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

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10 Y.Ai-Sahalia,A.W.Lo/Journal of Econometrics 94(2000)9-51 catastrophes,regulators,investment bankers,and chief executive officers have now embraced the notion of risk management as one of the primary fiduciary responsibilities of the corporate manager.Because financial risks often manifest themselves in subtle and nonlinear ways in corporate balance sheets and income statements,recent attention has focused on quantifying the fluctuations of market valuations in a statistical sense.These value-at-risk(VaR)measures lie at the heart of mo current risk management systems and protocols.For example JP Morgan's(1995)RiskMetrics system documentation describes VaR in the following way: Value at Risk is an estimate,with a predefined confidence interval,of how much one can lose from holding a position over a set horizon.Potential horizons may be one day for typical trading activities or a month or longer for e methods described in our documentation useis volatilities and correlations that are then used to estimate the market risk.These statistics can be applied across a set of asset classes covering products used by financial institutions,corporations,and insti- tutional investors. By modeling the price fluctuations of securities held in one's portfolio,an estimate and confidence interval of how much one can lose is readily derived from the basic principles of statistical inference.However,in this paper we argue that statistical notions of value-at-risk are,at best,inco omplete measures of the true risks facing investors.In particular,while statistical measures do provide some information about the range of uncertainty that a portfolio exhibits,they have little to do with the economic valuation of such uncertainty.For example, a typical VaR statistic might indicate a 5%probability of a $15M loss for a S10M portfolio over the next month,which seems to be a substantial risk exposure at first glance.But if this 15%loss occurs only when other investments of similar characteristics suffer losses of 25%or more,such a risk may seem rather mild after all.This simplistic example suggests that a one-dollar loss is not always worth the same,and that circumstances s urrounding the loss can affect its economic valuation,something that is completely ignored by purely statistical measures of risk In this paper,we propose an alternative to statistical VaR(henceforth S-VaR) that is based on economic valuations of value-at-risk,and which incorporate many other aspects of market risk that are central to the practice of risk management.Our alternative is based on the seminal ideas of Arrow (1964)and Debreu(1959),who first formalized the economics of uncertainty by introducing elementary securities each paying $1 in one specific state of nature and nothing

catastrophes,1 regulators, investment bankers, and chief executive o$cers have now embraced the notion of risk management as one of the primary "duciary responsibilities of the corporate manager. Because "nancial risks often manifest themselves in subtle and nonlinear ways in corporate balance sheets and income statements, recent attention has focused on quantifying the #uctuations of market valuations in a statistical sense. These value-at-risk (VaR) measures lie at the heart of most current risk management systems and protocols. For example, JP Morgan's (1995) RiskMetrics system documentation describes VaR in the following way: Value at Risk is an estimate, with a prede"ned con"dence interval, of how much one can lose from holding a position over a set horizon. Potential horizons may be one day for typical trading activities or a month or longer for portfolio management. The methods described in our documentation use historical returns to forecast volatilities and correlations that are then used to estimate the market risk. These statistics can be applied across a set of asset classes covering products used by "nancial institutions, corporations, and institutional investors. By modeling the price #uctuations of securities held in one's portfolio, an estimate and con"dence interval of how much one can lose is readily derived from the basic principles of statistical inference. However, in this paper we argue that statistical notions of value-at-risk are, at best, incomplete measures of the true risks facing investors. In particular, while statistical measures do provide some information about the range of uncertainty that a portfolio exhibits, they have little to do with the economic valuation of such uncertainty. For example, a typical VaR statistic might indicate a 5% probability of a $15M loss for a $100M portfolio over the next month, which seems to be a substantial risk exposure at "rst glance. But if this 15% loss occurs only when other investments of similar characteristics su!er losses of 25% or more, such a risk may seem rather mild after all. This simplistic example suggests that a one-dollar loss is not always worth the same, and that circumstances surrounding the loss can a!ect its economic valuation, something that is completely ignored by purely statistical measures of risk. In this paper, we propose an alternative to statistical VaR (henceforth S-VaR) that is based on economic valuations of value-at-risk, and which incorporates many other aspects of market risk that are central to the practice of risk management. Our alternative is based on the seminal ideas of Arrow (1964) and Debreu (1959), who "rst formalized the economics of uncertainty by introducing elementary securities each paying $1 in one speci"c state of nature and nothing 1For example, the multimillion-dollar losses su!ered by Gibson Greetings, Metallgesellschaft, Orange County, Proctor and Gamble, Barings Securities, etc. 10 Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51

