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 [11] formula

BS C (X, t, K, T, σ):=XN(d1) − KN(d2)

where √ log(X/K) σ T − t d1,2 := √  , σ T − t 2 and N is the cumulative normal distribution function. Conversely, given a time-t asset price X and option price C(K, T ), define the (Black–Scholes) implied volatility I(K, T ) as the unique solution to

C(K, T )=CBS(X, t, K, T, I(K, T )) .

This definition holds, regardless of whether X actually follows a constant-volatility diffusion. The empirical evidence against the Black–Scholes assumption of constant volatility has prompted the development of models where volatility varies randomly. In particular, the deterministic volatility function (DVF) class of models assumes that volatility is a deterministic function σ(X, t) of asset price and time:

(1) dXt = µtXtdt + σ(Xt,t)XtdBt . The DVF approach, including discrete-time implementations known as “implied tree” models, is treated in Dupire [29], Derman and Kani [21], Rubinstein [59], Derman, Kani, and Zou [24], Dupire [30], Jackwerth [46], Derman, Kani, and Chriss [23], Chriss [16], and Berle and Cakici [9]. Having introduced no new sources of randomness into the Black–Scholes setting, the DVF model retains market completeness, and preference-free option pricing is possible. If volatility is a deterministic function, then, given a time-t asset price X, Dupire [29] shows that C(K, T ) satisfies the PDE 1 C − σ2(K, T )K2C =0 (2.1) T 2 KK with initial condition C(K, t)=(X−K)+ ≡max(X − K, 0). December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 49

Conversely, given a cross-section of option prices C(K, T ) at all maturities and strikes, we define the local volatility function L(K, T )by

2 2CT(K, T ) L(K, T )= 2 . (2.2) K CKK(K, T ) We emphasize that this definition holds regardless of what kind of process — DVF or other — actually governs the evolution of volatility. If volatility is in fact a deterministic function of asset price and time, then the local volatility implied by any cross-section of option prices would coincide with that deterministic function — but we make no such assumption. Just as it is possible to quote the Black–Scholes implied volatility of an option, regardless of whether volatility is actually constant; so is it possible to quote the local volatilities implied by a set of options, regardless of whether volatility is actually a deterministic function of asset price and time. Thus both B-S implied volatility and local volatility are implied volatilities — the former with respect to a constant-volatility model, and the latter with respect to a deterministic-volatility-function model; however this paper will use “local volatility” in accordance with common practice. Note that (2.2) is a theoretical definition that abstracts from the fact that option prices are not actually available at a continuum of strikes and maturities. In practice the computation of L is an inverse problem, versions of which have been analyzed by Avellaneda, Friedman, Holmes and Samperi [2], Bouchouev and Isakov [12], Andersen and Brotherton-Ratcliffe [1], Coleman, Li, and Verma [17], Levin [51], and Carr and Madan [14]. Our local volatility calculations will be in the theoretical setting, and will not focus on the practical solution of the inverse problem. When we discuss the implied or local volatility surface or skew, we mean I or L viewed as a function of K and T . Depending on the context, skew may refer more specifically to I or L as a function of K or, after a change of variable, a function of log-moneyness m := log(X/K)forXgiven. A smile isaskewthathasaU-shape when plotted against m; where necessary, we will make this more precise. In the stochastic volatility setting, C, L,andIwill also depend on the volatility σ, in addition to X, t, K and T . We will often suppress some of these arguments.

2.2. Review of option pricing under stochastic volatility We review some elements of option pricing under stochastic volatility. Although this material is standard and expositions can be found in, for example, Frey [35] or Lewis [52], we take care here to distinguish among the notions of “volatility risk premium” that exist in the literature. On a probability space (Ω, F, P) with a filtration {Ft}, suppose that the asset 2 price Xt and its the squared volatility Vt = σt follow the Itˆo processes (1) dXt = µtXtdt + σtXtdBt (2.3)

V dVt = αtdt + βtdBt (2.4) December 28, 2000 15:0 WSPC/104-IJTAF 00087

50 R. W. Lee

(1) V where B and B are {Ft}-Brownian motions with constant correlation ρ.Itis useful to write BV in the form p V (1) − 2 (2) Bt = ρBt + 1 ρ Bt , where B(2) is a standard Brownian motion uncorrelated with B(1). Assume that the market admits no arbitrage. Under mildly stronger conditions, also to be assumed, there exists an equivalent local martingale measure: a probabi- lity measure P˜ equivalent to P such that the discounted prices of all assets are local martingales under P˜. Sufficient conditions are given in the line of work that begins with Harrison and Kreps [41]; see for example the “no free lunch with vanishing risk” condition of Delbaen and Schachermayer [19]. (1) It follows from Girsanov’s theorem that there exist {Ft}-adapted processes λ and λ(2) such that (B˜(1), B˜(2)) is a two-dimensional standard Brownian motion under P˜,where Z Z t t ˜(1) (1) (1) ˜(2) (2) (2) Bt := Bt + λs ds and Bt := Bt + λs ds . (2.5) 0 0 Furthermore, P˜ and P are related by !  Z Z Z  P˜ t t t d − (1) (1) − (2) (2) − 1 (1) 2 (2) 2 Et P = exp λs dBs λs dBs (λs ) +(λs ) ds d 0 0 2 0

(1) (1) where Et denotes the conditional expectation E( ·|Ft). We call λ the B - risk premium or the asset risk premium.Wecallλ(2) the B(2)-risk premium or sometimes the volatility risk premium — the latter only in contexts where it is clear which volatility risk premium is meant; we return to this point momentarily. As a local martingale, X must have a drift that vanishes under P˜, so 2.3 implies that

(1) λt = µt/σt . The dynamics of X alone do not provide a similar restriction on λ(2), and hence do not determine the prices of derivative assets, since distinct choices of λ(2) cor- respond to distinct local martingale measures and, consequently, distinct pricing functions. This is a manifestation of the incompleteness of the market in which only X is traded: perfect replication of derivatives is not necessarily possible, because two sources of randomness exist, but only one risky asset is available as a hedging instrument. Pricing will depend on investors’ risk preferences, which are encapsulated in λ(2). Suppose we want to calculate the price Ct an option paying at date T some function h(XT ) of the underlying asset price. Under P˜ we have ˜(1) dXt = σtXtdBt p − V ˜(1) − 2 ˜(2) dVt =(αt λt βt)dt + ρβtdBt + 1 ρ βtdBt December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 51

where p V (1) − 2 (2) λt := ρλt + 1 ρ λt (2.6) is what we will call the BV -risk premium. Now assume that αt,βt and Ct are all functions of (Xt,Vt,t). If C(x, v, t)is smooth then Itˆo’s rule gives its drift under P˜, which must be zero: 1 √ 1 C + vx2C + ρx vβC + β2C +(α−λVβ)C =0, (2.7) t 2 xx xv 2 vv v where we omit the arguments (x, v, t). The terminal condition is

C(x, v, T )=h(x).

Solving this PDE to obtain an option price usually requires numerical methods. Alternatively, one can approximate using asymptotic expansions, as we do in Sec. 5. To motivate our choice of risk premium terminology, return to the objective measure P: p (1) − 2 (2) dC =[(σtXtCx+ρβCv)λt + 1 ρ βCvλt ]dt p (1) − 2 (2) +(σtXtCx +ρβCv)dBt + 1 ρ βCvdBt (2.8)

(1) (1) dXt = σtXtλt dt + σtXtdBt . (2.9) Holders of risky assets require compensation in the form of excess drift proportional to their exposure to the risk. For example, the option and the underlying asset have (1) B -risk exposures of, respectively, σtXtCx + ρβCv and σtXt. Multiplying these exposures by the B(1)-risk premium λ(1) gives the excess drifts associated with B(1)-risk. p (2) 2 Similarly, the option’s exposure to the B -risk is 1 − ρ βCv. Multiplying this by the B(2)-risk premium λ(2) gives the excess drift associated with B(2)-risk. The λ(2) is one notion of volatility risk premium. A second notion of volatility risk premium is λV as defined in 2.6. Writing

(1) V (1) V dC =(σtXtCxλt + βCvλt )dt + σtXtCxdBt + βCvdBt , justifies our terminology BV -risk premium. A third notion of volatility risk premium is defined by

β V λt := βλt , which allows us to restate (2.8) as

(1) β (1) V dC =(σtXtCxλt + Cvλt )dt + σtXtCxdBt + Cv(βdBt ). Thus λβ can be viewed as a risk premium for exposure to the diffusion component V βdBt of squared volatility. December 28, 2000 15:0 WSPC/104-IJTAF 00087

52 R. W. Lee

All three of these distinct concepts appear in the literature. Including also the disagreements in sign convention, at least five different interpretations of “market price of volatility risk” or “volatility risk premium” have appeared. Table 1 relates other authors’ terminology to our notation. Since this paper will make use of both λV and λ(2), we will either avoid the “volatility risk premium” terminology, or else make clear from the context what is meant.

Table 1. Volatility risk premium definitions compared.

Authors Their term Our notation

Scott (1987) [60] RPV λβ Melino-Turnbull (1990) [53] RP, PVR λV Stein-Stein (1991) [62] MPVR λV or λ(2) Heston (1993) [43] PVR λβ Ball-Roma (1994) [4] MPVR λV or λ(2) Ghysels-Harvey-Renault (1996) [39] VRP λV or λ(2) Bates (1996) [5] VRP λ(2) Renault-Touzi (1996) [57] VRP λV or λ(2) Hobson (1996) [44] MPVR −λ(2) Frey (1996) [35] MPVR −λ(2) Guo (1998) [40] RP λβ Kapadia (1998) [50] VRP λβ Poteshman (1998) [56] MPvR λβ Fouque-Papanicolaou-Sircar (1998) [31] MPVR λV Sircar-Papanicolaou (1999) [61] MPVR λV Fouque-Papanicolaou-Sircar (1999) [32] MPVR, VRP λ(2) Benzoni (1999) [8] RPvR λβ Chernov-Ghysels (1999) [15] MPVR λ(2) Chernov-Ghysels (1999) [15] VRP λβ Pan (2000) [54] VRP −λβ Lewis (2000) [52] HRP λ(2) Lewis (2000) [52] VRP, PVR λV Fouque-Papanicolaou-Sircar (2000) [33] MPVR λ(2)

(M)P(V/v)R = (market) price of/for (volatility/variance) risk (V)RP = (volatility) risk premium RPV = risk premium associated with the volatility parameter RPvR = risk premium on variance risk HRP = hedging risk premium

Note that λV = λ(2) in the case ρ = 0, which is the only case some papers consider. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 53

3. Prices and Volatilities via Expectations In Secs. 3 and 4, we assume that P˜ is an equivalent martingale measure. Write E˜ for expectation with respect to the measure P˜, under which all asset prices are martingales. Existence of a martingale measure is guaranteed by conditions mildly stronger than those for the existence of a local martingale measure. Also, in Secs. 3 and 4, let us take the current time be time 0, in order to simplify notation.

