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Complete Models with Stochastic Volatility

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Complete Models with Stochastic Volatility Complete Mo dels with Sto chastic Volatility 1 David G. Hobson Scho ol of Mathematical Sciences, University of Bath and 2 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. 1 Research supp orted in part byRecordTreasury Management 2 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 t 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 t satis es the SDE dP t (2) = (P )dB + dt P t t P t where is a function of P .For example in their Constant ElasticityofVariance (CEV) P (1 ) mo del Cox and Ross (1976) take (x) x : This is a di usion mo del for the price P pro cess and following Harrison and Pliska (1981) the mo del is complete with unique, preference-indep endent option prices.
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