December 28, 2000 15:0 WSPC/104-IJTAF 00087 International Journal of Theoretical and Applied Finance Vol. 4, No. 1 (2001) 45–89 c World Scientific Publishing Company IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY ROGER W. LEE∗ Department of Mathematics, Stanford University, Stanford, CA 94305, USA Received 8 February 1999 Accepted 4 June 1999 For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some na¨ıve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a sym- metric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou’s asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatil- ity skew. Also we compare the slow-variation asymptotics against what we call small- variation asymptotics, and against Fouque, Papanicolaou, and Sircar’s rapid-variation asymptotics. We apply the slow-variation asymptotics to approximate the biases of two na¨ıve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied- volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk. Keywords: Stochastic volatility, local volatility, implied volatility, volatility risk pre- mium, skew, smile. 1. Introduction 46 2. Models 48 2.1. Implied and local volatilities . .................... 48 2.2. Review of option pricing under stochastic volatility ........... 49 3. Prices and Volatilities via Expectations 53 3.1.Pricesviaexpectations....................... 53 3.2. Local volatilities via expectations . ................ 54 ∗E-mail: [email protected] 45 December 28, 2000 15:0 WSPC/104-IJTAF 00087 46 R. W. Lee 4. Local and Implied Volatility Smiles 55 4.1.Smilesymmetry:Non-parametriccharacterizations........... 55 4.2. Smile symmetry under stochastic volatility .............. 56 5. Asymptotic Expansions for Implied and Local Volatility 57 5.1.Slow-variationasymptotics..................... 57 5.2.Small-variationasymptotics..................... 60 5.3.Discussion............................ 61 5.4.Figures.............................. 62 5.5.Slowvariationversusrapidvariation................. 62 6. Asymptotic Valuation Errors 72 6.1. Implied volatilities: The Ad-Hoc,“StickyStrike”method........ 72 6.2. Local volatilities: The DVF,“StickyImpliedTree”method....... 75 6.3. Two notions of volatility risk premia . ............... 77 6.4.Comparisons........................... 77 7. Asymptotic Hedging Errors 78 7.1. Hedge at the stochastic-volatility delta . ............... 79 7.2.Hedgetominimizeinstantaneousvariance............... 80 7.3.HedgeattheBlack–Scholesdelta.................. 83 7.4. Two notions of volatility risk premia, revisited . .......... 84 7.5.Comparisons........................... 84 8. Further Research 86 Acknowledgments 86 References 86 1. Introduction For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium- based explanations of the biases in some na¨ıve pricing and hedging schemes. Section 2 reviews implied (Black–Scholes) volatilities, local volatilities and option pricing under stochastic volatility, taking care to distinguish among three different notions of “volatility risk premium.” Section 3 treats these ideas proba- bilistically, representing option prices and local volatilities in terms of expectations. Section 4 deals with the symmetric implied and local volatility smiles that arise in the case where fluctuations in price and volatility are uncorrelated. It begins by showing the general equivalence of symmetry in Black–Scholes implied volatility and symmetry in local volatility. Then it applies the expectations formula to show that local volatility as a function of log-moneyness has the shape of a symmetric smile. In the correlated case, Sircar and Papanicolaou [61] derive for implied volati- lity an approximation linear in time to maturity and in log-moneyness, under the assumption that volatility is a slowly-varying stochastic process. Section 5 extends to the next order this asymptotic expansion, finding a term that is quadratic in December 28, 2000 15:0 WSPC/104-IJTAF 00087 Implied and Local Volatilities Under Stochastic Volatility 47 time-to-maturity and log-moneyness. We expand local volatilities analogously, ver- ifying the rule of thumb that the local volatility skew has roughly twice the slope of the implied volatility skew. Changing the volatility assumption from slow-variation to small-variation, we obtain some alternative expansions for implied and local volatility. The slow and small variation asymptotics are closely related. For example, in both expansions, the local volatility skew, as a function of log-moneyness, has approximately twice the slope of the Black–Scholes implied volatility skew; likewise, the local volatility term structure also has approximately twice the slope of the implied volatility term structure, in both expansions. In contrast, the slow-variation development and the rapid-variation development of Fouque, Papanicolaou and Sircar [33] are in many ways diametrically opposed. We highlight these points of opposition; but we also discuss the possibility of reconciling the two approaches — a project which can be viewed in terms of bridging the gap between the short-horizon domain of validity of the slow-variation asymptotics, and the long-horizon domain of validity of the rapid-variation asymptotics. Turning our attention to applications, we approximate in Sec. 6 the biases of na¨ıve pricing strategies, assuming that slowly-varying stochastic volatility is the true description of the underlying price process. Each na¨ıve model attempts to price options using a local or implied volatility function calibrated to old (say, one week old) market prices. The sticky strike or Ad Hoc (AH) model assumes that implied volatility as a function of strike and maturity has not changed. The sticky implied tree or Deterministic Volatility Function (DV F)modelassumesthatlocal volatility as a function of strike and maturity has not changed. Our calculations offer theoretical explanations for empirical observations, by Dumas, Fleming and Whaley [28], and/or by Rosenberg [58], that the AH and DV F valuations are biased downward, and the DV F bias is worse. The signs and relative magnitudes of the valuation biases appear to be consistent with a positive risk premium λ(2) for the portion of volatility risk that is uncorrelated with asset risk. As another application, we approximate in Sec. 7 the biases of various strategies that attempt to hedge a call under stochastic volatility, using only the underlying stock and the riskless security. The strategies are to hedge at the stochastic volatility delta, to hedge at the ratio that minimizes instantaneous variance of the portfolio, or to hedge at the Black–Scholes delta evaluated at the implied volatility. We also propose how to implement these strategies without specifying or estimating any particular stochastic volatility model. Our calculations indicate that Bakshi, Cao and Chen’s [3] empirical observa- tions, of an upward bias in the minimal instantaneous variance strategy, can be explained again by a positive λ(2). Moreover, under the defensible assumptions of a positive λ(2) and a negative premium λV for a second notion of volatility risk, the slow-variation and small-variation expansions suggest that the stochastic-volatility delta hedge is biased low, the minimal instantaneous variance hedge is biased high, and the Black–Scholes delta hedge is somewhere in between. December 28, 2000 15:0 WSPC/104-IJTAF 00087 48 R. W. Lee 2. Models 2.1. Implied and local volatilities Throughout this paper we will for convenience assume a riskless rate of 0. Generali- zation to non-zero but deterministic interest rates is straightforward; for example we could take the riskless security as numeraire, or express prices as forward prices. If some asset price X follows a diffusion (1) dXt = µtXtdt + σXtdBt with constant volatility σ, then the time-t no-arbitrage price of a European-style call struck at K and expiring at T is given by the Black–Scholes