14 Y.Ai-Sahalia,A.W.Lo/Journal of Econometrics 94(2000)9-51 Under suitable assumptions for preferences and endowment shocks,it is well-known that market completeness allows us to introduce a representative investor with utility function U [see,for example,Constantinides (1982)] Assuming that he can consume at date t and at the fixed future date T(allowing intermediate consumption or labor income does not affect the basic results),and that he receives one share of the stock as endowment at date t,the representative nvestor adjusts the dollar amount invested in the stock at each intermediary date t to solve the Merton(1971)optimization problem max E[U(W] {q:|t≤s≤T} subject to dWs=rWs +gsu(Ss,s)-r)ds +gso(Ss,s)dzs W.≥0，t≤s≤T where W,denotes his wealth at date s In equilibrium,the investor optimally invests all his wealth in the risky stock at every instant prior to T(W,=S,for all tss sT)and then consumes the terminal value of the stock at T(Cr=Wr=ST).Let J(W,S,t)denote the investor's indirect utility function.The first-order condition for the investor's problem takes the form aJ(W,Ss,母 =e-n-nel(W S.D aw (2.2) with terminal condition at date s=T U(WT)=e-HT-OU(WT (2.3) where =ep.-）) (2.4) Clearly,Eq.(2.2)implies that the right-hand-side of Eq.(2.4)must be path independent,i.e,it depends only on the values of the stock price at t and s and not on the path taken by the stock price in between these dates.The price at date t of a security with a single date-T liquidating payoff of(Wr)is then given by E,[(WT)M.T] (2.5)
Under suitable assumptions for preferences and endowment shocks, it is well-known that market completeness allows us to introduce a representative investor with utility function ; [see, for example, Constantinides (1982)]. Assuming that he can consume at date t and at the "xed future date ¹ (allowing intermediate consumption or labor income does not a!ect the basic results), and that he receives one share of the stock as endowment at date t, the representative investor adjusts the dollar amount q q invested in the stock at each intermediary date q to solve the Merton (1971) optimization problem max M qs @ txsxTN E[;(=T )] subject to d=s "Mr=s #q s (k(S s , s)!r)N ds#q s p(S s , s) dZs , =s *0, t)s)¹ where =s denotes his wealth at date s. In equilibrium, the investor optimally invests all his wealth in the risky stock at every instant prior to ¹ (=s "S s for all t)s)¹) and then consumes the terminal value of the stock at ¹ (CT "=T "ST ). Let J(=, S, t) denote the investor's indirect utility function. The "rst-order condition for the investor's problem takes the form RJ(=s , Ss , s) R= "e~r(s~t) RJ(=t , S t , t) R= f s (2.2) with terminal condition at date s"¹ ;@(=T )"e~r(T~t);@(=t )f T (2.3) where f s ,expGP s t A k(S u , u)!r p(S u , u) B dZu !1 2P s t A k(S u , u)!r p(S u , u) B 2 du H . (2.4) Clearly, Eq. (2.2) implies that the right-hand-side of Eq. (2.4) must be path independent, i.e., it depends only on the values of the stock price at t and s and not on the path taken by the stock price in between these dates. The price at date t of a security with a single date-¹ liquidating payo! of t(=T ) is then given by Et [t(=T )Mt,T ] (2.5) 14 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 where MU(W)U(W)is the stochastic discount factor or marginal rate of substitution(MRS)between dates t and T.If we define the state-price density to be f*(Sr)=fS)×E[5rlS,Sr]=fS)×Sr (since is a function of(S ST))and recall that Wr=ST,then we can rewrite Eq.(2.5)as E[(Wr)M.r]= (WT) U(Wf(SpdWr U'(W) =e-mr-n[ (WT)f (ST)dWT =eT-E*[(Wr门. (2.6) This version of the Euler equation shows that the price of any asset can be expressed as a discounted expected payoff,discounted at the riskless rate of rest.However,the expectation must be take en with respect toan MRS- weighted probability density function,not the original probability density function f of future consumption. This density f*is called the state-price density(SPD)and it is the continuous- ness,f*is unique.In particular,Arrow(1964)and Debreu (1959)showed that if there are as many state-contingent claims as there are states.then the price of any security can be expressed as a weighted average of th prices ofthe state-contingent claims,now known as Arrow-Debreu state prices.In a continu- ous-state setting,f*satisfies the same property-any arbitrary security can be priced as a simple expectation with respect to f*. This underscores the importance off*for risk management:the SPD aggre- gates all economically pertinent information regarding investors'preferences endowments,asset price dynamics,and market clearing,whereas purely statist- ical descriptions of the DGP of prices do not. Within certain restrictions,it is possible to characterize the class of DGP of prices that are compatible with an equilibrium model (see,for example Bick 1990;Wang,1993;He and Leland,1993).Fixing the utility function,however,is not sufficient to identify uniquely the DGP of the price process.If parametric restrictions are imposed on the DGP of asset prices,e.g.,geometric Brownian motion,the SPD may be used toinfer the preferences of the representative agent
