Delta-Hedged Gains and the Negative mkt vol risk premium

Delta-Hedged Gains and the Negative mkt vol risk premium -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Delta-Hedged Gains and the Negative Market Volatility Risk Premium Gurdip Bakshi and Nikunj Kapadia  April 9, 2001  Bakshi is at Department of Finance, Smith School of Business, University of Maryland, College Park, MD 20742, and Kapadia is at Department of Finance, School of Management, University of MassachusettsAmhert, MA 01003. Bakshi can be reached at Tel: 301-405-2261, Email: gbakshi@rhsmith.umd.edu, Website: www.rhsmith.umd.edu/ nance/gbakshi/; and Kapadia at Tel: 413-545-5643, Email: nkapadia@som.umass.edu. For helpful comments, we thank Doron Avramov, Charles Cao, Bent Christensen, Sanjiv Das, Stephen Figlewski, Christopher Jones, Hossein Kazemi, Leonid Kogan, Dilip Madan, George Martin, Vasant Naik, Jun Pan, Jay Patel, Allen Poteshman, N. R. Prabhala, Rangarajan Sundaram, Bob Whaley, and Gregory Willette. Parts of the article build on Kapadia's thesis written at New York University. Earlier versions of the paper were presented at Boston University, University of Massachusetts and Virginia Polytechnic Institute. Conference participants at the 1998 WFA (Monterey) and 2001 AFA (New Orleans) meetings provided many useful suggestions. Nick Bollen (AFA discussant) and Je Fleming (WFA discussant) provided extremely constructive comments. The reports of Bernard Dumas (the Editor) and two anonymous referees have substantially improved this paper. Kristaps Licis has provided excellent research assistance. Bent Christensen graciously shared his option dataset. Kapadia acknowledges nancial support from the Center of International Derivatives and Securities Markets. The 1998 version of the paper was circulated under the title "Do Equity Options Price Volatility Risk?" 0 Delta-Hedged Gains and the Negative Market Volatility Risk Premium Abstract We investigate whether the volatility risk premium is negative by examining the statistical properties of delta-hedged option portfolios (buy the option and hedge with stock). Within a stochastic volatility framework, we demonstrate a correspondence between the sign and magnitude of the volatility risk premium and the mean delta-hedged portfolio returns. Using a sample of S&P 500 index options, we provide empirical tests that have the following general results. First, the delta-hedged strategy underperforms zero. Second, the documented underperformance is less for options away from the money. Third, the underperformance is greater at times of higher volatility. Fourth, the volatility risk premium signi cantly a ects delta-hedged gains even after accounting for jump-fears. Our evidence is supportive of a negative market volatility risk premium. 1 The notion that volatility of equity returns is stochastic has a rm footing in nancial economics. However, a less than understood phenomenon is whether volatility risk is compensated, and whether this compensation is higher or lower than the risk-free rate. Is the risk from changes in market volatility positively correlated with the economy-wide pricing kernel process? If so, how does it a ect the equity and the option markets? Evidence that market volatility risk premium may be non-zero can be motivated by three empirical ndings: 1. Purchased options are hedges against signi cant market declines. This is because increased realized volatility coincides with downward market moves (French, Schwert and Stambaugh (1987) and Nelson (1991)). One economic interpretation is that buyers of market volatility are willing to pay a premium for downside protection. The hedging motive is indicative of a negative volatility risk premium; 2. At-the-money Black-Scholes implied volatilities are systematically and consistently higher than realized volatilities (Jackwerth and Rubinstein (1996)). A potential explanation for this puzzling empirical regularity is that the volatility risk premium is negative. Ceteris paribus, a negative volatility risk premium increases the risk-neutral drift of the volatility process and, thus, raises equity option prices; 3. Equity index options are non-redundant securities (Bakshi, Cao and Chen (2000) and Buraschi and Jackwerth (2001)). Index option models omitting the economic impact of a market volatility risk premium may be inconsistent with observed option pricing dynamics. This article investigates, both theoretically and empirically, whether the volatility risk premium is negative in index option markets. This is done without imposing any prior structure on the pricing kernel, and without parameterizing the evolution of the volatility process. The setup is a portfolio of a long call position, hedged by a short position in the stock, such that the net investment earns the risk-free interest rate. The central idea underlying our analysis is that if option prices incorporate a non-zero volatility risk premium, then we can infer its existence from the returns of an option portfolio that has dynamically hedged all risks except volatility risk. If volatility is constant, or the price process follows a one-dimensional Markov di usion, then our theoretical analysis implies that the net gain, henceforth \delta-hedged gains," on the deltahedged portfolio is precisely zero. A similar conclusion obtains when volatility is stochastic, but volatility risk is unpriced. In this particular case, we show that the distribution of the deltahedged gains has an expected value of zero. However, if volatility risk is priced, then the sign and magnitude of average delta-hedged gains are determined by the volatility risk premium. 1 Our theoretical characterizations point to empirical implications that can be tested using discrete delta-hedged gains. First, in the time-series, the at-the-money delta-hedged gains must be related to volatility risk premium. Second, cross-sectional variations in delta-hedged gains (in the strike dimension) are restricted by the sensitivity of the option to volatility (i.e., the vega). Our framework allows us to di erentiate between the volatility risk premium and the jump-fear underpinnings of delta-hedged gains. We test model implications using options written on the S&P 500 index, over the period January 1988 through December 1995. Our empirical speci cations are supportive of the following general results:  The delta-neutral, positive vega, strategy that buys calls and hedges with the underlying stock signi cantly underperforms zero. On average, over all strike and maturity combinations, the strategy loses about 0.05% of the market index, and about 0.13% for at-themoney calls. This underperformance is also economically large; for at-the-money options, this amounts to 8% of the option value.  As predicted by theory, the underperformance is decreasing for options away from the money. Controlling for moneyness, the underperformance is greater when the hedging horizon is extended. These results are robust across time, and to the inclusion of put options.  At times of higher volatility, this underperformance is even more negative. The losses on long call positions persist throughout the sample period, and cannot be reconciled by a downward trending market volatility. Both the cross-sectional and time-series tests provide evidence that support the hypothesis of a non-zero volatility risk premium. In particular, the results suggest that option prices re ect a negative market volatility risk premium. One, the cross-sectional regression results indicate that delta-hedged gains are negatively related to vega, after controlling for volatility and option maturity. Consistent with our predictions, the hedged gains are maximized for at-the-money options. 1 Buraschi and Jackwerth (2001) and Coval and Shumway (2000) provide additional evidence on the possible existence of a non-zero volatility risk premium. For instance, the statistical examination in Buraschi and Jackwerth supports stochastic models with multiple priced factors. Our paper di ers from existing treatments in several respects. First, we provide an analytical characterization that links the distribution of the gains on a delta-hedged option portfolio to the underlying risk sources. Speci cally, we show a correspondence between the sign of the mean delta-hedged gains and the sign of the volatility risk premium. Economically, the magnitude of the market volatility risk premium is connected to the value of the option as a hedge. Second, our modeling framework provides an explicit set of hypotheses for testing whether the market volatility risk premium is negative. Instrumental to this thrust is whether priced volatility risk or jumps are the primary source of the underperformance of delta-hedged portfolios. For related innovations, we refer the reader to Bakshi, Cao and Chen (1997), Bates (2000), Buraschi and Jackwerth (2001), Benzoni (1999), Dumas, Fleming and Whaley (1998), Eraker, Johannes and Polson (2000), Jackwerth and Rubinstein (1996), Jones (2000), Pan (2000), and Poteshman (1998). 1 2 Two, in the time-series regressions, a higher volatility implies more negative delta-hedged gains for at-the-money options (both using the conventional volatility estimate and a GARCH volatility estimate). To con rm the hedging rationale underlying a negative volatility risk premium, we empirically estimate the option vega, and verify that it is strictly positive. Moreover, we show that options become more expensive (as measured by implied volatility) after extreme market declines. Three, we empirically examine whether measures of asymmetry and peakedness of the riskneutral distribution (jump-fear surrogates) are linked to losses on delta-hedged strategies. This investigation nds that, while risk-neutral skewness helps explain some portion of the delta-hedged gains for short-dated options, the volatility risk premium retains its explanatory ability even in the presence of risk-neutral skew and kurtosis. Speci cally, the volatility risk premium is the predominant explanatory factor for delta-hedged gains for short and medium maturities. Our empirical inquiry also nds that the qualitative patterns of delta-hedged gains persist in a precrash sample; the average excess return on the delta-hedged option portfolio is negative for both calls and puts. Finally, we make the observation that delta-hedged gains for options bought immediately after a tail event do not depart considerably between downward and upward daily return movements. In summary, the jump-fear explanation, although plausible, cannot be the sole economic justi cation for the systematic losses incurred on the delta-hedged portfolios. However, stochastic volatility models with a negative volatility risk premium show promise in reconciling this observation. The rest of the article is organized as follows. Section 1 formulates our theoretical analysis of gains on a delta-hedged option portfolio. Section 2 discusses the data and variable de nitions. The statistical properties of delta-hedged gains are described in Section 3. We conduct, in Section 4, an empirical exercise that relates delta-hedged gains and vega in the cross-section. Section 5 examines the time-series implications between delta-hedged gains and the volatility risk premium. In Section 6, we analyze whether volatility risk premium explains delta-hedged gains in the presence of jump factors (as re ected by the risk-neutral skew and kurtosis). Finally, concluding remarks are o ered in Section 7. All technical details are in the Appendix. 1 Delta-Hedged Gains and the Volatility Risk Premium This section describes the distribution of the gain on a portfolio of a long position in an option, hedged by a short position in the underlying stock, such that the net investment earns the riskfree interest rate. We call the gain on the hedged portfolio the \delta-hedged gains." We rst develop the relevant theory in Section 1.1 assuming that volatility of equity returns is constant, 3 and then relax this assumption in Section 1.2 by allowing volatility to be stochastic. The theoretical implications are then used in Section 1.3 to motivate empirical tests about the negative volatility risk premium. Our analysis also shows how the presence of jumps can contribute to the underperformance of delta-hedged equity portfolios. To formalize main ideas, let C (t;  ; K ) represent the price of a European call maturing in  periods from time t, with strike price K . Denote the corresponding option delta by (t;  ; K ). De ne the delta-hedged gains, t;t  , as the gain or loss on a delta-hedged option position, where the net (cash) investment earns the risk-free rate: + Zt t;t   Ct  Ct +  + t + Zt u dSu  + t r (Cu u Su ) du; (1) where St is the time t price of the underlying (non-dividend paying) stock and r is the constant risk-free interest rate. In equation (1), we have used the short-hand notation Ct  C (t;  ) and t  (t;  ) = @Ct=@St for compactness. The expected t;t  can be interpreted as the excess rate of return on the delta-hedged option portfolio. Relating delta-hedged gains to the volatility risk premium is our primary objective throughout. + 1.1 Delta-Hedged Gains Under Constant Volatility Let the stock price follow a geometric Brownian motion under the physical probability measure (with constant drift, , and constant volatility,  ): dSt =  dt +  dW : t St (2) 1 By Ito's lemma, we can write the call price as, Ct  = Ct + + Zt  + t u dSu + Zt ! @ Cu + 1  S @ Cu du: @u 2 u @Su  2 + 2 t 2 2 (3) Standard assumptions also show that the call option price is a solution to the Black-Scholes valuation equation, 1  S @ C + r S @C + @C r C = 0: (4) 2 @S @S @t Using equations (3) and (4), it follows that 2 2 2 2 Ct  = Ct + + Zt  + t u dSu + 4 Zt  + t r (Cu u Su) du; (5) which is the statement that the call option can be replicated by trading a stock and a bond. Combining equation (5) with the de nition of delta-hedged gains in equation (1), it is apparent that, with continuous trading, t;t  = 0 over every horizon  . More generally, it can veri ed that t;t  = 0 is a property common to all one-dimensional Markov Ito price processes: dStt = S t[St] dt + t[St] dWt . When the hedge is rebalanced discretely, t;t  will not necessarily be zero. However, Bertsimas, Kogan and Lo (2000) show that the delta-hedged gains have an asymptotic distribution that is symmetric with zero mean. Consider a portfolio of an option that is hedged discretely N times over the life of the option, where the hedge is rebalanced at each of the dates tn , = 0 1 1 (where we de ne t = t, tN = t +  ) and kept in place over the period, tn tn = =N . De ne the discrete delta-hedged gains, t;t  , as, + + 1 + n 0 ; ; :::N +1 + t;t   Ct + +  Ct N X 1 n tn (Stn +1 Stn ) =0 N X 1 n =0  r (Ct tn Stn ) N ; p R (6) C where tn  @Ctn =@Stn . From Bertsimas, Kogan and Lo (2000), N ) p tt   Su @@Suu dWu , where Wu is a Wiener process, independent of Wu . Thus, the asymptotic distribution of the discretely hedged option portfolio has a mean of zero, and is symmetric. Simulation results in Bertsimas, Kogan and Lo as well as those reported in Figlewski (1989) suggests that the distribution of t;t  is centered around zero for a wide range of parameters, and for low values of N (about 10). We show next that if we relax the geometric Brownian motion assumption for the stock price and allow for stochastic volatility outside of the one-dimensional Markov di usion context, then t;t  is centered around zero unless volatility risk is priced. Therefore, this setting allows us construct a test to examine whether the volatility risk premium is negative. 