《金融经济学原理》课程教学资源（文献资料）Nonparametric risk management and implied risk aversion.pdf_P11-P15

3.2. The Black}Scholes SPD For example, recall that under the hypotheses of Black and Scholes (1973) and Merton (1973),8 the date-t price H of a call option maturing at date ¹,t#q, with strike price X, written on a stock with date-t price Pt and dividend yield d t,q , is given by:9 HBS(S t , X, q, r t,q , d t,q ; p)"e~rt,qq P = 0 max[S T !X, 0] f H BS,t (S T ) dS T "St U(d 1 )!Xe~rt,qqU(d 2 ) (3.2) where d 1 ,log(S t /X)#(r t,q !d t,q #p2/2)q pJq , d 2 ,d 1 !pJq. (3.3) In this case the corresponding SPD is a log-normal density with mean ((r t,q !d t,q )!p2/2)q and variance p2q: f H BS,t (S T )"ert,qq K R2HBS RX2 K X/ST " 1 S T J2pp2q expC ![log(S T /S t )!(r t,q !d t,q !p2/2)q]2 2p2q D . (3.4) 8Frictionless markets, continuous trading, geometric Brownian motion for S t , speci"c bond price dynamics including as a special case constant interest rates, and investors' agreement on the value of p and the distributional characteristics, not necessarily including the asset's expected rate of return [see Merton (1973, Section 6)]. The assumption of geometric Brownian motion can be relaxed to allow for mean reversion without changing the functional form of the Black}Scholes formula (as long as the di!usion coe$cient p is constant), however, equilibrium prices (hence the SPD) will not be una!ected [see He and Leland (1993, Section 3) and Lo and Wang (1995) for further discussion]. 9 As discussed in AmKt-Sahalia and Lo (1998, Section 3), the empirical analysis is greatly simpli"ed by inferring dividend rates through observed prices on forward contracts written on the underlying asset. Let Ft,q "S t e(rt,q~dt,q)q denote the value at t of a forward contract written on the asset, with the same maturity q as the option. At the maturity of the futures, the futures price equals the asset's spot price. Thus a European call option on the asset has the same value as a European call option on the futures contract with the same maturity. As a result, we will often rewrite the Black}Scholes formula as HBS(Ft,q , X, q, r t,q ; p)"e~rt,qq(Ft,q U(d 1 )!XU(d 2 )), with d 1 ,(log(Ft,q /X)#(p2/2)q)/(pJq) and d 2 ,d 1 !pJq. Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51 19

20 Y.Ai-Sahalia,A.W.Lo/Journal of Econometrics 94(2000)9-51 This expression shows that the SPD can depend on many quantities in general,and is distinct from but related to the PDF of the terminal stock price Sr.More generally,while sufficiently strong assumptions on the underlying asset price dynamics can often characterize the SPD uniquely,in most cases the SPD cannot be computed in closed form and numerically intensive methods must be used to calculate it.It is clear from Eq.(3.4)that the SPD is inextricably linked to the parametric assumptions underlying the Black-Scholes option pricing model.If those parametric assumptions do not hold,e.g.,if the dynamics of{S,contain Poisson jumps,then Eq.(3.4)will yield incorrect prices,prices that are inconsistent with the dynamic equilibrium model or the hypothesized stochastic process the general lack of success in fitting highly parametric models to financial data (see,for example,Campbell et al.,1997, Chapters 2 and 12),combined with the availability of the data and the large effects of differences in specification,it is quite natural to focus on nonparamet- ric methods for estimating SPDs 3.3.Nonparametric SPD estimators Ait-Sahalia and Lo(1998)propose to estimate the SPD nonparametrically by exploiting Breeden and Litzenberger's (1978)insight that *(S)= suggest using market prices to estimate an option-pricingformula ()nonparametrically,which can then be differentiate twice with respect to X to obtain o-H()/X.They use kernel regression to construct A().10 Assuming that the option-pricing formula H to be estimated is a an arbitrary nonlinear function of a vector of option characteristics or "explanatory"variables,Y=[F.is the forward price o the asset. In practice,they propose to reduce the dimension of the kernel regression by using a semiparametric a parameter for that option is a nonparametric function a(X/F.): H(S X,t,ri.e .)=HBs(Fi X,t,ri.ti a(X/Fit)). (3.5) We assume that the function H defined by Eq.(3.5)satisfies all the required conditions to be a 'rational'option-pricing formula in the sense of Merton ee Hardle(19)and Wand and Jones(1995)fora more detailed discussion of nonparametric regression.There are other alternatives to Eq.(3.5)that can be used to obtain option-pricing s:see Derman and Kani (1994),Dupire (1994).Hutchins on e a19