where Mt,T ,;@(=T )/;@(=t ) is the stochastic discount factor or marginal rate of substitution (MRS) between dates t and ¹. If we de"ne the state-price density to be f H t (S T ),f t (S T )]E[f T D S t , S T ]"f t (S T )]f T (since f T is a function of (S t , S T )) and recall that =T "S T , then we can rewrite Eq. (2.5) as Et [t(=T )Mt,T ]"P = 0 t(=T ) ;@(=T ) ;@(=t ) f t (S T ) d=T "e~r(T~t) P = 0 t(=T ) f H t (S T ) d=T "e~r(T~t)EH[t(=T )]. (2.6) This version of the Euler equation shows that the price of any asset can be expressed as a discounted expected payo!, discounted at the riskless rate of interest. However, the expectation must be taken with respect to f H, an MRSweighted probability density function, not the original probability density function f of future consumption. This density f H is called the state-price density (SPD) and it is the continuousstate counterpart to the prices of Arrow}Debreu state-contingent claims that pay $1 in a given state and nothing in all other states. Under market completeness, f H is unique. In particular, Arrow (1964) and Debreu (1959) showed that if there are as many state-contingent claims as there are states, then the price of any security can be expressed as a weighted average of the prices of these state-contingent claims, now known as Arrow}Debreu state prices. In a continuous-state setting, f H satis"es the same property } any arbitrary security can be priced as a simple expectation with respect to f H. This underscores the importance of f H for risk management: the SPD aggregates all economically pertinent information regarding investors' preferences, endowments, asset price dynamics, and market clearing, whereas purely statistical descriptions of the DGP of prices do not. Within certain restrictions, it is possible to characterize the class of DGP of prices that are compatible with an equilibrium model (see, for example Bick, 1990; Wang, 1993; He and Leland, 1993). Fixing the utility function, however, is not su$cient to identify uniquely the DGP of the price process. If parametric restrictions are imposed on the DGP of asset prices, e.g., geometric Brownian motion, the SPD may be used to infer the preferences of the representative agent Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51 15
16 Y.Ait-Sahalia,A.W.Lo/Journal of Econometrics 94(2000)9-5 in an equilibrium model of asset prices (see,for example,Bick,1987).Alterna- tively,if specific preferences are imposed,e.g.logarithmic utility,the SPD may be used to infer the DGP of asset prices.Indeed,two of the following imply the third:(1)the representative agent's preferences;(2)asset price dynamics;and(3)the SPD. 2.2.No-arbitrage models The practical relevance of SPDs for derivative pricing and hedging applica- tions has also become apparent in no-arbitrage or dynamically complete-markets models in which sophisticated dynamic trading strategies involving a set of "fundamental"securities can perfectly replicate the payoffs of more complex "derivative"securities.For example,suppose that we observe a set of n asset prices following Ito diffusions driven byindependent Brownian motions dS,=u dt +a:dz (2.7) where S,and u,are (n x 1)-vectors.Z,is an (n2 x 1)-vector of independent Brownian motions,a,i 2.Supp oose that there derivative securities on an asset with payoff function (ST)are spanned by certain dynamic trading strategies,i.e,derivatives are redundant assets hence they may be priced such applications the asset price dynamics are specified explicitly and conditions are imposed to ensure the existence of an SPD and dynamic completeness of markets(Harrison and Kreps,1979;Duffie and Huang,1985:Duffie,1996). For example,the system of asset prices S,in Eq.(2.7)supports an SPD if and only if the system of linear equations=admits at every date a solutior A,such that exp d2 has finite expectation,and oxpd nL973.19920r 197
in an equilibrium model of asset prices (see, for example, Bick, 1987). Alternatively, if speci"c preferences are imposed, e.g., logarithmic utility, the SPD may be used to infer the DGP of asset prices. Indeed, in equilibrium, any two of the following imply the third: (1) the representative agent's preferences; (2) asset price dynamics; and (3) the SPD. 2.2. No-arbitrage models The practical relevance of SPDs for derivative pricing and hedging applications has also become apparent in no-arbitrage or dynamically complete-markets models in which sophisticated dynamic trading strategies involving a set of `fundamentala securities can perfectly replicate the payo!s of more complex `derivativea securities. For example, suppose that we observe