by Naoto Kunitomo Graduate School of Economics, University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, JAPAN Phone and Fax : 03-5841-5614, E-mail: kunitomo@e.u-tokyo.ac.jp and Yong-Jin Kim Faculty of Business Administration 2-17-1 Fujimi, Chiyoda-ku, Tokyo 102-8160, JAPAN October 2004 (Edited Version May 2005)
Abstract
We investigate the effects of stochastic interest rates and volatility of the underlying asset price on the contingent claim prices including futures and options prices. The futures price can be decomposed into the forward price and an additional term, and the options price can be decomposed into the Black-Scholes formula and several addi- tional terms by applying the asymptotic expansion approach of the small disturbance asymptotics developed by Kunitomo and Takahashi (1995, 1998, 2001, 2003a, 2003b). The technical method is based on a new application of the Malliavin-Watanabe Calcu- lus or the Watanabe-Yoshida Theory on Malliavin Calculus in stochastic analysis. We illustrate our new formulae and their numerical accuracy by using the modified CIR processes for short term interest rates and stochastic volatility. We discuss implications of our results for financial economics.
Key Words
Stochastic Interest Rate, Stochastic Volatility, Contingent Claims, Futures, Options, Asymptotic Expansion Approach, Malliavin-Watanabe Calculus, Near Completeness. JEL Classification: G13 Mathematics Subject Classification (1991): 90A09, 60H07, 60G44 1
1
Introduction
In the past two decades a considerable number of studies in finance have been devoted to the generalizations of the standard theory of financial contingent claims. Because the option theory originally developed by Black and Scholes (1973) assumed that the underlying asset price follows the geometrical Brownian Motion and there is a constant risk-free interest rate, among several other assumptions, there have been some attempts to relax these aspects of the standard Black-Scholes theory. One of the important characteristics of many asset prices is the phenomenon that the volatility of asset returns does not seem to be constant and changes randomly over time. This empirical observation has led to the observation that volatility should be incorporated into the analysis as a state variable. In options pricing models, the effects of stochastic volatility have been investigated by Hull and White (1987, 1988), Johnson and Shanno (1987), Wiggins (1987), Scott (1987), Stein and Stein (1991), Heston (1993), and Ball and Roma (1994). Another important aspect in the analysis of contingent claims has been the fact that the spot interest rate is a fundamental economic variable in the economy and it cannot be treated as a constant. The feature that interest rates are stochastic has been incorporated into the modeling of contingent claims. Merton (1973), Rabinovitch (1989), Turnbull and Milne (1991), and Amin and Jarrow (1992) are representative of such research. Recently Kim and Kunitomo (1999) have also developed option pricing theory in a direction closely related to the present study. However, attempts to construct the stochastic model for contingent claims that allow the randomness of both the volatility and interest rates have not been made in abundance. See Amin and Ng (1993), Bakshi and Chen (1997), Baily and Stulz (1989), Helmer and Longstaff (1991), and Scott (1997) for this line of investigation. Due to the fact that we need to treat both stochastic volatility and interest rate processes at the same time, past valuation methods became complicated in the general case and their results can hardly be analytical except some special cases of the underlying stochastic processes. For instance, the closed-form expressions obtained by Bakshi and Chen (1997), and Scott (1997) contain the Fourier inversion formula even in simple cases. The methods based on the inversion formulae have been further developed by Bakshi and Madan (2000), and Duffie, Pan, and Singleton (2000). In this paper, we shall develop a general framework of the analysis of European-type contingent claims including the futures price, forward price, and options prices that in- corporate both the stochastic volatility and stochastic interest rates. Our method is based on the asymptotic expansion approach called the small disturbance asymptotics in which we consider a sequence of stochastic processes when the diffusion parameters of some stochastic processes are small under the probability measure Q , which is equiv- alent to the historical probability measure P . The small disturbance asymptotics under Q has been recently developed by Kunitomo and Takahashi (1995, 1998, 2001, 2003a, 2003b, 2004), Takahashi (1999), Yoshida (1992), Takahashi and Yoshida (2001a,b). It has been mathematically justified by the Malliavin-Watanabe Calculus in the rigorous manner, which is an infinite dimensional analysis of Wiener functionals and the gener- alized Wiener functionals on the abstract Wiener space, and it can be also called the Watanabe-Yoshida theory on Malliavin Calculus.  (See Watanabe (1987), Chapter V of Ikeda and Watanabe (1989), Shigekawa (1998), and Yoshida (1992) for the basics). The details of this theory, the mathematical notations, and applications to derivative pricing problems have been explained by Kunitomo and Takahashi (1998, 2001, 2003a). Our formulation of stochastic processes in this paper differs mathematically from that of Kunitomo and Takahashi (1998, 2003a) because the limiting stochastic processes are the solution of stochastic differential equations and we need a non-degeneracy condi- tion for the partial Malliavin covariance of certain random variables. In this sense this paper gives a new application and a real example of Malliavin-Watanabe Calculus to financial derivative analysis. As we shall show in this paper, our approach has the advantage that an explicit formula is used for the theoretical values of contingent claims which can be decomposed into the leading term plus several additional terms when both the stochastic volatility and spot interest rate follow the general class of continuous processes. The leading term is the well-known formula when both volatility and interest rate are constant and hence it allows us to explicitly and analytically investigate the effects of stochastic volatility and stochastic interest rates on contingent claims. When the volatility function is stochastic, the underlying financial market is incom- plete. We shall adopt an analogous argument on market completion by Romano and Touzi (1997) whereupon it is possible to give the financial (or economic) meaning of our results. Also we shall show that it is straightforward to incorporate more complicated stochastic models into our framework, including the model for the HJM term structure of interest rates model. In Section 2 we provide the general framework of our analysis that explains when the spot interest rate and the volatility of security price follow diffusion processes. Then in Section 3 we present the main results on the futures price, forward price, and European options prices. Section 4 provides a numerical example with some modified CIR-type interest rate and volatility processes and gives some evidence on the numerical accuracy of our formulae. Section 5 offers discussions on some related issues including an interpretation of our framework, the risk premium functionals, the change of measure problem, and an extension to the HJM-type term structure of interest rates.  Also some concluding remarks are given in Section 5.3. Section 6 is the mathematical appendix for some proofs omitted in Sections 2 and 3.
2
Asymptotic Behavior of the Underlying Asset Price
Process
Let (Ω, F , { Fi}t€[0.T ]; Q ) be a complete filtered probability space ¹ with the probability measure Q . We consider a continuous time economy where some securities and bonds are traded in the interval [0, T] ( T < +∞) without any friction and we assume that there does not exist any default risk nor any transaction costs associated with bonds and securities. Let S{c,s} (0 < t < T ) be the price of the underlying security at t with two parameters 0 < ε, δ ≤ 1 . In this section we focus on the situation that this security pays no dividends and the price process follows the stochastic differential equation
もwhere r ′ ( ε ′ is the instantaneous spot interest rate at t with the parameter 0 < ε ≤ 1 σ (δ) is the instantaneous volatility at t with the parameter 0 < δ ≤ 1, and W 1t is the Brownian motion under Q . The non-negative stochastic process σ ξ δ ) follows the stochastic differential equation :
where W 2 t is the second Brownian motion under Q . We note that when the volatility function is not a traded asset, the markets for the contingent claims on the underlying asset { ${€,6} could be incomplete. For the interest rate processes, we assume that there exists a locally riskless money market and the money market account (accumulation factor) is given by M f c 1 = exp( f t r ' ε ' ds) . We also consider the situation when there also exist bond markets in the economy and let P(c)(s,t) (0 ≤ s ≤ t) be the discount bond price at s with the maturity date t . We assume that the non-negative (instantaneous) spot interest rate process r ( é ) , which is consistent with the money market and discount bond markets, follows the stochastic differential equation :
where W3t is the third Brownian motion under Q . As the simplest case, we adopt the situation when all discount bond prices P(c)(s, t) (0 ≤ s ≤ t ≤ T) are solely determined by the single factor { r { c } . As the more general case, we shall discuss the HJM term structure of interest rates model in which the spot interest rate is not necessarily Markovian. (See (5.73) and (5.74).) In (2.1)-(2.3) we consider the situation when three Brownian motions under Q are correlated and their instantaneous correlations are given by