This expression shows that the SPD can depend on many quantities in general, and is distinct from but related to the PDF of the terminal stock price S T . More generally, while su$ciently strong assumptions on the underlying asset price dynamics can often characterize the SPD uniquely, in most cases the SPD cannot be computed in closed form and numerically intensive methods must be used to calculate it. It is clear from Eq. (3.4) that the SPD is inextricably linked to the parametric assumptions underlying the Black}Scholes option pricing model. If those parametric assumptions do not hold, e.g., if the dynamics of MS t N contain Poisson jumps, then Eq. (3.4) will yield incorrect prices, prices that are inconsistent with the dynamic equilibrium model or the hypothesized stochastic process driving MSt N. Given the general lack of success in "tting highly parametric models to "nancial data (see, for example, Campbell et al., 1997, Chapters 2 and 12), combined with the availability of the data and the large e!ects of di!erences in speci"cation, it is quite natural to focus on nonparametric methods for estimating SPDs. 3.3. Nonparametric SPD estimators AmKt-Sahalia and Lo (1998) propose to estimate the SPD nonparametrically by exploiting Breeden and Litzenberger's (1978) insight that f H t (S T )"exp(r t,q q)R2H( ) )/RX2. They suggest using market prices to estimate an option-pricing formula HK( ) ) nonparametrically, which can then be di!erentiated twice with respect to X to obtain R2HK( ) )/RX2. They use kernel regression to construct HK( ) ).10 Assuming that the option-pricing formula H to be estimated is a an arbitrary nonlinear function of a vector of option characteristics or `explanatorya variables, Y,[Ft,q X q r t,q ]@ where Ft,q is the forward price of the asset. In practice, they propose to reduce the dimension of the kernel regression by using a semiparametric approach. Suppose that the call pricing function is given by the parametric Black}Scholes formula (3.2) except that the implied volatility parameter for that option is a nonparametric function p(X/Ft,q ,q): H(S t , X, q, r t,q , d t,q )"HBS(Ft,q , X, q, r t,q ; p(X/Ft,q , q)). (3.5) We assume that the function H de"ned by Eq. (3.5) satis"es all the required conditions to be a &rational' option-pricing formula in the sense of Merton 10 See HaKrdle (1990) and Wand and Jones (1995) for a more detailed discussion of nonparametric regression. There are other alternatives to Eq. (3.5) that can be used to obtain option-pricing formulas: see Derman and Kani (1994), Dupire (1994), Hutchinson et al. (1994), and Rubinstein (1994). For an extension to American options and the nonparametric estimation of the early exercise boundary, see Broadie et al. (1996). For tests of the volatility models, see Dumas et al. (1995). 20 Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51

