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An Alternative Valuation Model for Contingent Claims

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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 Classication 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-specic 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-
specic 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 articial 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 articial 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 dened 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 dened shortly. Conditional expectations, Et (), are dened 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 qualier \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 modied.

                                                               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 modications. 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 dened 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 dened 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 specically,
                  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 specied 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 specication 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.
    Specically, 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-specic 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 specied 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 specied 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-specic 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 specied 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-specic 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 Articial Economy
The purpose of this section is two-fold. First, using an articial 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 signicant 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 signicant 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
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                                              33
                   Table 1: Structural Parameters in the Articial 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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