where we denote dW t = (dW 1 t, dW 2 t, dW 3 t) ′ In Sections 2 and 3 we treat both the interest rate process and the volatility function of the asset returns as Markovian processes under the probability measure Q . The form of the stochastic differential equations in (2.1)-(2.3) should be interpreted as the representation under the probability measure Q , (alternatively we write Q(σ(δ), r(ε))), which is equivalent to the historical probability measure P for the observed price process and the interest rate process. The probability measure Q is a martingale measure in the sense that the discounted security price process
is a martingale under Q given the volatility process and the interest rate process. We shall further discuss the related interpretation in Section 5.1, which is similar to the one by Romano and Touzi (1997). Let G( S ( ξ, δ ) be the non-negative payoff function of the European contingent claim at T on the underlying security { S{ e, δ } } . The contingent claims are regarded as Wiener functionals in the Wiener space that are not necessarily smooth in the standard math- ematical sense. In this paper we try to analyze the theoretical values of the contingent claims based on the payoff, which is a function of time T security price. The immediate examples of this type are the forward contract, the futures contract, and the standard European options contracts. For our purpose, we need to define the theoretical values of this type of contingent claim ² . Definition 2.1  : The theoretical value of the European contingent claim with the ter- minal payoff function G( S f ∈ δ ) at time t is defined by
provided that the expected value is finite, where the expectation operator is taken with respect to a probability measure Q given the σ-field F t . We shall analyze the effects of the stochastic volatility and the stochastic interest rates on the theoretical value of the contingent claims when both ε and δ are small. In order to develop the asymptotic expansion approach when both δ and ε are small, we need to have some regularity conditions so that the solutions of (2.1)-(2.3) are well- behaved and the stochastic expansions of the stochastic processes { r { ε } } and { σ { ε } } can be allowed. Assumption I : ( i ) Given 0 < ε, δ ≤ 1 the drift functions μr( r (ε ), t, ε), μσ(σ (δ ), t, δ ) (R × [0, T] × [0, 1) → R ) and the diffusion functions w r ( r ζ β , t ), w σ (σ ζ β , t ) (R × [0, T] + R ) are progressively measurable such that r ( ε ) and σ i δ ) are F t -measurable. (ii) For the (finite) terminal period T > 0 the stochastic processes r(c) and σ( δ) satisfy the conditions 0 < f T (σξδ) 2 ds  < +∞ (a.s.) and (2.7) Assumption II : (i) The drift functions are continuously twice differentiable and their first and second derivatives are bounded uniformly in ε and δ . The diffusion func- tions are continuously differentiable and their first derivatives are bounded uniformly in ε and δ.  (ii) For any 0 < t ≤ T there exist unique solutions { r t } and { σ t } for the ordinary differential equations
and
(iii) Suppose δ is a function ³  of ε with the notation δ = δ(ε). There exists a positive constant c (0 < c < ∞) such that
The uniquness and existence for the stochastic differential equations of (2.2) and (2.3) follow from Assumption II-(i). A set of standard conditions for them are the Lipschitz type conditions and the growth condition on the drift functions and diffusion functions for { r { e } and { σ { e } that have been discussed by Chapter IV of Ikeda and Watanabe (1989) or Nagai (1999), for instance. We need an integrability condition given by Also we need some smooth Assumption I-(ii) for the security price process 5(€,δ) conditions of underlying stochastic processes with respect to the parameters ε and δ . By applying Ito's lemma, the solution of (2.1) can be expressed as
Under Assumption I-(ii) the first two parts of (2.11) consist an exponential martingale and also the security price S ( ε,δ) is square-integrable. In the rest of this section, we shall investigate the asymptotic behavior of the security price process in the situation 4  when ε ↓ 0 and δ ↓ 0 under Assumption II. It should be noted that Assumption II can be relaxed with some additional complications. We shall derive the explicit form of S ( ε, δ ) for any 0 < t ≤ T in this small disturbance asymptotic approach. Let (2.12) where r t = r (0) is the solution satisfying the ordinary differential equation (2.8). By substituting r ( ε ) = r t + ε A (ε ) ( t ) into (2.3), we have
Then by using Assumption II-(i), we can find positive constants c 1 and c 2 such that for
Also by using the standard arguments in stochastic analysis, martingale inequality, and the Gronwall inequality, we can find positive constants c 3 and c 4 such that
uniformly with respect to ε . Hence we confirm the convergence in probability that r{€) → r t uniformly with respect to t as € $ 0 . Let
