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An Alternative Valuation Equation for Contingent Claims Gurdip S. Bakshi and Zhiwu Chen September 27, 1995 Bakshi is at Department of Economics and Finance, College of Business Administration, University of New Orleans, LA 70148, Tel: (504)-286-6096, email: gsbef@uno.edu; and Chen at Department of Finance, College of Business, Ohio State University, 1775 College Road, Columbus, OH 43210, Tel: 614-688-4107, email: chen@cob.ohio- state.edu. For helpful comments and suggestions, we would like to thank Charles Cao, Craig Holden, Louis Scott, and Ren Stulz. Any remaining errors are ours alone. e 0 An Alternative Valuation Equation for Contingent Claims Abstract The fundamental valuation equation of Cox, Ingersoll and Ross is expressed in terms of the indirect utility of wealth function. As closed-form solution for the indirect utility is generally unobtainable when investment opportunities are stochastic, existing contingent claims models involving general asset price processes are almost all derived under the restrictive log utility assumption. An alternative valuation equation is proposed here that depends only on the di- rect utility function. To show the wider applicability of this alternative, we derive closed-form solutions for bonds, bond options and stock options under both power utility and exponen- tial utility functions. Allowable stochastic processes for the aggregate endowment and state variables are quite general and empirically plausible. Our option pricing model with stochas- tic volatility and stochastic interest rates has most existing models nested in it. This means most existing models also hold for economies with power or exponential utility functions. The option pricing model is also shown to have the ability to reconcile certain puzzling empirical regularities such as the volatility smile. JEL Classication Numbers: G10, G12, G13 Keywords: fundamental valuation equation, stock options, interest rates, interest rate deriva- tives, contingent claims 1 The well-known fundamental valuation equation of Cox, Ingersoll and Ross (henceforth, CIR) (1985a) has been the basis for most recent models on contingent claims valuation and on the term structure of interest rates. Their work has provided the necessary push for the development of more general and empirically plausible valuation models beyond Black and Scholes (1973). Like the Merton (1973) intertemporal asset pricing model, however, their fundamental valuation equation is expressed in terms of the indirect utility of wealth or the value function. As such, one has to follow two steps in order to apply their valuation equation to price stocks, bonds or any contingent claims: (i) solve explicitly for the indirect utility function the Hamilton-Jacobi-Bellman equation from the investor's consumption-portfolio problem, and (ii) substitute the indirect utility into the fundamental valuation equation and solve for the contingent claim price. According to Merton (1971), unfortunately, one typically cannot nd a closed-form solution for the indirect utility function, unless the investor has a log period utility or the investment opportunities are non-stochastic. For this reason, virtually all existing equilibrium valuation models for contingent claims as well as models for the term structure of interest rates assume a log utility function.1 The few exceptions are the discrete-time models of Amin and Ng (1993), Brennan (1979), Rubinstein (1976), Sun (1992), and Turnbull and Milne (1991) where they typically allow the representative agent to have a power utility function. On the surface, it appears that one can derive contingent claims pricing models in a more general setting using a discrete-time framework than using its continuous-time counterpart, clearly contradictory to the fact that a continuous-time framework usually oers more, rather than less, convenience. The purpose of this paper is to propose an alternative fundamental valuation equation that depends solely on the direct utility of consumption, so that one only needs to solve a single partial dierential equation (PDE) for contingent claim prices, rather than two PDEs in two steps. The basic idea is quite simple. From the well known Breeden (1979) consumption CAPM equation, the equilibrium risk premium for any claim is equal to the relative risk aversion in consumption times the covariance between the claim's return and aggregate consumption growth. In Breeden's type of equilibrium, however, consumption is endogenously determined. Consequently, in order to literally use Breeden's pricing equation to value contingent claims, one would need to rst solve for the optimal consumption process in closed-form before going any further. But, that would essentially lead one back to the same diculty as with the fundamental valuation equation of CIR, because we would have to know the exact indirect utility function in order to nd the optimal consumption plan and vice versa. To avoid this diculty completely, we resort to the Lucas (1978) pure exchange economy setup in which the aggregate output, and hence the optimal aggregate consumption 1 For a partial list, see Bailey and Stulz (1989), Chen and Scott (1992), CIR (1985b), Longsta (1990), Longsta and Schwartz (1992), and Scott (1995). 1 process is exogenously given. Then, substituting the exogenous aggregate output into Breeden's pricing relation and applying Ito's lemma to the contingent claim price results in the desired alternative valuation equation. Just like the CIR fundamental valuation equation is a synthesis of Merton's intertemporal CAPM and Lucas' exchange economy, our alternative valuation equation is a synthesis of Breeden's consumption CAPM and Lucas' pure exchange economy. Without either Breeden's pricing relation or Lucas' equilibrium concept, the convenience that comes with our valuation equation would not be there. Besides, as the aggregate output is exogenous in a pure exchange economy, it aords researchers the exibility to choose any empirically and technically plausible stochastic process for output. To show the wider applicability of the alternative valuation equation, we examine pricing issues for bonds, bond options and individual stock options under the two most widely-used classes of utility functions: the power and the exponential utility class. Economically, these two classes are interesting because the power utility functions form the constant relative risk aversion (CRRA) class while the exponential utility functions represent the constant absolute risk aversion (CARA) class. For the example economy that we examine, there are two systematic state variables that each follow a mean-reverting square-root process as in, for instance, Longsta and Schwartz (1992). But, unlike in Longsta and Schwartz (1992), the aggregate endowment process has both its drift and volatility being a function of the two state variables. The resulting endogenous interest rates and bond price volatility are also functions of both state variables. The term structure model developed here contains most elements of Longsta and Schwartz's (1992) model, and in comparison our model has at least two additional features: (i) the term structure is now a function of the agent's risk aversion and (ii) the term premium in our model has more desirable properties. First, as the agent becomes more risk averse (in either the absolute or the relative sense), its impact on interest rates is ambiguous, depending on where the risk aversion level is. The reason is that even though a higher risk aversion implies, on the one hand, lower interest rates, the intertemporal elasticity in consumption substitution decreases at the same time, which produces, on the other hand, a positive impact on interest rates. Thus, as risk aversion increases, interest rates may increase or decrease, re ecting the joint working of the two opposite eects. A similar point is made by Wang (1995) in a dierent context.2 Second, as Constantinides (1992) argues, the term premium in the CIR (1985b) single-factor term structure model can only be either monotonically increasing or monotonically decreasing in the term to maturity, whereas in practice other term premium shapes are also observed. In contrast, like Constantinides' (1992) 2 Wang (1995) examines the impact of investor heterogeneity on the term structure of interest rates. For the case of investors with the same power utility function, he also nds a closed-form term structure. But, the focus and issues involved are quite dierent between his work and ours. 2 model, the term premium here can take any desired shape. The closed-form bond option pricing formula is also a two-factor model that depends on the risk aversion level. The structure of this formula resembles those in Heston (1993) and Scott (1995) in that the probabilities that the option expires in the money are recovered by inverting their respective characteristic functions, and it is dierent from the formulas in CIR (1985b) and Longsta and Schwartz (1992) where the probabilities are obtained by integrating over a bivariate chi-square distribution function. In deriving the stock option pricing model with stochastic volatility and stochastic interest rates, we allow the volatility of the underlying stock's return to be driven by both a systematic and an idiosyncratic state variable, as is done in Amin and Ng (1993). This volatility structure re ects the growing empirical evidence that individual stock volatility is