Pro of

Since the functions and are assumed to b e Lipschitz, the existence and uniqueness

of a non-explosive solution to (8) and (9) is guaranteed (see Rogers and Williams (1987,

p132)). Moreover the Markov prop erty also follows (Rogers and Williams (1987, p162)). 10

(m)

The sto chastic calculus is easier with the following equivalent de nition for S :

(m)

u m t

= e (Z Z ) du e S

t u

k u mk

= (Z ) e (Z ) du:

t u

k =0

Then

(m) (m)

t t

e S dt + e dS

t t

k (k 1)

u mk k 1 k 2

e (Z ) du k (Z ) dZ + = (Z ) dhZ i

u t t t t

k =0

t mk k

+e (Z ) dt(Z )

t t

m 1

u (m1)(k 1) k 1

= m e (Z ) du (Z ) dZ

u t t

k 1

k =1

m(m 1)

m 2

u m2(k 2) k 2

+ e (Z ) du (Z ) dhZ i

u t t

k 2

k =2

m(m 1)

(m1) (m2)

dZ + = e mS dhZ i S :

t t

4 Option pricing

4.1 The General Theory

For any mo del of sto ck prices a key feature is the price equation for options. The aim

of this section is to price a Europ ean Contingent Claim with exercise date T and payo

q (P ). For simplicitywe assume that n = 1 in (8) so that the volatility dep ends only on

(1)

the rst order o set S . As a result we can simplify notation in this section by writing

(1)

S as a shorthand for S . Note that (9) collapses to dS = dZ S dt and that we

t t t

can combine (8) and (9) to conclude that S is an autonomous di usion satisfying the

sto chastic di erential equation

(10) dS = (S )dB +((S ) S )dt:

t t t t t 11

Then

log fP e gZ = Z +(S S )+ S du:

t t 0 t 0 u

Note also that S is adapted to the ltration F of B .

t t

(S )du. De ne (S )= (S )+f(S )= (S )g and consider the pro cess B B +

u t t

Let P b e Wiener measure for B . De ne a new measure P by sp ecifying that on F

Z Z

t t

=exp (S )dB (S ) du :

u u u

0 0

See App endix A for a discussion of the conditions under which P is a probability measure,

and under which P is equivalentto P.

If we assume that these conditions are satis ed then we can rewrite (10) as

(11) (S ) + S )dt; dS = (S )dB (

t t t t t

~ ~ ~

where B is a P Brownian motion. Then under P the discounted price pro cess e P is

a martingale, and we can apply the elegant martingale pricing theory of Harrison and

Kreps (1979) and Harrison and Pliska (1981) to conclude that the option price can b e

written

r (T t)

f (P ;S ;T t)= e (12) E [q (P )jF ]:

t t T t

Sub ject to integrability conditions on q (:), the Brownian martingale representation Theo-

rem implies that the contingent claim can b e replicated using a previsible trading strategy.

By the Feynman-Kac formula (Karatzas and Shreve (1988, p366)) f satis es the

partial di erential equation

2 2

1 1 1

0=(rpf rf sf f )+ (13) (s) f + p f + f + pf

p s t s pp ss sp

2 2 2

sub ject to the b oundary condition

(14) f (p; s; 0) = q (p):

Assuming that E (jq (P )j) < 1, the recip e (12) de nes a solution to the partial di erential

equation (13) with b oundary condition (14), the necessary smo othness of the f so de ned

b eing assured byHormander's Theorem (for a statement, see, for example, Rogers and

Williams (1987, Theorem V.38.16)). In general uniqueness requires a further argument,

but in the sequel we consider a situation which can b e reduced using put-call parityto

the case where q is b ounded, and then the optional sampling theorem guarantees that 12

all b ounded solutions can b e expressed via (12) and it is clear that there is a unique

b ounded solution to the PDE.

2 2

Note that if the sto ckvolatility is constantsothat dhZ i =dt (S ) = and f

t t

dep ends only on P and t we recover

2 2

p f 0=rpf rf f +

pp p t

which is the standard p de for the Black-Scholes option price.