where 5  Ar(t) = p lime10 A(e)(t) . Then by substituting r(e) = rt + cAr(t) + c2B(e)(t) into (2.3), we can use a similar argument recursively to lead that E ♀ ‖ B ( ε ) (t) | 2 ‖ is bounded uniformly with respect to t and ε and we have the uniform convergence of Afe)(t) to Ar(t) with respect to t as ε ↓ 0 in probability. We need similar arguments on the existence and convergence of random variables A r ( t ), which we have omitted. By using the above arguments under Assumption II, the stochastic expansion of the instantaneous interest rate r ( ε ) can be expressed by
as ε ↓ 0, where the remainder term R 1 is in the order o ρ ( ε ). Then by using (2.13) and convergence arguments of its each terms, A r ( t ) can be regarded as the solution of the stochastic differential equation :
where we denote (2.19) and (2.20)
In order to have a concise representation for Ar(t), let Yjr be the solution of dYjr = ∂µr(r t , t , 0)Y r dt with the initial condition Y r = 1. Then (2.18) can be solved as
Similarly, under Assumption II we can expand the integral equation (2.2) with re- spect to δ . By using the same argument as r ( ε ) , the stochastic expansion of stochastic volatility σ ( δ ) can be also expressed by
as δ ↓ 0 , where the leading term σ t is the solution of the ordinary differential equation (2.9), the second term is given by A σ (t) = p lim δ 10 A(δ 1 (t) with A(δ 1 (t)(= [σξ δ ] − σε ]/δ), and the remainder term R 2 is of the order o p ( δ ). Let Y 2 be the solution of dY 2 = ∂µσ(σ t, t, 0)Y q dt with the initial condition Y q = 1. Then, because A σ(t) is the solution of the corresponding stochastic differential equation as (2.18) for {σ{δ}}, we can express A σ ( t ) as
where ∂w σ (σ s, s, 0) and ∂ δ μ σ (σ s, s, 0) are defined in the same ways as (2.19) and (2.20). In order to state the asymptotic behavior of the security price process S ( ε,δ) as ϵ ↓ 0 and δ ↓ 0, we need the new mathematical device of the Malliavin-Watanabe Caluculus recently developed in stochastic analysis.  Let the H-differentiation be defined by DF h (w) = lim ε 0 (1/ε)[F(w + εh) − F(w)] for a Wiener functional F(w) and h ∈ M, where M is the Cameron-Martin subspace of continuous functions with square- integrable derivatives in the Wiener space W. Then the Malliavin covariance is given by σмc(F) =< DF(w), DF(w) >н, where < · >н is the inner product of M space. We summarize the first result on the asymptotic behavior of the security price process S ξ ( ,δ ) as ε ↓ 0 and δ ↓ 0 in the next lemma. The proof is given in Section 6.1. Lemma 2.1 : (i) Under Assumptions I and II,
as ε ↓ 0 and δ ↓ 0, where St is the solution of the stochastic differential equation
dSt = rtStdt + σtStdW1t , and {r i} and {σ i} are the solutions of the ordinary differential equations (2.8) and (2.9). (ii) Let σмc(S(e,δ) be the Malliavin covariance of the stochastic process S(e,δ) and ς(ε,δ) ∈ D 2,1 (R) 6 . Then we have
Next we shall derive the asymptotic expansion of the security price process S{e,s} as ε ↓ 0 and δ ↓ 0 . For this purpose we insert (2.17) and (2.22) into (2.11).  By evaluating probability order calculations, we obtain the expressions for the first term of the exponential part in (2.11) as
and for the second term of the exponential part in (2.11) as
where R i (i = 3,4) are the remaining terms of the higher orders. If we set the leading term as (2.28)
we can write (2.11) as
and
where R 5 and R 6 are the remaining terms of higher orders. Then we can obtain a stochastic expansion of the price process of the security at time t with respect to ε and δ , which can be summarized in the next lemma. Lemma 2.2 :  Under Assumptions I and II, an asymptotic expansion of the price process of the security S ( ε,δ) at any particular time point t as ε ↓ 0 and δ ↓ 0 is given by
where z it (i = 1,2,3) are the random variables with E Q [ z it | X 1t ] = 0, E Q [ z it ] < ∞ (i = 1, 2, 3), (2.32)
9
and R 5 is the remainder term of the order o p (ε, δ). Let X{€,8} = log(S{€,8}) and X{€,8} = ft σ{δ}W1s . Then Assumption I implies the boundedness of some expectations such that EQ[| X ( c , δ ) | 2 ] < +∞ and EQ[exp( X ( c , δ ) )] < +∞ by using the Cauchy=Schwartz inequality and Theorem 5.3 of Ikeda=Watanabe (1989) for exponential martingales.  The remaining proof of Lemma 2.2 is given in Section 6.2. We should note that we have the non-degeneracy of the random variable X 1 t for any 0 < t ≤ T as the key condition from our Assumption I-(ii). Let also σ MC (X{e,δ}) be the Malliavin covariance of X{e,δ}) , which is the partial Malliavin co- variance of the three dimensional process in (2.1)-(2.3). Then the integrated volatility function σ(t)² can be interepreted as the limit of the corresponding partial Malliavin covariance σмс ( X (ε,δ) as ε ↓ 0 . By evaluating inequalities and other related arguments, it is possible to take the expectation operator and its asymptotic expansion in terms of ε and δ in (2.6) with respect to the probability measure Q when
for some constant K 1 . Because the derivative securities, such as forward contracts, futures contracts, and standard option contracts, in the next section satisfy this condi- tion, we can take the expectation operators and their asymptotic expansions formally in our discussions.