1 2 + 2 2 2 2 1 + + 1.2 Delta-Hedged Gains Under Stochastic Volatility Consider a (two-dimensional) price process that allows stock return volatility to be stochastic (under the physical probability measure), dSt =  [S ;  ] dt +  dW ; ttt tt St dt = t[t ] dt + t[t] dWt ; (7) (8) 1 2 where the correlation between the two Weiner processes, Wt and Wt , is . It may be noted that volatility, t , follows an autonomous stochastic process; the drift coecient, t [t], and the 1 5 2 di usion coecient, t[t ], are functionally independent of St. Therefore, by Ito's lemma, Z t  @Cu Z t  @Cu Zt  Ct  = Ct + bu du; @S dSu + @ du + + + t + + t u (9) t u C C @ where de ning bu  @Cu + u Su @@Suu + u @@uu +  u u Su @SuCu u . The valuation equation @u @ that determines the price of the call option is: 1 2 2 2 1 2 2 2 2 2 2 2 1  S @ C + 1  @ C +    S @ C + r S @C + (  [ ]) @C + @C r C = 0; t t @S@ t t 2 t @S 2 t @ @S @ @t 2 2 2 2 2 (10) 2 2 2   where t [t]  Covt dmtt ; dt represents the price of volatility risk, for a pricing kernel process m mt, and Covt(; :) is a conditional covariance operator divided by dt. In general, the volatility risk premium will be related to risk aversion and to the factors driving the pricing kernel process. Re-arranging (10), it follows that bu is also equal to:   bu = r Cu Su @Cu (u [u] u [u ]) @Cu : @Su @u (11) Substituting out bu in the stochastic di erential equation (9), we obtain Z t  @Cu Z t  Z t  @Cu @Cu ) ( [  [) @Cu  du: r(Cu Su @S Ct  = Ct + u u @Su dSu + t @u du + t @u t u + + + + (12) This equation can be further simpli ed by substituting for du in the second integral to give, Zt   Z t  @Cu @ Cu S  du + Z t   @Cu du+ Z t   @Cu dW : (13) Ct  = Ct + u @ u @ u @Su dSu + t r Cu @Su u t t t u u + + + + + 2 We are now ready to prove the following relationships between delta-hedged gains, the volatility risk premium and the option vega. Proposition 1 Let the stock price process follow the dynamics given in equations (7)-(8). Moreover, suppose the volatility risk premium is of the general form t[t]. Then, 1. The delta-hedged gains, t;t  , is given by + Zt t;t  = t + +  Zt  u[u] @Cu du + u [u ] @Cu dWu ; @ @ + t u and from the martingale property of the Ito integral Et(t;t  ) = + Zt  + t  u @C  Et u [u ] u du; @ u 6 2 (14) (15) where @Ctt represents the vega of the call option, and Et (:) is the expectation operator under @ the physical probability measure. 2. If volatility risk is not priced in equilibrium, i.e., t[t ]  0, then Et (t;t  ) = O(1=N ); (16) + where the discrete delta-hedged gains, t;t  , is as de ned previously in equation (6). + The proposition states that, with continuous trading, if volatility risk is not priced, the deltahedged gains are, on average, zero. In practice, the hedge is re-balanced discretely over time, and this may bias the average t;t  away from zero. However, in (16), we show that this bias is small. That is, even if we allow for discrete trading, for both the Black-Scholes model and the stochastic volatility model, the mean delta-hedged gains is zero, up to terms of O(1=N ). If volatility risk is priced, equation (15) shows that Et(t;t  ) is determined by the price of volatility risk, t, and the vega of the option, @Ct=@t. The statistical tests in Buraschi and Jackwerth (2001) support a non-zero volatility risk premium. There are two speci c testable implications that follow from equation (15). First, as the vega is positive, a negative (positive) t implies that Et (t;t  ) will be negative (positive). In particular, a negative volatility risk premium is consistent with the notion that market volatility often rises when the market return drops. To see this, consider a Lucas-Rubinstein investor that is long the market portfolio, and has a coecient of relative risk aversion . Under this particular assumption,   the pricing kernel mt = St . An application of Ito's lemma yields t[t] = Covt dStt ; dt , so S that a negative correlation between the stock return and the volatility process implies a negative t. In our modeling paradigm, there is a one-to-one correspondence between the sign of t and the sign of mean delta-hedged gains. Economically, purchased options are hedges against market declines because increased realized volatility tends to occur when market falls signi cantly. Consequently, in the stochastic volatility setting, the underperformance of the delta-hedged portfolio is tantamount to the existence of a negative volatility risk premium. Our framework allows one to determine the sign of the volatility risk premium without imposing any strong restrictions on the pricing kernel process; it also does not rely on the identi cation or the estimation of the volatility process. The quantitative strategy (6) is relatively easy to implement in option markets. + 2 + + 2 The distribution of the delta-hedged gains can be described in terms of single and multiple Ito integrals. It is dicult to represent multiple Ito integrals in increments of their component Wiener processes (Milstein (1995)). p Therefore, unlike the Black-Scholes case, the asymptotic distribution of Nt;t+ cannot be described succinctly. 7 Second, as the option vega is largest for near-the-money options, the absolute value of Et (t;t  ) is also largest for near-the-money options. If in addition, the volatility risk premium is negative, as we have hypothesized, the underperformance of the delta-hedged portfolio should decrease for strikes away from at-the-money. We may note that, as option vega's are negligible especially for deep in-the-money options, these options have little to say about the nature of the volatility risk premium. Although our focus has been on calls, all the results also apply to puts. + 1.3 Testable Predictions Before we can derive precise empirical implications from equation (15), we need to simplify the R right-hand term Et tt  u @Cu=@u du, in terms of the contemporaneous stock price and the level of volatility. To streamline discussion, de ne g (St; t)  t[t ] @Ctt , and consider the Ito-Taylor @ R expansion of tt  g (Su; u ) du (Milstein (1995)): + + Zt + t  g(Su; u) du = g (St; t) Zt  + Z t  Z ut + + t t 1 Z t Z u L[g(Su0 ; u0 )] du0 du t t  Zu [g (Su0 ; u0 )] dW + [g (Su0 ; u0 )] dW du; du + + 1 t 2 2 @ @ @ @ where the (in nitesimal) operators are de ned as, L[:] = @t [:]+ t St @St [:]+ t @t [:]+ t St @St [:]+ @ @ @ t @t [:] + t Stt @S@t@t [:], [:] = tSt @St [:] and [:] = t @t [:]. Using the martingale property of the Ito integral and equation (15), we, therefore, have 1 2 1 2 2 2 2 2 2 2 2 2 1 2 Z t Z u Et (t;t  ) = g (St; t)  + Et L[g(Su0 ; u0 )] du0 du; t t 1 X  n n @Ct = L [t @ ]; t n (1 + n)! + (17) + 1+ (18) =0 by a recursive application of the Ito-Taylor expansion. Observe that Et (t;t  ) is abstractly related to the current stock price and volatility, and the parameters of the option price especially the maturity and moneyness. To develop testable empirical speci cations, we now exploit certain option properties and derive the functional form of each term in equation (18). In the discussions that follow, we assume that all parameters of the option price are held xed. For a broad class of option models, the call price is homogeneous of degree one in the stock price and the strike price (Merton (1973)). So, for a xed moneyness, the call price scales with the price of the underlying asset St . In this case, the option vega, @Ct=@t, also scales with St . We may therefore separate g (St; t) = t (t;  ; y ) St, for option moneyness y , and t (:) independent of St. If @Ct=@t, and, therefore, g (St; t), scales with St and volatility risk is priced, then we assert + 8 that t;t  also scales with St , where the scaling factor is a function of t (and other parameters of the option contract). To prove this, we make use of equation (18), the standard assumption that the stock price, St , follows a proportional stochastic process and the following property of L as it operates on a function g : + Lemma 1 Consider g(St; t) = t(t;  ; y) S , for any  2 R, where t is at most a function of maturity and volatility. If St obeys a proportional stochastic process, then Ln [g (S;  )] is also proportional to St , for all n 2 f1; 2; 3:::g. The lemma, which is proved in the Appendix, shows that if g (St; t) scales with St , so does It follows from equation (18) that t;t  is also proportional to St. Thus, we can represent Et (t;t  ) as Et (t;t  ) = St  ft [t ;  ; y ]; (19) Ln [g]. + + + for some ft [ that is determined by the functional dependence of t and @Ct=@t on t , and the parameters of the option price, in particular, the option moneyness and maturity. That Et (t;t  )=St varies in the time-series with physical volatility, t , and in the cross-section (for a xed t ) with the option moneyness, y , forms the basis of the empirical tests. To derive the cross-sectional test, we keep t as xed, and write Et (it;t  )=St = ft [ ; yi;t], for moneyness corresponding to strike price Ki . It is important to keep t as xed, as the option price is non-linear in t for away from the money strikes. In the absence of any information regarding the form of the non-linearity, it is dicult to specify a model and the corresponding econometric test that allows both t and yi;t to vary simultaneously. Given a suitable model for ft [ ; y ], we can then test the relation between Et (it;t  )=St and yi;t . Because the vega of the option and thus the absolute value of Et(it; ) is, ceteris paribus, maximized for at-the-money option, and decreases for strikes away from at-the-money, it follows that ft [y ] must also be of such a functional form (controlling for volatility). We can reject the hypothesis of a non-zero volatility risk premium if we do not nd this hypothesized relation between Et(it;t  )=St and yi;t . Thus, the cross-section of delta-hedged gains contains information about the volatility risk premium. Next, to develop the time-series relation between E(t;t  )=St and t, consider (19) applied to at-the-money options. It has been noted elsewhere (Stein (1989)) that the short-term at-themoney call is almost linear in volatility. If Ct is linear in t , @Ct=@t will be independent of t, and the functional dependence of t;t  on t will be determined only by t and the underlying stochastic volatility process. Given the functional form of t[t ] and the underlying volatility process, we can infer the functional form of t;t  . For at-the-money options, we may specialize ^ ft [t;  ; y ] = f t [y ;  ] ft[t]. To make this point precise, we develop the functional form of at-themoney t;t  for the Heston (1993) model. In his model, the volatility risk premium is linear in + + + + + + + + 9 volatility (see also the set of assumptions in Bates (2000), Eraker, Johannes and Polson (2000) and Pan (2000)). Proposition 2 Consider the special case of the stock price process (7)-(8), where [t] =  t and  [t] =  . Speci cally, dSt =  [S ;  ] dt +  dW ; ttt tt St dt =  t dt +  dWt ; (20) 1 (21) 2 and the volatility risk premium is linear in volatility, as in t[t] =  t. Let the call option vega be proportional to St and independent of t, as in @Ct=@t = t ( ; y ) St. Then, the delta-hedged gains for near-the-money options must be: Et (t;t  ) =  't( ) St t ; (22) + where 't( ) > 0 is de ned in the Appendix. At-the-money delta-hedged gains are negative only if  < 0. Speci cally for at-the-money options, Proposition 2 shows that if t is proportional to t , so is the scaled delta-hedged gains, Et (t;t  )=St. Although not done here, it is straightforward to extend the analysis to other models, in which case, more generally, Et(t;t  )=St may be a polynomial in t (i.e., Hull and White (1987)). We can, thus, construct a time-series test relating the scaled at-the-money delta-hedged gains to physical return volatility (or equivalently the volatility risk premium). We can reject the hypothesis of a zero volatility risk premium if we nd a relation between at-the-money Et (t;t  )=St and any functional of physical volatility. In summary, our theoretical results indicate that the bias in t;t  from discrete hedging is small relative to the impact of a volatility risk premium (as suggested by Proposition 1). Moreover, the mean at-the-money delta-hedged gains (normalized by the stock price) is approximately linear in the level of physical volatility. We veri ed both these results via simulations. More exactly, the delta-hedged strategy typically underperforms (overperforms) zero with negative (positive) volatility risk premium. Additionally, the negative bias is related to the change that occurs because a negative volatility risk premium increases the option price. In large part, the level of underperformance is greater with higher volatility. The details are provided in Appendix B. Before we operationalize and implement the cross-sectional and time-series tests using options data, one question remains unresolved: How is the performance of delta-hedged strategies a ected by jumps? To address this question, we appeal to a jump-di usion model for the equity price + + + + 10 (Bates (2000), Merton (1976) and Pan (2000)). Consider dSt =  [S ;  ] dt +  dW + (ex 1) dq    dt; ttt tt t JJt St 1 (23) where the volatility dynamics are as displayed in (8). This framework allows for both stochastic volatility