3.1. Prices via expectations By the martingale property, a European call with strike K and expiry T has price

+ C = E˜((XT − K) ) . We review some ways to rewrite this as the expectation of the Black–Scholes formula evaluated at random arguments. Letα ˜ := α − λV β, so that the stock price and its squared volatility follow the processes ˜(1) dXt = σtXtdBt ˜V dVt =˜αtdt + βtdBt where B˜(1) and B˜V are P˜-Brownian motions with correlation ρ. Hence we can write p B˜(1) = ρB˜V + 1 − ρ2B˜X where B˜V and B˜X are independent standard Brownian motions. Now assume that the squared volatility’s driftα ˜ and diffusion β are functions of (V,t), but not of X. Defining Z T 2 ¯ 1 2 σ¯ := V := σt dt T 0 and applying the representation of an option price as the discounted risk-neutral expectation of the payoff, Hull and White [45] show that in the case where ρ =0 the call price can be expressed as

+ + BS C = E˜((XT − K) )=E˜(E˜((XT − K) |{Vt}06t6T )) = E˜(C (¯σ)) by iterated expectations and the lognormality of the asset price, conditional on the path of V . Willard [65] extends this formula to the correlated case ρ =6 0 by writing Z Z Z ! 1 T T p T − 2 ˜V − 2 ˜X XT = X0 exp σt dt + ρ σtdBt + 1 ρ σtdBt 2 0 0 0 Z Z ! 1 − ρ2 T p T − 2 − 2 ˜X = X0ξT exp σt dt + 1 ρ σtdBt 2 0 0 December 28, 2000 15:0 WSPC/104-IJTAF 00087

54 R. W. Lee

where Z Z ! 2 T T −ρ 2 ˜V ξT := exp σt dt + ρ σtdBt . (3.1) 2 0 0

V Conditional on the path of B˜ , therefore, log XT has the normal distribution   1 N log X ξ − (1 − ρ2)¯σ2T, (1 − ρ2)¯σ2T , 0 T 2 which implies that the call price C under stochastic volatility is given by p BS 2 C = E˜(C (X0ξT , 1 − ρ σ¯)) . (3.2)

3.2. Local volatilities via expectations Here we express local volatilities using an expectation formula in which the random variables do not depend on the stock price X. This formula helps us, in Sec. 4.2, to relate the shapes of the I and L curves to properties of the underlying volatility process.

Proposition 3.1. We have E˜(V W ) L2(K, T )= T T , (3.3) EW˜ T where   1 (log(X ξ /K))2 (1 − ρ2)¯σ2T W := exp − 0 T − . T σ¯ 2(1 − ρ2)¯σ2T 8

Proof. Substitute the expected–Black–Scholes formula (3.2) into the defini- tion (2.2) of local volatility. Use Itˆo’s rule, and simplify to obtain the desired formula.

Remark 3.1. Derman and Kani [22] derive the following interpretation of local volatility as a conditional expectation:

2 L (K, T )=E˜(VT|XT =K). P˜| Comparing their formula to ours leads us to the following conjecture: Let X =K be P˜ conditioned on XT = K.Then ! P˜| d X=K WT E˜T = . dP˜ EW˜ T

If true, this would give an interpretation of WT , normalized, as the Radon–Nikodym derivative of the terminal-spot-conditional martingale measure with respect to the martingale measure. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 55

4. Local and Implied Volatility Smiles Renault and Touzi [57] showed that in stochastic volatility models where ρ =0, the implied volatility is a symmetric smile. We prove analogous statements for local volatility: that it is symmetric, and that it forms a U-shaped smile when plotted against log-moneyness. First, though, we show that symmetry in implied volatility is essentially equiva- lent to symmetry in local volatility. This result is non-parametric in the sense that it takes as primitives the volatility curves, making no explicit assumptions about the dynamics of the underlying.

4.1. Smile symmetry: Non-parametric characterizations

Given an asset price X0 at time t = 0, and option prices C(K, T ) such that the implied and local volatilities exist, let us rewrite C, I and L as functions of log-moneyness m := log(X0/K) instead of K, by defining

−m −m −m C˜(m, T ):=C(X0e ,T); I˜(m, T ):=I(X0e ,T); L˜(m, T ):=L(X0e ,T). The next two propositions do not assume that the underlying follows a stochastic- volatility (or any other kind of) diffusion.

Proposition 4.1. If I˜ is even in m then L˜ is even in m.

Proof. Changing variables in (2.2) gives 2C˜ L˜2 = T . (4.1) C˜mm + C˜m and a change of variables in the Black–Scholes formula, together with the definition of implied volatility, gives
 ˜2 −m m I T/2 C˜ = X0(N(d1) − e N(d2) where d1,2 := √ . I˜ T A direct calculation now shows I˜4 +2TI˜3I˜ L˜2 = T (4.2) ˜− ˜ 2 ˜3˜ − 1 2 ˜4 ˜ 2 (I mIm) +TI Imm 4 T I (Im) which is even in m.

Conversely we have the following:

Proposition 4.2. Assume L˜ is even in m and satisfies the usual conditionsa for uniqueness of a solution to (4.1) with boundary conditions corresponding to no-arbitrage. Then I˜ is even in m.

aThe parabolicity, continuity and H¨older conditions as specified in Friedman [36] suffice. December 28, 2000 15:0 WSPC/104-IJTAF 00087

56 R. W. Lee

Proof. By definition of I˜,

BS −m BS C˜(m, T )=C (X0,X0e ,T,I˜(m, T )) =: C˜ (x, m, T, I˜(m, T )) . Define

R BS C˜ (m, T ):=C˜ (X0,t,m,T,I˜(−m, T )) and let L˜R denote the local volatility associated with C˜R. Using (4.2) and the evenness of L˜,

L˜R(m, T )=L˜(−m, T )=L˜(m, T ) .

By uniqueness of the solution to (4.1), we have C˜R = C˜. Therefore I˜(−m, T )=I˜(m, T ).

Remark 4.1. Propositions 4.1 and 4.2 clarify a comment by Carr, Ellis and Gupta [13] who derive static hedges for exotic options, assuming a symmetric smile (or frown). They warn that “the assumed smile [symmetry] is in the local volatility as opposed to the Black model implied volatility.” In fact the two notions of symmetry are equivalent in their setting.

4.2. Smile symmetry under stochastic volatility Returning to the stochastic volatility model, consider the case ρ = 0. Proposition 3.1 simplifies to E˜(V f(V¯)) L2(K, T )= T , (4.3) Ef˜ (V¯) where   1 (log(X /K))2 vT f(v):=√ exp − 0 − , v 2vT 8 which is useful for characterizing the shape of the smile. In fact we have two ways of proving the symmetry of local volatility in the uncorrelated-volatility case.

Proposition 4.3. If ρ =0then the local volatility L˜ is even in m.

Proof 1. Renault and Touzi’s Proposition 3.1 shows that implied volatility is even in m. Our result follows from Proposition 4.1. Proof 2. The result follows from Proposition 3.1 by rewriting Eq. (4.3) in terms of m. The following proposition is the local volatility analogue of Renault and Touzi’s proof that B-S implied volatilities smile. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 57

Proposition 4.4. Assume that ρ =0and the process V has the property that the conditional expectation E˜(VT |V¯ = v) is an increasing function of v. Then the local volatility is a smile. That is, for all m ≷ 0, we have L˜m ≷ 0. Equivalently, for all K ≷ X0, we have LK ≷ 0.

Proof. Let M = m2. Define

g(v):=E˜(VT |V¯=v)

h(v):=1/v .

Differentiate (4.3) with respect to M, passing the derivatives through the expec- tations. This is justifiable because elementary calculus shows that f(v)/v can be bounded, independently of v, by a continuous function of M>0; it follows that, around any M>0, there is an interval in which f(V¯ )/V¯ and f(V¯ )VT /V¯ are uniformly integrable. Therefore

2 sign (∂L /∂M)=sign[E˜(f(V/¯ V¯)E˜(f(V¯)VT)−E˜(f(V¯)VT/V¯)Ef˜ (V¯)]

=sign[E˜(f(V¯)h(V¯))E˜(f(V¯ )g(V¯ )) − E˜(f(V¯ )g(V¯ )h(V¯ ))E˜(f(V¯ ))]

by iterated expectations. To see that this quantity is non-negative, note that f is positive, g is increasing and h is decreasing, and re-arrange the expression ZZ f(x)f(y)(g(x) − g(y))(h(x) − h(y))dxdy 6 0 ,

where the integrals are with respect to the law of V¯ under P˜ given Ft. So ∂L2/∂M > 0, which implies the result.

Usually the implied and local volatility skews in equity prices are found empiri- cally to be asymmetric, so these symmetry results indicate that such equity price processes are not of the uncorrelated-volatility class.