a set of n 1 asset prices following Ito( di!usions driven by n 2 independent Brownian motions: dS t "l t dt#r t dZt (2.7) where S t and l t are (n 1 ]1) -vectors, Zt is an (n 2 ]1)-vector of independent Brownian motions, r t is an (n 1 ]n 2 )-matrix, and n 1 *n 2 . Suppose that there exists a riskless asset with instantaneous rate of return r. Then path-independent derivative securities on an asset with payo! function t(S T ) are spanned by certain dynamic trading strategies, i.e., derivatives are redundant assets hence they may be priced by arbitrage.6 In such applications the asset price dynamics are speci"ed explicitly and conditions are imposed to ensure the existence of an SPD and dynamic completeness of markets (Harrison and Kreps, 1979; Du$e and Huang, 1985; Du$e, 1996). For example, the system of asset prices S t in Eq. (2.7) supports an SPD if and only if the system of linear equations r t ) k t "l t admits at every date a solution k t such that expCP T t k q ) k q dq/2D has "nite expectation, and expC !P T t k q dZq !P T t k q ) k q dq/2D 6 Additional assumptions are, of course, required such as frictionless markets, unlimited riskless borrowing and lending opportunities at the same instantaneous rate r t,q , a known di!usion coe$cient, etc. See Merton (1973, 1992) for further discussion. 16 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 1 has finite variance.In the presence of an SPD,markets are complete if and ony if rank()=n2 almost everywhere.Then the SPD can be characterized explicit ly without reference to preferences.In the particular case of geometric Brownian motion,with constant volatility o,interest rate r and dividend yield over dS=.x-ò，Sdt+oSdZ which is a lognormal distribution with mean((r.t-8.)-2/2)t and variance 石2T More generally,denote by S,the price of an underlying asset and by f*(Sr) the SPD of the asset price ST at a future date T,conditioned on the current price S,[for simplicity,we leave implicit the dependence off*on(S.,t,r.]. Consider now a European-style derivative security with a single liquidat- equal to e-me't (ST)f*(ST)dSr (2.8) Jo For example,a European call option with maturity date T and strike price X has a payoff function (Sr)=max[ST-X,0]hence its date-t price H,is simply ”+00 H(S X,t,r1.)=e rat naxST-X.Of,*(Sr）dSr (2.9) Even the most complex path-independent derivative security can be priced and hedged according toE(8). 3.Economic VaR The relevance of the SPD for risk management is clear:the MRS-weighted probability density ically complete mea of value-at-risk-economic value-than the probability density function fof the DGP.Therefore,f*can be used in the same ways that statistical VaR measures such as standard deviation,95%confidence intervals,tail probabilities,etc.are used.To distinguish the more traditional method of risk management from this 7See also Cox and Ros(1976),Goldberger(1991),Madan and Milne(1994)and Rady (1994)
has "nite variance. In the presence of an SPD, markets are complete if and only if rank(r t )"n 2 almost everywhere. Then the SPD can be characterized explicitly without reference to preferences. In the particular case of geometric Brownian motion, with constant volatility p, interest rate r t,q and dividend yield d t,q over the interval (t, t#q), the SPD or risk-neutral pricing density is given by the conditional distribution of the risk-neutral stochastic process with dynamics dSH t "(r t,q !d t,q )SH t dt#pSH t dZt which is a lognormal distribution with mean ((r t,q !d t,q )!p2/2)q and variance p2q. More generally, denote by S t the price of an underlying asset and by f H t (S T ) the SPD of the asset price S T at a future date ¹, conditioned on the current price S t [for simplicity, we leave implicit the dependence of f H t on (S t , q, r t,q , d t,q )]. Consider now a European-style derivative security with a single liquidating payo! t(S T ). To rule out arbitrage opportunities among the asset, the derivative and a risk-free cash account, the price of the derivative at t must be equal to e~rt,q>q P `= 0 t(S T ) f H t (S T ) dS T . (2.8) For example, a European call option with maturity date ¹ and strike price X has a payo! function t(S T )"max[S T !X, 0] hence its date-t price Ht is simply H(S t , X, q, r t,q )"e~rt,qq P `= 0 max[S T !X, 0] f H t (S T ) dS T . (2.9) Even the most complex path-independent derivative security can be priced and hedged according to Eq. (2.8).7 3. Economic VaR The relevance of the SPD for risk management is clear: the MRS-weighted probability density function f H provides a more economically complete measure of value-at-risk } economic value } than the probability density function f of the DGP. Therefore, f H can be used in the same ways that statistical VaR measures such as standard deviation, 95% con"dence intervals, tail probabilities, etc. are used. To distinguish the more traditional method of risk management from this 7 See also Cox and Ros (1976), Goldberger (1991), Madan and Milne (1994) and Rady (1994). Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51 17