as well as random jumps to a ect delta-hedged gains. The set-up is brie y as follows. First, in (23), the variable qt represents a Poisson jump counter with volatility-dependent intensity J t. Denote the physical density of the jump-size, x, by q [x]. Second, we posit that x and qt are orthogonal to each other and to all sources of uncertainty. In addition, if we assume that the mean of ex 1 is J , the compensator is J J t dt, which is the nal term in (23). Lastly, to isolate the impact of jump-size and jump intensity on delta-hedged gains, for now, we assume that only jump-size is priced. The jump risk premium will therefore introduce a wedge between the physical density, q [x], and the risk-neutral density, q  [x]. Speci cally, assume that the risk-neutral mean of ex 1, is  . J In the stochastic environment of (23), the delta-hedged gains are equal to (see the Appendix): @Cu  du +   Z t  E  @Cu  S  du Et (t;t  ) = Et u [u ] @ t @S u u JJ t t u Z 1  Zt  Z 1u J u du Cu (Su ex ) q[x] dx Cu (Su ex) q[x] dx : (24) Zt   + + + + 1 t 1 The rst term is a consequence of the volatility risk premium, and the other two terms are a consequence of jumps. When u = 0, equation (24) imparts the intuition that delta-hedged gains are negative provided the mean jump size is negative (i.e.,  < 0), and there are occasional price J discontinuities (i.e., J > 0). In theory, the fatter left-tails of the equity price distribution can lead to the underperformance of delta-hedged portfolios (the sign of return skewness is determined by the sign of the mean jump size and J controls excess kurtosis). Equation (24) suggests the e ect of jumps on delta-hedged gains is most pronounced for in-the-money options. In our extended framework, the bias in delta-hedged gains is partly due to priced volatility risk and partly due to jump exposures. Observe that the nal double integral term in (24) is typically negative. This is because the option price evaluated at the risk-neutral density of the jump-size is generally higher than under the physical density. Moreover, when the jump risk premium is volatility dependent, as is the case here, the component of delta-hedged gains due to jump risk is related to variations in volatility. In particular, higher the physical volatility, the more negative are the total deltahedged gains. Now if one additionally assumes that jump intensity is priced (J gets altered to  ), the expression for Et (t;t  ) must be modi ed. Speci cally, the last two terms must J + 11 R   R R u be replaced by:   tt  Et @Cu u Su du  tt  u 1 fCu (Su ex ) Cu (Su )g q [x] dxdu + J 1 R t   R 1 fCJ (S ex) C @S )g q[x] dxdu. JThis analysis suggests that both forms of jump J t u 1 u u u (Su risk will lead to the underperformance of delta-hedged portfolios. As we will see, equation (24) provides the impetus for empirically di erentiating between the negative volatility risk premium and the jump fear explanations for negative delta-hedged portfolio returns. + + + 2 Description of Option Data and Variable De nitions All empirical tests employ daily observations on S&P 500 index options. This equity option contract is European, and traded on the Chicago Board Options Exchange. The option prices consists of time-stamped calls and puts, and correspond to the last bid-ask quote reported before 3:00 pm CST. Rubinstein (1994) and Jackwerth and Rubinstein (1996) have suggested that the pre-crash and the post-crash index distributions di er considerably. The initial sample date was accordingly chosen to begin from January 1, 1988 to avoid mixing pre-crash and post-crash options (see also Christensen and Prabhala (1998)). Our option sample ends on December 30, 1995. The option universe is constructed in the following way. First, the option data is screened to eliminate option prices that violated arbitrage bounds. Speci cally, we exclude call options whose price is outside of the range: (Se z e r K; Se z ), for dividend yield z . Second, to minimize the impact of recording errors, we discard all options that have Black-Scholes implied volatilites exceeding 100%, or less than 1%. Third, we deleted options with maturity less than 14 days. In addition, all options with maturity longer than 60 days are eliminated. Our present focus on short-term options allows us to reduce the impact of stochastic interest rates. Finally, deep in-the-money option prices can be unreliable due to the lack of trading volume. As in Jackwerth and Rubinstein (1996) and Buraschi and Jackwerth (2001), deep away from the money options are omitted. Our option sampling procedure results in 36,237 calls and 35,030 puts. We require a series for dividends, interest rates and index return volatility. First, following a prevalent practice, we assume that daily dividends are known over the life of the option contract. That is, we take the actual dividend pay-out (from the S&P 500 Bulletin) and subtract the present discounted value of dividends from the contemporaneous stock price. The adjusted stock price is employed in our empirical tests throughout. The interest rates are computed using the procedure outlined in Jackwerth and Rubinstein (1996) and Buraschi and Jackwerth (2001). Each day, we infer the interest rate using the put-call parity. Speci cally, we use a strike price and maturity matched pairs of puts and calls, quoted   within a 1-minute interval. The borrowing rate is computed as: rb = (1= ) log (S e z C a + P b )=K .   Likewise, the lending rate is: rl = (1= ) log (S e z C b + P a )=K , for each pair of bid-ask 12 call and put quotes, (C b ; C a) and (P b ; P a). The daily rate used in the tests is the mid-point of rb and rl , averaged across all strikes of a speci c maturity. For robustness, we adopt two measures of return volatility. One, we estimate a GARCH(1,1) model using daily S&P 500 returns over the entire period: Rt = R + t ; t = a + a t + a t ; and; t = t t; t  i.i.d. N (0; 1); ;t 1 2 0 1 2 1 2 2 1 (25) (26) (27) where the  -period return is de ned as Rt;t   log(St  =St) and t is the conditional volatility. Relying on the GARCH model estimates, the  -period GARCH volatility estimate is: + + v u t g  u 252 X  ; VOLt t  ^n (28) 2 nt = where n is the tted value obtained from the GARCH estimation. We experimented with other ^ GARCH speci cations and obtained similar volatility estimates. The GARCH volatility measure also allows us to construct a daily volatility series for estimating the hedge ratio in equation (6). The other volatility measure is the estimate of the sample standard deviation, as in: v u t h = u 252 X Rn VOLt t  nt ;n 1  R; 2 (29) = where R is now the average daily return. This rolling estimation procedure produces volatility estimates, with estimation error serially uncorrelated through time for non-overlapping periods. To construct an empirical test design that limits overlapping observations, we will sometimes appeal to a sample of options with constant maturity (for example, 30 days and 44 days). Over our sample period, the S&P 500 index options have continual option quotes available only for the two near months. Thus, to build as large a series as possible and yet limit overlap, we employ options of maturity no more than 60 days. De ne the option moneyness as y  Se r z  =K . Consequently, a call (put) option is classi ed as out-of-the-money if it has moneyness corresponding to y < 1 (y > 1). For reasons already discussed, our empirical work is restricted to the 10% moneyness range. While out-of-money puts are sometimes used to sharpen our results, for tractability, much of our analysis centers on calls.   13 3 Statistical Properties of Delta-Hedged Gains We compute the discrete delta-hedged gains for each call option in two steps. hedge ratio, t, re-computed daily at the close of the day price. The total delta-hedged gains for each option up to the maturity date is then calculated as: t;t  = max(St + +  K; 0) Ct N X 1 n tn (Stn Stn ) +1 =0 N X 1 n =0  rn (Ct tn Stn ) N where t = t, tN = t +  is the maturity date, and tn is the hedge ratio at tn . In our implementation procedure, the interest rate is updated on a daily basis. For tractability, tn is computed as the Black-Scholes hedge ratio, tn = N [d (Stn ; tn )], where N [ is the cumulative normal distribution, and 0 1 d   1 p log(yn ) + 1 t;t  pn : 2 t;t  n 1 + + (30) All our delta-hedged calculations allow for time-varying volatility, as re ected by the use of GARCH volatility in equation (30). Although the Black-Scholes hedge ratio is a reasonable estimate of the true hedge ratio when volatility is not correlated with the stock return process, it will be biased otherwise. In a later sectionn, we will examine the impact of a mis-speci ed delta. Panel A of Table 1 provides descriptive statistics for delta-hedged gains grouped over maturity and moneyness combinations. Speci cally, we report the averages for (i) dollar delta-hedged gains t;t  , (ii) delta-hedged gains scaled by the index level t;t  =St (in %), and (iii) delta-hedged gains scaled by the call price t;t  =Ct (in %). For at-the-money calls, and for each maturity, the delta-hedging strategy loses money. On average, over all moneyness and maturities, the strategy loses about 0.05% of the index level, and for at-the-money calls (i.e., y 2 [ 2:5%; 2:5%]), the strategy loses about 0.10%. Moreover, the mean t;t  =Ct over the full 8 years sample is -12.18%. It may be noted that the reported standard errors, computed as the sample standard deviation divided by the square-root of the number of options, are relatively small. The delta-hedged gains are statistically signi cant in all moneyness and maturity categories. The average loss on the delta-hedged strategy of about $0.43 for at-the-money options also appears high compared with the mean bid-ask spread of $ . This nding implies that the buyer of the call (\long" volatility) is paying the seller of the call (\short" volatility) a premium of about 43 cents per call. The economic impact of this premium is substantial, given the large volume of S&P 500 contracts traded. The S&P 500 trading volume in 1991 was about 11 million contracts, so that the dollar impact of this premium could be as high as $500 million. The cumulative impact + + + + 3 8 14 over the eight year period is of the order of several billion dollars. We can make two additional empirical observations that appear broadly consistent with a volatility risk premium. First, the mean delta-hedged gains for away from the money strikes are mostly negative, and less so relative to at-the-money calls. Consider options with moneyness y 2 [ 7:50%; 5%) versus options with moneyness y 2 [ 2:50%; 0%). In the \All" category, we can observe that the dollar delta-hedged gains is $-0.28 versus $-0.42. Because the vega for away from the money options is small, the impact of the volatility risk premium should be small. Second, the losses on delta-hedged portfolios generally deepen when the hedging horizon is extended from 14-30 days to 31-60 days. For at-the-money options, the dollar loss over the 31-60 days maturity is almost twice than the loss in the 14-30 day maturity. This empirical nding tallies with the theoretical prediction that delta-hedged gains should become more negative with maturity (because the vega is increasing with maturity). Overall, the delta-hedged gains are negative except for deep in-the-money options. That deep in-the-money calls have positive delta-hedged gains is anomalous. We will reconcile this result shortly. Next, to ensure that the documented results are not driven by extremes, we also examine the relative outcomes of positive and negative delta-hedged gains. The last column displays the 1< statistic that measures the frequency of negative delta-hedged gains (consolidated over all maturities). For at-the-money (out-of-the money) options, it is assuring that 68% (76%) of the observations have negative gains. Therefore, the observation that the mean delta-hedged gains are negative on average, appears robust. Moreover, the frequency of negative delta-hedged gains rise (fall) monotonically when options go progressively out-of-the-money (in-the-money). If deep in-the-money calls are excluded, then as much as 72% of the remaining call sample have negative delta-hedged gains. As seen from Panel B of Table 1, the results are robust across subsamples (the standard errors are small and suppressed). In the rows marked SET 1 and SET 2, we report the mean deltahedged gains over the 88:01-91:12 and the 92:01-95:12 sample periods, respectively. Clearly, the underperformance of the delta-hedged strategy is more pronounced over the second subsample. In yet another exercise, we examined the sensitivity of our conclusions to any unexpected declines in index volatility (the delta-hedged portfolios su er losses when volatility declines). The deltahedged gains for at-the-money options are negative in 7 out of 8 years. Therefore, the persistent losses on the delta-hedged portfolios cannot be attributable to any secular declines in index volatility. Finally, to verify the results from a di erent options market, we examined delta-hedged gains using options on the S&P 100 index (the details were reported in an earlier version). Reassuringly, the mean delta-hedged gains are also negative for S&P 100 index options. Our conclusions are robust across sample periods, as well as across both index option contracts. 