Y.Ait-Sahalia,A.W.Lo Journal of Econometrics 94 (2000)9-51 in any other state.Now known as Arrow-Debreu securities,they are widely recognized as the fundamental building blocks of all modern financial asset pricing theories,including the CAPM,the APT,and the Black and Scholes (1973)and Merton(1973)option-pricing models. By construction,Arrow-Debreu prices have a probability-like interpretation -they are nonnegative and sum to unity but since they are market prices determined in equilibrium by supply and demand,they contain much more information than statistical models of prices.Arrow-Debreu prices are deter- amics,informa- ined by the combination of investors pckec nn and the sitio general equilibrium.Moreover,we shall show below that under certain special condi tions,Arrow-Debreu prices reduce to the simple probabilities on which statist- ical VaR measures are based,hence the standard measures of value-at-risk are cial case s of the Arrov w-Debreu f amework. The fact that the market prices of these Arrow-Debreu securities need not be equal across states implies that a one-dollar gain need not be worth the same in every state of nature-indeed,the worth of a one-dollar gain in a given state is w-Debreu pri ice of that security.Therefo Debreu sites to msur ono VaR he we prop e to use the Despite the fact that pure Arrow-Debreu securities are not yet traded on any organized exchange,2 Arrow-Debreu prices can be estimated from the prices of traded financial recently developed no rictechniques such as kernel regression,artificial neural networks,and implied binomial trees. Nonparametric techniques are particularly useful for value-at-risk calculations because departures from standard parametric assumptions,e.g.,normality,can have dramatic consequences for tail probabilities Using such techniques,we compare the performance of s-VaR and E-VaR measures and develop robust statistical methods to gauge the magnitudes of their differences. Moreover,to provide an economic interpretation for the differences between S-VaR and E-VaR,we show how to con nbine S-VaR and E-VaR to yield a measure of the aggregate risk aversion of the economy,i.e,the risk aversion the representative investor in a standard dynamic asset-pricing model.We propose to extract (unobservable)aggregate risk-preferences,what we call mplied risk aversion,from (observable)market prices of traded financial secur- ities.n particur,the aggregate preferences that are compatible with the pair of option and index values. 2This is changing as derivatives markets become more sophisticated.For example,it is now le to construct a limited set of Arrow-Debreu securitie (97 whih s be man(19 tes ios o Hakan markete recently by 31e a Forther em of thes of mcthods in fnance,Ait-Sahalia 1996b)and Stutzer (1996)

in any other state. Now known as Arrow}Debreu securities, they are widely recognized as the fundamental building blocks of all modern "nancial assetpricing theories, including the CAPM, the APT, and the Black and Scholes (1973) and Merton (1973) option-pricing models. By construction, Arrow}Debreu prices have a probability-like interpretation } they are nonnegative and sum to unity } but since they are market prices determined in equilibrium by supply and demand, they contain much more information than statistical models of prices. Arrow}Debreu prices are determined by the combination of investors' preferences, budget dynamics, information structure, and the imposition of market-clearing conditions, i.e., general equilibrium. Moreover, we shall show below that under certain special conditions, Arrow}Debreu prices reduce to the simple probabilities on which statistical VaR measures are based, hence the standard measures of value-at-risk are special cases of the Arrow}Debreu framework. The fact that the market prices of these Arrow}Debreu securities need not be equal across states implies that a one-dollar gain need not be worth the same in every state of nature } indeed, the worth of a one-dollar gain in a given state is precisely the Arrow}Debreu price of that security. Therefore, we propose to use the prices of Arrow}Debreu securities to measure economic VaR (henceforth E-VaR). Despite the fact that pure Arrow}Debreu securities are not yet traded on any organized exchange,2 Arrow}Debreu prices can be estimated from the prices of traded "nancial securities using recently developed nonparametric techniques such as kernel regression, arti"cial neural networks, and implied binomial trees. Nonparametric techniques are particularly useful for value-at-risk calculations because departures from standard parametric assumptions, e.g., normality, can have dramatic consequences for tail probabilities.3 Using such techniques, we compare the performance of S-VaR and E-VaR measures and develop robust statistical methods to gauge the magnitudes of their di!erences. Moreover, to provide an economic interpretation for the di!erences between S-VaR and E-VaR, we show how to combine S-VaR and E-VaR to yield a measure of the aggregate risk aversion of the economy, i.e., the risk aversion of the representative investor in a standard dynamic asset-pricing model. We propose to extract (unobservable) aggregate risk-preferences, what we call implied risk aversion, from (observable) market prices of traded "nancial securities. In particular, we are inferring the aggregate preferences that are compatible with the pair of option and index values. 2This is changing as derivatives markets become more sophisticated. For example, it is now possible to construct a limited set of Arrow}Debreu securities by forming portfolios of `digitala or `binarya options. See, also, the `supersharesa security proposed by Garman (1978) and Hakansson (1976) which has been test-marketed recently by Leland, O'Brien, and Rubinstein Associates, Inc. 3For other examples of the use of nonparametric methods in "nance, see AmKt}Sahalia (1996a, 1996b) and Stutzer (1996). Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51 11