5. Asymptotic Expansions for Implied and Local Volatility 5.1. Slow-variation asymptotics Following Sircar and Papanicolaou ([61]; S-P hereafter), we assume that volatility varies slowly in time, under the objective measure. After reviewing S-P’s calculation of an asymptotic expansion for implied volatility, we extend it to the next order; then we calculate the analogous expansion for local volatility. Consider the class of models discussed in Sec. 2.2, “slowed-down” in the following way. Let the asset price process satisfy

(1) dXt = µtXtdt + σtXtdBt , December 28, 2000 15:0 WSPC/104-IJTAF 00087

58 R. W. Lee

2 where the squared volatility Vt = σt follows a stationary process √ V dVt = εα(Vt)dt + εβ(Vt)dBt (5.1)

depending on a small parameter ε>0, where β(Vt) > 0 and the correlation between B(1) and BV is a constant ρ. Assume that the dBV -risk premium process depends V V only on the volatility, so that it can be written as λt = λ (Vt). The motivation for the specification 5.1 is that Vt = Uεt where

∗ dUt = αdt + βdBt , so V canbeviewedasUmodified to run on a time scale slower by a factor of ε. The clearest way to proceed, notationally, is to change variables from v to σ. Let

a(σ):=α(σ2)/(2σ) − β(σ2)/(8σ3)

b(σ):=β(σ2)/(2σ)

b V 2 2 λ (σ):=λb(σ):=λ (σ )β(σ )/(2σ) ,

so a and b can be interpreted, by Itˆo’s rule, as the drift and diffusion of the volatility process √ dσ = εa(σ)dt + εb(σ)dBV ,

and λb = bλV can be understood as a premium for exposure to volatility-of-volatility risk, in the same way that our λβ = βλV was a premium for exposure to volatility- of-variance risk. By (2.7) the value C of an option struck at K and maturing at T satisfies the PDE 1 √ 1 √ C + σ2x2C + ερxσbC + εb2C +(εa − ελb)C =0. (5.2) t 2 xx xσ 2 vv σ To solve asymptotically for the option price, write √ C = C0 + εC1 + O(ε) , (5.3)

and substitute into (5.2). Examining the O(ε0)termsandtheO(ε1/2) terms yields two PDEs, whose solutions are

BS C0 = C 1 1 C = ρbxσ(T − t)CBS − λb(T − t)CBS , 1 2 xσ 2 σ where CBS and its partials are evaluated at (x, t, K, T, σ). To construct an implied volatility expansion √ 3/2 I = I0 + εI1 + εI2 + O(ε ) , (5.4) December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 59

use the definition CBS(I)=C to obtain

I0 = σ (5.5)   ρσb λ ρb K I = − b (T − t)+ log , (5.6) 1 4 2 2σ x which is what S-P find. Extending their calculation to O(ε), we obtain the next term of the implied volatility expansion:    b(σ)b0(σ) 5b(σ)2 b(σ)2 K I = − ρ2 + log2 2 6σ2 12σ3 6σ3 x      b(σ)2 b(σ)b0(σ) b0(σ)λ (σ) b(σ)λ0 (σ) b(σ)λ (σ) − − ρ2 + b + b − b ρ 24σ 6 6σ 6σ 4σ2    K σb(σ)2 σ2b(σ)b0(σ) b(σ)λ (σ) σb0(σ)λ (σ) × (T − t)log + + ρ2 − b + b x 12 24 8 12   σb(σ)λ0 (σ) λ (σ)2 λ (σ)λ0 σb(σ)2 + b ρ + b + b b − (T − t)2 12 24σ 6 24    b(σ)2 b(σ)b0(σ) a(σ) b(σ)2 + − ρ2 + − (T − t) . 24σ 6 2 12σ Now expand local volatility as √ 3/2 L = L0 + εL1 + εL2 + O(ε ) , (5.7) by using the definition (2.2) and lengthy calculation to find

L0 = σ   ρσb ρb K L = − λ (T − t)+ log 1 2 b σ x    b(σ)b0(σ) b(σ)2 b(σ)2 K L = − ρ2 + log2 2 2σ2 σ3 2σ3 x     b(σ)b0(σ) b0(σ)λ (σ) b(σ)λ0(σ) b(σ)λ (σ) K + ρ2 − b + − b ρ (T − t)log 2 2σ 2σ 2σ2 x     σb(σ)2 σ2b(σ)b0(σ) b(σ)λ (σ) σb0(σ)λ (σ) σb(σ)λ0 (σ) + + ρ2− b + b + b ρ 4 8 4 4 4     λ (σ)λ0 (σ) σb(σ)2 b(σ)2 b(σ)b0(σ) + b b − (T − t)2 + − ρ2 + a(σ) (T − t) . 2 8 2σ 2 For j =0,1,2, then, Ij and Lj are degree-j polynomials in log(K/x)andT−t. December 28, 2000 15:0 WSPC/104-IJTAF 00087

60 R. W. Lee

5.2. Small-variation asymptotics As an alternative to the slow-variation assumption, we could assume that the volatility process has small variation under the martingale measure. √ √ ˜ ˜V dσ = εa˜(σ)dt + εb(σ)dBt . √ The motivation for this specification is that dσ = εdUt where ˜ ˜V dUt =˜adt + bdBt , √ so σ can be viewed as U modified to have fluctuations dampened by a factor of ε. The small-variation skew expansions can be derived from the slow-variation series, and vice-versa. For implied volatility, the small-variation series is (5.4) with

I0 = σ ! ρσ˜b(σ) a˜(σ) ρ˜b(σ) K I = + (T − t)+ log , 1 4 2 2σ x " ! # ˜b(σ)˜b0(σ) 5˜b(σ)2 ˜b(σ)2 K I = − ρ2 + log2 2 6σ2 12σ3 6σ3 x " ! ! # ˜b(σ)2 ˜b(σ)˜b0(σ) ˜b0(σ)˜a(σ) ˜b(σ)˜a0(σ) ˜b(σ)˜a(σ) − − ρ2 − + − ρ 24σ 6 6σ 6σ 4σ2 " ! K σ˜b(σ)2 σ2˜b(σ)˜b0(σ) ˜b(σ)˜a(σ) σ˜b0(σ)˜a(σ) × (T − t)log + + ρ2 + + x 12 24 8 12 ! # σ˜b(σ)˜a0(σ) a˜(σ)2 a˜(σ)˜a0(σ) σ˜b(σ)2 + ρ + + − (T − t)2 12 24σ 6 24 " ! # ˜b(σ)2 ˜b(σ)˜b0(σ) ˜b(σ)2 + − ρ2 − (T − t) . 24σ 6 12σ

For local volatility the small-variation series is (5.7) with

L0 = σ ! ρσ˜b(σ) ρ˜b(σ) K L = +˜a(σ) (T−t)+ log 1 2 σ x " ! # ˜b(σ)˜b0(σ) ˜b(σ)2 ˜b(σ)2 K L = − ρ2 + log2 2 2σ2 σ3 2σ3 x " ! # ˜b(σ)˜b0(σ) ˜b0(σ)˜a(σ) ˜b(σ)˜a0(σ) ˜b(σ)˜a(σ) K + ρ2 + + − ρ (T − t)log 2 2σ 2σ 2σ2 x December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 61 " ! ! σ˜b(σ)2 σ2˜b(σ)˜b0(σ) ˜b(σ)˜a(σ) σ˜b0(σ)˜a(σ) σ˜b(σ)˜a0(σ) + + ρ2 + + + ρ 4 8 4 4 4 # " ! # a˜(σ)˜a0(σ) σ˜b(σ)2 ˜b(σ)2 ˜b(σ)˜b0(σ) + − (T − t)2 + − ρ2 (T − t) . 2 8 2σ 2 √ The slow-variation expansions of implied and local volatility to O( ε) ignore the objective volatility drift a, which does not appear until the√ O(ε) term. In the small-variation expansions, however, a does enter into the O( ε) term, becausea ˜ is the objective drift plus a risk adjustment. The small-variation expansions are useful when the objective drift cannot be assumed to be O(ε)√ over the relevant time horizon. Also the small-variation approximation L0 + εL1 matches exactly the local volatility approximation recently derived by Gatheral [38] using a different method, which indicates some robustness in this expansion. Our discussion of pricing errors will use the slow-variation approximation. In our analysis of hedging errors, the approximations will be justifiable by either slow-variation or small-variation.

5.3. Discussion To a first approximation in both the slow-variation and small-variation expansions, we observe the following when we fix t, x and v:

(1) For each fixed maturity T , the implied and local volatility surfaces exhibit a skew:

IK ≷ 0andLK ≷0accordingasρ≷0.

This agrees with the downward-sloping strike-structures in implied and local volatility that are observed in markets where changes in stock price are negatively correlated with changes in volatility. (2) For each fixed K, the term structure of implied and local volatilities slopes upward or downward, according to the sign of ρσ − 2λV . In particular, in a market where ρ<0, an upward-sloping term structure of volatility would suggest a negative BV -risk premium. (3) The local volatility surface varies with time and with log-moneyness twice as fast as the implied volatility surface does. This agrees with a “Rule of Thumb” proposed by Derman, Kani, and Zou [24]. Moreover, it is intuitively sensible; if one thinks of the squared implied volatility as an average of the squared local volatility, taken over the time from the present to the maturity date — which is literally correct in the case of time-dependent but deterministic volatility — then the averaging will cause the implied volatility to vary only one-half as quickly as a linear local volatility function. December 28, 2000 15:0 WSPC/104-IJTAF 00087

62 R. W. Lee

5.4. Figures Figures 1–6 illustrate the effects of truncating the small-variation expansions after one, two, and three terms. These approximations improve in accuracy as time to maturity decreases. This is consistent with the intuition that the fundamental time scale in option pricing is the time to maturity; and the shorter the time to maturity, the more valid the assumption that volatility has slow or small variation, relative to this time scale. We expand on this point in the next section. Moreover, the approximate and true implied and volatilities diverge from each other as log(x/K) approaches ∞, where the expansions (5.4) and (5.7) are not valid. In the near-the-money region of interest to us, however, the figures show a close fit. In these examples the base stochastic volatility model, under the martingale measure, is p − ˜V dVt = κ(θ Vt)dt + β VtdBt , (5.8) √ with κ =1.15,θ =0.04,β =0.39,ρ = −0.64 and today’s volatility V0 =0.19, which is Heston’s [43] model with average parameters estimated by Bakshi, Cao and Chen [3], from a sample of daily S&P 500 options data from June 1988 through to May 1991. Under the invariant distribution of the square-root process (5.8), the covariances are given by β2θ Cov(V ,V )= e−κ|s−t|. s t 2κ To quantify the rate of decorrelation, we can state the typical correlation time of this process, which is 1/κ =0.87. One could also think in terms of the typical half-life of a volatility shock, which would be log 2 times the correlation time. We also consider two alternative processes: one in which the fluctuations of the base process are halved, and one in which those fluctuations are doubled, as described in the individual captions of the figures. The approximate skews are calculated using the small-variation expansions. On each page we include the one-term, two-term and three-term small-variation ap- proximation to the volatility surface, and we show that surface at two time slices: one month and two months.