0 15 Although the conventional estimates of the cross-sectional standard errors are small in both the full sample and the subsamples, these standard errors may not account for the fact that the theoretical distribution of t;t  depends on option moneyness and maturity. We attack this problem on two fronts. First, we construct representative option time-series that are homogeneous with respect to moneyness and maturity. Speci cally, we take at-the-money call options with a xed maturity of 30 days, 44 days, and 58 days, and delta-hedged them until maturity. For 30 days calls, we get a mean  = $0:47 with a t-statistic of -2.34. Similarly, the mean  for 44 (58) days options is -0.53 (-0.63), with a t-statistic of -2.90 (-2.80). Therefore, inferences based on a homogenous time-series of delta-hedged gains (and standard t-tests) also reject the null hypothesis of zero mean delta-hedged gains. Second, the standard deviation of discrete delta-hedged gains in the context of one-dimensional di usions is known from Bertsimas, Kogan and Lo (2000). For Black-Scholes, this standardi dehR  = p viation equals (see their Theorems 1 and 3): K (1 u ) = exp  u = u St =K du . Even though analytical, the above expression requires estimates of the expected rate of return, , and the volatility, . We set =11.6% and  =11%, to match the average annual index return and volatility in our sample. For a given strike K, we compute the standard deviation of t;t  at each date t. Standardizing each t;t  by the corresponding standard deviation results in a variable with unit variance. Adhering to a standard practice, we then compute the t-statistic as the average standardized  multiplied by the square-root of the number of observations. The resulting t-statistics are -5.29, for 30 days options, and -7.30 (-9.32) for 44 days and 58 days options. That the standard deviation of the distribution of t;t  decreases with maturity when the hedge ratio is updated daily is to be expected (see the simulations in Table 1 of Bertsimas, Kogan and Lo). Reinforcing our earlier results, under Black-Scholes, we can easily reject the hypothesis that Et (t;t  ) = 0. We tried other combinations of  and  , and obtained similar results. It would be of interest to extend this analysis by theoretically characterizing the distribution of delta-hedged gains under stochastic volatility. Now return to the result that the delta-hedged gains are typically positive for deep in-themoney options, with moneyness greater than 5%. The relative illiquidity for in-the-money calls may upwardly bias the mean delta-hedged gains. Because there is not much trading activity, the market makers often chooses not to update in-the-money call prices in response to small changes in the index level (Bakshi, Cao and Chen (2000)). We believe that illiquidity of in-the-money options may lead to positive delta-hedged portfolio returns. To verify this conjecture, and to understand the sources of this phenomena, we examine, in Table 2, delta-hedged gains for out-of-money puts, which are equivalent to in-the-money calls. Relative to in-the-money calls, the out-of-money puts are more actively traded. Supportive of our conjecture, and in contrast to the empirical results + 1 2 0 2 12  2 2+log  12 2 1+  + + + + 16 from in-the-money calls, the out-of-money put delta-hedged gains are now strongly negative:  The average delta-hedged gains are $-1.03 and $-0.82 for put options with moneyness y 2 [5%; 7:5%) and y 2 [7:5%; 10:0%), respectively. It is evident that the losses on the deltahedged put portfolios is robust to samples restricted by strikes, maturity and time periods;  When deep in-the-money calls (beyond 5%) are combined with deep out-of-the-money puts, the mean dollar delta-hedged gains are $-0.14, and of absolute magnitude less than that for all at-the-money calls and puts. To sum up, when we combine the results from calls and puts, for the vast majority of the options that are actively traded, the delta-hedged gains are overwhelmingly negative. This evidence on the underperformance of delta-hedged portfolios, among calls and puts, is strongly supportive of a negative volatility risk premium. In a spirit similar to ours, Coval and Shumway (2000) corroborate that (long volatility) at-the-money S&P 500 straddles produce average losses of about 3% per week. That the sign of the market volatility risk premium is negative is in agreement with the parametric approach adopted in Eraker, Johannes and Polson (2000) and Pan (2000) (i.e., the negative volatility risk premium increases the risk-neutral drift of the volatility process). Exploiting the spanning properties of options, the results in Buraschi and Jackwerth (2001) suggest the possibility of a non-zero volatility risk premium. Although stochastic volatility option models have been shown to reduce tting errors (Bakshi, Cao and Chen (1997)), a negative volatility risk premium o ers the further potential for reconciling option prices. Under the premise that higher volatility implies a more negative volatility risk premium, is it empirically true that higher volatility translates into greater underperformance of delta-hedged portfolios? To investigate this issue, we constructed the two measures of volatility outlined in (28) and (29), and binned the at-the-money delta-hedged gains into 7 volatility groups (< 8%, 8-10%, 10-12%, 12-14%, 14-16%, 16-18%, and >18%). To save on space, we maintain focus on the conventional measure of volatility displayed in (29). The pattern of average delta-hedged gains in Table 3 validates several results of economic relevance. First, consistent with a volatility risk premium, the returns on delta-hedged portfolios are inversely proportional to volatility: during times of higher return volatility, the delta-hedged gains get even more negative. This is true irrespective of whether the underperformance is measured in dollar terms or as a fraction of the index level. Second, the ndings are invariant to outliers. In most volatility groups, the median delta-hedged gains is more negative than the mean delta-hedged gains, and generally decline with increase in volatility. Volatility is an important source of the underperformance of delta-hedged portfolios. 17 4 Delta-Hedged Gains and Option Vega in the Cross-Section We consider next the cross-sectional implication of the volatility risk premium. Following Section 1.2 (equation (19)), for a xed t , t;t  =St must be related to the option vega, such that mean delta-hedged gains decrease in absolute magnitudes for strikes away from at-the-money. We test this implication by adopting the econometric speci cation, + GAINSit = + VEGAit + eit ; 0 i = 1;    ; I; 1 (31) where GAINSit  t;t  =St and VEGAit is the option vega (indexed by moneyness i = 1;    ; I ). While controlling for volatility and option maturity, equation (31) models the proportionality of delta-hedged gains in the option vega. The null hypothesis that volatility risk is not priced corresponds to = 0. For estimating equation (31), we require a proxy for VEGAit , and a procedure for controlling for volatility. To demonstrate robustness of the cross-sectional regression estimates, the option vega is approximated in two di erent ways: + 1 8 < exp d =2 VEGA = : jy 1j Black-Scholes Vega, Absolute Moneyness, 2 1 (32) where d is as presented in equation (30). Two points are worth emphasizing about (31)-(32). First, because exp( d =2) reaches a maximum when the strike is at-the-money, a negative (positive) volatility risk premium corresponds to < 0 ( > 0). Furthermore, the magnitude of + is approximately the mean delta-hedged gains for at-the-money options. Note that the average volatility embedded in d serves simply as a scaling factor for log(y ) and governs the rate of change in exp( d =2), as the option moneyness moves away from the money. For example, for a 30 days option evaluated at 12% volatility, the impact of the risk premium on a 4% away from the money option is half that for at-the-money options. This rate of decrease is slower for higher levels of volatility. Second, the function jy 1j reaches a minimum for at-the-money options. In this case, the hypothesis of a negative (positive) volatility risk premium corresponds to < 0 and > 0 ( > 0 and < 0). In this model, the mean delta-hedged gains for at-the-money options is  precisely . Both approximations, exp d =2 and jy 1j, plausibly characterize the behavior of the option vega. It is necessary that the sample for each estimation of equation (31) consists of a panel of deltahedged gains where the historically measured volatility is approximately constant. To achieve 1 2 1 1 0 1 1 1 2 1 0 0 1 0 2 1 18 1 this, we divide the sample period into intervals of 2%; within each sample, we include all dates where the volatility is within one of these intervals. Therefore, we assume the constancy of the volatility risk premium within a volatility classi cation. To increase the power of the test, and because the sensitivity of the vega (and, thus, the delta-hedged gains) to moneyness is more pronounced at shorter maturities, we estimate equation (31) for 30 and 44 days options. With two vega surrogates, we thus have 28 distinct panels, with volatility approximately ranging from 6% to 20%, and with panel size ranging from 46 to 283 observations. When implementing (31), one econometric issue arises. As there are multiple observations of option prices on each date within a volatility sample, it is possible that there is a date-speci c component in t;t  that needs to be explicitly modeled. We follow standard econometric theory (see, for example, Greene (1997)) and allow for either a date-speci c xed e ect, or a date-speci c random e ect. In the xed e ects model, we replace in equation (31) by ;t . In the random e ects model, we allow for a component of the disturbance to be date-speci c, as modeled by eit = ut + vti. We conduct speci cation tests on our samples, and, in the majority of the samples, the Hausman test of xed versus random e ects and a Lagrange Multiplier test of random e ects versus OLS favors the random e ects speci cation. As a consequence, all reported results are based on the random e ects model, where the coecients are estimated by Feasible Generalized Least Squares panel regression (hereby FGLS). Table 4 supports the central implication that the volatility risk premium is negative. Consider rst 30 days options and vega measured by exp( d =2). In this case, as hypothesized, the coef cient is persistently negative. The regression coecient ranges between -0.67 and -0.06, and implies a negative volatility risk premium. For 5 out of 7 volatility levels, the coecient is statistically signi cant with a minimum (absolute) z-statistic of 2.95 (shown in square brackets). The estimate of + are roughly in line with the ndings in Table 1 and Table 3: the mean delta-hedged gains are more negative for higher volatility regime versus lower volatility regimes. For instance, the estimate of + is -0.13% in the 8-10% volatility grouping in comparison to -0.41% in the 14-16% volatility grouping. Based on the Wald test, the hypothesis + = 0 is rejected at the usual signi cance level (for most groups). Since the R is not particularly instructive for panel regressions, it has been excluded. The results for the 44 days options are comparable with 5 out 7 signi cantly negative coecient. Therefore, for both maturities, the absolute value of delta-hedged gains are maximized for at-the-money options, and decrease with the option vega. When vega is proxied by jy 1j, there is evidence for the joint hypothesis that < 0 and > 0. For 30 days options, varies from a low of 1.02 to a high of 6.78, and statistically signi cant in 5 out of 7 estimations. The estimated coecient and the associated t-statistics + 0 0 2 1 1 1 0 1 0 1 0 1 2 1 0 1 1 0 19 allow us to reject the hypothesis that Et(t;t  =St) is zero (in 5 out of 7 volatility groups). As before, the results from 44 days options are consistent with those from 30 days options. Both sets of estimations verify that mean delta-hedged gains decrease in absolute magnitudes for strikes away from at-the-money. Our evidence supports the cross-sectional implication of a negative market volatility risk premium. In Table 5, we provide additional con rmatory evidence for 30 days options. First, in Panel A, we report the results from a panel regression when (31) is altered to: GAINSit = + VOLh  t i + ei (i = 1;    ; I ). In this speci cation test, we also allow the mean delta-hedged gains to VEGAt t vary with volatility. For example, the time t at-the-money delta-hedged gains are now represented by + VOLt. As observed, the results reported in Panel A of Table 5 and those reported in Table 4 are mutually consistent. Second, Panel B of Table 5 substantiates that similar results can be found in the subsamples. Therefore, our key ndings are robust across subsamples and to modi cations in the test speci cations. To summarize, the cross-sectional regressions support three main empirical results. The rst conclusion that emerges is that we can formally reject the hypothesis that Et [t;t  =St] = 0. Moreover, the signs of the estimated coecients are compatible with a negative volatility risk premium. Finally, the delta-hedged gains are maximized for at-the-money options, and decrease in absolute value for moneyness levels away from at-the-money. Each nding is consistent with the theoretical predictions. A negative market volatility risk premium has the interpretation that investors are willing to pay a premium to hold options in their portfolio, or that a long position in an index option acts as a hedge to a long position in the market portfolio. We illustrate this point from two di erent angles. First, we directly examine how option prices react to volatility. For a xed option maturity, we build a monthly time-series of at-the-money call option prices (divided by the index level) and regress it on historical volatility (as estimated in equation (29) for  = 30 days): + 0 0 1 1 + 30 Days: Ct=St = 0.004 +0.05VOLh +0.44 Ct =St t 1 1 1 1 + et , R =43.16%, DW=2.01, 2 [3.36] [3.23] [4.12] 44 Days: Ct=St = 0.003 +0.06VOLh +0.52 Ct =St + et, R =65.23%, DW=2.28. t [2.26] [3.89] [5.18] Controlling for movements in the index level through time, these regressions show, as would be expected, that call prices respond positively to volatility. To put the estimated slope coecient in perspective, we note that, in our sample, the average C/S is 1.70%, and the average volatility is 11%. An increase in the level of volatility from 11% to 12% will increase C/S from 1.70% to 20 2 1.79%. This increase is the order of magnitude as that implied by the 30 days at-the-money BlackScholes vega. Given the extensive evidence on the negative correlation between stock returns and volatility, the positive estimate of the empirical vega con rms the hedging role of options. To highlight the value of the option as a hedge during signi cant market declines, we contrast the change in the relative value of index options for the largest 20 negative and positive daily returns (roughly a 3 standard deviation event). On the day prior to a tail event, we buy a nearestto-the-money short-term call option, and compute the Black-Scholes implied volatility. Proceeding to the day after the tail event, we re-compute the Black-Scholes implied volatility for the prevailing nearest-to-the-money calls. For each of the largest extreme movement, Table 6 reports (i) the (annualized) implied volatilities, and (ii) the corresponding change in implied volatility as a fraction of the implied volatility of the option bought. We can observe that the average change (relative change) in the implied volatility is 1.71% (10.58%) to a downward movement versus -0.84% (1.53%) to an upward movement. Holding everything else constant, the index options become more expensive during stock market declines (in 18 out of 20 moves, the implied volatility increases). On the other hand, when the market has a strong positive return, the e ect on option values is not as striking. These ndings further support our assertion that equity index options are desirable hedging instruments. 