4.2 A Sp eci c Example: Smiles and Skews

The purp ose of this section is to calculate the option price bynumerically solving the

partial di erential equation (13) sub ject to the b oundary condition (14). This results

in an option price surface and, to facilitate comparison with the standard Black-Scholes

mo del, Black-Scholes implied volatilities are calculated. A numerical solution is necessary

b ecause even in the simple example presented b elow it seems dicult to derive prop erties

of the solution analytically; see however the end of this section for a few observations.

The sp eci c example considered is p otentially one of the simplest non-trivial cases,

namely to price a Europ ean call with the interest rate r taken to b e zero (or equivalently

working with forward prices) and dynamics for the discounted log-price pro cess Z given

(s)= (15) 1+s ^ N;

for some large constant N . The function has the useful prop erties of b eing even (see the

remarks at the end of this section) and b ounded. While more complicated functions

dep ending on higher order o set functions could b e studied, it will b e demonstrated b elow

that even the simple example sp eci ed by (15) e ectively accounts for the p ossibilities

of smiles and skews.

The intuition that we hop e to capture with this mo del is that if the current price

di ers greatly from a past average, then the volatility is high. Moreover, by basing (s)

1+s + s ^ N ), on a quadratic in s with non-zero linear co ecient (so that (s)=

then we could mo del markets in whichvolatilitychanges are correlated with price changes.

Thus for example, it is p ossible to construct a mo del in whichvolatility is higher when

the current (log)-price is b elow the past average, than when it is ab ove the past average

by the same amount. Finally note that for suitable choice of the drift , the o set S is

mean reverting (see (10)), and this prop erty will b e inherited by the volatility . 13

By assumption and are Lipschitz continuous and time-homogeneous so that the

SDEs (10) and (11) eachhave a pathwise unique strong solution. Moreover, since (S )

is b oth b ounded ab ove and b ounded away from zero, S is non-explosive under b oth of

the probability measures P and P,andby Theorem 7.19 in Liptser and Shiryayev (1977)

these measures are equivalent. Thus the pricing theory of the previous section applies.

Note that if the volatility sp eci cation (15) was replaced by (s)= 1+s , then the

di usion S would explo de with p ositive probability under the candidate pricing measure

~ ~

P,thus contradicting the assumption that P and P are equivalent.

The option valuation formula b ecomes a function of the current price P , the current

o set S and the time to go (T t), as well as the parameters K (the strike), , and

.Taking (for the moment) K = 1, and using the transformation P e , U Z S ,

the option pricing problem simpli es to solving for V V (Z ;U ;T t) V (Z ;U ;T

t t t t

t; ;;), where V satis es the partial di erential equation

V = (16) (z u) [V V ]+(z u)V

t zz z u

sub ject to the b oundary condition

Z +

(17) V (Z; U; 0) = (e 1) :

Since the numerical solution of (16) sub ject to (17) is calculated over nite regions of the

z and u variables, the choice of N do es not a ect the solution V .Furthermore, in order

to guarantee existence and uniqueness of the solution it is b etter to solve (16) sub ject to

Z +

the b oundary condition V (Z; U; 0) = (1 e ) . The price of a call can then b e deduced

using put-call parity.

Now V gives the price of a call with xed unit strike as a function of the asset

price at time 0, but it is a simple exercise using scaling to then calculate the price

~ ~

V V (K ; S; T ) of a call with arbitrary strike assuming, as we shall from now on, that

the initial asset value satis es P = 1. (Here the analysis dep ends critically on the fact

that in our example is a function of the o set S alone, and is otherwise indep endent

of the price level.) Denote by p ( ) the Black-Scholes option price as given by

p p

2 2

p ( ) = [( ln K + T=2)=( (18) T )] K [( ln K T=2)=( T )]:

Then we can de ne a mo del implied volatilityvia

(K; S; T )=p (V (K ; S; T )):

BS 14 0.1

0.08

0.06

0.04

0.02

0.9 0.95 1 1.05 1.1

Figure 2: Europ ean Call Prices: For an option with time to maturity T =0:25,

and strike K = 1 the upp er curvegives the Black-Scholes price for a volatilityof

0.21, the lower curve the Black-Scholes price for a volatility of 0.2, and the middle

curve the price from the mo del prop osed in (13) with =0:2, =5, = 1 and

Z U S =0:1. The intrinsic value of the option is also shown.