(1973, 1992).11 In this semiparametric model, we only need to estimate nonparametrically the regression of p on a subset YI of the vector of explanatory variables Y. The rest of the call pricing function H( ) ) is parametric, thereby considerably reducing the sample size n required to achieve the same degree of accuracy as the full nonparametric estimator. We partition the vector of explanatory variables Y,[YI@ Ft,q r t,q ]@ where YI contains dI nonparametric regressors. As a result, the e!ective number of nonparametric regressors d is given by dI. In our empirical application, we will consider YI,[X/Ft,q q]@ (dI"2) and form the Nadaraya}Watson kernel estimator of E[p D X/Ft,q ,q] as: p((X/Ft,q , q)" +n i/1k X@F A X/Ft,q !Xi /Fti,qi h X@F B k qA q!q i h q B p i +n i/1k X@FA X/Ft,q !Xi /Fti,qi h X@F B k qA q!q i h q B (3.6) where p i is the volatility implied by the option price Hi , and the univariate kernel functions k X@F and k q and the bandwidth parameters h X@F and h q are chosen to optimize the asymptotic properties of the second derivative of HK( ) ), i.e., of the SPD estimator. We then estimate the call pricing function as: HK(S t , X, q, r t,q , d t,q )"HBS(Ft,q , X, q, r t,q , d t,q ; p((X/Ft,q , q)). (3.7) The SPD estimator follows by taking the second partial derivatives of HK( ) ) with respect to X: fKH t (S T )"ert,qq C R2HK(S t , X, q, r t,q , d t,q ) RX2 DKX/ST . (3.8) We give the asymptotic distribution of this estimator in Section A.1 of Appendix A. 3.4. Other estimators of the SPD Several other estimators of the SPD have been proposed in the recent literature [see AmKt-Sahalia and Lo (1998) for a more detailed discussion and an empirical comparison]. Hutchinson et al. (1994) employ several nonparametric 11 In particular, see Merton 1992, Section 8.2). These conditions imply that p(X/F, q) cannot be an arbitrary function but must yield an HBS(Ft,q , X, q, q t,q ; p(X/Ft,q , q)) that satis"es all the conditions of a rational option-pricing formula. Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51 21

22 Y.Ai-Sahalia,A.W.Lo Journal of Econometrics 94(2000)9-5 techniques to estimate option-pricing models that they describe collectively as learning networks-artificial neural networks,radial basis functions,and projec- tion pursuit-and find that all these techniques can recover option-pricing models such as the Black-Scholes model.Taking the second derivative of their option-pricing estimators with respect to the strike price vields an estimator of the spD Another estimator is Rubinstein's(1994)implied binomial tree,in which the risk-neutral probabilities)associated with the binomial terminal stock price S,are estimated by minimizing the sum of squared deviations between and a set of prior risk-neutral probabilities )subject to the restrictions that correctly price an existing set of optic ns and the underlying stock,in the sense that the optimal risk-neutral probabilities yield prices that lie within the bid-ask spreads of the options and the stock [see also Jackwerth and Rubinstein(1996) for smoothness criterial. This approach is similar in spirit to Jarrow and Rudd's(198)and Longstafs (1995)method of fitting risk-neutral density functions using a four-parameter Edgeworth expansion.However,Rubinstein(1994)points out several important limitations of Longstaff's method when extended to a binomial model,including the possibility of negative probabilities.Derman and Kani(19)and Shimko (1993)have proposed related estimators of the SPD. There are several important differences between kernel estimators and im- neutral probabilities;kernel estimators do not.Implie inomial trees are typically estimated for each cross-section of options;kernel estimators aggregate options prices over time to get a single SPD.This implies that implied binomial tree is completely consistent with all option prices at each date,but is not ss time.In con ntrast,the ke coistent across time.but there may be some dateD estimator fits the cross-section of option prices poorly and other dates for which the SPD estimator performs very well. Whether or not con tency over time is a useful property depends on how well the economic variables used in constructing the kernel SPD can account for time variation in risk-neutral probabilities.In addition,the kernel SPDs take advantage of the data temporally surrounding a given date.Tomorrow's and yesterday's SPDs contain information about today s SPD-this information is ignored by the implied binomial trees but not by kernel-estimated SPDs Finally,and perhaps most importantly,statistical inference is virtually im- possible with learning-network estimators and implied binomial trees,because of the rec nature of the former approach(White,1992), and the e non stationarities inherent in the latter approach(recall that implied binomial trees are estimated for each cross-section of option prices).In contrast,the statistical inference of kernel estimators is well developed and computationally quite tractable