3
Futures, Forward, and Options Prices
3.1
Futures and Forward Prices
First, we consider the theoretical values of the futures contract written on the security Sf(c,s) that matures at time T . The standard financial theory asserts that the futures price at time zero, Fo, is determined by
where the expectation is taken with respect to the probability measure Q. (See Chapter 8 of Duffie (1996) for the simple arbitrage free arguments for the forward contract and the futures contract, for instance.) By applying Lemma 2.1 and using the moment rela- tions that for X 1 T ～ N (0 , σ (T) 2 ) we have E [exp(X 1 T )] = exp(σ (T) 2 /2), E [X exp(X 1 T )] = σ(T)² exp(σ(T)²/2), and E[X i T exp(X 1T )] = σ(T)²[σ(T)² + 1] exp(σ(T)²/2). Then we immediately obtain the following result.
10
Theorem 3.1 : Under Assumptions I and II, an asymptotic expansion of the theo- retical value of the futures contract at time zero with the delivery time T, F 0 , is given by
as ε ↓ and δ ↓ 0, where Σ(r) (T) and λ r (T) are given by (2.33) and (2.34), respectively. We should notice that there is no effect due to the stochastic volatility in (3.39) up to the order of o(ε,δ), although there might be some effects on the higher order terms. From this result, we can predict that the effect of volatility is considerably smaller than the effect of stochastic interest rate on the futures price. Next we consider the theoretical value of the forward contract written on the security S ( c, 8) that matures at time T . Standard financial theory asserts that the forward price at time zero, f 0 , is determined by
By using the fact E[exp( aX )] = exp( ξΣ ) when X ～ N [0, Σ] for any constant α , the denominator of (3.40) can be written as where R 6 is the remainder term of the order o p ( ε ). Then we have the following result on the forward price. Theorem 3.2 : Under Assumptions I and II, an asymptotic expansion of the theoret- ical value of the forward contract at time zero with the delivery time T, fo, is given by
as ε ↓ 0 and δ ↓ 0 , where λ r (T) is given by (2.34). Let the first term in (3.39) be FD and set the coefficient of e in (3.39) to be Fr. Then Theorem 3.1 asserts that the futures price can be decomposed into the futures price under constant interest rate and the adjustment terms as
where F C is the futures price under constant interest rate r 0 and volatility σ 0 . The second term in (3.42) represents the adjustment value induced by the deterministic interest rate. The third term is the adjustment value induced by incorporating the stochastic interest rate. Because Y r in Σ ( r ) ( T ) takes only positive values, we have the following result: Corollary 3.1 : Under Assumptions I and II, we have the relations depending on ρ r such that
There have been some discussions on the relation between the futures price and the forward price when the short term interest rate is stochastic. First, when the interest rate is independent of the underlying asset, the futures price is equal to the forward price, which is also the direct result from (3.40). Second, when the interest rate has positive (negative) correlation with the underlying asset, the futures price is greater (smaller) than forward price. This result has been also presented in Equation (26) of Cox, Ingersoll, and Ross (1981). From our analysis, we have found that the relations between the futures price and the forward price mentioned above hold even when the volatility of underlying asset is stochastic in the small disturbance asymptotics sense.