0 0 0 15

By de nition (K ; S; T )isthevolatilitywhich, when substituted into the Black-Scholes

pricing formula (18) together with the strike K and the time to exercise, gives the cal-

culated option price.

Implied volatilityprovides a convenient measure for expressing the price of deriva-

tive securities in a language which can b e applied accross di erent securitypayo s. The

results presented b elow are expressed in terms of the implied volatility whichisa

function of the in-the-moneyness of the option (given by the strike K ; recall that the

currentvalue of the asset is unity), the time to maturity T and the initial value of the

o set S .To b egin with we set T to b e 0.25, though later we consider implied volatilities

as a function of time.

Figure 3 shows the calculated implied volatility as a function of the rst order

o set S and the strike K , for parameter values =0:2, = 5 and =5. The

immediate conclusion is that the presence of the non-constantvolatility term has the

e ect of increasing implied volatility, and that this e ect is strongest when the initial

o set is non-zero, and the option is not at-the-money.

Atany moment in time there may trade a family of options with di erent degrees of

in-the-moneyness, or corresp ondingly di erent strikes. Thus it is illuminating to consider

cross-sections of this implied volatility surface, each cross-section corresp onding to a

di erentvalue of S . In principle, with the b ene t of historical data, the value of S is

0 0

an observable so that the options trader will knowwhich of the p ossible regimes for S

describ es the current situation. In practice it may b e that the options trader will use

current options prices to infer the value of S .

For all values of S the implied volatilityisa convex function of the strike. Moreover

skew e ects are also plainly visible: the smile has a pronounced p ositiveskew for p ositive

values of S and a negativeskew when the current log-price is b elow a past average. This

is in agreement with an intuitive understanding of the b ehaviour of the asset pro cess.

Consider the situation of an out-of-the-money option with initial o set S > 0.

This option will b e worthless until the asset price rises to the strike price, and the larger

S , the so oner this will happ en. Given that the strike price is reached b efore exercise,

the o set at that time will always be positive, and typically at least as large as S .

The volatility will then b e quite high, in ating the option price. If wenow consider what

happ ens to this argumentaswe allow S to get smaller, and then go negative, weseethat

it b ecomes more dicult (at least assuming that S is not to o large) for the price of the

asset to reach the strike b efore exercise, since the volatility has got smaller. Also, if S

0 16

Figure 3: The Volatility Surface: The volatility surface is a function of the initial

value of the o set, here ranging from -0.2 to +0.2, and shown from left to right,

and the logarithm of the strike, whichvaries from -0.2 (in the money) in the

foreground, to 0.2 (out of the money) at the back. The vertical scale showing the

implied volatility ranges from 0.2 to 0.218. 17 Implied Volatility 0.21

0.208

0.206

0.204

0.202

Strike

0.9 1 1.1 1.2

Figure 4: Sectional Smiles: A plot of Implied Volatilityversus Strike; for the

example in Figure 3 with the three curves representing S = 0:1, S = 0 and

0 0

S =0:1 taken from top to b ottom on the left. In each case there is an observable

smile, and the skew varies with the initial o set. 18

is not to o large, once the price reaches the strike, S will b e nearer zero; the instantaneous

volatility will therefore b e low, and so the price of the option will b e reduced. Of course,

if S gets to b e large then S will still b e large when the exercise price is reached, and

so the option price will again b e larger. This gives a qualitative explanation of the curves

seen in Figure 4.