5.5. Slow variation versus rapid variation Fouque, Papanicolaou and Sircar [33] develop a theory of pricing and hedging under rapidly mean-reverting stochastic volatility. This is in direct contrast with our pri- mary assumption of slowly varying volatility. Here we comment on the differences, and suggest how these two lines of development can be reconciled. A given volatility process can be said to vary rapidly or slowly, depending on the time horizon in the application at hand. If the goal is to price a sufficiently December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 63

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Actual One term Two term Three term 0 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 log(K/x) (a)

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Actual One term Two term Three term 0 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 log(K/x) (b)

Fig. 1. Implied volatilities: base parameters for V .(a)T=0.083 and (b) T =0.167. √ • − ˜V − The stochastic volatility√ model here is dVt =1.15(0.04 Vt)dt +0.39 Vtdbt , with ρ = 0.64 and today’s volatility V =0.19. The correlation time is 0.87 years. • The small-variation approximations are more accurate at one month to maturity than at two months to maturity. December 28, 2000 15:0 WSPC/104-IJTAF 00087

64 R. W. Lee

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Actual One term Two term Three term 0 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 log(K/x) (b)

Fig. 2. Implied volatilities: V fluctuation size halved. (a) T =0.083 and (b) T =0.167 √ • − ˜V − The stochastic volatility√ model here is dVt =0.58(0.04 Vt)dt +0.20 VtdBt , with ρ = 0.64 and today’s volatility V =0.19. The correlation time is 1.74 years. • The small-variation approximations are more accurate at one month to maturity than at two months to maturity. • The result of halving the V fluctuations is, as expected, that the fits improve. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 65

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Actual One term Two term Three term 0 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 log(K/x) (b)

Fig. 3. Implied volatilities: V fluctuation size doubled. (a) T =0.083 and (b) T =0.167 √ • − ˜V − The stochastic volatility√ model here is dVt =2.30(0.04 Vt)dt +0.78 VtdBt , with ρ = 0.64 and today’s volatility V =0.19. The correlation time is 0.44 years. • The small-variation approximations are more accurate at one month to maturity than at two months to maturity. • The result of doubling the V fluctuations is, as expected, that the fits begin to deteriorate, particularly at the two-month time horizon. December 28, 2000 15:0 WSPC/104-IJTAF 00087

66 R. W. Lee

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Fig. 4. Local volatilities: base parameters for V .(a)T=0.083 and (b) T =0.167 √ • − ˜V − The stochastic volatility√ model here is dVt =1.15(0.04 Vt)dt +0.39 VtdBt , with ρ = 0.64 and today’s volatility V =0.19. The correlation time is 0.87 years. • The small-variation approximations are more accurate at one month to maturity than at two months to maturity. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 67

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Fig. 5. Local volatilities: V fluctuation size halved. (a) T =0.083 and (b) T =0.167 √ • − ˜V − The stochastic volatility√ model here is dVt =0.58(0.04 Vt)dt +0.20 VtdBt , with ρ = 0.64 and today’s volatility V =0.19. The correlation time is 1.74 years. • The small-variation approximations are more accurate at one month to maturity than at two months to maturity. • The result of halving the V fluctuations is, as expected, that the fits improve. December 28, 2000 15:0 WSPC/104-IJTAF 00087

68 R. W. Lee

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Actual One term Two term Three term 0 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 log(K/x) (b)

Fig. 6. Local volatilities: V fluctuation size doubled. (a) T =0.083 and (b) T =0.167 √ • − ˜V − The stochastic volatility√ model here is dVt =2.30(0.04 Vt)dt +0.78 VtdBt , with ρ = 0.64 and today’s volatility V =0.19. The correlation time is 0.44 years. • The small-variation approximations are more accurate at one month to maturity than at two months to maturity. • The result of doubling the V fluctuations is, as expected, that the fits begin to deteriorate, particularly at the two-month time horizon. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 69

Table 2. Slow variation versus rapid variation: A summary.

Section 5.1 here F-P-S [33]

Mean reversion assumption slow rapid Region of validity T − t small T − t large ≈ − log(X/K) Implied volatility linear in log(X/K)and(T t) T −t Leading-order dependence on today’s volatility yes no on long-run volatility no yes

long-dated option, then the volatility process will appear to vary rapidly; it will have many fluctuations over the long lifetime of the option. If, however, the goal is to price a sufficiently short-dated option, then that same volatility process will not have much time to vary, and thus it will appear to vary slowly, relative to the option’s lifetime. It follows that the F-P-S fast-mean-reversion model is best suited to long-dated options, and the slow-variation model to short-dated options. The leading terms in the implied volatility expansions reflect this observation. The slow-variation expansion’s coefficients a0, a1 and a2 have no dependence on the level to which volatility reverts, because there is not enough time to revert to that mean; what matters more is the level of today’s volatility. In rapid-variation approximation, however, today’s volatility is irrelevant, because its influence can be expected to decay rapidly; what appears in b0 and b1 is instead the ergodic mean. The breakdown of rapid mean-reversion in the near-maturity region is evident also in the log(X/K)/(T − t) term, which explodes as T − t → 0. To illustrate the distinct regimes of validity of slow-variation and fast-variation asymptotics, consider the term structure of the slope of the at-the-money implied volatility skew, where “slope” means the derivative of implied volatility with respect to log(K). According to the slow-variation approximation, the skew slope does not depend on time-to-maturity; but according to the rapid-variation approximation, this slope should decay as 1/(T − t). To take a specific example, we calculate these approximations under the Heston volatility process (5.8) as follows. ∂I ρβ Slow : ≈ (5.9) ∂(log K) 4σ ∂I ρβ4 Γ(2κθ/β2 +3/2) 1 Rapid : ≈ (5.10) ∂(log K) (2κ)5/2θ2 Γ(2κθ/β2) T − t The approximation (5.9) can be read off from (5.6), while the approximation (5.10) comes from applying the results of F-P-S [33] to the Heston process. The F-P-S approximation involves expectations with respect to the stationary distribution of (5.8), which is why the mathematical gamma function appears. In Fig. 7, we have calculated the term structure of the actual at-the-money slope of implied volatility, for the particular stochastic volatility model 5.8 with December 28, 2000 15:0 WSPC/104-IJTAF 00087

70 R. W. Lee

0 Exact Slow−variation approximation Rapid−variation approximation

−0.5 Slope of ATM implied volatility

−1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time to maturity (years)

Fig. 7. Term structure of skew slope: correlation time 0.87 years. √ • − ˜V − The stochastic volatility√ model here is dVt =1.15(0.04 Vt)dt+0.39 VtdBt , where ρ = 0.64 and today’s volatility V =0.19. Thus the correlation time is 0.87 years. • The exact implied volatility slope ∂I/∂(log K) is plotted as a function of T − t, and compared to the approximate slopes under the slow and fast mean reversion approximations. For this volatility process, the slow-variation assumption gives the better approximation, over the entire range of maturities shown.

parameters that yield a correlation time of 0.87 years. The true term structure is juxtaposed with the slow-variation and rapid-variation approximations. For time horizons of up to several months, the true slope is well approximated by the flat term structure implied by the slow-variation assumptions. At longer time horizons, the 1/(T − t) behavior implied by the rapid-variation assumptions becomes more reasonable, but even at a one-year horizon, the rapid-variation approximation is not very close to the true skew slope. In Fig. 8, we perform the same calculation, except we adjust the parameters of the stochastic volatility process to obtain a typical correlation time of only 0.22 years. What we observed in Fig. 7 still holds here: the slow-variation approxima- tion works best at short maturities and the rapid-variation approximation at long maturities. However, it is apparent that the small-variation approximation is valid here in a smaller region than in Fig. 7, while the rapid-variation approximation is valid here in a larger region than in Fig. 7. This is what one would expect: reducing December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 71

0 Exact Slow−variation approximation Rapid−variation approximation

−0.5 Slope of ATM implied volatility

−1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time to maturity (years)

Fig. 8. Term structure of skew slope: correlation time 0.22 years. √ • − ˜V − The stochastic volatility√ model here is dVt =4.60(0.04 Vt)dt+0.78 VtdBt , where ρ = 0.64 and today’s volatility V =0.19, which has the same invariant distribution as the model from Fig. 7. The difference is that the correlation time is 0.22 years, only one-fourth as long as before. • As a result, the fast-variation approximation here is appropriate over a wider range of maturities than in Fig. 7. It gives the better skew slope approximation at maturities longer than about 0.6 years, but is still inferior at shorter maturities.

the correlation time should make fast-variation applicable in a wider domain, while making slow-variation applicable in a smaller domain. One area for further research is how to bridge the two regimes of validity. For example, in the case of the Heston model, Gatheral [38] finds that the decay of the at-the-money skew slope should behave approximately as   1 1−exp(−τ) G(τ):= 1− , τ τ where τ is proportional to T − t.NotethatsinceG0(τ)→0asτ→0, and G(τ)= O(τ−1)asτ→∞, this decay does in fact interpolate between a flat term structure and a 1/τ term-structure. It is an open question whether this can be extended beyond the Heston model and given a full general justification. A second direction of research would be to introduce a second factor into the stochastic volatility model, making it a multidimensional process incorporating both a rapidly-varying component and a slowly-varying component. Gallant, Tauchen December 28, 2000 15:0 WSPC/104-IJTAF 00087

72 R. W. Lee

and Hsu [37], Bates [7] and Pan [54] give empirical evidence, indicating that both scales are present in volatility. One way to pursue this would√ be to analyze asymp- totically a model that includes√ one component on an O( ε) timescale and a second component on an O(1/ ε) timescale.