5 Delta-Hedged Gains and the Volatility Risk Premium: TimeSeries Evidence Following Proposition 2, we now consider the time-series implications of the volatility risk premium for at-the-money options. Fixing option maturity, we estimate the time-series regression: GAINSt = + VOLt + GAINSt + t ; 0 1 2 1 (33) where GAINSt represents the dollar delta-hedged gains for at-the-money options divided by the index level, and VOLt is the estimate of historical volatility computed over the 30 calendar day period prior to t (see equation (28) for VOLg , and equation (29) for VOLh ). In the time-series t t setting of equation (33), testing whether volatility risk is not priced is equivalent to testing the null hypothesis = 0. Observe that we have added a lagged value of GAINS to correct for the serial correlation of the residuals. The estimation is done using OLS, and the reported t-statistics are based on the Newey-West procedure (with a lag length of 12). As a check, we also estimate the model using the Cochrane-Orcutt procedure for rst-order autocorrelation. Since the results are virtually the same, they are omitted to avoid duplication. 1 21 To ensure that the regression results are not an artifact of option maturity, we perform regressions at the monthly frequency using delta-hedged gains realized over (i) 30 days, (ii) 44 days and (iii) 58 days. Although the 30 days series for delta-hedged gains is non-overlapping, a partial overlap exists with 44 days and 58 days series. To begin, consider the 30 days series for VOLh . t The results of Table 7 show that the OLS estimates of the volatility coecient, , are negative and statistically signi cant in all the samples. Over the full sample, the estimated is -0.032 with a t-statistic of -4.39. In addition, the serial correlation coecient, , is negative with a t-statistic of -3.47. The inclusion of GAINSt leads to residuals that show little autocorrelation, as is evident from the Box-Pierce statistic with 6 lags (denoted as Q ). The coecient is comparable across maturities, and is signi cantly negative throughout. The empirical t of the regressions is reasonable, with the adjusted R higher for each of the two subsamples. Furthermore, the magnitude of the coecient is an order smaller than that of . Overall, our results seem to indicate that variations in at-the-money delta-hedged are related to variations in historical volatility. This result also holds when volatility is measured by VOLg . t h . This indicates that a measure of However, the adjusted R 's are consistently higher with VOLt volatility that puts more weight on the recent return history has greater explanatory power, and is more informative about delta-hedged portfolio returns. Our repeated nding that < 0 has the implication that the market volatility risk premium is negative. Is the magnitude of the risk premium indicated by economically signi cant? Consider again the 30 days series for VOLh . Evaluating (33) at the estimated parameter values, we estimate the t e ect of the volatility risk premium as measured by the implied dollar delta-hedged gains (at three representative volatility levels): 1 1 2 1 6 1 2 0 1 2 1 1 1. On August 19, 1992, the volatility level was 8.05% with at-the-money call price and index level of $5.44 and 418.67, respectively. The volatility risk premium is -3.63% of the call option value; 2. Now consider July 19, 1989, where the volatility level was 12.04% with at-the-money call price and index level of $6.19 and 334.92, respectively. The volatility risk premium is -11.18% of the call option value; 3. Finally, on November 20, 1991, the volatility level was 15.86% with at-the-money call price and index level of $6.94 and 378.80, respectively. In this case, the volatility risk premium is -19.60% of the value of the call. Overall, the magnitudes of the volatility risk premium embedded in at-the-money delta-hedged gains are plausible and economically large. The impact of the volatility risk premium is more 22 prominent during times of greater stock market uncertainty. As emphasized in the previous section, this e ect may be related to demand for options as hedging instruments. 5.1 Robustness of Findings Several diagnostic tests are performed to examine the stability of . First, we re-estimated the regression using the variance as an explanatory variable, with no material change in the results. This last conclusion is not surprising as the standard deviation and variance are highly correlated. In fact, a model with both variables included performs worse than a model with either of these variables. This suggests that not much can be gained by modeling t;t  =St as a polynomial in volatility. Second, to evaluate whether the results are sensitive to a trending stock market, we reestimated the model using dollar delta-hedged gains, t;t  . Again, the results were invariant to this change in speci cation. Third, we explored the possibility that volatility may be nonstationary. To investigate the impact of non-stationarity on the parameter estimates, we performed an OLS estimation in rst di erences rather than in levels. This extended speci cation again points to a negative (these results are available upon request). The principal nding that the market volatility risk premium is negative is robust under alternative speci cations. A natural question that arises is: How sensitive are the results to the mis-measurement of the hedge ratio? Extant theoretical work suggests that the Black-Scholes hedge ratio can depart from the stochastic volatility counterpart when volatility and stock returns are correlated. Guided by this presumption, we now examine (i) whether a negative correlation biases the estimate of t;t  , and, if so, (ii) whether our conclusions about the negative volatility risk premium are robust. For each maturity, we assemble a time-series of at-the-money calls where the return, Rt;t  , is positive. For this sample, it is likely that underhedging (overhedging) results in higher (lower) delta-hedged gains. We estimate the regression: GAINSt = + VOLh + GAINSt + Rt;t  + t , with the t additional variable added to capture the e ect of systematic mis-hedging. In a trending market, we expect > 0, if the call is consistently underhedged, and < 0, if it is overhedged. Although not reported in a table, two ndings are worth documenting. First, even when we explicitly account 1 + + 1 + + 3 0 1 3 2 1 3 + 3 The logic behind this exercise can be explained as follows. Suppose that the di erence between the true hedge ratio and the Black-Scholes hedge ratio is t (t ; y), where  > 0 if Black-Scholes underhedges and negative otherwise. t+ the de nition of delta-hedged gains, it immediately follows that the bias in its estimate is equal to: R From R R t;t+ = t u dSu tt+ ru Su du, which has an expected value of tt+ ( r)Su du, where  is the drift of the price process and  represents the expectation of  (assuming  is independent of the entire path of Su ). Thus, the expected delta-hedged gains is of the order of the market risk premium. If  r > 0, and Black-Scholes underhedges the call, then the estimated delta-hedged gains is biased upwards. 3 23 for the impact of under or over-hedging, the coecient , is signi cantly negative. Second, the coecient, , is positive, and hence t;t  is upwardly biased. However, in none of the regressions is statistically signi cant. That is signi cantly negative appears robust to errors in hedging arising from a correlation between the stock return process and the volatility process. One interpretation is that the hedge ratio takes into consideration time-varying GARCH volatility and is therefore less mis-speci ed. That a mis-speci ed hedge ratio cannot account for the large negative delta-hedged gains that are observed for at-the-money options is also the conclusion of our simulation results below. One nal cause of concern is that the theoretical distribution of delta-hedged gains may vary across the sample set in a complex manner. Therefore, standard procedures adopted in estimating (33) may not fully account for changes in the covariance matrix of t;t  =St . To explore this, we repeated our estimation using generalized method of moments (Hansen (1982)). The instrumental variables are a constant and three lags of volatility. For options of maturity 30 days, the estimated is -0.045 with a t-statistic of -5.08 (using Newey-West with 12 lags). The minimized value of the GMM criterion function, which is distributed  (2), has a value of 2.77 and a p-value of 0.24. The results are similar for options of 44 and 58 days. Thus, we do not reject the empirical speci cation in equation (33). The volatility risk premium coecient, , is signi cantly negative in line with our earlier ndings. 1 3 + 3 1 + 1 2 1 5.2 Simulation Evidence Since the empirical tests reject the null hypothesis that volatility risk is unpriced, we pose two additional questions using simulated data: (i) How severe is the small sample bias?, and (ii) What is the impact of using Black-Scholes hedge ratio as the approximation for the true hedge ratio? For this arti cial economy exercise, our null hypothesis is that volatility is stochastic, but not priced. Therefore, we set [t] = 0, so that the dynamics of t requires no measure change conversions. We simulate the paths of f(St; t) : t = 1;    ; T g, according to (50)-(51). To be consistent with our empirical work, the simulated sample path is taken to be 8 years (2880 days). At the beginning of the month, an at-the-money call option is bought and delta-hedged discretely over its lifetime. Proceeding to the next month, we repeat this delta-hedging procedure. The option price is given by the stochastic volatility model of Heston (1993). For comparison, the delta-hedged gains are computed using the hedge ratio from the true stochastic volatility option model as well as using the Black-Scholes model. Across each simulation run, we generate 96 observations on delta-hedged gains and the prior 30 days volatility. Using the simulated sample, we estimate (33). In Table 8, we report the sample distribution of estimated coecients over 1000 24 simulations for two option maturities, 30 days and 44 days (the mean, and the mean absolute deviation in curly brackets). The rst point to note is that with unpriced volatility risk, the mean delta-hedged gains are virtually zero. Under the stochastic volatility model, the magnitude of =S is of several orders lower than those depicted in Table 1. Second, the use of Black-Scholes delta imparts a negligible bias. For example, for 30 days options, the mean =S is -0.0018% with the stochastic volatility hedge ratio versus 0.0022% with Black-Scholes hedge ratio. In conclusion, the simulations show that the use of Black-Scholes hedge ratio does not perversely bias the magnitude of delta-hedged gains. Now shift attention to the sample rejection level of the estimated coecients from the simulated data. First, given the theoretical p-value benchmark of 5%, the null hypothesis = 0 should be rejected only occasionally. Again consider stochastic volatility model with option maturity of 30 days. Inspection of Table 8 shows that the frequency of t( ) < 2 is 3.94%. Moreover, the frequency of t( ) > 2 is 1.21%. Therefore, when combined, there is only a small over-rejection of the null hypothesis. If the hedge ratio is replaced with the BS delta, the simulated rejection frequency is again close to the theoretical 5%. Because the 44-day options allow for some overlap in the data, we expect to see autocorrelation and, therefore, worse small sample properties. The simulations con rm that the frequency of the rejection of the hypothesis of = 0 is slightly higher at 7.1%. However, the overlap does not a ect the estimate of the mean =S (which is 0.0017%); neither does it worsen the t with the BlackScholes hedge ratio. Overall, the simulation evidence suggests that small sample biases are not large, and that the use of the Black-Scholes hedge ratio has negligible e ect on the estimations. Having said this, we can now proceed to examine the jump-fear foundations of negative deltahedged portfolio returns. 