To date the analysis has concentrated on options with maturity T =0:25. We

now relax this condition. Figure 5 displays plots of implied volatility as a function of

b oth time to maturity and strike. There are two graphs corresp onding to di erent initial

values of the o set. An immediate observation is that the magnitude of the skews and

smiles decreases with time. (Note that the main reason why the smile app ears more

pronounced when the initial o set is zero, is that the vertical scales in the two plots

are di erent.) In b oth plots there are cross sections of constant strike along which the

implied volatility increases with time, and cross sections along which it decreases with

time. Both these observations are a corollary of the fact that the implied volatilityisa

measure of the average instantaneous volatilityover the lifetime of the option, and this

averaging leads to smo othing e ects.

To complete the discussion of this numerical example we consider the sensitivityof

the implied volatilities to changes in the parameters of the mo del.

There is a strong qualitative similaritybetween the families of smiles for each

parameter pair (; ) as illustrated byFigure6. However parameter choice is seen to

a ect the magnitude of the smiles; the size of the smiles is directly related to and

is inversely related to . The rst of these relationships is immediate from (15). The

explanation for the second observation is that large values of are asso ciated with a

shorter half-life for the lo okback p erio d in the de nition of the past average. Typically

this will decrease the values of the o set function S ,(see(9)),which in turn reduces the

volatility as de ned via (15). Thus the nature of dep endence on b oth the parameters

and follows from the particular sp eci cation of the dynamics for the price pro cess.

Finally we make some observations concerning the shap e of the option price surface

V V (Z; U; T ). If (S )isaneven function then it is easy to verify from the partial

di erential equation (16) and the b oundary condition (17) that

z z

V (z ; u; t)= e V (z; u; t)+ (e 1):

This prop erty is shared by the Black-Scholes formula p (z; t):

z z

p (z; t)= e p (z; t)+(e 1)

BS BS 19

Initial Offset=0.0

0.208

0.206

0.204

implied volatility implied 0.2 0.202 0.5

0.4 0.2 0.3 time to maturity 0.1

0.2 0

ln(strike) 0.1 -0.1

-0.2

Initial Offset=0.1

0.215

0.21

implied volatility implied

0.205 0.2 0.5

0.4 0.2 0.3 time to maturity 0.1

0.2 0

ln(strike) 0.1 -0.1

-0.2

Figure 5: Term Structure of Volatility: two plots of implied volatility as a function

of the logarithm of the strike and time to maturity of the option, with di erent

initial values of the o set S .

0 20 0.21 0.22

0.2175

0.208 0.215

0.2125 0.206

0.21

0.204 0.2075

0.205 0.202

0.2025

0.9 1 1.1 1.2 0.9 1 1.1 1.2

0.202

0.2035

0.2015 0.203

0.2025

0.201 0.202

0.2015

0.2005 0.201

0.2005

0.9 1 1.1 1.2 0.9 1 1.1 1.2

Figure 6: Sectional Smiles for di erentvalues of the parameters and . Note

the changes in magni cation of the y -scale. Each of the four plots displays a

triple of volatility smiles for the initial o sets S = 0:1; 0; 0:1, in a manner

similar to Figure 4. The parameter has the value 5 for the upp er two graphs,

and one for the lower pair; has value 5 on the left and one on the right. 21

where p is considered as a function of the current log-price z , and time to go t, assuming

a unit strike. As a corollary, (see Renault and Touzi (1995) for details of this argument

in a similar context),

(K; S; T )= (K ; S; T ):

BS BS

Thus the implied volatility of an in-the-money option is equal to that of an out-of-the-

money option sub ject to the sign of the current rst-order o set b eing switched. This

explains why in Figures 4 and 6 the skewed volatility smiles corresp onding to S = 0:1

intersect at unit strike, and why in Figure 5 wehave not shown a plot with initial o set

S = 0:1.

5 Summary

In this pap er wehaveintro duced a new class of mo dels for asset prices. The chosen

dynamics of the price pro cesses are motivated by a desire to have a mo del which satis es

two criteria. Firstly the instantaneous volatility should b e related to the recent history

of the price pro cess, and secondly, through the mechanism of options replication, there

should b e preference-indep endentcontingent-claim prices.