12 Y.Ai-Sahalia,A.W.Lo Journal of Econometrics 94(2000)9-5 When applied to daily S&P 500 option prices and index levels from 1993,our nonparametric analysis uncovers substantial differences between S-VaR and E-VaR(see Fig.2).A comparison of S-VaR and E-VaR densities shows that aggregate risk aversion is not constant across states or maturity dates,but changes in important nonlinear wavs (see Fig.4). Of course,risk management is a complex process that is unlikely to be driven by any sine risk measure E-VaR-Vak.In particular.despite the fact that E-VaR provides information not contained in S-VaR,both measures belong in the arsenal of tools that risk managers can bring to bear on assessing and controlling risk.For certain purposes-regulatory reporting requirements. quick summaries of corporate exposure,r comparisons across companies and industries-S-VaR may be a simpler measure to compute and interpret. Nevertheless,E-VaR brings a new dimension into the risk management process and should be integrated into any complete risk management system. In Se ection 2,we present a brief review of the theoretical underpinnings of Arrow-Debreu prices and their relation to dynamic equilibrium models of financial markets.In Section 3,we formally introduce the notion of economic value-at-risk,describe its implementation,and propose statistical inference procedures that can quantify its accuracy and relevanc over statistical VaR.An explicit comparison of E-VaR with S-VaR,along with the appropriate statistical inference,is described and developed in Section 4.We construct an estimator of implied risk aversion in Section 5 and propose tests for risk neutrality and for ic preferences based on this estimato trate the empirical relevanc of E-VaR,we apply our estimators to daily500 options data in Section6. We conclude in Section 7 and collect our technical assumptions and results in the Appendix. 2.DGP,SPD,MRS,and VaR Denote by S,the price at time t of a security or portfolio of securities whose risk we wish to manage and let u,=log(S,/S,)denote its return between t and .The usual statistical VaR measures are based on the probability distribu- tion ofFor example,one common VaR measure is the standard deviatior of returns u.Another is the 95%confidence interval of u centered at its historical mean.More sophisticated VaR measures incorporate conditioning information and dynamics in specifying and estimating the probability distribu- tion of ur.ie they are based on conditional probabilities obtained from the data-generating process DGP)ofS. +See.for example.Smithson et al.(1995)

When applied to daily S&P 500 option prices and index levels from 1993, our nonparametric analysis uncovers substantial di!erences between S-VaR and E-VaR (see Fig. 2). A comparison of S-VaR and E-VaR densities shows that aggregate risk aversion is not constant across states or maturity dates, but changes in important nonlinear ways (see Fig. 4). Of course, risk management is a complex process that is unlikely to be driven by any single risk measure, E-VaR or S-VaR. In particular, despite the fact that E-VaR provides information not contained in S-VaR, both measures belong in the arsenal of tools that risk managers can bring to bear on assessing and controlling risk. For certain purposes } regulatory reporting requirements, quick summaries of corporate exposure, or comparisons across companies and industries } S-VaR may be a simpler measure to compute and interpret. Nevertheless, E-VaR brings a new dimension into the risk management process and should be integrated into any complete risk management system. In Section 2, we present a brief review of the theoretical underpinnings of Arrow}Debreu prices and their relation to dynamic equilibrium models of "nancial markets. In Section 3, we formally introduce the notion of economic value-at-risk, describe its implementation, and propose statistical inference procedures that can quantify its accuracy and relevance over statistical VaR. An explicit comparison of E-VaR with S-VaR, along with the appropriate statistical inference, is described and developed in Section 4. We construct an estimator of implied risk aversion in Section 5 and propose tests for risk neutrality and for speci"c preferences based on this estimator. To illustrate the empirical relevance of E-VaR, we apply our estimators to daily S&P 500 options data in Section 6. We conclude in Section 7 and collect our technical assumptions and results in the Appendix. 2. DGP, SPD, MRS, and VaR Denote by S t the price at time t of a security or portfolio of securities whose risk we wish to manage and let u t,q ,log(S t`q /S t ) denote its return between t and t#q. The usual statistical VaR measures are based on the probability distribution of u t,q .4 For example, one common VaR measure is the standard deviation of returns u t,q . Another is the 95% con"dence interval of u t,q centered at its historical mean. More sophisticated VaR measures incorporate conditioning information and dynamics in specifying and estimating the probability distribution of u t,q , i.e., they are based on conditional probabilities obtained from the data-generating process (DGP) of MS t N. 4 See, for example, Smithson et al. (1995). 12 Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51