6. Asymptotic Valuation Errors We apply the slow-variation approximations of Sec. 5.1 to examine the errors asso- ciated with na¨ıve pricing strategies based on implied and local volatilities. We offer explanations for the underpricing biases reported in papers that have tested these strategies empirically, and for the stronger bias in the local volatility method DV F as compared to the implied volatility method AH. The models involve the calibration of a local or implied volatility surface at some time t0, and the out-of-sample use of that surface to price options at time t1 := t0 +∆t for some fixed time interval ∆t. In D-F-W and in Rosenberg’s empirical experiments, ∆t is essentially one week. The error at time t1 can be measured in dollars, as the predicted option price less the actual option price, or in implied volatility, as the B-S volatility implied by the predicted price less the B-S volatility implied by the actual price. We will approximate the expectations of the implied-volatility-denominated error and a normalized dollar-denominated error. The approximations suggest that the sign and the relative magnitude of the empirically negative biases in models AH and DV F can be explained by a positive B(2)-risk premium. All expectations in this section are with respect to the objective probability measure.

6.1. Implied volatilities: The Ad-Hoc, “Sticky Strike” method The ad-hoc model AH, to use D-F-W’s terminology, prices each option (K, T )at time t1 using the implied volatility observed at time t0 for that same contract (K, T ): ˆAH BS C (Xt1 ,t1,K,T):=C (Xt1 ,t1,K,T,I(Xt0,t0,K,T,σt0)) This is what Derman [20] calls the sticky strike model, what Rosenberg [58] calls the volatility-by-strike model, and what Jackwerth and Rubinstein [47] call the absolute smile method. Expressed in terms of implied volatility, the pricing error is −∆I,where∆Iis the change, from time t0 to time t1, in the implied volatility of the contract (K, T ). To approximate ∆I,notethatbyItˆo’s rule, √ V dσt = εatdt + εbtdB

and   σ2 d(log X )= µ − t dt + σ dB(1) . t t 2 t December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 73

So by (5.4) and an Itˆo-Taylor expansion,

∆I := I(Xt ,t1,K,T,σt )−I(Xt ,t0,K,T,σt ) 1 1 0 ! 0 √ ρσ b λb √ ρb − − t0 t0 − t0 − − t0 − = σt1 σt0 ε (t1 t0) ε (log Xt1 log Xt0 )+O(ε) 4 2 2σt0   √   p V ρ (1) 1 2 (2) = εbt ∆B − ∆B + 1 − ρ λ ∆t + O(ε) , (6.1) 0 2 2 t0 where − ∆B := Bt1 Bt0 .

The t0-conditional expectation of the change in implied volatility for a given contract is p 2 √ 1−ρ (2) Et (∆I)= ∆t εbt λ + O(ε) , 0 2 0 t0 so by stationarity the unconditional expectation is p   1−ρ2 √ E(∆I)= ∆t×E εbλ(2) + O(ε) . (6.2) 2 The expected pricing error, measured in implied volatility, is −E(∆I). The dollar valuation error is ˆAH − BS C C = C (Xt1 ,t1,K,T,I(Xt0,t0,K,T,σt0)) − BS C (Xt1 ,t1,K,T,I(Xt1,t1,K,T,σt1)) − BS × = Cσ (Xt1 ,t1,K,T,σt0) (∆I)+O(ε).

For fixed log-moneyness m,thet0-conditional expected valuation error, expressed as a percentage of the underlying’s price, is therefore p − 2 √ AH 1 ρ (2) Et ((Cˆ − C)/Xt )=− ∆t εbt λ 0 1 2 0 t0 × BS −m Cσ (Xt1 ,t1,Xt1e ,T,σt0)/Xt1 + O(ε) .

Note that the “normalized vega” factor is random only through σt0 ;theXt1 cancels out entirely. The unconditional expectation is p 1−ρ2 E((CˆAH − C)/X)=− ∆t 2 √ × (2) × BS −m E( εbλ Cσ (X, t1,Xe ,T,σ)/X)+O(ε) (6.3) by stationarity. Thus a positive volatility risk premium λ(2) would lead us to expect negative valuation errors, whereas a negative λ(2) would lead us to expect positive valuation errors. In empirical tests of the AH model on S&P 500 options data from 1988-1993, D-F-W report a mean outside error of −0.054, where outside error is defined as the December 28, 2000 15:0 WSPC/104-IJTAF 00087

74 R. W. Lee

Table 3. Pricing errors for sticky strike and sticky implied tree models: D-F-W.

Sticky strike (AH) Sticky implied tree (DV F)

Mean outside error −0.054 −0.066

• D-F-W estimate a local volatility surface L(K, T) once per week over the period June 1988 through December 1993, using a sample of S&P 500 options within 10% of at-the-money, and between 6 and 100 days to expiration. • On each Wednesday t0 they observe option prices C(K, T) and infer the S&P forward Xt0 . Then they estimate parametrically a local volatility function

2 2 DV F : L(K, T) = max(0.01,a0 +a1K+a2K +a3T +a4T +a5KT)

via least squares. Parameters ai are chosen such that the solution U(K, T)of

1 2 2 + UT − K L (K, T)UKK =0,U(K, t0)=(Xt −K) , 2 0 P − 2 minimizes the time-t0 total squared error (K,T )(U(K, T) C(t0,K,T)) . They test how well L values options one week later at time t1, by solving

DV F 1 2 2 DV F DV F + Cˆ − K L (t0,K,T)Cˆ =0, Cˆ (K, t1)=(Xt −K) . T 2 KK 1 The error statistic we report is D-F-W’s mean outside error, defined as X 1 DV F MOE(DV F):= OE(Cˆ (t1,K,T)) , #of(K, T)att1

where OE(Cˆ) is zero if Cˆ lies within the time-t1 bid and ask prices for option (K, T); otherwise it is the signed amount by which Cˆ exceeds the ask or undershoots the bid. • D-F-W compare the DV F model against an “Ad Hoc” model that uses the earlier week’s implied volatilities to price the later week’s options. At time t0 they fit

2 2 AH : I(K, T) = max(0.01,a0 +a1K+a2K +a3T +a4T +a5KT)

to a cross-section of option prices. One week later at time t1, they price options using ˆAH BS C (K, T)=C (Xt1 ,t1,K,T,I(Xt0,t0,K,T)) . and calculate the mean outside error MOE(AH). They report the average error across all pairs (t0,t1) of consecutive weekly cross-sections in the sample.

dollar amount by which the AH model price overshoots the ask price or undershoots the bid price. For more details see Table 3. This negative bias suggests a positive premium for B(2)-risk, the part of volatility risk uncorrelated with asset risk. Similarly, Rosenberg tests “volatility-by-strike,” his implementation of the AH method, on S&P 500 options data from 1988-1997. As we detail in Table 4, his results support the finding of negative bias, measured in dollar terms as −0.08, and in terms of implied volatilities at −0.31%. One of his conclusions is that this model “exhibits substantial prediction bias.” Our approximations (6.2) and (6.3) suggest that the negative signs on both measures of error can be explained, again, by a positive B (2)-risk premium. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 75

Table 4. Pricing errors for sticky-strike models: Rosenberg.

Measure of error Mean Standard deviation T-statistic for mean = 0

Implied volatility error −0.31% 1.92% 13.06 Dollar pricing error −0.08 1.02 6.76

• Rosenberg (2000) estimates an implied volatility smile I(K) once per week, over the period 1988 through 1997, using a sample of S&P 500 futures options data. On each Wednesday t0,

he observes the S&P futures level Xt0 and option prices C(K, T) at a single expiration: the nearest date, between 10 and 100 calendar days away, on which both the futures contracts and futures options expire. Using the parameterization

2 AHr : I(K)=a0+a1K+a2K ,

he estimates the parameters a0,a1 and a2 using a polynomial regression.

One week later at time t1, Rosenberg observes the futures price Xt1 and prices options using the following estimate, adjusted for early exercise: ˆAHr BS C (K, T)=C (Xt1 ,t1,K,T,I(Xt0,t0,K)) . He then calculates the pricing errors in terms of dollars and in terms of implied volatility, and he reports the mean errors across all 6577 observations, where each observation consists of a particular option contract’s valuation error for a particular pair (t0,t1) of consecutive weeks. • Rosenberg finds that the AHr model’s estimates are biased downward, as measured in dollars and in implied volatility. This agrees with the downward bias observed by D-F-W. We argue that this bias is consistent with positive λ(2).

Table 5. Na¨ıve pricing models: A summary.

Sticky implied tree Sticky strike

Assumes stable local volatilities implied volatilities p 1 p Bias (in vol) ≈ E[vol − of − vol × ∆t ×··· − 1−ρ2λ(2) − 1 − ρ2λ(2) 2 Empirical bias negative ($) negative ($, vol) in work by D-F-W D-F-W, Rosenberg Possible explanation λ(2) > 0 λ(2) > 0

Of course our idealized concept of pricing errors differs in places from the way these authors chose to conduct their tests; for example they report dollar errors instead of our percentage errors. Nonetheless, the same bias appears in the implied- volatility-denominated errors, where there is no normalization issue.