1 1 1 1 6 Delta-Hedged Gains and Jump Exposures While the body of evidence presented so far appears consistent with a volatility risk premium, the losses on the delta-hedged portfolios may also be reconciled by the fear of stock market crashes. The underlying motivation is that option prices not only re ect the physical volatility process and the volatility risk premium, but also the potential for unforeseen tail events (Jackwerth and Rubinstein (1996)). Jump fears can therefore dichotomize the risk-neutral index distribution from the physical index distribution, even in the absence of a volatility risk premium. Indeed, empirical evidence indicates that the risk-neutral index distribution is (i) more volatile, (ii) more left-skewed, and (iii) more leptokurtotic, relative to the physical index distribution (Bakshi, Kapadia and Madan (1999), Jackwerth (2000) and Rubinstein (1994)). As our characterization of delta-hedged 25 gains shows in (24), these distributional features can induce underperformance of the delta-hedged option strategies. If, in addition, the jump risk premium surfaces more prominently during volatile markets (Bates (2000), Eraker, Johannes and Polson (2000) and Pan (2000)), then it can account for the accompanying greater delta-hedged losses. To empirically distinguish between the e ects of stochastic volatility and jumps on deltahedged gains, two decisions are made at the outset. One, in the tradition of Bakshi, Kapadia and Madan (1999), Bates (2000) and Jackwerth and Rubinstein (1996), we assume that jump fears can be surrogated through the skewness and kurtosis of the risk-neutral index distribution. In the modeling framework of (23), the mean jump-size governs the risk-neutral skew, and the jump intensity is linked to kurtosis. For instance, the fear of market crashes can impart a left-skew, and shift more probability mass towards low probability events. Two, the risk-neutral skews and kurtosis are recovered using the model-free approach of Bakshi, Kapadia and Madan (1999). They show that the higher-order risk-neutral moments can be spanned and priced using a positioning in out-of-money calls and puts. In what follows, the relative impact of jump fears on delta-hedged gains is gauged from three perspectives. First, we modify the time-series speci cation (33) to include a role for risk-neutral skew and kurtosis, as shown below: GAINSt =  +  VOLh +  GAINSt +  SKEW +  KURT +  ; t t t t 0 1 1 2 3 4 (34) where VOLh is the historical volatility, SKEW is the risk-neutral index skewness and KURT is t t t the risk-neutral index kurtosis. For convenience, the exact expressions for skew and kurtosis are displayed in (45) and (46) of the Appendix. Speci cally, the risk-neutral skewness and kurtosis re ect the price of the cubic contract and the kurtic contracts, respectively. As before, we include a lagged value of delta-hedged gains to correct for serially correlated residuals. The estimation is by OLS, and the t-statistics are computed using the Newey-West procedure with 12 lags. The main idea behind the empirical speci cation (34) is to investigate whether physical volatility loses its signi cance in the presence of such jump fear proxies as risk-neutral skews and kurtosis. We also employed the slope of the volatility smile and the Bates skewness premium measure as alternative proxies for jump fear, and obtained similar conclusions (details are available from the authors). To maintain the scope of the investigation, these extended measures are excluded from the main body of the paper. Before we discuss the estimation results presented in Table 9, it must be stressed that there is substantial evidence of jump fear in the post-crash risk-neutral distributions. Over the entire sample period, the average risk-neutral skewness is -1.38 and the risk-neutral kurtosis is 7.86, for 26 30 day distributions. These numbers are roughly comparable to those reported in Jackwerth and Rubinstein (1996) for longer-term options, and in Bakshi, Kapadia and Madan (1999) for the S&P 100 index options. The most important point that emerges from Table 9 is that historical volatility continues to signi cantly a ect variations in delta-hedged gains. The coecient on volatility ranges between -0.111 to -0.041, and is statistically signi cant in all the nine estimations. The evidence on the role of skew and kurtosis is less conclusive. Although skew enters the regression with the correct sign, it is only marginally signi cant. The positive estimate of  indicates that a more negatively skewed risk-neutral distribution makes delta-hedged gains more negative from one month to the next. In addition, the sign of kurtosis is contrary to what one might expect. While not reported, skewness (kurtosis) is not individually signi cant when volatility and kurtosis (skewness) are omitted as explanatory variables in (34). Finally, comparing the empirical t between Table 7 and Table 9, the inclusion of skew and kurtosis only modestly improves the adjusted-R (by about 3%). In summary, this exercise suggests that volatility may be of rst-order importance in explaining negative delta-hedged gains. In the second exercise, we study the behavior of delta-hedged option portfolios for a hold-out sample when jump-fears are much less pronounced. For this purpose, we selected the six-month interval from January 1987 through June 1987 (option data provided by Bent Christensen). What is unique about this pre-crash period is that risk-neutral index distributions are essentially lognormal. Especially suited for the task at hand, the jump-fears are virtually lacking during this pre-crash sample (Jackwerth and Rubinstein (1996)). Table 10 reports the mean delta-hedged gains for out-of-money calls and puts. The average delta-hedged gains for near-the-money 14-30 days calls (puts), is $-0.65 (-$0.82). In fact, the delta-hedged gains are strongly negative in all the 16 moneyness and maturity categories. Furthermore, the majority of the options have  < 0, as seen by the large 1< statistics. The delta-hedged gains are negative in both the pre-crash and the post-crash periods. While not displayed, the average implied volatility for at-the-money options is higher than the historically realized volatility suggesting that the well-known bias between the implied and the realized volatility pre-dates crash-fears and option skews. In the nal evaluation exercise, we compute the average delta-hedged gains for the largest downward and upward market movements. Intuitively, if fears of negative jumps are the predominant driving factor in determining negative delta-hedged gains, then there should be a strong asymmetric e ect on the delta-hedged gains (see (24)), with large positive index returns not necessarily resulting in large negative delta-hedged gains. In contrast, it may be argued that a negative volatility risk premium would cause large negative delta-hedged gains, irrespective of the sign of the market return. To brie y examine this reasoning, consider closest to at-the-money short-term calls bought on the day subsequent to a tail event. Respectively for the largest 10 3 2 0 27 (20) tail events, the average scaled delta-hedged gains,t;t  =St, are -0.52% (-0.43%) on positive returns dates, compared with -0.86% (-0.51%) on negative return dates. The evidence indicates that delta-hedged gains become more negative for both extreme negative and positive returns. In e ect, this evidence from the extremes is consistent with the regression results. One overall conclusion that can be drawn is that priced volatility risk is a more plausible characterization for negative delta-hedged gains. While it is possible that if some extremely low probability event is included, the resulting large positive gain may wipe out all cumulative losses. However, this low probability event (of the required magnitude) has yet not occurred in our sample. Our key nding that the market volatility risk premium is, on average, negative is mutually consistent with other evidence reported in Benzoni (1999), Jones (2000), Pan (2000) and Poteshman (1998). + 7 Final Remarks Is volatility risk premium negative in equity index option markets? We argue that the central implication of a non-zero volatility risk premium is that the gains on a delta-neutral strategy that buys calls and hedges with the underlying stock are non-zero, and determined jointly by the volatility risk premium and the option vega. Speci cally, we establish that the volatility risk premium and the mean discrete delta-hedged gains share the same sign. It is shown that this implication can be tested by relatively robust econometric speci cations in either the cross-section of option strikes, or in the time-series. These tests do not require the identi cation of the pricing kernel, nor the correct speci cation of the volatility process. Using S&P 500 index options, our empirical results indicate that the delta-hedged gains are non-zero, and consistent with a non-zero volatility risk premium. The main ndings of our investigation are summarized below: 1. The delta-hedged call portfolios statistically underperform zero (across most moneyness and maturity classi cations). The losses are generally most pronounced for at-the-money options. 2. The underperformance is economically signi cant and robust. When out-of-money put options are delta-hedged, a similar pattern is documented. The documented underperformance of delta-hedged option portfolios is consistent with a negative volatility risk premium. 3. Controlling for volatility, the cross-sectional regression speci cations provide support for the prediction that the absolute value of delta-hedged gains are maximized for at-the-money options, and decrease for out-of-the-money and in-the-money options. 28 4. During periods of higher volatility, the underperformance of the delta-hedged portfolios worsens. As suggested by the hypothesis of a negative volatility risk premium, time-variation in delta-hedged gains of at-the-money options are negatively correlated with historical volatility. This nding is robust across subsamples, and to mis-measurement of the hedge ratio. 5. Finally, volatility signi cantly a ect delta-hedged gains even after accounting for jump-fears. Jump risk cannot fully explain the losses on the delta-hedged option portfolios. In economic terms, a negative volatility risk premium suggests an equilibrium where equity index options act as a hedge to the market portfolio, and is consistent with prevailing evidence that equity prices react negatively to positive volatility shocks. Thus, investors would be willing to pay a premium to hold options in their portfolio, and this would make the option price higher than its price when volatility is not priced. The empirical results of this paper indeed strengthen the view that equity index options hedge downside risks. There are two natural extensions to this paper. First, given that volatilities of individual stocks and the market index co-move highly, one could examine whether the volatility risk premium is negative in individual equity options. The cross-sectional restrictions on delta-hedged gains and the volatility risk premium can be tested in the cross-section of individual equity options. Second, volatility risk is of importance in almost every market. The analysis conducted here can be directly applied to include other markets such as foreign exchange and commodities. Much more remains to be learned about how volatility risk is priced in nancial markets. 29 Appendix A: Proof of Results Proof of Proposition 1: We need to show that Et(t;t  ) = O(1=N ), when t[t]  0. First, without loss of generality, assume r = 0, t  @Ct=@St and vt  t @Ct=@t. Second, let the + period corresponding to the time to expiration, t = 0 to t = t +  , be divided equally into N periods, corresponding to dates, tn , n = 0; 1;    ; N , where t = 0, tN = t +  , and tn tn = h. Consider the delta-hedged gains over one period, from tn to tn . If the volatility risk premium is zero, then from equation (13), 0 1 +1 Zn Cn = Cn + u dSu + +1 n +1 Zn +1 n vu dWu ; (35) 2 @ @ where, for brevity, we intend n to mean tn . De ne the operators, L[:] = @t [:] + t St @St [:] + @ @ @ @ @ t @t [:] + t St @St [:] + t @t [:] + tStt @S@t @t [:], = tSt @St [:] and = t @t [:]. Appealing to an Ito-Taylor expansion, 1 2 2 2 2 2 Cn +1 1 = Cn + + 2 2 2 2 Zn +1 n 2 1 Zn  Zu 2 Zu  Zu L[t]dt + [t]dSt + [t]dt dSu n n Zu Zu Zu vn + L[vt] dt + [vt ] dSt + [vt]dt du : +1 n n + 1 2 0 n n n 1 (36) 2 With an additional Ito-Taylor expansion to include all terms up to O(h), we can re-write this as, Cn Zn +1 Zn Zu +1 Zn +1 = Cn + n dSu + n Sn [n] dWt dWu + vn dWu n n n n Zn Zu Zn Zu + n Sn n [n ] dWt dWu + n [vn] dWt dWu +1 2 2 1 +1 2 + n Sn n [vn ] 2 Zn Zn n u +1 n R n 2 1 2 1 1 +1 n 1 n dWt dWu + A ; 1 2 2 (37) 0 R 2 2 R R n u n u where A consists of terms such as n n g (St; t; t) dt du, and n n h(St ; t; t) dWtj ds; j = S;  . Under generally accepted regularity conditions (Lemma 2.2 of Milstein (1995)), E (A ) = O(h ), and E (A ) = O(h ). It follows from Theorem 1.1 in Milstein (1995) that the order of accuracy of the above discretization over the N steps in the interval, t = 0 to t = t +  , is h = =N , so that it is of O(1=N ). Rearranging (37), we can write t;t  as, 0 +1 +1 0 2 2 0 3 + t;t +   N X 1 n N X Cn +1 Cn n (Sn = 1 n =0 [tn Stn [tn ] 2 2 Sn ); +1 =0 1 Zn Zu +1 tn tn dWt dWu + vtn 30 1 1 Zn +1 tn dWt + 2 + tn Stn tn [tn ] 2 + tn Stn tn [vtn ] Zn Zu +1 Z tt Z ts 2 n n+1 n tn tn dWt dWu + tn [vtn ] 2 1 2 1 Zn Zu +1 tn tn dWt dWs 2 2 dWt dWs ] + O(1/N): 1 (38) 2 As the expected value of the Ito integrals is zero, the proposition is proved. 2 Proof of Lemma 1: The proof is by induction. To x ideas, we prove the case where t( ). The extension to t(t ;  ) is straightforward. Consider L[ t ( ) S , @ S  +  S @S  + 1  S @ S  t @S 2t @S  @ 1 ( 1) ( )   S  = @ t =@ +  t( ) t + 2 t t L[ ( ) S = 2 2 (39) 2 (40) 2 by assuming dSt = t St dt + t St dW . This implies that L[L[ t ( )S ] is again proportional to S  . By induction, Ln [g], for any n 2 f1; 2; 3; :::g, are proportional to S . 2 Proof of Proposition 2: The proof relies on evaluating each term in the expansion of equation (18). We have E(t;t  ) = g (St; t)  +  L[g (St; t)] +  L [g (St; t)] +   , where g (St; t) = 1 + 1 2 2 3 2 6 t @Ct=@t. Here, the vega is proportional to St, and so @Ct=@t = t( ; y ) St. Under the maintained assumption that t =  t, g (St; t) =  t( ; y ) St t . Whence, L [g] = 1 (@ t=@ ) St t +  t  St(@St=@St) t +  t St (  t )(@t=@t); =  ' St t; 1 (41) (42) where '  @ =@ +  . From Lemma 1, successive Ln [g ] inherit the same form as (42), P as in  'n St t . Therefore, E(t;t  ) =  't( ) St t , where 't ( )  1  nn 'n . 2 n 1 1+ + =0 1+ ! Proof of Equation 24: Using (23) and applying Ito's lemma, the call option satis es the dynamics: Ct + Z t  @Cu Z t  @Cu Zt  = Ct + dSu + du + bu du;  @Su @u t t t Zt  Z1 J u (C (Su ex ) C (Su )) q [x] dx du; + + + + t (43) 1 where C (Su ex ) implies that the option price is evaluated at Su ex . In (43), q [x] is the physical C C @ density of the jump-size, x, and bu  @Cu + u Su @@Suu + u @@uu +  u u Su @SuCu u . The call @u @ 1 2 2 31 2 2 2 1 2 2 2 2 2 price is a solution to the partial integro-di erential equation, 1  S @ C + 1  @ C +    S @ C + (r    ) S @C + (  [ ]) @C t t @S@ t t JJt 2 t @S 2 t @ @S @ Z1 @C r C +   x ) C (S )) q [x] dx = 0; (C (Su e (44) + @t Jt u 2 2 2 2 2 2 2 2 1 for risk-neutral density q [x]. Combining (43) and (44) and using the de nition of t;t  , we get (24). 