The new class of complete mo dels with sto chastic volatility prop oses a causal link

between current asset price movements and future volatility. In contrast in the level-

dep endentvolatility mo dels, the volatility dep ends solely up on the asset price pro cess,

and in the sto chastic volatility mo dels the volatility is frequently taken to b e an au-

tonomous pro cess. The class of level-dep endentvolatilitymodelsmay b e thoughtofasa

sp ecial case of the new class of complete mo dels with sto chastic volatility. It shares the

prop erty of preference-indep endent options prices, but there is no opp ortunity for sho cks

in volatility to p ersist through time. In this sense the new class of mo dels is similar in

spirit to the class of sto chastic volatility mo dels.

Smiles and skews in the implied volatility of traded options are a common phe-

nomenon. By considering one of the simplest non-trivial examples wehave shown that

smiles and skews arise naturally through the new mo del. For the parameters used in x4.2

the magnitude of the smile and skew of a three month option ranges from almost nothing

to 10%. These values agree with the magnitudes of smiles found in empirical tests, for

example byFung and Hsieh. Moreover without changing the underlying dynamics of the

price pro cess the shap e of the smile maychange as the o set changes.

In the mo del used in x4.2 there is a simple qualitative relationship b etween the 22

shap e of the smile and the recent history of the price pro cess. In particular there is a

negativeskew if and only if the asset price is b elow its recentaverage value. With a more

complicated sp eci cation of the volatility in (15) other more complicated relationships

between the shap e of the implied volatility curveandthevalues of the rst and higher

order o sets will arise. Suchvolatility mo dels have the p otential to describ e and explain

b oth the prevailing implied volatility smiles and the dynamic changes of these smiles over

time.

A App endix

Recall that in Section 4.1 weintro duced a new probability measure P under which the

discounted price pro cess was a martingale, and we assumed that P was equivalentto P.

The purp ose of this app endix is to discuss that assumption.

Supp ose that P is a probability measure on ( ; fF g ; F ) and supp ose that

0 t 0tT T

we attempt to de ne a probability measure P by

where Y is a non-negativecontinuous P -lo cal martingale with Y =1. We require

0 0

conditions on Y which will guarantee that Y is a true martingale, and hence that P is

well de ned.

Lemma A.1 De ne =inffu : Y >ng^ T , and set Y = Y (t ^ ). De ne the

n u n

n n n n n

probability measures P via (dP =dP )j = Y so that for A 2F, P [A]= E [Y I ].

0 F t 0 A

1 1 t 1 T

Then the fol lowing areequivalent:

(i) Y is a martingale;

(ii) E (Y )= 1;

0 T

(iii) P (

If any of these conditions holds then P is wel l de nedandP is absolutely continuous

1 1

with respect to P .

Pro of 23

First let us remark that byFatou's Lemma Y is a sup ermartingale and E (Y ) 1.

0 T

Clearly (i) implies (ii) and the converse is easily shown using the sup ermartingale prop erty

and pro of bycontadiction.

For the remaining implications observe that Y is a true martingale, and

n n

1=E [Y ] = E [Y ; = T ]+ E [Y ;

0 0 T n 0 n

T T

= E [Y ; = T ]+ P [

0 T n n

Now if (iii) holds then E [Y ; = T ] ! 1sothatE [Y ]=1.Conversely if E [Y ]=1

0 T n 0 T 0 T

then from Do ob's submartingale inequalitywe deduce that P [

0 n

E [Y ; = T ] ! 1.

0 T n

Finally for A 2F we can de ne P (A)=E [Y I ]andifP (A) = 0 then neces-

T 1 0 T A 0

sarily P (A)= 0 also.

There is little more to b e said in such a general setting, but when P and P are the

0 1

laws of the solutions of sto chastic di erential equations there is hop e of further progress.