not only stochastic over time, but it is also correlated with systematic or market-wide volatility. The resulting stock option pricing model has three factors (in addition to its dependence on the underlying stock price), which means that we can expect the model to capture more variations in option prices, both across dierent strikes or maturities and over time. In particular, the decomposition of volatility into a systematic and a rm-specic component allows one to analyze both how an equity option price will respond to a change in systematic volatility versus to a change in the rm- specic volatility component and how options with distinct risk characteristics may dier in value. Like in the bond option formula, the probabilities in the stock option formula are obtained by inverting their characteristic functions. Even though in a discrete-time framework Amin and Ng (1993) also assume such a volatility structure and a power utility function for the representative agent, their option pricing model is expressed as a conditional expectation of the Black-Scholes formula and hence they do not have a closed-form solution for option prices. In contrast, our stock option formula is not only given in closed-form (which makes its applications straightforward), but also it applies to economies with exponential utility. To see both how risk aversion may aect interest rates and option prices and what empirically appealing features our option valuation model may exhibit, we study an articial economy in which the structural parameter values are all chosen so that the initial term structure is in line with some known features of real-life term structures. First, it is shown that interest rates in general have a hump-shaped relation with risk aversion: interest rates initially increase with risk aversion and then decrease as risk aversion increases beyond a critical point. Second, note that risk aversion in uences stock option prices mostly through its impact on interest rates. It turns out that equity option prices also have a hump-shaped response to a changing risk aversion level. Third, our equity option model can resolve the volatility smile puzzle that has been empirically documented for all option markets [e.g., Bates (1995b) and Rubinstein (1985, 1994)]. More precisely, when 3 option prices determined by our option valuation model are used as input into the Black-Scholes model to back out the implied volatility, the implied volatilities exhibit a U-shaped pattern across strike prices. This is particularly true for short-term options. This empirically plausible feature means that our model will likely perform better than most existing ones in explaining real-life option prices. The paper is organized as follows. Section 1 introduces the basic continuous-time economy and derives the alternative valuation equation for contingent claims. Section 2 rst describes an example economy with power utility investors and then solves for the term structure of interest rates and bond option prices. Continuing with the economy in Section 3, we develop a pricing model for individual stock options with stochastic volatility and stochastic interest rates. In Sec- tion 4, we show that all results from Sections 2 and 3 still hold even in economies with exponential utility. Section 5 is devoted to using an articial economy to study properties of the stock option pricing model. Concluding remarks are oered in Section 6, and proof and derivation of results are given in the Appendix. 1 An Alternative Valuation Equation Consider a continuous-time pure exchange economy of the Lucas (1978) type in which a sole perishable consumption good is produced and markets are dynamically complete. This economy has a nite time horizon [0; T ]. As in CIR (1985a) and Lucas (1978), assume that there is a representative agent who is an expected utility maximizer with preferences given below:3 ZT u(c) E0 e t U (c(t))dt; (1) 0 for a given consumption plan fc(t) : t 2 [0; T ]g, a time preference parameter , and a thrice continuously dierentiable period utility U : < ! <, where c(t) is the amount of the good consumed and Uc > 0 and Ucc < 0, with the subscripts on U denoting partial derivatives.4 In this economy there are N production rms whose capital stock (with dividends included) 3 In this paper, all random variables are dened on a given complete probability space ( ; F ; P r). A stochastic process c is then a collection of random variables fc(t) : t 2 [0; T ]g on ( ; F ; P r). Throughout the paper we use the standard ltration fFt : t 2 [0; T ]g generated by the standard (N + M + 1)-dimensional vector Brownian motion (!q ; !1 ; : : : ; !N ; !X1 ; : : : ; !XM ), which is dened shortly. Conditional expectations, Et (), are dened according to this ltration. Every time-t variable, such as c(t) and y(t), is taken to be Ft -measurable. For simplicity, we suppress the qualier \almost surely" when taking equality between two random variables. 4 For our general discussion, the period utility function can be state-dependent as in CIR (1985a) and all of our characterizations in this section will not be aected. It may even be allowed to exhibit other forms of state- or path-dependence such as habit formation as in Constantinides (1990). In those latter cases, however, some of the characterizations may have to be modied. 4 Sn, for n = 1; : : :; N , evolves according to the following diusion processes: dSn(t) = n (q; S; X; t) dt + n(q; S; X; t) d!n(t); (2) where S = (S1 ; : : :; SN )0 with each Sn (0) > 0, X = (X1; : : :; XM )0, and !n is a standard Brownian motion. Throughout the paper we maintain the assumption that the drift and diusion terms in each stochastic dierential equation, such as n (q; S; X; t) and n (q; S; X; t), are all time-t measurable functions and that they satisfy the local Lipschitz and growth conditions [e.g., footnote 4 of CIR (1985a)], implying that there is a unique solution to each stochastic dierential equation. In writing n (q; S; X; t) and n (q; S; X; t), we allow the investment growth process of each rm to depend on the aggregate output q , the M state variables Xm , and each rm's capital stock Sn . Each rm has one equity share issued. In other words, the representative agent holds one share of each rm in his initial endowment. Implicit in the above setup is the assumption that, unlike in CIR's (1985a, 1985b) production economy, each rm's production decision is exogenous to our model.5 Furthermore, assume that each rm's production decisions are such that the aggregate output of the economy, q , also follows a diusion process: d q(t) = q (q; S; X; t) dt + q (q; S; X; t) d !q(t); (3) where !q is again a standard Brownian motion, and the expected change, q (q; S; X; t), and the volatility of changes, q (q; S; X; t), in aggregate output can both be functions of the state of the economy as well as the state of each rm. Each state variable Xm , for m = 1; : : :; M , is also assumed to follow a diusion process: d Xm(t) = m (X; t) dt + m (X; t) d !Xm (t); (4) where the drift and diusion terms, m (X; t) and m (X; t), are functions of only the general state of the economy. Consumption smoothing in this economy is achieved through trading in (i) the equity shares issued by the rms, (ii) one (real) discount bond, and (iii) M nancial claims whose net supply is zero. To describe the agent's consumption-portfolio problem, let Fm (t) be the time-t price of 5 As emphasized by Sun (1992), a pure exchange economy can be supported by a CIR-type production economy, and vice versa. We chose a pure exchange economy since it oers us the exibility to directly assume any desirable exogenous process for the aggregate production output, rather than deriving the aggregate output process endoge- nously from within the model (as one would need to within the CIR production economy framework). Typically, the endogenously determined processes for aggregate output may not render it possible to solve for such variables as interest rates and asset prices in closed-form. 5 the m-th nancial claim and R(t) the time-t instantaneous interest rate. Further, let n (t) be the number of shares held of rm n and m (t) the number of units held of the m-th nancial claim. Then, the agent's problem is to solve max u(c); c; (5) subject to the budget constraint X N X M " X N X M # d W (t) = n(t)dSn (t)+ m (t)dFm (t)+ W (t) n(t)Sn(t) m (t)Fm(t) R(t)dt c(t)dt; n=1 m=1 n=1 m=1 (6) P with W (0) = N Sn (0) > 0, where W (t) is the time-t wealth (in terms of units of the consump- n =1 tion good) generated by the plan (c; ). Together, the processes in (q; S; X ) are taken to be jointly Markov. Take any contingent claim whose value may depend on the aggregate output, the state of the economy and the state of each rm. That is, we can write F (q; S; X; t) as the time-t price of such a claim. Assume that F is at least twice continuously dierentiable in every argument. For now, we further assume that the optimal consumption process c from the optimization problem in (5) is also a diusion process. Then, from the existing literature on consumption-based asset pricing, it is known that the rst-order conditions for the problem in (5) lead to the following restrictions on R(t) and expected returns: c Ucc (t) 1 c Uccc (t); 2 R(t) = + Uc c 2 Uc 2 c (7) [see equation (21) of Breeden (1986)], where c (t) and c (t) are respectively the conditional ex- pected value and standard deviation of instantaneous consumption growth, and c Ucc dF (t) dc(t) F (t) R(t) = Uc Covt F (t) ; c(t) ; (8) [see, e.g., equation (17) of Breeden (1979) or equation (30) of CIR (1985a)], where F (t) is the conditional expected rate of return on the nancial claim and Covt (; ) the conditional covariance operator. These two equations form the core of the consumption-based asset pricing theory. In this Lucas-type representative-agent economy the perishable aggregate output is exoge- nously given. In equilibrium, the agent's optimal holdings should be: n (t) = 1 for n = 1; : : :; N and m (t) = 0 for m = 1; : : :; M . Consequently, the optimal consumption process is equal to the aggregate output process: c(t) = q (t). These equilibrium conditions together with equations (7) and (8) lead to the following alternative fundamental valuation equation. 