6.2. Local volatilities: The DVF, “Sticky Implied Tree” method By (5.7) and an Itˆo–Taylor expansion, the change in local volatility is − ∆L := L(Xt1 ,t1,K,T,σt1) L(Xt0,t0,K,T,σt0) √ p V − (1) − 2 (2) = εbt0 [(∆B ρ∆B )+ 1 ρ λt0 ∆t]+O(ε), December 28, 2000 15:0 WSPC/104-IJTAF 00087

76 R. W. Lee

which has conditional and unconditional expectations p √ − 2 (2) Et0 (∆L)= 1 ρ ∆t εbt0 λt0 + O(ε) p √ E(∆L)= 1−ρ2∆t×E( εbλ(2))+O(ε). (6.4) We will find that the dollar valuation error to leading order is just the Black– Scholes vega times the negative of the change in local volatility. The analogous statement for the implied volatility model AH followed directly from an Itˆo–Taylor expansion, but to see this in the local volatility case requires some more computa- tion, because the DV F valuation of the contract (K, T ) is not simply a function of L(K, T ); it comes from a PDE involving the local volatility surface over a whole strip in (K, T )-space. The DV F method prices options at time t1 using the local volatility surface DV F calculated at time t0.ThusCˆ satisfies 1 CˆDV F − K2L2(t ,K,T)CˆDV F =0 (6.5) T 2 0 KK ˆDV F − + with initial condition C (K, t1)=(Xt1 K) . We seek an expansion √ CˆDV F = CˆDV F0 + εCˆDV F1 + O(ε) . When ε = 0, Eq. (6.5) reduces to the forward counterpart to the Black–Scholes

PDE with volatility parameter σt0 . Hence ˆDV F0 BS C = C (Xt1 ,t1,K,T,σt0). √ To obtain Cˆ DV F1, equate coefficients of ε in (6.5). Using (5.7), this reduces to the PDE

DV F1 1 2 2 DV F1 2 BS Cˆ − K σ Cˆ =2K σt I1(Xt ,t0,K,T,σt )C (Xt ,t1,K,T,σt ) T 2 t0 KK 0 0 0 KK 1 0 with zero initial condition. The solution ˆDV F1 BS 2 − C = Cσ (Xt1 ,t1,K,T,σt0)I1(Xt0/Xt1 , 2t0 t1,K,T,σt0) can be verified by direct substitution. Thus the DV F dollar valuation error is indeed ˆDV F − BS − BS C C = C (Xt1 ,t1,K,T,σt0) C (Xt1 ,t1,K,T,σt1) √ BS × 2 − + εCσ (Xt1 ,t1,K,T,σt0) [I1(Xt0/Xt1 , 2t0 t1,K,T,σt0) − I1(Xt1,t1,K,T,σt1)] + O(ε) √ BS × 2 − = εCσ (Xt1 ,t1,K,T,σt0) [σt0 +I1(Xt0/Xt1 , 2t0 t1,K,T,σt0) − − σt1 I1(Xt1,t1,K,T,σt1)] + O(ε) − BS × = Cσ (Xt1 ,t1,K,T,σt0) (∆L)+O(ε). Denominated in terms of implied volatility, this error is −∆L + O(ε). December 28, 2000 15:0 WSPC/104-IJTAF 00087

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At each fixed log-moneyness m, the conditional and unconditional expectations of the normalized valuation errors are, by stationarity, p √ ˆDV F − − − 2 (2) Et0 ((C C)/Xt1 )= 1 ρ ∆t εbt0 λt0 × BS −m Cσ (Xt1 ,t1,Xt1e ,T,σt0)/Xt1 + O(ε) (6.6) p E((CˆDV F − C)/X)=− 1−ρ2∆t √ × (2) × BS −m E( εbλ Cσ (X, t1,Xe ,T,σ)/X)+O(ε). (6.7) Thus, to leading order the DV F valuation, like the AH valuation, has negative bias if the volatility risk premium λ(2) is positive, and positive bias if λ(2) is negative. As described in Table 3, D-F-W report that in their sample the aggregate mean outside error for the DVF model is −0.066, which again suggests a positive premium for B(2) risk.

6.3. Two notions of volatility risk premia The distinction between λ(2) and λV is relevant here, because evidence indicates that λV is negative in equity markets, according to work by Kapadia [50], Benzoni [8] and Pan [54]. We have argued that D-F-W and Rosenberg’s data samples contain a positive λ(2). To reconcile intuitively the opposite signs on the two concepts of volatility risk premium, recall that p V (1) − 2 (2) λt = ρλt + 1 ρ λt . To the extent that asset risk B(1) carries a positive premium and volatility risk BV is negatively correlated with asset risk, the BV risk premium will have a negative component, namely the first term on the right-hand side. So even if λ(2) > 0, it is (1) V possible that ρλt is negative enough to make the sum λ also negative. In other words, it is plausible that investors do not require positive compensation to bear BV risk, because of its anticorrelation with asset price shocks; this is similar to points raised in Kapadia [50] and Bates [7]. At the same time it is plausible that investors do demand positive compensation for bearing the portion of BV -risk that is orthogonal to asset price shocks.

6.4. Comparisons According to (6.2) and (6.4), the DV F method has approximately twice the AH method’s expected error, as measured in implied volatility: E(−∆L)=2E(−∆I)+O(ε). Consequently, as shown in (6.3) and (6.7), DV F method has approximately twice the AH method’s expected normalized dollar error: E((CˆDV F − C)/X)=2E((CˆAH − C)/X)+O(ε). December 28, 2000 15:0 WSPC/104-IJTAF 00087

78 R. W. Lee

If λ(2) < 0 then, to leading order,

E((CˆDV F − C)/X)

Thus a negative λ(2) may explain not only the negative biases in both methods, but also the underperformance of the DV F method in this regard, as reported by D-F-W and summarized in Table 3. Intuitively, the DV F model’s error is determined by the shift, from time-t0 to time-t1, in local volatilities L(K, T ), while the AH model’s error is determined by the shift in implied volatilities I(K, T ). Each of these√ shifts decomposes√ into three components, as indicated in the approximations L0 + εL1 and I0 + εI1. First is the movement in volatility σ, which has the effect of shifting vertically the surfaces I(K, T )andL(K, T ). Second is the movement in the log of the asset price (or equivalently, the asset’s log-moneyness with respect to any fixed strike); this has the effect of shifting horizontally the I(K, T )andL(K, T ) surfaces, in a direction parallel to the K axis. Third is the decrease in time-to-maturity of every option, which has the effect of shifting horizontally the I(K, T )andL(K, T ) surfaces by ∆t units in a direction parallel to the T axis. Because the local volatility surface has twice the slope of the implied volatility surface, the horizontal shifts have twice the effect on local volatility as they do on implied volatility, which could account for local volatility’s greater absolute bias.

7. Asymptotic Hedging Errors Consider the problem of hedging one call option with a portfolio of the underlying stock and the riskless bond, where the stock satisfies

(1) dXt = µtXtdt + σtXtdBt

2 Vt = σt

V dVt = α(Vt)dt + β(Vt)dBt

V V V under the objective measure, and the dB -risk premium process λt = λ (Vt) depends only on the volatility. Here β>0, and the correlation between B(1) and BV is a constant ρ. The call, expiring at time T , has at time t ∈ [0,T] the market price C(t)= C(x, v, t), given by (2.7). The hedging portfolio at time t will hold Θt shares of the stock Xt,andC(t)−ΘtXt units of the bond, where the strategy Θt is yet to be specified. The value of the portfolio at any time is identical to that of the call C(t), but the portfolio is not self-financing. Define the hedging error Z t Π(t):=C(t)−C(0) − ΘudXu 0 December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 79

which is the cumulative amount that has to be deposited into (or withdrawn from) the portfolio over the time interval (0,t), not including the initial investment of C(0). Alternatively, we could view Πt as the profit or loss in a self-financing portfolio consisting of a long position of one call, a short position of Θt shares of stock, and an initial short position of C(0) − Θ0X0 in bonds. Under Black–Scholes dynamics, the delta-hedge Θt = Cx(X, t) makes the cost process identically zero. Under stochastic volatility, however, it is not generally possible to replicate the option with a self-financing portfolio of the stock and bond. One measure of the error in this context, for any particular choice of Θt,is the strategy’s mean error EΠ(t) (or the hedging bias). For certain variants on delta-hedging, we will calculate or approximate this statistic, and relate it to volatility risk premia. For each of these strategies we also derive implementation schemes that use only today’s implied volatility skew, and do not require the specification or estimation of any particular volatility model. All expectations in this section are with respect to the objective probability measure.

7.1. Hedge at the stochastic-volatility delta First consider the strategy

sv Θt := Cx(Xt,Vt,t)

so that the hedge ratio is the stochastic-volatility analogue of the Black–Scholes delta hedge. By Itˆo’s rule and (2.7) we have   1 1 dC = C dX + C dV + C + vx2C + β2C dt x v t 2 xx 2 vv

V V = CxdX + Cv(αdt + βdB ) − (α − λ β)Cvdt

V V = CxdX + βCvdB + λ βCvdt ,

hence β β dΠsv = dC − C dX = C dBV + λV C dt . (7.1) x 2σ σ 2σ σ Since the martingale part has expectation zero, we have Z  t sv V EΠ = E λ βCvdu . 0 which is equivalent to the result obtained by Kapadia [50]. In particular, if the volatility risk premium λV is always negative, then the hedging bias is always negative. December 28, 2000 15:0 WSPC/104-IJTAF 00087

80 R. W. Lee

This strategy can be implemented without directly estimating the parameters of a stochastic volatility model. Multiply the equations

BS BS Cx = Cx + Cσ Ix

BS BS CK = CK + Cσ IK by X and K respectively; then sum. Since C and CBS are both homogeneous of degree 1 in X and K, it follows that K I = − I , x X K hence K C = CBS − CBS I , (7.2) x x σ X K

so the level I and slope IK of the implied volatility skew at strike K contain enough information to determine the stochastic-volatility delta. Using a sample of S&P 100 options data from 1988 to 1996, Kapadia tests this delta-hedging strategy. On each day, for each active call contract in his sample, he commences a stochastic-volatility delta-hedge, and rebalances it daily. The average hedging error at maturity, over the 61172 observations, is −0.39 dollars, from which Kapadia concludes that λV is negative, and that the magnitude of the bias is significant both statistically and economically. Note that our implementation scheme (7.2) is a simple alternative to Kapadia’s BS multi-step process: assuming that ρ = 0, he approximates Cx as Cx evaluated at his estimate of the historical volatility; then he argues that his conclusions are robust to non-zero ρ.