2 + Expressions for Risk-Neutral Skew and Kurtosis Used in Section 6: The model-free estimates of risk-neutral return skewness and kurtosis are based on Bakshi, Kapadia and Madan (1999). Speci cally, the risk-neutral skewness, SKEW (t;  ), is given by n o Et (Rt;t  Et[Rt;t  ]) SKEW (t;  )  n o= Et (Rt;t  Et[Rt;t  ]) r r = e W (t;  ) r 3(t;  )e V (t;  ) + 2(t;  ) [e V (t;  ) (t;  ) ] = 3 + + 2 + 32 + 3 (45) 232 and the risk-neutral kurtosis, denoted KURT (t;  ), is r r W r KURT(t;  ) = e X (t;  ) 4(t;  )e r (t;  ) + 6e (t;  ) V (t;  ) 3(t;  ) ; (46) [e V (t;  ) (t;  ) ] 2 4 22 where V (t;  ) = K Z 1 2(1 ln h S i) St K t 2 C (t;  ; K ) dK + Z S 2(1 + ln h S i) K t t K 0 2 P (t;  ; K ) dK (47) and the price of the cubic and the quartic contracts are K K Z 1 6 ln h S i 3(ln h S i) W (t;  ) = C (t;  ; K ) dK S hS K i hS i Z 2 t 2 t St 0 X (t;  ) = t 6 ln Kt + 3(ln Kt ) P (t;  ; K ) dK; Z 1 12(ln 2 hK i K 4(ln St ) C (t;  ; K ) dK K St Z St 12(ln h St i) + 4(ln h St i) K K + P (t;  ; K ) dK: K St ) hK i 2 2 (48) 3 2 2 3 2 0 32 (49) Each security price can be formulated through a portfolio of options indexed by their strikes. In addition, (t;  )  er 1 er V (t;  ) er W (t;  ) er X (t;  ). 2 2 6 24 Appendix B: Simulation Experiment To implement the simulation experiment, the stock return and volatility process are discretized as (h is some small interval): p = St +  St h + t St t h;   p = t +   t h +  t t h: h St t + h 1 2 + 2 2 (50) (51) 2  0, Simulate a time series of two independent, standard normal processes: t ; t 0 where t = 0 1 01 1 1 A t A = H @ t A. The 1; 2;    ; T . De ne H = @ , and generate a new vector: @ 1 t t transformed vector is a bivariate normal process with zero mean and a variance-covariance matrix of H, where t and t have a correlation of . Construct the time series of St and t , t = 1; 2;    ; T , based on equations (50)-(51) and using the simulated t and  . The initial stock price is set to be 100, and the initial value of volatility is chosen to be 10%. We initially assume that  = 2:0,  = 0:01,  = 0:1, and  =-0.50. For the calculations involving delta-hedged gains, the risk-neutralized variance process is: 1 2 1 1 1 2 2 1 2 2 1 2 2   p = t +   t h +  t t h h t 2 + 2 2 (52) 2 where t[t ] = t , so that  and  are related to the physical parameters of the variance process by the relations  =  + , and  = =( + ). Suppose we set  = 0:20, then each path corresponds to 73 observations of fSt; tg. The deltahedged gains, t;t  , over the period,  , is calculated using equation (6), for h call of strike 100 and a n R 1 Re e iu K f u i duo initial maturity of 0.2 years. The call price is computed as: St +  u n R 1 Re h e iu K f u i duo, where the characteristic functions, f iand f , are disr Ke + iu played in Heston (1993, equation (17)). For simplicity, the interest rate and the dividend yield are assumed to be zero. 2 2 + 1 2 1 2 ln 1  2  ln 1 1 0 33  1  0 2 References [1] Bakshi, G., C. Cao, Z. Chen, 1997, \Empirical performance of alternative option pricing models," Journal of Finance 52, 2003-2049. [2] Bakshi, G., C. Cao, and Z. Chen, 2000, \How often does the call move in the opposite direction to the underlying?," Review of Financial Studies 13, 549-584. [3] Bakshi, G., N. Kapadia, D. Madan, 1999, \Stock return characteristics, skew laws, and the di erential pricing of individual equity options," Conditionally accepted at the Review of Financial Studies. [4] Bates, D., 2000, \Post-87 crash fears in S&P 500 futures options," Journal of Econometrics 94, 181-238. [5] Benzoni, L., 1999, \Pricing options under stochastic volatility: An econometric analysis," mimeo, University of Minnesota. [6] Bertsimas, D., L. Kogan, A. Lo, 2000, \When is time continuous," Journal of Financial Economics (55) 2, 173-204. [7] Buraschi, A., and J. Jackwerth, 2001, \The price of a smile: Hedging and spanning in option markets," Review of Financial Studies 14, No. 2, 495-527. [8] Christensen, B. and N. Prabhala, 1998, \The relation between implied and realized volatility," Journal of Financial Economics 50, 125-150. [9] Coval, J., and T. Shumway, 2000, \Expected option returns," Journal of Finance (forthcoming). [10] Dumas, B., J. Fleming, R. Whaley, 1998, \Implied volatility functions: Empirical tests," Journal of Finance, 53 (6), pp 2059-2106. [11] Eraker, B., M. Johannes, N. Polson, 2000, \The impact of jumps in volatility and returns," mimeo, University of Chicago and Columbia University. [12] Figlewski, S., 1989, \Options arbitrage in imperfect markets," Journal of Finance, 44 (5), 1289-1311. [13] French, K., W. Schwert, R. Stambaugh, 1987, \Expected stock returns and volatility," Journal of Financial Economics, 19, 3-29. 34 [14] Greene, W., 1997, Econometric Analysis, Third Edition, MacMillan Publishing Company: New York. [15] Hansen, Lars, 1982, \Large sample properties of generalized method of moments estimators," Econometrica 50, 1029-1084. [16] Heston, S., 1993, \A closed form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6(2), 327-343. [17] Hull, J., and A. White, 1987, \The pricing of options with stochastic volatilities," Journal of Finance, 42, 281-300. [18] Jackwerth, J., 2000, \Recovering risk aversion from options prices and realized returns," Review of Financial Studies 13 (2), 433-451. [19] Jackwerth, J., and M. Rubinstein, 1996, \Recovering probability distributions from option prices," Journal of Finance 51, 1611-1631. [20] Jones, C., 2000, \The dynamics of stochastic volatility: Evidence from options and the underlying market," mimeo, University of Rochester. [21] Merton, R., 1973, \Theory of rational option pricing," Bell Journal of Economics and Management Science 4, 41-83. [22] Merton, R., 1976, \Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics 3, 125-144. [23] Milstein, G., 1995, Numerical Integration of Stochastic Di erential Equations, Kluwer Academic Publishers, Boston. [24] Nelson, D., 1991, \Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59 (2), 347-370. [25] Pan, J., 2000, \The jump-risk premia implicit in options: Evidence from an integrated timeseries study," Journal of Financial Economics (forthcoming). [26] Poteshman, A., 1998, \Estimating a general stochastic variance model from option prices," mimeo, University of Illinois. [27] Rubinstein, M., 1994, \Implied binomial trees," Journal of Finance, 49, 771-818. [28] Stein, J., 1989, \Overreactions in the options market," Journal of Finance, 44 (4), 1011-1023. 35 t;t+ Table 1: Delta-Hedged Gains for S&P 500 Index Calls = max(St+ K; 0) Ct N1 X n=0 tn (Stn+1 Stn ) N1 X n=0 ( rn Ct  tn Stn ) N ; We compute the gain on a portfolio of a long position in a call option, hedged by a short position in the underlying stock, such that the net investment earns the risk-free interest rate. The discretely rebalanced delta-hedged gains, t;t+ , are computed as 1 N 2.5% to 5% 0% to 2.5% 4811 5530 5752 14-30 31-60 -0.11 (0.01) -0.09 (0.00) -0.09 (0.00) -0.10 (0.01) All -2.93 (0.51) -0.31 (0.19) 0.93 (0.10) 1.03 (0.07) -97.37 (3.81) -70.71 (3.81) -34.43 (2.57) -6.98 (1.54) -4.44 (0.31) -1.34 (0.15) 0.62 (0.09) 1.51 (0.07) -56.34 (3.50) -38.13 (2.08) -11.88 (1.54) -7.95 (0.72) 14-30 31-60 -3.88 (0.27) -0.96 (0.12) 0.74 (0.07) 1.33 (0.05) -68.30 (2.80) -48.05 (1.87) -20.19 (1.36) -7.59 (0.73) All Panel A: Full Sample Period -0.13 (0.01) -0.10 (0.01) -0.11 (0.01) -0.13 (0.01) -0.11 (0.01) -0.05 (0.01) 0.04 (0.00) 0.11 (0.00) (in %) All -0.06 (0.01) -0.06 (0.00) -0.06 (0.01) -0.06 (0.01) -0.13 (0.01) -0.07 (0.01) 0.03 (0.01) 0.12 (0.01) =C 14-30 31-60 -0.31 (0.02) -0.28 (0.01) -0.32 (0.02) -0.42 (0.02) -0.07 (0.01) -0.01 (0.01) 0.05 (0.01) 0.08 (0.01) (in %) -0.37 (0.03) -0.32 (0.02) -0.38 (0.02) -0.52 (0.03) -0.45 (0.02) -0.19 (0.02) 0.20 (0.02) 0.47 (0.02) =S -0.18 (0.02) -0.19 (0.01) -0.21 (0.02) -0.26 (0.04) -0.55 (0.03) -0.26 (0.02) 0.18 (0.02) 0.54 (0.02) (in $) -0.29 (0.04) -0.05 (0.03) 0.24 (0.02) 0.36 (0.02)  26 41 60 68 68 72 78 87 1<0 % where the interest rate, rn, and the option delta, tn , are updated on a daily basis. The option delta is computed as the Black-Scholes hedge ratio evaluated at the GARCH volatility. The rebalancing frequency,  =N , is set to one day. We report (i) the dollar delta-hedged gains (t;t+ ), (ii) the delta-hedged gains normalized by the index level (t;t+ =St ) and (iii) the delta-hedged gains normalized by the option price (t;t+ =Ct). All delta-hedged gains are averaged over their respective moneyness and maturity category. The moneyness of the option is de ned as y Ser z =K . The standard error, shown in parenthesis, is computed as the sample standard deviation divided by the square root of the number of observations. 1<0 is the proportion of delta-hedged gains with  < 0, and N is the number of options. Results are shown separately for options with maturity 14-30 days and 31-60 days; \All" combines the delta-hedged gains from both maturities. There are 36,237 call option observations on the S&P 500 index. Subsample results are displayed in Panel B: SET 1 corresponds to 1988:01-1991:12, and SET 2 corresponds to 1992:01-1995:12 (standard errors are small and omitted in Panel B). y Moneyness -10% to -7.5% 1284 -7.5% to -5% 3619 -5% to -2.5% 5684 5% to 7.5% 3647 -2.5% to -0% 5903 7.5% to 10% 36 Panel B of Table 1: Delta-Hedged Gains Across the 88:01-91:12 and 92:01-95:12 Subsamples -2.39 -0.93 -1.48 -9.84 -12.17 -11.31 -0.15 -0.13 -19.96 -7.39 -12.19 -0.08 -0.06 -44.94 -14.84 -25.66 63 71 63 70 70 73 1<0 % -0.96 -5.71 59 60 All -1.46 -6.31 -0.51 -1.24 50 35 (in %) -0.09 -0.08 -0.15 -0.12 -0.10 -4.69 -0.83 -1.65 0.39 0.96 38 16 =C -0.09 -0.03 -0.09 -0.07 -0.16 -0.13 0.02 -0.52 0.21 0.88 0.95 1.60 All 14-30 31-60 86 92 -10% to -7.5% SET 1 -0.45 -0.39 -0.34 -0.28 -0.06 -0.07 -0.06 -0.04 -0.07 -0.05 0.68 1.10 1.12 1.76 (in %) -0.17 -0.13 -97.34 -46.83 -65.01 -0.03 -0.03 -97.82 -77.86 -76.73 78 79 -0.29 -0.16 -0.26 -0.23 -0.67 -0.54 -0.03 -0.09 0.02 0.06 0.71 1.30 =S -0.06 -0.02 -0.16 -0.12 -68.44 -18.98 -38.60 -0.05 -0.04 -75.88 -54.62 -58.85 SET 2 3375 -0.18 -0.31 -0.28 -0.20 -0.72 -0.61 -0.01 -0.02 0.00 0.05 0.07 0.13 All 14-30 31-60 -0.47 -0.36 -0.14 -0.13 -0.07 -0.03 -2.5% to -0% SET 1 2240 SET 2 3668 -0.08 -0.42 -0.19 -0.12 -0.31 -0.22 0.04 0.07 0.09 0.14 (in $) N -0.18 -0.10 -0.45 -0.36 -0.20 -0.18 SET 1 2211 SET 2 3540 -0.02 -0.08 0.06 0.29 0.06 0.10  Moneyness Sample 989 295 -0.22 -0.11 0% to 2.5% SET 1 2119 SET 2 3410 0.02 0.27 0.26 0.62 14-30 31-60 SET 2 1930 1 SET 2 1688 y -7.5% to -5% SET 1 2.5% to 5% 0.14 0.31 0.30 0.69 2308 5% to 7.5% SET 1 1882 SET 2 2928 0.19 0.50 -5% to -2.5% SET 1 7.5% to 10% SET 1 1523 SET 2 2123 37 Table 2: Delta-Hedged Gains for Out-of-Money Puts This table reports the delta-hedged gains for out-of-money puts on the S&P 500 index. Put options correspond to moneyness, y, greater than 1. We compute the gain on a portfolio of a long position in a put option, hedged by a short position in the underlying stock, such that the net investment earns the risk-free interest rate. As before, the discretely rebalanced delta-hedged gains, t;t+ , are computed as: t;t+ = max(K St+ ; 0) Pt N1 X n=0 ^ t (St +1 n n Stn ) N1 X n=0 rn Pt ^ t n Stn  N ^ where t is the Black-Scholes put option delta evaluated at GARCH volatility; and rn is the nominal interest rate. The rebalancing frequency,  =N , is set to one day. Reported are (i) dollar delta-hedged gains (t;t+ ), and (ii) delta-hedged gains normalized by the put price (t;t+ =Pt). All delta-hedged gains are averaged over their respective moneyness and maturity categories. 1<0 is the proportion of delta-hedged gains with  < 0. N represents the number of put options. There are 20,216 out-of-money puts. SET 1 refers to the 1988:01-1991:12 subsample, and SET 2 refers to the 1992:01-1995:12 subsample. Standard errors are small, and omitted. n Moneyness Sample N y 1  (in $) =P (in %) 14-30 30-60 All 14-30 31-60 1<0 All % 0% to 2.5% FULL 5342 SET 1 2116 SET 2 3226 -0.55 -0.16 -0.79 -0.77 -0.69 -16.95 -13.01 -14.47 0.10 0.00 0.06 4.82 3.03 -1.27 -1.09 -27.10 -23.27 -24.68 74 62 81 2.5% to 5% FULL 5695 SET 1 2011 SET 2 3684 -0.80 -0.66 -0.88 -1.37 -1.16 -55.40 -41.72 -46.82 -0.80 -0.75 -23.83 -13.95 -17.63 -1.68 -1.38 -72.61 -56.89 -62.75 89 81 94 5% to 7.5% FULL 5364 SET 1 2042 SET 2 3322 -0.60 -0.66 -0.56 -1.30 -1.03 -73.57 -66.82 -69.41 -1.18 -0.97 -45.06 -39.23 -41.56 -1.38 -1.07 -92.25 -83.11 -86.53 94 88 98 FULL 3815 7.5% to 10% SET 1 1568 SET 2 2247 -0.43 -0.57 -0.33 -1.06 -0.82 -91.59 -82.66 -86.14 -1.12 -0.90 -81.72 -60.90 -69.17 -1.02 -0.76 -98.71 -97.54 -97.99 97 94 99 38 Table 3: Delta-Hedged Gains for Near Money Calls, by Volatility Regimes Each date is classi ed into one of seven di erent volatility regimes based on the annualized estimate of historical volatility (i.e., VOLh). Delta-hedged gains are computed as described in the note to Table 1. We report (i) the dollar delta-hedged gains, (ii) delta-hedged gains normalized by the index price, and (iii) the delta-hedged gains normalized by the call price, averaged over each volatility regime. Both the mean and the median are displayed. The results are reported for two near money categories: -2.5% to 0% and 0 to 2.5%. The standard errors are small and therefore omitted. T represents the number of days in each volatility classi cation. y VOLh(%) T (%) <8 428 Mean  2 [-2.5%,0] =S ($) (%) 0.11 0.04 Median -0.21 -0.04 =C y  2[0,2.5%] =S =C (%) ($) (%) 5.64 -0.25 -0.04 -7.81 -0.38 -0.08 (%) -1.69 -3.87 399 Mean -0.54 -0.10 -13.52 -0.37 -0.06 Median -0.72 -0.16 -19.63 -0.57 -0.13 -3.03 -5.92 10 to 12 450 Mean -3.02 -4.92 12 to 14 295 Mean -1.78 -4.27 14 to 16 147 Mean -1.01 -0.29 -20.90 -1.14 -0.32 -11.97 8 to 10 -0.35 -0.07 -6.68 -0.38 -0.08 Median -0.54 -0.14 -13.18 -0.51 -0.13 -0.44 -0.11 -8.52 -0.22 -0.06 Median -0.50 -0.14 -11.28 -0.41 -0.11 Median -1.12 -0.31 -21.84 -1.13 -0.32 -12.33 16 to 18 99 > 18 Mean -1.32 -0.41 -24.93 -1.31 -0.40 -12.87 Median -1.38 -0.42 -27.36 -1.28 -0.38 -12.34 125 Mean -1.57 -0.51 -22.90 -1.51 -0.47 -13.41 Median -1.42 -0.48 -25.53 -1.62 -0.51 -14.58 39 Table 4: Delta-Hedged Gains and Option Vega: Cross-Sectional Tests i i = 0 + 1 VEGAt + t ; = ut + !ti ; We estimate a FGLS random e ect panel regression of delta-hedged gains on the VEGA: i GAINSt i t 0.029 (0.53) 0.018 (0.26) 0.046 (1.22) 0.057 (1.65) 0 -0.53 [-10.83] -0.19 [-3.94] -0.06 [-1.02] -0.18 [-4.15] -0.10 [-2.95] 1 -0.054 [-0.75] -0.142 [-3.32] -0.068 [-1.81] 78 65 -0.450 6.00 55 [-5.67] [10.83] -0.179 [-3.25] 3.88 [1.79] -0.704 6.78 [-12.66] [8.21] 0 0.421 [1.96] 0.283 [1.95] 0.025 [0.13] 2.44 125 0.062 [4.16] [0.81] 1.02 219 0.070 [1.27] [0.75] 2.73 247 0.195 [4.60] [2.38] 2.68 300 0.067 [7.35] [1.72] N i i 2 where GAINSt t;t+ =St (for strike Ki , i;    ; I ). We use two proxies for the option vega. First, VEGA is de ned as exp( d1), where d1 p 1 1 p   log(y) + 2   , where yi is option moneyness corresponding to strike Ki . Second, VEGA is the absolute level of moneyness, jy 1j. The data consists of monthly observations of calls over the period January 1988 to December 1996. We focus on options with maturity of 30 and 44 days. The sample is chosen such that the volatility is within a pre-de ned interval, VOLh . N is the number of observations. Since the estimation method is not least squares, the coecient of determination is omitted. Numbers in square brackets show the z-statistics (Greene (1997)). 