6 Theorem 1 The equilibrium price for any contingent claim, F , is a solution to the following partial dierential equation: @F @F (q; S; X; t) + X @F (q; S; X; t) + X @F (X; t) R(t) F + @q q N M @t n m n=1 @Sn m=1 @Xm 1 @ 2F 2(q; S; X; t) + 1 X X @ 2F Cov (dS ; dS 0 ) + 1 X X @ 2F Cov (dX ; dX 0 ) + 2 @q 2 q N N M M 2 n=1 n0 =1 @Sn @Sn0 t n n 2 m=1 m0 =1 @Xm@Xm0 t m m X @2 N X @ 2F M X X @ 2F N M + @S F Covt (dSn ; dq ) + Covt (dXm; dq) + Covt (dSn ; dXm) n=1 n @q m=1 @Xm @q n=1 m=1 @Sn @Xm ( 2 ) = q (t) Uqq q (q; S; X; t) @F + X Cov dS ; dq @F + X Cov dX ; dq @F ; N M (9) Uq q(t) @q n=1 t n q @Sn m=1 t m q @X m subject to the relevant boundary conditions for the contingent claim as determined by the terms of the contract, where ! R(t) = + q (t)Uqq q (q; S; X; t) 1 q2 (t)Uqqq q (q; S; X; t) 2 : (10) Uq q(t) 2 Uq q(t) The valuation PDE in (9) applies to any contingent claim satisfying the smoothness condition, whether interest rate-sensitive or equity-sensitive. The left-hand side of the equation is obtained via applying Ito's lemma to F (q; S; X; t), while the right-hand side determines the equilibrium risk compensation for the claim. This fundamental valuation equation is the Lucas-type pure exchange economy counterpart to the fundamental valuation equation of CIR (1985a). The two valuation equations are equivalent to the extent that a pure exchange economy can be supported by a production economy and vice versa or even to the extent that Breeden's (1979) CAPM is equivalent to Merton's (1973) ICAPM. Operationally, however, the two alternatives are quite dierent. Note that in CIR's valuation equation, the PDE is expressed in terms of the indirect utility of wealth or value function that is a solution to the Hamilton-Jacobi-Bellman equation for the investor's consumption-portfolio choice problem. In contrast, the PDE in (9) involves only the direct utility of consumption, U (c), so that one does not need to solve the Hamilton-Jacobi- Bellman equation in advance and can directly use our valuation PDE to solve for the contingent claim price. Our valuation equation also diers from Breeden's (1979) consumption CAPM in a subtle way. In his case, aggregate consumption is endogenously determined re ecting individual rms' production outputs and investors' consumption-production decisions, which means that the pro- cess followed by aggregate consumption cannot be given outside of the model. This leads us back 7 to the diculties of the CIR fundamental valuation equation, because one cannot solve for the representative agent's optimal consumption in closed form unless he is able to solve explicitly for the indirect utility of wealth function. This feature makes it equally hard for one to literally apply Breeden's consumption CAPM to value contingent claims since, even if one can solve for the endogenous consumption process, closed-form expressions for contingent claim prices may still be unobtainable unless the endogenous aggregate consumption process has certain structure. In contrast, the aggregate output/consumption process in the Lucas-type exchange economies is ex- ogenously given, which oers researchers much exibility in choosing the \right" stochastic process for aggregate output so that a closed-form solution to the PDE in (9) can be found. Finally, the solution to the PDE in (9) can be given in the same functional form as in either Lemma 3 or Lemma 4 of CIR (1985a), with appropriate modications. Take their Lemma 4 as an example. We rst need to adjust the drift of the contingent claim's return process using the risk premium expression in (8) and without involving the indirect utility function, and then construct the equivalent martingale measure using the partial derivative Uc . This equivalent martingale probability measure is thus dened in terms of the direct utility function. Next, we replace the interest rate and the expectation operator in their Lemma 4 respectively by our interest rate equation in Theorem 1 and the expectation operator dened by this equivalent martingale measure. The resulting expression is the direct utility-based counterpart to their indirect utility- based risk-neutral valuation equation. Since the formula for our case looks virtually the same as in their Lemma 4, the repetition is omitted here. 2 Interest Rates and Interest Rate Derivatives in an Economy with CRRA Agents In the remainder of the paper, we demonstrate that the alternative valuation equation allows one to solve for contingent claim prices and the term structure of interest rates under more general utility functions. For this purpose, we rst work with an example economy in which the representative agent has a general power utility function: c1 U (c) = 1 ; (11) where > 0 is the coecient of relative risk aversion, and in which the output follows dq (t) = ( + x(t) + y (t)) dt + qx(t) d! (t) + qy (t) d! (t); (12) q(t) q x y q q q ^ q ^ 8 where the two state variables, x and y , respectively follow a mean-reverting square-root process: q d x(t) = x (x x(t)) dt + x x(t) d!x (t) (13) q d y(t) = y (y y (t)) dt + y y (t) d!y (t); (14) where again all the parameters are positive constants. The four standard Brownian motion pro- cesses, (!x ; !y ; !q ; !q ), are mutually independent except that !x may be correlated with !q while ^ !y may be correlated with !q^. Let Covt(d!q ; d!x) = q;x and Covt(d!q^; d!y ) = q;y , for constants ^ q;x and q;y . Clearly, (q; x; y ) are jointly Markov. ^ The output process in (12) closely resembles the one assumed in Longsta and Schwartz (1992): expected output growth depends on x(t) and y (t), except that here the conditional volatility of output growth is a linear function of both state variables. In this economy with stochastically changing production and investment opportunities and power utility function, it is known that one may not be able to explicitly solve for the indirect utility of wealth function or for the optimal consumption plan [Merton (1971)]. This is the reason that Longsta and Schwartz (1992), among others, had to rely on the log utility function, which is a special case of the power utility in (11). Solving the stochastic dierential equation in (12), we have the time- output given by Z 1 2) x(t) + ( 1 2 ) y (t) dt q( ) = q(0)exp q + (x 2 q y 2 q ^ Zq0 Z q +q x(t) d!q (t) + q^ y (t) d!q^(t) ; (15) 0 0 with q (0) > 0. In this economy, the equilibrium risk premium for any contingent claim F is q q F (t) R(t) = q x(t)Covt dF((tt)) ; d!q (t) + q^ y(t)Covt dF((tt)) ; d!q^(t) ; F F (16) which is obtained by substituting the utility function in (11) and the processes in (12)-(14) into equation (8). Let's rst examine a pure discount bond that pays 1 unit of consumption in periods and whose time-t price is B (t; ). Then, following a variational argument and using the utility function in (11), B (t; ) is determined in equilibrium by q B (t; ) = e Et t+ : (17) qt Substituting (15) into this equation and realizing that the processes in (!q ; !q ; !x ; !y ) are standard ^ Brownian motions, we conclude that the above conditional expectation can only be a function of 9 x(t) and y (t). Write B(t; ; x; y ). Substituting B(t; ; x; y ) into the fundamental valuation equation in (9) and specializing it to the present context leads to the following PDE for the discount bond:6 1 2 x @ 2 B + [ ( + ) x] @B + 2 x @x2 x x x x q q;x @x 1 2 y @ 2 B + [ ( + ) y ] @B @B R B = 0; (18) 2 y @y 2 y y y y q q;y ^ ^ @y @ subject to the boundary condition: B (t + ; 0; x; y ) = 1. Using a standard separation-of-variable technique, the solution for the bond price is: B (t; ) = exp [ ( + q) x( ) y ( ) %x( ) x(t) %y ( ) y(t)] ; (19) r n o where, letting x xq q;x, y y q q;y , #x (x + x)2 + 2 x x 2 1 (1 + )q , and 2 r ^ ^ 2 n o #y (y + y )2 + 2 y y 2 1 2 (1 + )q , 2 ^ ( " # ) 2 x x ln 1 + (1 e #x )(x + x #x ) + 1 [# ] x ( ) = 2 x 2 #x 2 x x x ( " # ) 2 y y ln 1 + (1 e #y )(y + y #y ) + 1 [# ] y ( ) = 2 y 2 #y 2 y y y n o 2 x 1 (1 + )q 1 e #x 2 %x( ) = 2# + ( + # ) (1 e #x ) 2 (20) x n x1 x 2xo 2 y 2 (1 + )q 1 e #y %y ( ) = 2# + ( + # ) 1 e #y : ^ (21) y y y y Intuitively, x and y can be thought of as the equilibrium factor risk premiums respectively for state variables x and y . The two functions, %x( ) and %y ( ), can be interpreted as the -period bond's sensitivity (or exposure) to, respectively, x risk and y risk. As can be checked, both %x ( ) and %y ( ) are increasing in the term to maturity , which means that longer-term bonds tend to be more sensitive to both x and y risk. By Ito's lemma, the bond price dynamics are described by dB (t; ) = fR(t) % ( )x(t) % ( )y (t)g dt % ( )qx(t)d! (t) % ( )qy (t)d! (t): B (t; ) x x y y x x x y y y (22) 6 Sun (1992) derives a PDE from the conditional expectation in (17) for bond prices, whereas the PDE in (18) is obtained by applying the fundamental valuation equation in (9), which is clearly a more convenient alternative. 