7.2. Hedge to minimize instantaneous variance The strategy

minv ρβ Θt := Cx(Xt,Vt,t)+ Cv(Xt,Vt,t) σXt arises in different contexts. It is what Bakshi, Cao and Chen [3] call the “minimum- variance” hedge, because the diffusion component of dΠ=dC − ΘdX is

(1) V (Cx − Θ)σXdB + βCvdB , (7.3) so the instantaneous variance of the hedging error is

2 2 2 [(Cx − Θ)σX + ρβCv] +(1−ρ )(βCv) , which is minimized at each t bychoosingΘ=Θminv. Note that this stra- tegy minimizes instantaneous variance under both the objective and the risk- neutral measures, because a Girsanov change of measure does not affect diffusion coefficients. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 81

It is, moreover, identical to the strategy that, as Bertsimas, Kogan and Lo [10] prove, minimizes the risk-neutral expectation of the squared terminal hedging error. In fact, in the case r = 0, it solves the more general optimization problem where the hedger is free to choose not just the dynamic strategy Θ but also the initial investment: the optimal hedge is still Θ = Θminv, and the optimal initial investment is still the call price C(0) implied by that pricing measure. A related strategy is what Heath, Platen and Schweizer [42] call the local risk-minimizing strategy. It coincides with Θminv in the case λ(2) = 0, but not in general. Substituting Θminv into (7.3) shows that thisp strategy eliminates all dB(1)-risk, minv 2 (2) leaving dΠ with a diffusion component of 1 − ρ βCvdB . Therefore p p minv (2) 2 2 (2) dΠ = λ 1 − ρ βCvdt + 1 − ρ βCvdB and the expected hedging error is Z  t p minv (2) 2 EΠ = E λ 1 − ρ βCvdu . (7.4) 0 In particular, if the volatility risk premium λ(2) is always positive, then the hedging bias is always positive. To implement this hedge, without estimating a stochastic volatility model, note that   minv ρβ BS ρβ BS Θt = Cx + Cv = Cx + Ix + 2 Iσ Cσ . (7.5) σXt 2σ Xt Now under either the slow-variation or small-variation volatility asymptotics, the relevant statics of the implied volatility skew are approximated as ρβ I ≈ + O(ε) (7.6) x 4σ2X

Iσ ≈ 1+O(ε). (7.7) So, for a simple implementation of minimal-instantaneous-variance hedging, use K Θminv ≈ C − CBSI = CBS + CBS I . (7.8) x σ x x σ X K For an implementation that does not rely on slow-variation or small-variation assumptions, note that (7.6) is a good approximation for sufficiently short-dated near-the-money options, regardless of the speed of variation in volatility; this was discussed and illustrated in Sec. 5.5. So a more robust implementation of the strategy (7.5) is minv ≈ BS − BS Θ Cx +[Ix(K, T ) 2Ix(X, t+)Iσ(K, T )]Cσ , (7.9) where the two arguments of I that we have chosen to display are the strike and the maturity date, so that (X, t+) indicates the at-the-money short-dated skew slope. The robust implementation faces the hurdle that Iσ cannot simply be read off of the skew in the same way that Ix or IK can. The asymptotic approximation Iσ ≈ 1 December 28, 2000 15:0 WSPC/104-IJTAF 00087

82 R. W. Lee

is likely to be an overestimate at longer maturities, because the effects of volatility mean reversion tend to make longer-dated implied volatilities less responsive to volatility shocks. This issue can be addressed without estimating a stochastic volatility model, because practitioners know Iσ(T ) as the “bucket vega weighting” of the T -maturity option, and Taleb [63], for example, suggests ways to estimate it. Thus our robust implementation scheme does not require development of new machinery. In defense of the simple implementation, however, note that (7.9) shows that the error in the simple implementation’s approximation of Iσ tends to act in the direction that mitigates the error in its approximation of IK . In fact, if the decay in T of the effect Iσ of a volatility shock on implieds were to match thedecayin Tof the skew slope IK , then for at-the money options, the robust implementation (7.9) would just reduce to the simple implementation (7.8). Also note that the “robust” implementation is robust to departures from slow-variation, not necessa- rily to departures from the continuous-diffusion assumption; reliance on the very short-dated skew Ix(X, t+) could cause jump effects to come into play. The simple implementation avoids this issue. Bakshi, Cao and Chen calculate the hedging errors that arise from implement- ing the minimum instantaneous variance strategy on daily S&P 500 index data from June 1988 through May 1991. Table 6 shows that the average hedging errors are positive. B-C-C do not attempt to explain the positive hedging bias. We suggest that the bias can be interpreted as the effect of a positive premium for dB(2) risk,the

Table 6. Average dollar hedging errors: Bakshi, Cao and Chen.

Days to Expiration Moneyness < 60 60-180 > 180 < 0.94 NA 0.03 0.01 0.94 − 0.97 0.03 0.02 0.02 0.97 − 1.00 0.04 0.02 0.00 1.00 − 1.03 0.04 0.02 0.00 1.03 − 1.06 0.05 0.02 0.02 > 1.06 0.05 0.03 0.01

• Bakshi, Cao and Chen calculate the hedging errors that arise from implementing the minimum instantaneous variance strategy on daily S&P 500 index data from June 1988 through May 1991. • On each day they use options data to estimate the parameters of a Heston model. For each day and each option in the sample, they construct the portfolio of the call, the stock, and the riskless security, according to the minimal instantaneous variance strategy, and on the following day they calculate the hedging error. Note that B-C-C state their results in terms of portfolios that are short the call and long the stock, which is the opposite of our convention, so we have multiplied the numbers in their Table 7 by −1. • We argue that the positive bias is consistent with positive λ(2). December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 83

portion of volatility risk uncorrelated with asset risk. This is consistent with (7.4), which assumes stochastic volatility, but does not make any assumptions about the rate at which volatility varies.

7.3. Hedge at the Black–Scholes delta A more readily implementable strategy is to ignore stochastic volatility; instead let us continuously update a Black–Scholes implied-volatility hedge. Specifically, at each t ∈ [0,T], find the current Black–Scholes implied volatility for the contract being hedged, and set the hedge ratio according to the Black–Scholes delta, evalu- ated at the implied volatility: bs BS Θt := Cx (Xt,t,K,T,I(t)) ,

where I(t)=I(Xt,t,K,T,Vt) is directly obtainable from the market price of the option. The cost process is bs sv − BS dΠ = dΠ +(Cx Cx )dX

sv BS = dΠ +(Cσ Ix)dX

sv BS (1) BS (1) = dΠ +(Cσ Ix)λ σXdt +(Cσ Ix)σXdB   β β = λ(1)σXI + λV I CBSdt + σXI CBSdB(1) + I CBSdBV . x 2σ σ σ x σ 2σ σ σ Therefore the mean error is Z    t bs (1) V β BS EΠ (t)=E λ σXIx + λ Iσ Cσ du . 0 2σ In either the slow-variation or small-variation asymptotic regime, we can use the implied volatility expansion (5.4) to approximate the integrand as     β −ρβ β λ(1)σXI + λV I CBS ≈ λ(1)σX + λV CBS x 2σ σ σ 4σ2X 2σ σ   1 β = λV − ρλ(1) CBS 2 2σ σ   1 ≈ λV − ρλ(1) βC , 2 v √ wherewehaveabsorbedthe√ εfactor into β and leave it implicitly understood that β is O( ε). Therefore Z    t bs V 1 (1) EΠ (t)=E λ − ρλ βCvdu + O(ε) . 0 2 Practical implementation of this hedge requires only the implied volatility of the option being hedged. December 28, 2000 15:0 WSPC/104-IJTAF 00087

84 R. W. Lee

7.4. Two notions of volatility risk premia, revisited Modulo notational differences, at least three papers have found in equity markets a negative risk premium λV , or (equivalently if β>0) a negative λβ. Kapadia [50] does so with S&P 100 data from 1988 to 1996, as we described in Sec. 7.1. Benzoni [8] gives several estimates of the risk premium, all negative: a simulated method of moments estimate using S&P 500 data from 1986 to 1996, as well as least-squares estimates that minimize the difference between S&P 500 option prices for three months in 1997 and stochastic volatility model prices. Pan [54] uses an “implied state” generalized method of moments approach on S&P 500 data from 1989 to 1996 to estimate a negative volatility risk premium in a Heston-style stochastic volatility model (although she rejects this model in favor of jump-diffusions). We have proposed that biases reported in three other papers can be explained by a positive risk premium λ(2). First were the negative pricing biases in D-F-W’s [28] sticky-strike and sticky-implied-tree tests. Second were the negative pricing biases in Rosenberg’s [58] sticky-strike tests. Third, as we just saw, were the positive biases in B-C-C’s [3] minimal-instantaneous-variance hedging tests. The relation between λ(2) and this hedging bias does not depend on asymptotic assumptions. Our globally valid implementation (7.2) of stochastic volatility delta hedging, and our asymptotic implementation (7.8) of minimal instantaneous variance hedg- ing, offer simple ways to conduct, in possible future empirical research, inference about λV and λ(2), without the need to specify a particular stochastic volatility model, and without assuming a particular functional form for λV .