241 N 44 Days Options 8 0.129 (1.61) -0.67 [-8.69] -0.586 [-2.27] -0.86 [-4.15] -0.66 [-5.90] -0.29 [-1.38] -0.41 [-8.35] -0.14 [-2.05] -0.36 [-10.57] -0.03 [-0.86] -0.522 [-3.59] -0.443 [-3.45] -0.292 [-2.04] -0.387 [-5.27] -0.079 [-0.87] 5.64 [6.01] 2.70 [1.34] 4.46 [8.52] 1.65 [1.98] -0.182 4.84 [-2.21] [11.17] -0.001 [-0.23] 5.22 [4.29] 30 Days Options Vega is exp( < 212 0.029 (0.47) -0.47 [-1.69] Vega is exp( d2 Vega is jy 1j 1 =2) 1 0 1 10-12 283 8-10 83 12-14 177 VOLh 0 14-16 49 -0.052 (-0.17) 2) Vega is jy 1j 1 16-18 46 2 d1= 18 1.69 [3.49] > 40 Table 5: Robustness Results, Delta-Hedged Gains and Option Vega (30 days Calls) In Panel A, we report the results from the following FGLS random e ect panel regression: GAINSit = i 0 + 1 VOLh  VEGAit + it, and it = ut + !ti . The dependent variable is GAINSit t;t+ =St t 1 2 ) or jy 1j, where d1 = p log(y )+ 1  p . As before, (i;    ; I ). VEGA is de ned as either exp( d1  2 y is the option moneyness. The data consists of monthly observations of calls over the period January 1988 to December 1995. N is the number of observations. Numbers in square brackets show z-statistics (Greene (1997)). In Panel B, we repeat the analysis of Table 4 for the 1992:01-1995:12 subsample. All results are for 30 day calls. Panel A: The Speci cation is, GAINSit = 0 + 1 VOLh  VEGAit + it t VEGA is exp( d2) VEGA is jy 1j 1 h (%) VOL N 0 1 0 1 8 < 8-10 10-12 12-14 14-16 16-18 > 18 158 0.087 [2.08] 212 0.040 [1.04] 283 0.26 [0.37] 177 0.026 [0.47] 83 0.129 [1.60] 49 0.024 [0.34] 44 -0.061 [-0.20] -0.22 [-3.93] -0.18 [-3.77] -0.07 [-1.27] -0.14 [-3.82] -0.36 [-10.97] -0.39 [-8.60] -0.21 [-1.63] -0.073 4.30 [-1.58] [5.86] -0.136 2.81 [-3.30] [4.35] -0.060 1.09 [-0.85] [1.48] -0.177 1.86 [-3.22] [4.07] -0.450 4.02 [-5.70] [10.98] -0.701 3.89 [-13.24] [8.18] -0.589 1.82 [-2.33] [1.85] Panel B: Subsample Results for 1992:01-1995:12 VEGA is exp( d2) VEGA is jy 1j 1 h (%) VOL N 0 1 0 1 8 < 8-10 10-12 12-14 158 0.084 [1.97] 184 0.049 [1.43] 148 -0.013 [-0.28] 177 0.029 [0.53] -0.13 [-3.55] -0.26 [-4.15] -0.27 [-7.02] -0.19 [-3.94] 41 -0.071 [-1.55] -0.210 [-5.78] -0.293 [-6.24] -0.179 [-3.25] 2.82 [5.60] 3.73 [8.85] 3.60 [7.01] 2.44 [4.16] Table 6: Changes in Valuation of Index Options During Large Moves We proxy valuation changes in index calls by the corresponding change in Black-Scholes implied volatility. This is done in two steps. First, on the day prior to a large daily move, we buy a call option and compute the Black-Scholes implied volatility. Second, proceeding to the day after the large move, we re-compute the Black-Scholes implied volatility. We report four set of numbers (i) the price movement (in %) (ii) the prior day implied volatility (denoted as \Prior IMPL."), (iii) the subsequent day implied volatility (denoted as \Subs. IMPL."), and (iv) the corresponding change in implied volatility as a fraction of the implied volatility of the option bought (i.e., the relative change). The sample period is 1988 through 1995. In each implied volatility calculation, the index level is adjusted by the present discounted value of dividends. Only short-term call options with strikes that are closest to at-the-money are considered. We display the results from the largest 20 percentage price movements. Largest Negative Price Movements Date Largest Positive Price Movements Price Prior Subs. Relative Date Price Prior Subs. Relative Move IMPL. IMPL Change Move IMPL. IMPL Change 880108 891013 880414 911115 900823 900806 880120 901009 900122 900112 930216 910819 890317 900816 940204 900924 881111 880324 900821 910510 -7.00 -6.32 -4.44 -3.73 -3.04 -3.00 -2.72 -2.70 -2.61 -2.48 -2.41 -2.37 -2.27 -2.26 -2.24 -2.14 -2.13 -2.09 -2.03 -1.96 26.69 13.23 20.05 12.31 25.90 20.84 24.14 24.83 17.16 17.00 11.20 13.55 14.22 19.76 9.14 24.37 18.15 21.35 24.21 14.02 28.90 11.87 21.02 15.90 28.96 23.01 25.71 24.15 19.61 18.02 15.41 16.04 15.69 24.11 11.10 23.70 18.61 23.69 25.90 14.86 8.26 -10.32 4.84 29.16 11.83 10.37 6.51 -2.75 14.26 6.00 37.57 18.42 10.30 21.98 21.50 -2.77 2.50 10.94 6.99 6.01 910117 880531 900827 901001 910821 891016 880406 910211 911223 880115 880728 880608 880902 900511 880125 901018 880729 890512 910402 901019 Avg. -3.00 18.61 20.31 10.58 Avg. 42 3.66 3.41 3.18 2.90 2.90 2.75 2.66 2.58 2.50 2.49 2.39 2.37 2.37 2.37 2.31 2.31 2.24 2.22 2.19 2.18 27.69 19.21 28.96 23.27 16.04 11.87 20.16 19.07 13.99 29.03 17.27 20.45 19.06 16.60 25.10 25.18 17.58 14.30 16.24 25.25 21.29 19.25 24.21 21.61 13.05 20.48 19.36 19.58 12.80 25.38 18.79 21.17 17.40 16.93 25.38 24.26 17.48 14.27 16.43 20.40 -23.11 0.22 -16.40 -7.15 -18.67 72.60 -3.98 2.69 -8.49 -12.59 8.80 3.57 -8.69 1.96 1.10 -3.65 -0.59 -0.16 1.22 -19.19 2.60 20.32 19.48 -1.53 Table 7: Delta-Hedged Gains and Volatility Risk Premium: Time-Series Regressions The regression results are based on the following speci cation for delta-hedged gains and realized volatility: GAINSt = 0 + 1 VOLt + 2 GAINSt 1 + et ; where GAINSt t;tt+ . VOLt represents the prior month realized volatility. The null hypothesis is that S 1 = 0. We include a lagged dependent variable to correct for the serial correlation of the residuals (the Cochrane-Orcutt procedure yields similar inferences). The table reports the coecient estimate, the t-statistic (in square brackets), the adjusted R2 , and the Box-Pierce statistic with 6 lags (denoted Q6 ). The p-values for Q6 are in parenthesis. The t-statistics are based on the Newey-West procedure with a lag length of 12. FULL refers to the entire sample period of 1988:01-1995:12; SET 1 corresponds to the sub-sample of 88:1-91:12; and SET 2 corresponds to the sub-sample of 92:01-95:12. The results are reported for closest to at-the-money calls (with average moneyness of 1.004). For comparison, the regressions are performed using both the historical volatility, VOLh , and the GARCH volatility, VOLg . t t The GARCH estimation employs one year of daily return observations (for each of the 8 years from 1988 through 1995). All regressions use call options sampled monthly, with a constant maturity of 30 days, 44 days and 58 days, respectively. Historical Volatility, VOLh t  Sample 30 FULL Days SET 1 SET 2 44 FULL Days SET 1 SET 2 58 FULL Days SET 1 SET 2 0 (10 2 ) 1 0.22 [2.00] 0.87 [2.09] 0.32 [2.45] -0.032 [-4.39] -0.073 [-2.66] -0.051 [-3.92] 0.38 [2.53] 1.01 [4.37] 0.52 [4.31] 0.40 [1.92] 1.24 [3.07] 0.49 [4.69] 2 R2 GARCH Volatility, VOLg t 1 -0.199 10.80 1.78 [-3.47] (0.94) -0.137 15.35 1.10 [-1.36] (0.98) 0.058 14.70 4.45 [0.55] (0.62) 0.05 [0.41] 1.36 [2.04] 0.69 [1.62] -0.017 [-1.71] -0.101 [-1.07] -0.089 [-1.97] -0.282 6.74 2.34 [-6.04] (0.88) -0.361 12.55 1.91 [-7.77] (0.93) -0.067 3.81 4.20 [-0.71] (0.65) -0.045 [-4.27] -0.080 [-5.34] -0.070 [-5.23] 0.125 13.87 1.68 [1.70] (0.95) 0.077 24.75 3.06 [0.96] (0.80) 0.422 36.22 6.52 [7.04] (0.37) 0.15 [1.13] 1.91 [4.03] 0.99 [4.40] -0.023 [-2.23] -0.135 [-4.49] -0.118 [-4.97] 0.073 0.01 3.37 [1.09] (0.76) -0.093 13.00 5.43 [-1.23] (0.49) 0.317 24.21 9.34 [7.21] (0.16) -0.048 [-3.50] -0.099 [-4.21] -0.066 [-6.57] 0.217 11.81 2.81 [1.45] (0.83) 0.132 19.00 2.31 [0.71] (0.89) 0.510 36.56 2.18 [10.12] (0.90) 0.03 [0.17] 1.82 [2.40] 1.06 [2.83] -0.013 [-0.89] -0.131 [-2.76] -0.126 [-3.18] 0.199 2.49 3.09 [1.49] (0.80) 0.022 14.65 2.98 [0.15] (0.81) 0.311 24.79 2.30 [9.59] (0.88) 43 2 R2 0 (10 2 ) (%) Q6 (%) Q6 Table 8: Properties of Delta-Hedged Gains in Simulated Economies Simulated Delta-Hedged Gains =C Simulated Coecient Values 0.0050 0.0050 0.0120 0.0121 0.0710 0.0710 0.0420 0.0394 GAINSt = 0 + 1 VOLt + 2 GAINSt 1 + et Freq. for Freq. for t( 1) > 2 t( 1) < 2 R2 4.90% 4.90% 2.44% 2 0.149 0.149 -0.020 1 -0.615 -0.616 -0.555 2.43% 0.077 0.077 0.055 -0.019 f0.170g f1.412g f0.093g f1.74g 0.2039 0.1372 0.5151 -0.519 f0.1358g f0.171g f1.407g f0.093g f1.73g 0.0017 0.0021 (%) -0.1641 Hedge  =S Ratio ($) (%) SV -0.0024 -0.0018 f0.0235g f0.0015g f0.1346g 0.0022 0.0089 0.0022 f0.137g f0.922g f0.092g f3.17g SV BS 0.0036 f0.0233g f0.0015g f0.1629g f0.136g f0.923g f0.093g f3.17g BS f0.0305g f0.0021g f0.1576g 0.046 0 We simulate delta-hedged gains in an economy where volatility is stochastic, but not priced. The simulation experimentp Appendix (see 1 B for more details) is based on the following discretization of stock returns and volatility: St+h = St +  St h + t St t h and t2+h = 2 2 2p t +   t h +  t t h, where h is set equal to 1 day. For the simulation S0 = 100, 0=10%,  = 2,  = 0:01,  = 0:5, and  = 0:1. In each simulation run, the theoretical call option values are generated according to the stochastic volatility option pricing model of Heston (1993). Throughout, we assume that interest rate and the dividend yield are zero, and [t] = 0. The delta-hedged gains are computed in two di erent ways. First, we use the hedge ratio given by the stochastic volatility model (denoted as \SV"), and second using the Black-Scholes model (denoted as \BS"). We consider at-the-money options with a maturity of 30 days (and 44 days). Every 30 (44) days, the option is bought and delta-hedged. Over 8 years (2880 days), this produces 96 monthly observations on delta-hedged gains and prior 30 days volatility. For each simulation, we perform the time-series regression: GAINSt = 0 + 1 VOLt + 2 GAINSt 1 + et . The reported 0 , 1 , 2, and adjusted R2 are averages over 1000 simulations. The mean absolute deviation of the estimate is shown in curly brackets. We report the frequency of signi cant 1 (i.e., t( 1 ) > 2 and t( 1 ) < 2). We also show the mean delta-hedged gains across all simulations, for both the SV model and the BS model.  30 Days 44 Days f0.0297g f0.0015g 44 Table 9: E ect of Jumps on Delta-Hedged Gains We employ skewness and kurtosis of the risk-neutral distribution as proxies for jump-fear. The regression results are based on the following speci cation between delta-hedged gains, historical volatility, and the higher-order moments of the risk-neutral return distribution: GAINSt =  +  VOLh +  GAINSt 1 +  SKEW +  KURT + et ; 0 1 t 2 3 t 4 t where GAINSt t;tt+ . VOLh represents the historical volatility. To correct for the serial correlation of t S the residuals, we have included a lagged dependent variable (the Cochrane-Orcutt procedure yields similar inferences). We record the coecient estimate, the t-statistic (in square brackets), the adjusted R2 , and the Box-Pierce statistic with 6 lags (denoted Q6 ). The p-values for Q6 are in parenthesis. The t-statistics are based on the Newey-West procedure with a lag length of 12. FULL refers to the entire sample period of 1988:01-1995:12; SET 1 corresponds to the sub-sample of 88:1-91:12; and SET 2 corresponds to the sub-sample of 92:01-95:12. The results are reported for closest to at-the-money calls. All regressions use call options sampled monthly, with a constant maturity of 30 days, 44 days and 58 days, respectively. The model-free estimate of risk-neutral skewness, SKEW, and the risk-neutral kurtosis, KURT , are t t constructed as described in the Appendix. Sample  0 (10 2 ) 30 FULL 0.51 Days [2.86] SET 1 1.18 [1.97] SET 2 0.63 [3.55]  44 FULL Days SET 1 SET 2 58 FULL Days SET 1 SET 2  1  2 -0.041 [-3.44] -0.082 [-1.99] -0.064 [-4.74] -0.16 [-2.84] -0.10 [-0.94] 0.04 [0.47] 0.45 [2.03] 1.11 [2.84] 0.55 [3.34] -0.046 [-4.09] -0.085 [-3.70] -0.068 [-3.95] 0.19 [3.31] 0.118 [1.43] 0.42 [7.01] 0.23 [1.51] 0.04 [0.23] 0.12 [0.70] 0.32 [2.24] 0.05 [0.28] 0.17 [0.59] 17.26 0.60 [1.85] 1.71 [3.30] 0.46 [3.88] -0.055 0.22 [-3.08] [1.30] -0.111 0.10 [-3.35] [0.53] -0.064 0.50 [-6.88] [7.04] 0.34 [1.46] 0.50 [3.14] -0.05 [-0.43] 0.41 [1.79] 0.44 [4.25] -0.08 [-0.55] 12.42 45   R2 Q6 3 4 2 ) (10 3) (10 0.31 0.28 13.98 1.26 [1.82] [1.73] (0.97) 0.21 0.13 15.46 1.27 [0.90] [0.58] (0.97) 0.17 0.05 17.31 4.52 [1.40] [0.28] (0.61) 2.41 (0.91) 23.47 3.36 (0.76) 34.13 6.41 (0.38) 1.82 (0.94) 19.03 2.55 (0.86) 35.41 3.71 (0.72) Table 10: Delta-Hedged Gains for a Pre-Crash Period (January 1987 through June 1987, Calls and Puts) For a selected pre-crash period, we display the delta-hedged gains for out-of-money calls (i.e., y < 1) and out-of-money puts (i.e., y > 1). In the case of calls, the discretely rebalanced deltahedged gains, t;t+ , is computed as: t;t+ = max(St+ K; 0) Ct PN=01 t (St +1 St ) n PN 1  t St ) N , where the interest rate, rn, and the option delta, t , are updated on a n=0 rn (Ct daily basis. The option delta is computed as the Black-Scholes hedge ratio evaluated at the GARCH volatility. The rebalancing frequency,  =N , is set to one day. Reported in the table are (i) dollar delta-hedged gains (t;t+ ) and the (ii) delta-hedged gains normalized by the index level (t;t+ =St). All delta-hedged gains are averaged over their respective moneyness and maturity category. The standard errors, shown in parenthesis, are computed as the sample standard deviation divided by the square root of the number of observations. 1<0 is the proportion of delta-hedged gains with  < 0, and N is the number of options (in curly brackets). \All" combines the delta-hedged gains from maturities of 14-30 days and 31-60 days. n n n n n Moneyness  (in $) =S (in %) 1<0 1 14-30 31-60 All 14-30 31-60 All fN g -10% to -7.5% -0.12 -1.16 -0.73 -0.04 -0.40 -0.26 86% (0.08) (0.23) (0.18) (0.03) (0.08) (0.06) f21g y -7.5% to -5% -0.28 -0.77 -0.61 -0.10 -0.27 -0.22 (0.10) (0.19) (0.14) (0.04) (0.07) (0.05) 69% f85g Calls -5% to -2.5% -0.64 -1.11 -0.96 -0.22 -0.38 -0.33 73% (0.11) (0.16) (0.11) (0.04) (0.06) (0.04) f238g -2.5% to 0% 0% to 2.5% -0.63 -1.03 -0.87 -0.22 -0.36 -0.31 84% (0.11) (0.10) (0.08) (0.04) (0.03) (0.03) f274g 5% to 7.5% -0.23 -0.74 -0.53 -0.09 -0.26 -0.19 84% (0.09) (0.07) (0.06) (0.03) (0.02) (0.02) f245g 7.5% to 10% ALL -0.82 -0.87 -0.85 -0.29 -0.31 -0.30 74% (0.15) (0.14) (0.10) (0.05) (0.05) (0.03) f282g 2.5% to 5% Puts -0.65 -0.81 -0.75 -0.22 -0.28 -0.25 67% (0.12) (0.14) (0.10) (0.04) (0.05) (0.03) f276g -0.07 -0.67 -0.44 -0.03 -0.24 -0.16 92% (0.07) (0.04) (0.04) (0.02) (0.01) (0.01) f150g -0.25 -0.44 -0.37 46 -0.09 -0.15 -0.13 n ...
View Full Document

Ask a homework question - tutors are online