10 The implied -period yield-to-maturity is R(t; ) ln [B t; )] = + q + x( ) + y ( ) + %x ) x(t) + %y ) y (t) ( ( ( (23) which represents a two-factor term structure of interest rates. The instantaneous interest rate is given by R(t) lim0 R(t; ) = + [q + x x(t) + y y (t)] (1 2 ) [q x(t) + q^ y (t)]: ! + 2 2 (24) This two-factor term structure of interest rates specializes to the one given in Longsta and Schwartz (1992) if we take ! 1 and q = 0. Other single- and two-factor term structure models, such as the single-factor model of CIR (1985b) and Sun (1992), are nested within our model as well. This means that our model possesses most of the properties shared by the existing two-factor term structure models. However, since the utility function here is more general than assumed in most existing models and since the output volatility is driven by two state variables [rather than by one as in Longsta and Schwartz (1992)], our model has its own unique appealing features. First, the instantaneous interest rate is a function of the risk aversion level and of the current state of the economy. Depending on the magnitude of the structural parameters and the state of the economy, interest rates can be both higher and lower as the coecient increases. This is the case because the reciprocal of is also the intertemporal elasticity coecient. As increases, the intertemporal elasticity coecient decreases, which means the equilibrium interest rate has to increase in order to induce the agent to substitute future consumption for current consumption. This eect of an increase in is re ected by the second term on the right-hand side of equation (24). On the other hand, an increase in also means a higher level of risk aversion on the part of the agent, which tends to depress the interest rate. This negative eect is captured by the last term in equation (24). For this reason, the overall eect of a higher on R(t) is mixed. Compared to the log-utility economies in CIR (1985b) and Longsta and Schwartz (1992), therefore, economies with 6= 1 may have higher or lower interest rates. Second, the term premium in our model possesses more realistic properties. As noted by Constantinides (1992), the term premium in CIR-type single-factor models is either monotonically increasing or monotonically decreasing in the term to maturity, which is counterfactual. Like the term structure model of Constantinides (1992), ours allows the term premium to have many types of shapes, including the common monotonically increasing, decreasing, humped, or inversely 11 humped term premium shapes. More specically, 1 t; TP (t; ) dt Et dB((t; )) R(t) = x q q;x %x( ) x(t) y q^q;y %y ( ) y (t): (25) B ^ To see what happens within our model, divide the discussion into four cases. Case (i): q;x > 0 and q;y > 0. In this case, innovations in output growth are positively correlated with those in the ^ state variables. Thus, both x and y risks represent \positive" system risks that require a positive premium (i.e., both x and y must be positive). But, since bond prices in this case are negatively related to both x and y [see equation (22)], bonds in eect provide insurance against unfavorable movements in x and y and hence in output q . This explains why the term premium is negative in this case, regardless of the term to maturity . Furthermore, as %x( ) and %y ( ) are increasing in , longer-term bonds provide better insurance against unfavorable movements in q and are hence better hedging instruments, which implies that the term premium will be monotonically decreasing in . Now, consider Case (ii) in which q;x < 0 but q;y > 0. In this case, x represents ^ \negative" systematic risk. Since bond prices are negatively related to x, the component of risk in any bond which is due to x requires positive risk compensation, which is why the rst term on the right-hand side of equation (25) is positive under this scenario. In addition, the rst term is increasing in . On the other hand, for the same reason as given above, the second term in equation (25) is negative and decreasing in . As a result of these two opposite eects, the shape of the term premium in relation to can be anything but monotonic in this case, depending on the structural parameters and the state of the economy. Case (iii): q;x < 0 and q;y < 0, which is ^ just the opposite of Case (i). The term premium will then be monotonically increasing in . Case (iv): q;x > 0 and q;y < 0. This is similar to Case (ii), and non-monotonic term premium shapes ^ can arise. In addition, depending on the signs of the structural parameters and the state of the economy, an increase in risk aversion can mean a higher or lower term premium for any given . Observe that in the case of Longsta and Schwartz (1992), q = 0, which means the rst term on the right-hand side of equation (25) is zero. Then, as %y ( ) is monotone in , their model only permits either monotonically increasing or monotonically decreasing term premium shapes. Consider a European call option that matures in periods and is written on a -period pure ~ discount bond, B (t; ), where > . Let G(t; ) be the time-t price of the call. Using a standard ~ ~ argument, we have " # G(t; ) = e E q(t + ) max f0; B(t + ; ) K g ; ~ (26) t q(t) where K is the exercise price. By the production output process in (15) and the bond price 12 dynamics in (22), the conditional expectation in (26) is only a function of x(t) and y (t), which allows us to write G(t; ; x; y ). As G(t; ; x; y ) and B (t; ; x; y ) are both functions of x and y , the valuation PDE in (18) must also apply to the option price G(t; ; x; y ), except that the boundary condition becomes: G(t + ; 0; x; y ) = max f0; B (t + ; ) K g. Solving the resulting PDE, ~ we obtain the bond option price: G(t; ) = B(t; ) 1(t; ; x; y) K B(t; ) 2(t; ; x; y); ~ (27) with the two probabilities, 1 and 2 , given by " ~ # 1 + 1 Z 1 Re e i K fj (t; ; x; y ; ) d; j (t; ; x; y ; ) = 2 for j = 1; 2; (28) 0 i where Re[] stands for the real value part of the expression, i stands for imaginary numbers, ~ K ln[K ] + ( + q)(~ ) + x(~ ) + x (~ ), and the characteristic functions fj , j = 1; 2, are respectively provided in equations (45) and (46) in the Appendix. The bond option formula in (27) resembles many of the known bond option formulas in that they share the same functional form: the bond option price given in each model is determined by the price of a discount bond multiplied by a probability function minus the present value of the optimal exercise. Among these models, however, the probability functions can be quite dierent. For example, in Constantinides (1992), Jamshidian (1989), and Turnbull and Milne (1991), each probability is determined by a cumulative normal distribution function. In the single factor model of CIR (1985b) and the two-factor generalizations in Chen and Scott (1992) and Longsta and Schwartz (1992), the probabilities that the option expires in the money are determined by a bivariate non-central chi-square distribution function. Compared to these existing models, our bond option model in (27) has a few distinct features. First, the probabilities in (27) are easier to estimate than their counterpart in Chen and Scott (1992) and Longsta and Schwartz (1992). Recall that our setup is similar to theirs in that both interest rates and interest rate volatility are stochastic. But, in our case, the calculation of 1 and 2 involves taking the single integral over the characteristic function [see equation (28)], whereas in their cases computing the two probabilities involves evaluating the double integral over a bivariate noncentral chi-square distribution function. As Heston (1993) points out, integrating over the characteristic function numerically can be conducted quite eciently because the characteristic function declines rapidly in . Our bond option model therefore represents a practically more ecient alternative to the ones given in Chen and Scott (1992) and Longsta and Schwartz (1992). Second, the bond option formula in (27) is derived under a general power utility. It applies to 13 any economy with the assumed stochastic environment and with constant relative risk aversion. Economies considered in existing equilibrium bond option models, such as CIR (1985b), Chen and Scott (1992, 1995) and Longsta and Schwartz (1992), are thus nested in our economic setup. The option formula in (27) directly relates the price of a bond option to the level of relative risk aversion in the economy and it hence allows one to examine how bond option prices may change when risk-taking attitudes change, whereas such an exercise is not possible using existing models because of the log utility assumed. Finally, both the instantaneous interest rate and its volatility are linear functions of the un- observable state variables x(t) and y (t). As in Longsta and Schwartz (1992), one can use R(t) and its volatility to substitute out x(t) and y (t) in the bond and bond option formulas, so that for application purposes one can conveniently implement the model. For further details on this point, see, among others, CIR (1985b) and Longsta and Schwartz (1992). Thus far, we have focused on pure discount bond call options. Following the lines of discussion in Longsta (1990), Hull and White (1991), Turnbull and Milne (1991), and Chen and Scott (1992, 1995), our alternative valuation equation can also be applied to value other interest rate derivatives (e.g., options on coupon bonds, compound options, options on yields, and interest rate caps) and relate their prices to the level of risk aversion in the economy. Details are omitted here. 3 Valuing Individual Stock Options with CRRA Agents Continuing with the economy specied in the preceding section, we now turn to examining Eu- ropean options written on individual stocks. This part of the exercise serves two goals. First, we make the price process followed by a stock more consistent with the documented empirical evidence. Since the drift of the underlying asset price process will not enter the option valua- tion equation, special attention is given to nding an empirically plausible stochastic volatility structure. As is well known, the volatility specication is vital for explaining both skewness and kurtosis related biases found in existing option models. Second, we derive a closed-form formula for individual stock options that depends on the level of the representative agent's relative risk aversion. To the extent that all existing equilibrium option pricing models are derived by assum- ing the log utility function, our result should allow one to examine what happens to option prices when investors' risk aversion is dierent from that implied by the log utility. Specically, consider a generic rm (out of the N production rms) whose stock does not pay any dividends and whose stock price P (t), where for simplicity the rm index subscript is dropped, follows the process given below: 14 dP (t) = (t)dt + qy (t) d! (t) + qz(t) d! (t); (29) P (t) P P ^ P^ P P with the rm-specic state variable z (t) described by q d z(t) = z (z z (t)) dt + z z(t) d!z (t); (30) where P (t), to be determined in conjunction with the equilibrium, is the expected rate of return on the rm's stock, and the other parameters have the usual interpretation. The instantaneous correlation structure among the seven standard Brownian motions, (!q ; !q ; !x ; !y ; !z ; !P ; !P ), is ^ ^ given in the matrix below: 7 0 1 B 1 0 q;x 0 0 0 0 C B 0 B 1 0 q;y 0 0q;P C C B B q;x ^ C ^ ^ B 0 1 0 0 0 0 C C B B 0 C B q;y 0 1 0 0P;y C : C B B 0 ^ ^ C B 0 0 0 1 P;z 0 C C B B 0 C @ 0 0 0 P;z 1 0 C A 0 q;P ^ ^ 0 P;y ^ 0 0 1 We can note a few distinct features of the stock price process in (29). First, its implied instantaneous return volatility, VP (t), is given by VP (t) V art dP((tt)) = P y (t) + P z(t); P 2 ^ 2 (31) which is linear in both the systematic state variable y (t) and the rm's idiosyncratic state variable z(t).8 Here, the systematic volatility component is determined by the sensitivity coecient P : ^ the higher the coecient P , both the more systematic risk the stock has and the more highly ^ correlated its volatility is with the market-wide volatility factor. For instance, when P = 0, the stock has only systematic risk and no idiosyncratic risk; when P = 0, the stock has no ^ systematic risk but idiosyncratic risk. This assumed feature for stock price volatility is motivated by the considerable and growing empirical evidence that the volatility of an individual stock is 7 It is assumed here that each rm has its own vector of standard Brownian motions, (!P ; ! ^ ; !z ), that exhibit n Pn n the same correlation structure among these and the systematic Brownian motions as assumed in the above matrix for the generic rm. In addition, each rm's vector (!Pn ; !P^n ; !zn ) is independent of every other rm's counterpart. 8 We could clearly allow the stock's volatility to also depend on the state variable x(t), and our discussion would not change as a result of that. Without loss of generality, we chose to only let the systematic state variable y(t) be a determinant of VP (t). 15 not only stochastic over time, but it is also highly correlated with the overall market volatility. See the empirical literature on time-varying risk. Amin and Ng (1993), Bates (1995b), Merville and Pieptea (1989), and Wiggins (1987) have demonstrated that cross-correlation between rms' volatilities is positive and that rms' stock volatilities are highly correlated with market volatility. Over time, the stock's volatility follows a diusion process: n o q q dVP (t) = P y [y y (t)] + P z [z z(t)] dt + P y y (t) d!y (t)+ P z z(t) d!z (t); (32) 2 ^ 2 2 ^ 2 which is obtained by applying Ito's lemma to (31). This stochastic volatility process is dierent from those used in Heston (1993), Hull and White (1987), Scott (1995), Stein and Stein (1991), and Wiggins (1987). These authors assume that the stochastic volatility is driven by a single systematic risk source. The volatility process in (32) helps reconcile the empirical ndings of conditional excess kurtosis in stock returns. A high P and P , for instance, makes the stock ^ volatility process more volatile, and can bring the distributional properties of stock returns in line with its option pricing counterpart and vice-versa. Bates (1995b), in particular, discusses how the "volatility smile" can be treated as evidence of excess kurtosis and how this feature can be induced through option pricing models that incorporate stochastic volatility. Second, given the risk structure for the stock, its equilibrium expected rate of return is P (t) = R(t) + q^ P P;q^ y (t) ^ ^ (33) which is obtained by substituting equation (29) into the equilibrium pricing relation (8). Clearly, the idiosyncratic risk source z (t) is not priced in equilibrium. This way of determining the drift term P (t) guarantees that the underlying stock price process in (29) is consistent with the overall economic equilibrium. Third, the stock's volatility and the stochastic interest rate in this economy will in general be correlated with each other, but the correlation will be time-varying and not necessarily perfect, as can be seen by comparing the interest rate equation (24) and the volatility equation (31). In Heston (1993), he assumes that interest rate and underlying asset volatility are perfectly correlated. In practice it is unlikely for a stock's volatility to be perfectly correlated with interest rate. Finally, the covariance between volatility change and stock return is stochastic over time: dP (t) Covt P (t) ; dVP (t) = P;y P y y (t) + P;z P z z(t): ^ 3 ^ 3 (34) Bates (1995a,b) argues that such correlations are important for avoiding the skewness-related biases found in existing option pricing models. 16 Having specied an empirically plausible and internally consistent stock price process, we now consider a European call option written on the stock with strike price K and periods to expiration. The dollar payo to the holder at expiration is max f0; P (t + ) K g. Its time-t price, C (t; ), is given by " q(t + ) # C (t; ) = Et e max f0; P (t + ) K g : (35) q(t) By the output and technology shock dynamics respectively specied in (12) and (13) and the equity dynamics in (29), the conditional expectation in (35) is only a function of P (t), x(t), y (t) and z (t), which allows us to write C (t; ; P; x; y; z). Relying on this fact, we specialize the fundamental valuation equation in Theorem 1 to9 1 2 y + 2 z P 2 @ 2 C + R P @C + y P @ 2 C + z P @ 2 C 2 P ^ P @P 2 @P P y P;y @P@y P z P;z @P@z ^ ^ + 1 x x @ C + [x x (x + xq q;x) x] @C + 1 y y @ C 2 2 2 2 2 @x2 @x 2 @y 2 + [y y (y + y q q;y ) y ] @C + 1 z z @ C + [z z z z ] @C @C R C = 0; (36) 2 2 ^ ^ @y 2 @z2 @z @ where all time arguments are suppressed, subject to the boundary condition that C (t+; 0; P; x; y; z ) = maxf0; P (t + ) K g. The solution to this PDE is C (t; ) = P (t) 1 (t; ; P; x; y; z) K B(t; ) 2 (t; ; P; x; y; z) ; (37) where B (t; ) is the price of a -period discount bond as in (19) and the two probabilities, 1 and 2 , are determined by " # 1 + 1 Z 1 Re e i ln[K ] fj (t; ; ln[P ]; x; y; z ; ) d; j (t; ; P; x; y; z ) = 2 (38) 0 i for j = 1; 2, with the characteristic functions fj respectively given in equations (54) and (55) of the Appendix. As in the case of the bond option formula from the previous section, integrating over the characteristic functions to nd the two probabilities represents a practically convenient alternative to solving the original PDEs for 1 and 2 . This inversion method has been used by Bates (1995a), 9 Here, the state variables x, y and z can be thought of as corresponding to X1 , X2 and X3 in the general setup of Section 1, for M = 3. P (t) can be viewed as a substitute or sucient statistic for the corresponding rm's S (t) there. 