Y.Ait-Sahalia,A.W.Lo Journal of Econometrics 94 (2000)9-51 3 Although such VaR measures do capture important features of the uncer- tainty surroun fall short in one crucial respect:they are statistica evaluations of uncertainty,not economic valuations.In particular,one investor may be quite willing to bear a one-standard deviation drop in ur while another investor may be devastated by such an event.Therefore,although the dollar loss is the same for both investors,their personal valuationsof such a risk can differ dramatically.More importantly,the market valuation of this risk-the value assigned by the interactions of many heterogeneous investors in a market setting -can differ substantially from purely statistical measures. 2.1.Dynamic equilibrium models This distinction between the DGP and market valuations lies at the heart of dynamic equilibrium asset-pricing models in economics-beginning with Arrow (1964)and Debreu(1959)-in which the valuation of securities with uncertain ined by the inter ilibration of market forces and tionsIn such models theDOP for prices is not assumed but rather is derived from first principles as the (stochastic)sequence of prices that equates supply and demand at each point in time. More importantly,unlike a purely statistical model of prices,e.g,geometric Brownian motion,a DGP that is derived from prices contains an enormous amount of information about market conditions and investors'pref- erences that is critical for risk management.To see why,consider a standard ange economy (Lucas,197:Rubinstein,1976:Bick,19:He and which securities markets ars dynamically complete.there is a single consumption good,no exogenous income,and all investors seek to maximize at date t a state-independent utility function,subject to the usual budget constraints.There is one risky stock(the market portfolio,in total supply normalized to one share,and with price S,at date t)and one riskless bond (i zero net supply)with continuously compounded rate of return r,assumed constant for simplicity.Both are available for trading at any date between t and T.Assume that the stock price process follows the stochastic differential equa- tion ds=dS.)dt+as,d☑， (2.1) S. where Z denotes a standard Brownian motion,and let f(S)denote the condi- tional density of Sr given S,implied by Eq.(2.1). See Merton(1992)for a review of these and related models

Although such VaR measures do capture important features of the uncertainty surrounding u t,q , they fall short in one crucial respect: they are statistical evaluations of uncertainty, not economic valuations. In particular, one investor may be quite willing to bear a one-standard deviation drop in u t,q , while another investor may be devastated by such an event. Therefore, although the dollar loss is the same for both investors, their personal valuations of such a risk can di!er dramatically. More importantly, the market valuation of this risk } the value assigned by the interactions of many heterogeneous investors in a market setting } can di!er substantially from purely statistical measures. 2.1. Dynamic equilibrium models This distinction between the DGP and market valuations lies at the heart of dynamic equilibrium asset-pricing models in economics } beginning with Arrow (1964) and Debreu (1959) } in which the valuation of securities with uncertain payo!s is determined by the interaction and equilibration of market forces and market conditions.5 In such models, the speci"c DGP for prices is not assumed, but rather is derived from "rst principles as the (stochastic) sequence of prices that equates supply and demand at each point in time. More importantly, unlike a purely statistical model of prices, e.g., geometric Brownian motion, a DGP that is derived from equilibrium prices contains an enormous amount of information about market conditions and investors' preferences that is critical for risk management. To see why, consider a standard dynamic exchange economy (Lucas, 1978; Rubinstein, 1976; Bick, 1990; He and Leland, 1993) in which securities markets are dynamically complete, there is a single consumption good, no exogenous income, and all investors seek to maximize at date t a state-independent utility function, subject to the usual budget constraints. There is one risky stock (the market portfolio, in total supply normalized to one share, and with price S t at date t) and one riskless bond (in zero net supply) with continuously compounded rate of return r, assumed constant for simplicity. Both are available for trading at any date between t and ¹. Assume that the stock price process follows the stochastic di!erential equation dS t S t "k(S t , t) dt#p(S t , t) dZt (2.1) where Z denotes a standard Brownian motion, and let f t (S T ) denote the conditional density of S T given S t implied by Eq. (2.1). 5 See Merton (1992) for a review of these and related models. Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51 13

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