7.5. Comparisons In the usual case where ρ<0andλ(1) > 0, we have 1 p λV <λV − ρλ(1) <λV −ρλ(1) = λ(2) 1 − ρ2 . 2 If this holds almost surely, then in the asymptotic regime

EΠsv

Now combine this with the fact that

EΠsv < 0

assuming λ(2) > 0andλV <0, as defended in Sec. 7.4. The conclusion is 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. Thus the Black–Scholes hedge — the simplest of the three strategies — has an asymptotically smaller absolute bias than at least one and quite possibly both of the other two strategies. Table 7 summarizes these findings. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 85  du v βC  (1) ρλ 1 2 BS BS x x − C C V λ  t 0 Z  In between the other methods E ≈ ≈  K du I v 0 BS σ > βC v ) C C ) t (2) (1) λ ρβ ρλ σX K/X − + +( V x λ C ( t BS x 0 C Z  ≈ E K 0 Positive if I  < du BS σ v V C ) λ βC x Table 7. Hedging strategies: A summary. V C λ K/X t ( 0 Z −  BS E x C Stochastic volatility delta Minimize instantaneous variance Black–Scholes delta ” rely on the small-variation or slow-variation asymptotics. For all other entries the result is exact. ≈ Shares held Hedging bias Implementation via skew Hedging bias is empirically Negative (Kapadia) Positive (our reading of B-C-C) Hedging bias is theoreticallyEntries preceded Negative by if “ December 28, 2000 15:0 WSPC/104-IJTAF 00087

86 R. W. Lee

8. Further Research One area for further research is to analyze asymptotically an extension of the stochastic volatility model. For example, we could allow jumps in the price and volatility processes. Another possible extension would be introduce a second state variable into the volatility process to model the presence of both a slowly-varying factor and a rapidly-varying factor. These refinements would help to reduce the mis- specification present in single-factor volatility models; also a two-factor volatility model may help to unify these slow-variation asymptotics with the rapid-variation asymptotics of Fouque, Papanicolaou and Sircar, as we discussed in Sec. 5.5. A second area for research is to extend the pricing and hedging analysis. For ex- ample, we could generalize the hedging analysis to include path-dependent options, and to allow the use of other options as hedging instruments. Also we could consider alternative measures of error, such as mean squared error. Another possibility, as discussed in Sec. 7.4, is to apply empirically our suggested implementations of hedg- ing schemes, in order to extend the literature that draws inferences about volatility risk premia.

Acknowledgments I am grateful to, especially, my adviser George Papanicolaou, to my committee members Darrell Duffie, Tze Leung Lai, Peter Glynn and Michael Harrison, and to Jim Gatheral. This work was supported in part by an NDSEG Graduate Fellowship.

References [1]L.B.G.AndersenandR.Brotherton-Ratcliffe,The equity option volatility smile: an implicit finite-difference approach, J. Computational Finance 1 (1997) 5–37. [2] M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via relative-entropy minimization, Appl. Math. Finance 4 (1997) 37–64. [3] G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models,J.Finance52 (1997) 2003–2049. [4] C. A. Ball and A. Roma, Stochastic volatility option pricing, J. Financial and Quan- titative Analysis 29 (1994) 589–607. [5] D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Rev. Financial Studies 9 (1996) 69–107. [6] D. S. Bates, Testing option pricing models,inHandbook of Statistics, Vol. 14,eds. G. S. Maddala and C. R. Rao, Elsevier Science B. V. (1996) 567–611. [7]D.S.Bates,Post-’87 crash fears in the S&P 500 futures option market, J. Econom. 94 (2000) 181–238. [8] L. Benzoni, Pricing options under stochastic volatility: An econometric analysis (January 1999), Northwestern University. [9] S. Berle and N. Cakici, How to grow a smiling tree, J. Financial Engineering 7 (1998) 127–146. [10] D. Bertsimas, L. Kogan and A. Lo, Pricing and hedging derivative securities in incomplete markets: An -arbitrage approach (November 1997). NBER Working Paper 6250. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 87

[11] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy 81 (1973) 637–659. [12] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems 13 (1997) L11–L17. [13] P. Carr, K. Ellis and V. Gupta, Static hedging of exotic options,J.Finance53 (1998) 1165–1190. [14] P. Carr and D. Madan, Determining volatility surfaces and option values from an implied volatility smile (October 1998), Working paper. [15] M. Chernov and E. Ghysels, A study towards a unified approach to the joint estima- tion of objective and risk neutral measures for the purpose of options valuation (April 1999), Pennsylvania State University. [16] N. Chriss, Black–Scholes and Beyond: Option Pricing Models, McGraw-Hill (1997). [17] T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function (1998), Cornell University. [18] S. R. Das and R. K. Sundaram, Of smiles and smirks: A term-structure perspective, J. Financial and Quantitative Analysis 34 (1999) 211–239. [19] F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Mathematische Annalen 2 (1994) 61–73. [20] E. Derman, Regimes of volatility (January 1999), Goldman Sachs. [21] E. Derman and I. Kani, Riding on a smile,Risk7(1994) 32–39. [22] E. Derman and I. Kani, Stochastic implied trees: Arbitrage pricing with stochas- tic term and strike structure of volatility, International J. Theoretical and Applied Finance 1 (1998) 61–110. [23] E. Derman, I. Kani and N. Chriss, Implied trinomial trees of the volatility smile, J. Derivatives 3 (1996) 7–22. [24] E. Derman, I. Kani and J. Z. Zou, The local volatility surface: Unlocking the infor- mation in index option prices, Financial Analysts J., pp. 25–36. [25] E. Derman and J. Zou, Predicting the response of implied volatility to large index moves (November 1997), Goldman Sachs. [26] D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press (1996). [27] D. Duffie, J. Pan and K. Singleton, Transform analysis and option pricing for affine jump-diffusions (August 1998), Graduate School of Business, Stanford University. [28] B. Dumas, J. Fleming and R. E. Whaley, Implied volatility functions: Empirical tests, J. Finance 53 (1998) 2059–2106. [29] B. Dupire, Pricing with a smile,Risk7(1994) 18–20. [30] B. Dupire, Pricing and hedging with smiles,inMathematics of Derivative Securities, eds. M. A. H. Dempster and S. R. Pliska, vol. 15 of Publications of the Newton Institute, Cambridge University Press (1997) 103–112. [31] J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Asymptotics of a two-scale stochastic volatility model,inEquations aux Derive´es Partielles et Applications, Articles Dedi´es `a Jacques-Louis Lions, Gauthier-Villars (1998) 517–526. [32] J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Financial modeling in a fast mean- reverting stochastic volatility environment, Asia-Pacific Financial Markets 6 (1999) 37–48. [33] J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press (2000). [34] J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Mean-reverting stochastic volatility, International J. Theoretical and Applied Finance 3 (2000) 101–142. [35] R. Frey, Derivative asset analysis in models with level-dependent and stochastic volatility, CWI Quarterly 10. December 28, 2000 15:0 WSPC/104-IJTAF 00087

88 R. W. Lee

[36] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall (1964). [37] A. R. Gallant, C.-T. Hsu and G. Tauchen, Calibrating volatility diffusions and ex- tracting integrated volatility (1998), Duke University. [38] J. Gatheral, The structure of implied volatility in the Heston model (2000), Merrill Lynch. [39] E. Ghysels, A. C. Harvey and E. Renault, Stochastic volatility,inHandbook of Statis- tics, Vol. 14, eds. G. S. Maddala and C. R. Rao, Elsevier Science B. V. (1996) 119–191. [40] D. Guo, The risk premium of volatility implicit in currency options,J.Business& Economic Statistics 16 (1998) 498–507. [41] J. M. Harrison and D. Kreps, Martingales and arbitrage in multiperiod securities markets, J. Economic Theory 20 (1979) 381–408. [42] D. Heath, E. Platen and M. Schweizer, A comparison of two quadratic approaches to hedging in incomplete markets (December 1999), University of Technology Sydney, and TU Berlin. [43] S. L. Heston, A closed-form solution for options with stochastic volatility and appli- cations to bond and currency options, Rev. Financial Studies 6 (1993) 327–343. [44] D. G. Hobson, Stochastic volatility (October 1996). School of Mathematical Sciences, University of Bath. [45] J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance 42 (1987) 281–300. [46] J. C. Jackwerth, Generalized binomial trees, J. Derivatives 5 (1997) 7–17. [47] J. C. Jackwerth and M. Rubinstein, Recovering stochastic processes from option prices (October 1998), Working paper. [48] H. Johnson and D. Shanno, Option pricing when the variance is changing,J.Financial and Quantitative Analysis 22 (1987) 143–151. [49] M. Kamal and E. Derman, The patterns of change in implied index volatilities (September 1997), Goldman Sachs. [50] N. Kapadia, Do equity options price volatility risk? Empirical evidence (June 1998), University of Massachusetts, School of Management. [51] A. Levin, Recovering implied volatility and distribution from American futures option prices using the regularization method (August 1998), Working paper. [52] A. L. Lewis, Option Valuation under Stochastic Volatility, Finance Press (2000). [53] A. Melino and S. M. Turnbull, Pricing foreign currency options with stochastic volatil- ity, J. Econom. 45 (1990) 239–265. [54] J. Pan, Integrated time-series analysis of spot and options prices (April 2000), Stan- ford University. [55] G. Papanicolaou, Asymptotic analysis of stochastic equations,inStudies in Probability Theory, ed. M. Rosenblatt, Mathematical Association of America (1978) 111–179. [56] A. M. Poteshman, Estimating a general stochastic variance model from options prices (November 1998), University of Chicago. [57] E. Renault and N. Touzi, Option hedging and implied volatilities in a stochastic volatility model, Math. Finance 6 (1996) 279–302. [58] J. V. Rosenberg, Implied volatility functions: A reprise,J.Derivatives7(2000) 51–64. [59] M. Rubinstein, Implied binomial trees,J.Finance49 (1994) 771–818. [60] L. O. Scott, Option pricing when the variance changes randomly: Theory, estimation, and an application, J. Financial and Quantitative Analysis 22 (1987) 419–438. [61] K. R. Sircar and G. Papanicolaou, Stochastic volatility, smile & asymptotics, Appl. Math. Finance 6 (1999) 107–145. [62] E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: An analytic approach, Rev. Financial Studies 4 (1991) 727–752. December 28, 2000 15:0 WSPC/104-IJTAF 00087

Implied and Local Volatilities Under Stochastic Volatility 89

[63] N. Taleb, Dynamic Hedging: Managing Vanilla and Exotic Options, John Wiley & Sons (1997). [64] J. B. Wiggins, Option values under stochastic volatility, J. Financial Economics 19 (1987) 351–372. [65] G. A. Willard, Calculating prices and sensitivities for path-independent derivative securities in multifactor models, J. Derivatives, pp. 45–61. [66] P. Wilmott, Derivatives, John Wiley & Sons (1998).