17 Heston (1993) and Scott (1995) in pricing currency and stock index options. The individual equity option formula in (37) diers from existing option models in important ways. It admits both stochastic interest rates and stochastic volatility, with the latter consisting of a systematic volatility and an idiosyncratic volatility component. Further, it is eectively a three- factor option model, with the probabilities that the option will expire in-the-money depending on both economy-wide risk and rm-specic state variables. Consequently, many existing models, for either index or individual stock options and either general equilibrium-or partial equilibrium-based, are nested in our model. To appreciate this option valuation model, we oer a few observations. First, our valuation model for individual stock options diers from Amin and Ng's (1993) model in a crucial way. Even though in their discrete-time framework they also consider a general power utility function and a similar volatility structure for individual stocks, their option valuation model is given in terms of an expected Black-Scholes formula where the conditional expectation is taken with respect to the variance and interest rate processes. Hence, they do not have an exact closed-form pricing formula, and the option price can only be obtained via cumbersome numerical methods. In contrast, our option valuation formula in (37) is in closed-form and it holds even when individual stock prices follow quite general processes. Thus, our continuous-time framework, together with the alternative fundamental valuation equation, aords an advantage over Amin and Ng's discrete-time setup. Next, Heston's option model with stochastic volatility and stochastic interest rate is a special case of our model. This can be seen by letting x(t) = 1 and P = 0 in our model so that only the state variable y will be driving the interest rate as well as the stock's stochastic volatility. In his case, not only will the interest rate be perfectly correlated with the stock's volatility, but also will the stock have no idiosyncratic volatility component. Substituting x(t) = 1 and P = 0 into equation (37) results in a pricing formula for equity options as specied in Heston (1993). Further, note that in Heston's (1993) Section 2 the option formula with stochastic interest rate is not given in closed-form. Rather, one needs to rst numerically solve a system of ordinary PDEs and then invert the characteristic functions to obtain the probabilities. That can impose enormous computing constraints. In contrast, we have a closed-form option formula that applies to his setup as well. By the same logic, the option models in Hull and White (1987), Stein and Stein (1991), and Wiggins (1987) are also special cases. These authors assume stochastic volatility for the underlying asset but constant interest rate. Scott (1995) has a closed-form pricing model for stock options with stochastic volatility and stochastic interest rate. However, our model diers from his in two major aspects: (i) Like most existing equity option models with stochastic volatility, his model applies mostly to stock index options that have only systematic risk exposure, whereas ours also applies to individual stock 18 options with unsystematic risk and (ii) his model is derived only under the log utility assumption while our option formula obtains under both general power utility and exponential utility functions (the latter part to be shown shortly). The afore-mentioned dierences between existing option models and ours have many implica- tions for understanding and correctly pricing equity options. For example, consider two extreme types of stock options, type-A options written on stocks or stock indices with only systematic risk (i.e., P = 0 for those underlying assets) and type-B options written on stocks with only idiosyn- cratic risk (i.e., P = 0). Further, assume that the state variable x(t) is a constant (x(t) = 1) ^ so that y and z are the only two state variables. Then, for type-A options, the correct pricing formula is a version of equation (37) with only the systematic state variable y (t) as the driving factor (in addition to the stock price), whereas for type-B options the pricing formula has only the unsystematic state variable z (t) as the driving factor. Therefore, the two pricing formulas have completely dierent factor structures. Consequently, if one ts the same option pricing formula to both types of options, large pricing errors can result. This may explain why, for instance, Whaley (1982) nds that the Black and Scholes model leads to option pricing biases that dier across stocks of distinct risk characteristics. Since the pricing models in Heston (1993), Hull and White (1987), Stein and Stein (1991) and Scott (1995) are more suitable for type-A options and since individual stocks are bound to have exposures to both systematic and idiosyncratic risk sources, one can expect these existing option models with stochastic volatility to still generate pricing biases that dier across stocks of distinct sizes or risk characteristics. Given the complexity of the option formula in (37), clean comparative statics are dicult to obtain. It is easy to verify, however, that the call price is convex and increasing in P (t), convex and decreasing in K , and increasing in . Comparative statics with respect to the state variables are as follows. First, take the idiosyncratic risk variable, z (t). Clearly, z (t) will not aect the interest rate. From (31), a rise in z (t) will unambiguously increase the stock volatility and hence the call price. From equation (37), @C (t; ) = P (t) @ 1 KB(t; ) @ 2 > 0; @z @z @z where @ j = 1 Z 1 Re (i) 1e i ln[K ] @fj @z 0 @z d: Second, a change in y (t) leads to a change in both the interest rate and stock volatility. Its impact on C (t; ) is thus ambiguous.10 We divide the discussion into two cases. Case 1: y 2 (1+ )q > 0. 1 2 ^ 10 Bailey and Stulz (1989) show that in a general equilibrium a change in (the constant) systematic volatility can 19 In this case, a rise in y (t) causes a rise in both interest rates and stock return volatility. This in turn leads to a rise in the option price. Case 2: y 2 (1 + )q < 0. In this case, a rise in y (t) 1 2 ^ causes a rise in stock volatility but a decline in interest rates. Lower interest rates will lead to higher liability value, while higher volatility will result in higher asset value, for the call. Thus, the overall impact on the call price is not uniformly positive or negative. Finally, since x(t) aects the option price mainly through its impact on interest rates, a change in x(t) will produce a lower or higher call price, depending on whether this change increases or decreases interest rates. In Section 5, we resort to a calibration exercise to show how stock options may respond to a change in risk aversion. 4 Contingent Claims Valuation With Exponential Utility Using the alternative valuation equation, we can demonstrate that the results established in the two preceding sections also apply to economies with exponential utility functions. To do this, we try to maintain the same notation as before so that the formulas derived earlier for bonds, bond options and stock options carry over directly. Assume the representative agent has an exponential period utility: U (c(t)) = e c(t) (39) where is now interpreted as the coecient of absolute risk aversion. To arrive at the previous pricing formulas, we only need to change the output process in (12) to the following q q dq(t) = (q + x x(t) + y y (t)) dt + q x(t) d!q (t) + q^ y (t) d!q^(t); (40) and keep the processes for state variables x and y respectively as given in (13) and (14). Here, change in output, rather than growth in output, is driven by this linear combination of the state variables. With this output process and the exponential utility function, the equilibrium risk premium for any contingent claim F is the same as given in (16). This, together with the fact that x and y follow the same process as before, implies that the bond and bond option price equations must also be the same as in the previous economy with CRRA utility functions, except that is interpreted dierently here. Take the -period discount bond as an example. In this economy, E h (q(t+ ) q(t)) i B(t; ) = e t e imply an ambiguous impact on stock index options. They are concerned with index options that have exposure to only systematic risk, while individual stock options are the concern here. 20 Zt + = Etexp ( + q ) [x x(t) + y y (t)] dt Zt q t Z t q + + q x(t) d!q (t) q^ y(t) d!q^(t) : t t Thus, B (t; ) is still only a function of x(t) and y (t). As the risk premium for the bond and the processes for x and y remain unchanged from before, the PDE in (18) applies to this bond as well. As a result, the -period bond price in this exponential utility economy is the same as in (19). Similarly, the bond option price remains the same as determined in (27). Let the stock price process be as given in (29) and the rm-specic state variable z as in (30). Following the same reasoning, we conclude that the PDE in (36), and hence the closed-form pricing formula in (37), apply to a European call option written on a stock in this economy. In summary, the term structure and the bond and stock option pricing formulas remain un- altered, whether the representative agent has a power utility or an exponential utility function. This statement is, however, made only in terms of the functional forms for the respective pricing formulas. The two types of economy can be quite dierent from one another. For one thing, the output in each type of economy follows a distinct diusion process. 