techniques to estimate option-pricing models that they describe collectively as learning networks } arti"cial neural networks, radial basis functions, and projection pursuit } and "nd that all these techniques can recover option-pricing models such as the Black}Scholes model. Taking the second derivative of their option-pricing estimators with respect to the strike price yields an estimator of the SPD. Another estimator is Rubinstein's (1994) implied binomial tree, in which the risk-neutral probabilities MnH i N associated with the binomial terminal stock price S T are estimated by minimizing the sum of squared deviations between MnH i N and a set of prior risk-neutral probabilities Mn8H i N, subject to the restrictions that MnH i N correctly price an existing set of options and the underlying stock, in the sense that the optimal risk-neutral probabilities yield prices that lie within the bid-ask spreads of the options and the stock [see also Jackwerth and Rubinstein (1996) for smoothness criteria]. This approach is similar in spirit to Jarrow and Rudd's (1982) and Longsta!'s (1995) method of "tting risk-neutral density functions using a four-parameter Edgeworth expansion. However, Rubinstein (1994) points out several important limitations of Longsta!'s method when extended to a binomial model, including the possibility of negative probabilities. Derman and Kani (1994) and Shimko (1993) have proposed related estimators of the SPD. There are several important di!erences between kernel estimators and implied binomial trees. Implied binomial trees require a prior Mn8H i N for the riskneutral probabilities; kernel estimators do not. Implied binomial trees are typically estimated for each cross-section of options; kernel estimators aggregate options prices over time to get a single SPD. This implies that implied binomial tree is completely consistent with all option prices at each date, but is not necessarily consistent across time. In contrast, the kernel SPD estimator is consistent across time, but there may be some dates for which the SPD estimator "ts the cross-section of option prices poorly and other dates for which the SPD estimator performs very well. Whether or not consistency over time is a useful property depends on how well the economic variables used in constructing the kernel SPD can account for time variation in risk-neutral probabilities. In addition, the kernel SPDs take advantage of the data temporally surrounding a given date. Tomorrow's and yesterday's SPDs contain information about today's SPD } this information is ignored by the implied binomial trees but not by kernel-estimated SPDs. Finally, and perhaps most importantly, statistical inference is virtually impossible with learning-network estimators and implied binomial trees, because of the recursive nature of the former approach (White, 1992), and the nonstationarities inherent in the latter approach (recall that implied binomial trees are estimated for each cross-section of option prices). In contrast, the statistical inference of kernel estimators is well developed and computationally quite tractable. 22 Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51

4. Statistical VaR Having obtained an estimator fKH of the SPD, we can now gauge its importance for risk management by studying the behavior of the ratio of fKH to fK, where fKis an estimator of the conditional density of the DGP, i.e., S-VaR. If the ratio fK,fKH/fKexhibits substantial variation over its domain, this indicates that E-VaR measures contain important economic information that are not captured by their S-VaR counterparts. Of course, because of estimation error, fKwill never be constant in any given dataset even if f is. Therefore some measure of the statistical #uctuations inherent in fKis required, and we now propose estimators for f and fH and describe how to combine them to estimate and conduct statistical inference on fK. 4.1. Parametric S-VaR There are several methods of estimating the S-VaR or statistical distribution of the future price S T conditional on the current price S t . Currently, the most common approach is to assume that this distribution belongs to some parametric family, e.g., lognormal, which is characterized by a small number of parameters. Estimating the distribution is then reduced to estimating these parameters. This underlies virtually all of the S-VaR measures currently used in practice. However, there is considerable empirical evidence to suggest that the most popular parametric models are generally inconsistent with the behavior of recent "nancial data. Therefore, a nonparametric approach to S-VaR may be preferred. 4.2. Nonparametric S-VaR In keeping with the spirit of our nonparametric approach, we propose to estimate the S-VaR of S T without making any parametric assumptions. In particular, we collect the time series of the index values, calculate the q-period continuously compounded returns, u q ,log(S T /S t ), and construct a kernel estimator of the density function g( ) ) of these returns: g((u q ), 1 NHu N + i/1 k uA u q !u ti,q Hu B . (4.1) From the density of the continuously compounded returns we can then calculate Pr(S T )S)"Pr(S t euq)S)"Pr(u q )log(S/S t ))"P -0'(S@St) ~= g(u q ) du q (4.2) Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51 23