Consider the sp ecial case = C ([0;T]; R) as our sample space, with canonical

pro cess (X ) , and canonical ltration F (fX : u tg). For i =0; 1 consider

t 0tT u

the SDE

(19) dx = (t; x )dB + (t; x )

t t t i t

where :[0;T] R ! R and :[0;T] R ! R are measurable, and is everywhere p os-

itive. More generally we could consider a d-dimensional pro cess X ,andad-dimensional

SDE with everywhere invertible, but the one-dimensional case has the twin advantages

of notational simplicity and suciency for the example wehave in mind.

Assume in addition that for each i =0; 1 the SDE (19) has a pathwise unique

strong solution; a sucient condition for this assumption to hold is that and are

time homogeneous and Lipschitz continuous, and hence our example from Section 4.2 ts

into this setting. See Rogers and Williams (1987, Chapter V) for a discussion of concepts

of solutions of SDEs.

Nowfori =0; 1, let P b e the law of the solution of (19) with a given initial value

x 2 R. Consider P as a lawon( ; F ) and form the usual augmentation (F ) of

0 i t 0tT

(P + P ). ) with resp ect to (F

0 1 0tT

De ne = (t; X ) f (t; X ) (t; X )g.

t t 1 t 0 t

Prop osition A.1 The fol lowing areequivalent:

(i) P is absolutely continuous with respect to P ;

1 0 24

(ii) P [ ds < 1]=1.

By interchanging the roles of P and P we immediately obtain the following corol-

0 1

lary:

ds is nite almost surely Corollary A.1 P and P areequivalent if and only if

0 1

under both P and P

0 1

Pro of of Prop osition A.1

For i =0; 1de ne

i 1

dW = (t; X ) fdX (t; X )g:

t t i t

Then for each i, W is a (P ; F )-Brownian motion, and the two are related by

i t

0 1

dW = dW + dt:

t t

Supp ose that (i) holds. Then we can de ne the Radon-Niko dym derivative Y via

Z Z

t t

0 2

Y (t) = exp dW du

u u

0 0

Z Z

t t

1 2

= exp dW + du :

u u

0 0

Now Y is a non-negative P -lo cal martingale so that P (sup Y < 1) = 1, and byour

t 0 0 t

assumption of absolute continuity, P (inf (Y ) > 0) = 1. Moreover Y is a P -lo cal

1 t t 1

martingale; indeed

hM i ) Y =exp(M

t t

where M = dW and thus we can conclude that P [inf fM hM i g > 1]= 1.

t s 1 t t t

Hence P [sup hM i < 1]=1 which is condition (ii).

1 t

For the converse de ne

Z Z

t t

0 2

Y = exp dW du :

t u

u u

0 0

n o

R R R

t t T

2 1 2 1

Then Y = exp dW du and since by assumption P [ ds < 1]=

u 1

u u s

0 0 0

1wehavethatP [inf Y =0]=0.Nowlet and P b e de ned as in Lemma A.1

1 t n

Under P the canonical pro cess X solves the SDE

dX = (t; X )dB +( (t; X )I + (t; X )I )dt:

t t t 1 t 0 t

ft g ft> g

n n 25

Pathwise uniqueness implies that

1 n

< 1=n) # 0 (

n 1

t 1

0tT

so that by Lemma A.1, Y is a true (P )-martingale, such that (dP =dP )j = Y (t ^ )

0 0 F n

and in particular P is absolutely continuous with resp ect to P .

1 0

For many examples it is straightforward to check the condition

ds < 1 P almost surely.

For example if the functions , and do not dep end on t then the additive functional

0 1

a(X )du of the one-dimensional di usion X do es not explo de if and only if a is A =

u t t

lo cally integrable with resp ect to the sp eed measure m of X .

Finally, for the sp eci c example we considered in Section 4.2, let S , the rst order

o set, b e the canonical pro cess. Then ,asgiven by (15) is b ounded ab ove and b elow,

1 2

is Lipschitz (by Assumption 3.1 and Equation (10)), and given by ( + s)is

1 1

again Lipschitz. As a corollary is b ounded ab ove and b elow on compact intervals and

S is non-explosive under b oth P and P , or rather P and P in the notation of Section 4.2.

t 0 1

Hence by Corollary A.1, P and P are equivalent as desired. 26

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