5 Properties of Option Prices in an Articial Economy The purpose of this section is two-fold. First, using an articial economy, we seek to understand whether and how interest rates, term premiums and option prices may respond to a change in risk aversion. Given that existing models are often free of the risk aversion parameter, this calibration exercise should serve a special role. Second, we study whether our general option pricing model can reconcile certain dierences between existing option models and empirical regularities (most notably, the volatility smile associated with the Black-Scholes model). The values chosen for the structural parameters are reported in Table 1. The choice of these parameter values is made such that the resulting term structure of interest rates and stock price volatility are empirically plausible. For the given values in Table 1, for example, the resulting term structure of interest rates corresponding to = 2:0 is upward sloping, with R(t) = 6:9%, R(t; 0:50) = 7:12%, R(t; 1) = 7:32%, R(t; 2) = 7:69%, and R(t; 1) = 11:50%, and the steady- state standard deviation of the spot interest rate is 6:4%. The stock return standard deviation is 24%, which is not unreasonable. The fractions of stock volatility attributable to idiosyncratic and systematic risk sources are 74% and 26%, respectively. Furthermore, stock volatility is negatively correlated with stock returns, and interest rates are positively correlated with stock volatility. Let us rst look at the impact of risk aversion on interest rates. Note that when = 1 (i.e., 21 the often used log utility case), the term structure changes to R(t) = 4:0%, R(t; 0:50)=4.12%, R(t; 1) = 4:23%, R(t; 2) = 7:45%, and R(t; 1) = 6.66%, which is still upward-sloping but quite dierent from its counterpart at = 2. Figure 1 plots the response of the term structure to a change in risk aversion. One can see that the interest rate for any maturity has a hump-shaped relationship with : it rst increases and then decreases, with = 6 (approximately) being the turning point. This is exactly as expected. Recall that the elasticity of intertemporal substitution is the inverse of the risk aversion parameter . As increases, initially the (positive) eect of intertemporal substitution dominates and then the (negative) risk aversion eect on interest rates dominates. While this response pattern applies to every maturity, long-term interest rates are more sensitive (with a steeper slope), because long-term bonds are riskier and hence more responsive to changes in risk-taking attitudes. Figure 2 plots the term premiums in relation to . In general, the term premium, regardless of maturity, has an inverse hump-shaped response pattern to . In the range 2 (0:25; 4:0), the term premiums are decreasing and, beyond that, increasing in , even though virtually all term premiums are negligible for values below 5. This nding is a little surprising as one would expect the term premiums to be monotonically increasing in risk aversion. But, given the two opposite eects of on interest rates, this might not be as hard to see. For the stock call options reported in Figures 3 and 4, the spot stock price is $100. In Figure 3, we x the term-to-expiration for all the European calls at = 0:25 (quarter a year) and let the strike price change: K = $90, K = $100, and K = $110. For each call option, the reported prices are normalized (i.e., divided) by its price corresponding to = 1, which is done for ease of comparison. Observe the following features from Figure 3: (i) the call prices, regardless of strike price, have a hump-shaped response to risk aversion; (ii) the turning point for each hump shape is about = 6; (iii) the sensitivities of the call price dier substantially across strike prices, with in-the-money calls being the least sensitive and the out-of-the-money calls the most sensitive. Given the hump-shaped eect of risk aversion on interest rates, these results can be explained as follows: a hump-shaped response of interest rates causes an inverse hump-shaped response of bond prices, which in turn results in a hump-shaped response of the call price to changes in . In Figure 4, we x the strike price at K = 100 and allow the term-to-expiration to change. It is clear from this gure that longer-term calls are more sensitive to changes in risk aversion. This is true as longer-term options exhibit more uncertainty and as longer-term interest rates are more responsive to changes in . Our exercise thus shows that interest rates and option prices all depend on risk-taking attitudes in an economically signicant way. We now demonstrate that our option pricing model can generate a volatility smile. According to Rubinstein (1985) and Bates (1995b), when actual option prices are substituted into the Black- 22 Scholes model to back out the implied volatility, the implied volatility tends to vary across strike prices in a U-shaped manner. This volatility smile is particularly striking for short-term options and has been a persistent feature of all option markets [e.g., Rubinstein (1994)]. While it is beyond the scope of the present paper to subject our model to actual option data, we study the performance of our model in the following fashion. We rst obtain (theoretical) option prices using our equity option pricing model, and then substitute those prices into the Black-Scholes formula to numerically back out the implied volatility. The test hypothesis is that if our model exhibits empirically plausible features, the model-generated option prices should also lead to an implied volatility smile. For each term-to-expiration, Figure 5 plots the implied stock volatility against the strike price, whereas Figure 6 displays the term structure of implied volatilities from our model. In all the calculations, = 2:0. Several observations can be made from these gures: (i) the volatility smile is much stronger for call options that have shorter terms-to-expiration (30 days or less). For longer-term options, however, the implied volatilities are declining in strike price. Both of these features are consistent with the evidence presented in Rubinstein (1994, Figures 1 and 2); (ii) the implied volatilities for in-the-money options are substantially higher, compared to those for both at-the-money and out-of-the-money options. Take as an example options with 10 days to expiration (i.e., = 10=360 years). The implied volatility is 52% for K=$90, 19% for K=$100, and 45% for K=$110. Recall that the time-t standard deviation of the stock is 24%; (iii) the term structure of implied volatilities for both in-the-money options and out-of-the-money options also display a volatility smile, with 60 days being the turning point of the U-curve. But, for at-the- money call options, the implied volatilities are increasing in term-to- expiration: the longer it take to expire, the higher the implied volatility for at-the-money calls. For all the U-curves, however, the upper tail is virtually at. Note that these features observed above apply to option prices taken from one time point and obtained from one set of structural parameter values. As can be seen from our derivations, for instance, dierent time-t values for x, y and z may generate dierent volatility patterns across strike prices and across terms to expiration. In other words, the volatility patterns coming out of our model can be time-varying and economy-dependent, which is again consistent with the time- varying nature of the strike price biases and volatility smiles emphasized in Rubinstein (1985). Thus the ability of our model to produce option prices that exhibit the documented stylized features of equity option markets only reinforces the claim that our option pricing model can be expected to perform better empirically than existing ones. 23 6 Concluding Remarks The work here can be extended in dierent directions. As a rst example, one can clearly use the alternative valuation equation to study contingent claims valuation by assuming other classes of stochastic processes. The choice of the exact stochastic structure for the economy often depends on empirical plausibility and technical tractability. For instance, one can introduce systematic jumps along the lines of Amin and Ng (1993), Bates (1991, 1995a), Das (1995), Naik and Lee (1991, 1995), and Scott (1995). This type of extension is quite feasible. Given the existence of real-life jumps in asset prices and economic variables, this line of research is of signicant importance. As another example, one can subject the bond and stock option pricing models developed here to real-life data. In such empirical exercises, one can address issues related to the well-known biases associated with the Black-Scholes model as well as issues on how option prices dier across assets of distinct systematic and idiosyncratic risk characteristics. Finally, non-expected utilities can also be introduced into this framework. Typically, solving the Hamilton-Jacobi-Bellman equation for the indirect utility or value function is even more dif- cult when non-standard utility theories are applied [e.g., see Constantinides (1991), Due and Epstein (1992), Epstein and Zin (1991), and Sundaresan (1989)]. For these types of applications, it then becomes more important to develop contingent claims valuation equations without involving the value function. The exercise in this paper may oer a useful direction for nding a way to solve for contingent claim prices with non-standard or non-expected utility. 24 Appendix: Proof of Results Proof of Theorem 1 Applying Ito's lemma to any contingent claim, F(t, q,S,X), implies " X N XM @F d F (t) = @F + @F q (q; S; X; t) + @S n (q; S; X; t) + @X m (X; t) @t @q @F n=1 n m=1 m 1 @ 2F 2(q; S; X; t) + 1 X + 2 @q 2 q N X @ 2F N Covt (dSn; dSn 2 0) + 1X M M X @ 2F Cov (dX ; dX 0 ) 2 n=1 n0 =1 @Sn @Sn0 t m m m=1 m0 =1 @Xm@Xm0 X @ 2F N X @ 2F M X X @ 2F N M # + @S @q Covt (dSn ; dq ) + Covt (dXm; dq ) + Covt (dSn ; dXm) dt n=1 n m=1 @Xm @q n=1 m=1 @Sn @Xm X N X M +q (q; S; X; t) @F d !q (t) + n (q; S; X; t) @S d !n (t) + m (q; S; X; t) @X d !m (t) @q @F @F (41) n=1 n m=1 m Dividing (41) by F (t; q; S; X ) and inserting dF((tt)) into (8) results in (9). 