18 Y.Ai-Sahalia,A.W.Lo/Journal of Econometrics 94(2000)9-51 approach,we shall refer to the statistical measure of value-at-risk as 'S-VaR' well Now if the MRS in Eq.(2.5)were observable,implementing E-VaR measures and comparing them to S-VaR measures would be a simple matter.However. in practice obtaining f*can be quite a challenge,especially for markets that are more complex than the pure-exchange economy described in Section 2.1.Fortunately,several accurate and computationally efficient estimators of f*have been developed recently and we provide a brief review of these stimatorsin derive their.1 of Appendix A.With these estimators in hand,we show in Section 4 how to gauge the relative importance of E-VaR empirically by examining the ratio ff 3.1.SPDs and option prices Banz and Miller (1978),Breeden and Litzenberger(1978),and Ross (1976) were among the first to suggest that Arrow-Debreu prices may be estimated ices of traded financial securities.In particular ight that ontions can be uset to ereate pure Arrow-Debreu state-contingent securities,Banz and Miller(1978)and Breeden and Litzenberger(1978)provide an elegant method for obtaining an explicit expression for the SPD from option prices:the SPD is the second derivative (normalized to integrate to unity)of a call option pricing formula with respect to the strike price. To see why.consider the portfolio obtained by selling two call options struck at X and buying one struck at -sand one atX+Consider 1/e2 shares of this called because of the shape of its payof function (ST)which pays nothing outside the interval [X-8,X +&]Letting tend to zero,the payoff function of the butterfly tends to a Dirac delta function with mass at X,i.e,in the limit the butterfly becomes an elementary Ar- row-Debreu security payingf-and nothing otherwise.The limit ofits price as a tends to zero should therefore be equal to e'f*(X).Now denote by H(S,.X,t)the market price of a call option at time t with strike price X, time-to-maturity and underlying asset price S Then,by construction,the price of the butterfly spread must be -2HSnX,)+HS。X-6)+HS。X+6功 (3.1) which has,as its limit as→0，a2HS,X,t)/aX2
approach, we shall refer to the statistical measure of value-at-risk as &S-VaR' since it is based on a purely statistical model of the DGP, and call the SPD-based measure `E-VaRa since it is based on economic considerations as well. Now if the MRS in Eq. (2.5) were observable, implementing E-VaR measures and comparing them to S-VaR measures would be a simple matter. However, in practice obtaining f H can be quite a challenge, especially for markets that are more complex than the pure-exchange economy described in Section 2.1. Fortunately, several accurate and computationally e$cient estimators of f H have been developed recently and we provide a brief review of these estimators in Section 3.1 and derive their asymptotic distributions in Section A.1 of Appendix A. With these estimators in hand, we show in Section 4 how to gauge the relative importance of E-VaR empirically by examining the ratio f H/f. 3.1. SPDs and option prices Banz and Miller (1978), Breeden and Litzenberger (1978), and Ross (1976) were among the "rst to suggest that Arrow}Debreu prices may be estimated or approximated from the prices of traded "nancial securities. In particular, building on Ross's (1976) insight that options can be used to create pure Arrow}Debreu state-contingent securities, Banz and Miller (1978) and Breeden and Litzenberger (1978) provide an elegant method for obtaining an explicit expression for the SPD from option prices: the SPD is the second derivative (normalized to integrate to unity) of a call option pricing formula with respect to the strike price. To see why, consider the portfolio obtained by selling two call options struck at X and buying one struck at X!e and one at X#e. Consider 1/e2 shares of this portfolio, often called a &butter#y' spread because of the shape of its payo! function t(S T ) which pays nothing outside the interval [X!e, X#e]. Letting e tend to zero, the payo! function of the butter#y tends to a Dirac delta function with mass at X, i.e., in the limit the butter#y becomes an elementary Arrow}Debreu security paying $1 if S T "X and nothing otherwise. The limit of its price as e tends to zero should therefore be equal to ert,qqf H(X). Now denote by H(S t , X, q) the market price of a call option at time t with strike price X, time-to-maturity q, and underlying asset price S t . Then, by construction, the price of the butter#y spread must be 1 e2 [!2H(S t , X, q)#H(S t , X!e, q)#H(S t , X#e, q)] (3.1) which has, as its limit as eP0, R2H(S t , X, q)/RX2. 18 Y. An(t-Sahalia, A.W. Lo / Journal of Econometrics 94 (2000) 9}51