2 F Derivation of the Bond Option Formula in (27). Conjecture that the solution to the PDE (18) is given by (27). Then the probabilities, j for j = 1; 2, must respectively satisfy 1 2 x @ 2 1 + ( + ) x + 2 1 @B (t; ) x @ 1 + ~ 2 x @x2 x x x x q q;x x B (t; ) @x ~ @x 2 1 2 y @ 1 + ( + ) y + 2 1 @B (t; ) y ~ @ 1 @ 1 2 y @y 2 y y y y q q;y ^ ^ y B (t; ) @y ~ @y @ = 0; (42) and, 1 2 x @ 2 1 + ( + ) x + 2 1 @B (t; ) x @ 1 + 2 x @x2 x x x x q q;x x B (t; ) @x @x 2 1 2 y @ 1 + ( + ) y + 2 1 @B (t; ) y @ @ @y @ = 0; (43) 2 2 2 y @y 2 y y y y q q;y ^ ^ y B (t; ) @y where all time subscripts are suppressed and the PDEs must be solved subject to the terminal condition j (t + ; 0; x; y ) = 1 %x (~ )xt+ %y (~ )yt+ K ; ~ for j = 1; 2. As in Heston (1993), the characteristic functions corresponding to the two probabili- ties, fj (t; ; ) for j = 1; 2, must also satisfy the respective PDEs in (42) and (43) subject to the terminal condition fj (t + ; 0; ) = ei ( %x (~ )xt %y (~ )yt) : (44) Solving the respective PDEs, we have the characteristic functions for the bond option formula 25 given by ( " !# x f1 (t; ; x; y) = exp x 2 ln 1 [x x xq q;x (1 2 i)x %x(~ )](1 e ) + 2 x 2 x x " !# y y 2 ln 1 [y y y q^q;y (1 + i)y %y (~ )](1 e y ) ^ 2 y2 2y h1 2 i 2 2 x (1 + i)2%2 (~ ) + (x + xq q;x)(1 + i)%x(~ ) x 2 (1 + )q (1 e x ) x 1 2 + 2x [x (x + xq q;x + (1 + i)x%x (~ ))](1 e x ) 2 x(t) h 2 i 2 1 y (1 + i)2%2 (~ ) + (y + y q q;y )(1 + i)%y (~ ) y 1 (1 + )q (1 e y ) y 2 y (t) ^ ^ + 2 2 2 2y [y (y + y q q;y + (1 + i)y %y (~ ))](1 e y ) ^ ^ ^ ( + q )~ x (~ ) y (~ ) (1 + i)xx %x (~ ) (1 + i)y y %y (~ ) (1 + i)%x(~ )x(t) (1 + i)%y (~ )y (t) ln [B (t; )] ~ h x x i x q q;x (1 + i)x%x(~ ) 2 2 x x x ) y y h i y q q;x (1 + i)y %y (~ ) ; (45) 2 y2 y y ^ ^ and ( " !# f2(t; ; x; y ) = exp xx 2 ln 1 [x x xq q;x ix %x(~ )](1 e x ) 2 x2 2x " !# y y 2 ln 1 [y y y q^q;y iy %y (~ )](1 e y ) ^ 2 y2 2y h1 2 2 2 i 2 2 x (i) %x(~ ) + (x + xq q;x)i%x(~ ) x 2 (1 + )q (1 e x ) 1 2 + 2x [x (x + xq q;x + ix %x(~ ))](1 e x ) i 2 x(t) h 2 2 1 y (i)2%2 (~ ) + (y + y q q;y )i%y (~ ) y 2 (1 + )q (1 e y ) y 1 2 y (t) ^ ^ + 2 ^ 2y [y (y + y q q;y + iy %y (~ ))](1 e y ) ^ ^ 2 ( + q ) ix x %x(~ ) iy y %y (~ ) x h i%x (~ )x(t) i%y (~ )y(t) ln [B (t; )] x x x xq q;x ix %x(~ ) 2 i 2 i ) x y y h y y y y q^q;x iy %y (~ ) ; (46) 2 2 ^ where n o x = (x + xq q;x) + x(1 + i)%x(~ ) 2 2 1=2 2x (1 + i)(x + xq q;x)%x(~ ) + 1 x (1 + i)2%2 (~ ) x 2 (1 + )q 2 2 2 x 1 2 26 n o y = (y + y q q;y ) + y (1 + i)%y (~ ) 2 2 ^ ^ 1 2(1 + i)2%2 (~ ) 1 (1 + ) 21=2 2y (1 + i)(y + y q q;y )%y (~ ) + 2 y 2 ^ ^ y y 2 q^ n o2 x = (x + xq q;x) + xi%x(~ ) 2 1 2 (i)2%2 (~ ) 1 (1 + ) 21=2 2x i(x + xq q;x)%x(~ ) + x 2 x x 2 q n 2 o2 y = (y + y q^q;y ) + y i%y (~ ) ^ 2 1 (1 + ) 21=2 : 2y i(y + y q q;y )%y (~ ) + 1 y (i)2%2 (~ ) y 2 ^ ^ 2 2 y 2 q^ 2 Derivation of the Stock Option Formula in (37). The PDE in (36) can be written as 1 y + z @ C + R 1 y + z @C + y @ C + z @ C 2 2 2 2 2 2 2 2 P ^ P @g 2 2 P ^ P @g P y P;y @g@y P z P;z @g@z ^ ^ + 1 x x @ C + [x x (x + xq q;x) x] @C + 1 y y @ C 2 2 2 2 2 @x2 @x 2 @y 2 + [y y (y + y q q;y ) y ] @C + 2 z z @ C + [z z z z ] @C @C R C = 0; (47) ^ ^ @y 1 2 2 @z2 @z @ where, for convenience, g (t) ln[P (t)]. Conjecture that the solution is as given in (37). Substi- tuting it into (47) yields the PDEs respectively for 1 and 2 : 1 2 y + 2 z @ 2 1 + R + 1 2 y + 2 z @ 1 + y @ 2 1 + z @ 2 1 2 P ^ P @g 2 2 P ^ P @g P y P;y @g@y ^ ^ P z P;z @g@z + 1 x x @@x21 + [x x (x + xq q;x) x] @@x1 + 2 y y @@y 21 1 2 2 2 2 2 h i + ( + y y y y q q;y ^ ^ ) y @ 1 y P P;y ^ ^ @y + 1 z z @@z 21 + [z z (z z P z;P ) z ] @@z1 @ 1 = 0; 2 2 2 @ (48) and, 1 2 y + 2 z @ 2 2 + R 1 2 y + 2 z + 1 @B @ 2 2 P ^ P @g 2 2 P ^ P y P P;y B @y @g ^ ^ +P y P;y y @ 2 + P z P;z z @ 2 2 2 ^ ^ @g@y @g@z 27 1 2 x @ 2 2 + ( + ) x + 2 1 @B x @ 2 + 1 2 y @ 2 2 + 2 x @x2 x x x x q q;x x B @x y @y 2 @ 2@x 2 + y y (y + y q q;y y P P;y ) y + y B @B y @y ^ ^ ^ ^ 2 1 @y 1 2 z @ 2 2 + [ z ] @ 2 @ 2 = 0; + 2 z @z 2 (49) z z z @z @ subject to the respective terminal conditions: j (t + ; 0; g; x; y; z ) = 1gln[K ] ; for j = 1; 2. The characteristic functions corresponding to the two probabilities, fj (t; ; ) for j = 1; 2, must also satisfy the respective PDEs in (48) and (49) subject to the terminal condition fj (t + ; 0; ) = ei g : (50) The PDEs for the characteristic functions are 1 2 y + 2 z @ 2 f1 + R + 1 2 y + 2 z @f1 + y @ 2 f1 + z @ 2 f1 2 P ^ P @g 2 2 P ^ P @g P y P;y @g@y P z P;z @g@z ^ ^ + 1 x x @ f21 + [x x (x + xq q;x ) x] @f1 + 2 y y @@yf21 1 2 2 2 2 2 @x @x h i + y y (y + y q q;y y P P;y ) y @f1 ^ ^ ^ ^ @y + 2 z z @@zf1 + [z z (z z P z;P ) z ] @f1 1 2 @f1 = 0; 2 2 @z @ (51) and, 1 2 y + 2 z @ 2 f2 + R 1 2 y + 2 z + 1 @B @f2 2 P ^ P @g 2 2 P ^ P y P P;y B @y @g ^ ^ @ 2 f2 @ 2 f2 +P y P;y y @g@y + P z P;z z @g@z ^ ^ + 1 x x @@xf22 + x x (x + xq q;x) x + x B @B x @f2 + 1 y y @@yf22 2 1 2 2 2 2 2 @x @x 2 + y y (y + y q q;y y P P;y ) y + y2 1 @B y @f2 ^ ^ ^ ^ B @y @y + 1 z z @@zf2 + [z z z z ] @f2 @f2 = 0: 2 2 2 2 @z @ (52) Observe that the coecients for the PDE in (52) are not time-homogeneous and depend on the 28 parameters related to the bond price. To solve (52) make the substitution f^2 (t; ) = f2(t; ) B(t; ; x; y); and the resulting PDE becomes 1 2 y + 2 z @ 2 f2 + R 1 2 y + 2 z + 1 @B @ f^2 + y @ 2 f^2 ^ 2 P ^ P @g 2 2 P ^ P y P P;y B @y @g ^ ^ P y P;y @g@y ^ ^ @ 2 f^2 2 ^ ^ 2 ^ +P z P;z z @g@z + 1 x x @ f22 + [x x (x + xq q;x) x] @ f2 + 1 y y @@yf22 2 2 2 @x @x 2 h i @ f^2 1 2 @ 2 f2 ^ + y y (y + y q q;y y P P;y ) y @y + 2 z z @z 2 + [z z z z ] @ f2 ^ ^ ^ ^ @z @ f^2 R f^ = 0: (53) @ 2 The solution to the PDEs (51) and (53) is of the form f^j = exp [Aj ( ) + Hj ( ) x + Lj ( ) y + Dj ( ) z + i g ] ; where Aj ( ), Hj ( ), Lj ( ) and Dj ( ) must be determined subject to the boundary condition: Aj (0) = Hj (0) = Lj (0) = Dj (0) = 0, for j = 1; 2. The nal solution for each characteristic function is respectively given below: ( " x ! # q f1 = exp x 2 ln 1 [x x x2 q;x ](1 e ) + [x x xq q;x] x 2 x " x [ + (1 + i) ](1 e y ) !# y y 2 ln 1 y y y q q;y ^ ^ y P y;P ^ ^ y2 2y h y y + (1 + i) )i + i( + ) + iln[P (t)] y y y 2 y q q;y ^ ^ y P y;P ^ ^ q " ! # z z 2 ln 1 [z z + (1 + i)z P z;P ](1 e z ) + [ + (1 + i) ] z2 2z z z z P z;P h i 2i x 1 (1 + )q (1 e x ) 2 + 2 [ ](1 e x ) x(t) 2 xh x x x q q;x i 2 1 P [i + (i)2] + i y 1 (1 + )q (1 e y ) 2 2 + 2 2 [ ^ 2 ^ y (t) y y y y q q;y (1 + i)y P y;P ](1 e y ) ^ ^ ^ ^ ) i(1 + i)P (1 e z ) 2 + 2 [ + (1 + i) )](1 e z ) z (t) ; (54) z z z z P z;P 29 and ( " x ! # f2 = exp x 2 ln 1 [x x x2 q;x](1 e ) + [x x xq q;x] q x 2 2 x0 x 13 y y q q;y + iy ^ ^ ](1 e y ) y y 42 ln @1 [y ^ ^ P y;P A5 2 y 2 y y y h i y y y y q^q;y + iy P y;P + (i 1)( + q ) ln [B(t; )] + iln[P (t)] 2 ^ ^ ^ " ! # z z 2 ln 1 [z z + iz P z;P ](1 e z ) + [ + i ] z2 2z z z z P z;P h i 2(i 1) x 1 (1 + )q (1 e x ) 2 + 2 [ ](1 e x ) x(t) 2 x xh x x q q;x i 2(i 1) 1 iP + y 2 (1 + )q (1 e y ) 2 1 2 + 2 ^ ^ y (t) 2y [y y y q q;y + iy P y;P ](1 e y ) ^ ^ ^ ^ ) i(i 1)P (1 e z ) 2 + 2 [ + i ](1 e z ) z (t) ; (55) z z z z P z;P where 1 (1 + ) 21=2 x = (x + xq q;x) 2 2x i x 2 2 q 1 (1 + ) 2 + 1 i(1 + i) 2 1=2 y = y + yq^q;y (1 + i)y P y;P 2y i y 2 2 ^ ^ ^ 2 q^ 2 P^ h i= z = (z (1 + i)z P z;P )2 z i(1 + i)P 1 2 2 2 1=2 x = (x + xq q;x)2 2x (i 1) x 1 (1 + )q 2 2 2 1 (1 + ) 2 + 1 i 2 1=2 y = y + yq^q;y iy P y;P 2 2(i 1)y y 2 ^ ^ ^ 2 q^ 2 P ^ h i= z = (z iz P z;P )2 z i(i 1)P : 1 2 2 2 2 30 References Amin, K., and V. Ng, 1993, \Option Valuation With Systematic Stochastic Volatility," Journal of Finance 48, 881-910. Bailey, W., and R. Stulz, 1989, \The Pricing of Stock Index Options in a General Equilibrium Model," Journal of Financial and Quantitative Analysis 24, 1-12. Bates, D., 1991, \The Crash of 87: Was it Expected? The Evidence From Options Markets," Journal of Finance 46, 1009-1044. Bates, D., 1995a, \Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutschemark Options," Forthcoming in Review of Financial Studies. Bates, D., 1995b, \Testing Option Pricing Models," Working Paper, University of Pennsylvania. Black, F., and M. Scholes, 1973, \ The Pricing of Options and Corporate Liabilities," Journal of Political Economy 81, 637-659. Breeden, D., 1979, \An Intertemporal Asset Pricing Model With Stochastic Consumption and Investment Opportunities," Journal of Financial Economics 7, 265-296. 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Scott, L., 1995, \Pricing Stock Options in a Jump-Diusion Model With Stochastic Volatility and Interest Rates: Application of Fourier Inversion Methods," Working Paper, University of Georgia. Stein, E., and J. Stein, 1991, \Stock Price Distributions With Stochastic Volatility,"Review of Financial Studies 4, 727-752. Sun, T-S., 1992, \Real and Nominal Interest Rates: A Discrete-Time Model and Its Continuous- Time Limit," Review of Financial Studies 5, 581-612. Sundaresan, S., 1989, \Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth," Review of Financial Studies 2, 73-90. Turnbull, S., and F. Milne, 1991, \A Simple Approach to Interest Rate Option Pricing," Review of Financial Studies 4, 87-120. Wang, J., 1995, \The Term Structure of Interest Rates In a Pure Exchange Economy With Heterogeneous Investors," Forthcoming in Journal of Financial Economics. Wiggins, J., 1987, \Option Values Under Stochastic Volatilities," Journal of Financial Economics 19, 351-372. Whaley, R., 1982, \Valuation of American Call Options on Dividend Paying Stocks," Journal of Financial Economics 10, 29-58. 33 Table 1: Structural Parameters in the Articial Economy The time-t (initial) values are x(t) = 0.25, y(t) = 0.35, z (t) = 0.70, and P(t) = 100. Parameter Value 0.005 q 0.010 x 0.040 y 0.060 q 0.100 q^ 0.100 x 0.150 x 0.500 x 0.120 y 0.200 y 0.750 y 0.100 z 0.700 z 1.000 z 0.400 P 0.250 P^ 0.200 q;x 0.800 q;y ^ -0.200 P;y ^ 0.200 P;z -0.500 q;P ^ ^ 0.500 34

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