Options
3.2
We consider the theoretical value of the European call options contract written on the security S ( ε,δ) that matures at time T . For a given exercise price K at the expiry date T, the theoretical price of such options at the initial date, Vo, can be given by
under the probability measure Q , where [ -]+ denotes the function max[0, -] and By substituting (2.31) into Z ( ε,δ) and using (2.30) on the discount factor, we can obtain the expression for Z(€,δ) as
where

12
and R 7 is the remainder term of the order o p ( ε, δ). In the above expression the random variables z i T ( i = 1,2,3) have been defined by z i T = z i T K exp ( − f T r tdt ) and z i T = ziTS 0 exp(X 1T − ξσ (T) 2 ) (i = 2,3). Then we need to evaluate
where I( · ) is the indicator function of [0, +∞) . By the result of lengthy derivations as outlined in Section 6.3, we have finally obtained the theoretical value of the European call options contract as the next theorem. Theorem 3.3 : Under Assumptions I and II, an asymptotic expansion of the theo- retical value of the European call option with maturity T when the interest rate and volatility are stochastic, Vo, is given by
as ε, δ ↓ 0, where Φ( · ) is the distribution function of the standard normal variable, φ( · ) is its density function, d2 = d1 − σ(T)
and also Σ ( T ),Σ ( Z ( T ), λ r ( T ) and λ σ ( T ) are defined by (2.33), (2.35), (2.34), and (2.36), respectively. Let the first term in (3.49) be BSD and we set the coefficients of ε and δ to be BS r and BS σ , respectively. Then Theorem 3.3 asserts that the European stock call option price can be decomposed into the original Black-Scholes price and the adjustment terms as
where BS stands for the original Black-Scholes option price under the assumptions of constant interest rate and volatility. The second term in (3.51) represents the ad- justment value induced by the deterministic interest rate, which in itself relies on the assumed interest rate and volatility model. The third term and the fourth term are the adjustment values induced by the stochastic interest rate and the stochastic volatil- ity, respectively.  When the interest rate is stochastic and volatility is constant, for instance, the valuation problem in this case has been investigated by Kim and Kunit- omo (1999). Also when r s = r 0 and ε = 0, the model corresponds to those for many stochastic volatility models, which have been studied by several researchers. Hence our results include many previous studies as special cases in the sense of small disturbance asymptotics. By using a similar procedure for the valuation of European call options, we can derive the theoretical value for the European put options whose payoff function is given by [ K − S ( ε,θ ) ]+ at the maturity time T . By deriving the asymptotic expansion of the random variable [ − Z ( ε, δ )]+, we have the next result: Theorem 3.4 : Under Assumptions I and II, an asymptotic expansion of the theo- retical value of the European put option with maturity T when the interest rate and volatility are stochastic, V o , is given by
as ε, δ ↓ 0, where Φ( · ) is the distribution function of standard normal variable, φ( · ) is its density function, and d i (i = 1,2), Σ( γ (T), Σ( γ (T), σ(T), λ r(T), and λ σ(T) are the same as in Theorem 3.3 .
4
Numerical Examples and Related Analysis
4.1
CIR-Type Interest Rates and Volatility
In this section, we give some numerical examples to illustrate our theoretical results in Section 3.  For this purpose we assume that the spot interest rate and volatility processes are of the CIR square-root process originally proposed by Cox, Ingersoll, and Ross (1985b) for the spot interest rate model. Let the state variables ( Z ( δ ) , Z ( ξ ) follow
and
under the probability measure Q . We take a sufficiently large K 2 and set r, < K 2 ) , σ ( δ ) = Z 2 δ ) ( σ ( δ ) < K 2 ), and r ξ ε ) = K 2 , σ ( δ ) = K 2 otherwise. We notice that (4.54) is the special case of (2.3) when we take μ r ( r ξ ), s, ε ) = κ r ( r + ϵ ν, − r ( c ) and w r (r ξ c ), s) = √ r ξ c ), where κ r , ñ , and ε are positive constants and K 2 = + ∞ . Also (4.53) is also a special case of (2.2). The assumption of the CIR type processes for two state variables could be justified by the reason that both the nominal short term interest rates and the volatility functions take non-negative values with the mean-reversion properties. 7 . In our example it is possible to give some simple formulae for the valuation problems. The solution of the ordinary differential equation (2.8) for the spot interest rate is given by r t = roe-krt+r(1-e-krt) and for the volatility process the solution of the differential equation (2.9) is provided by σ t = σ oe-kst + σ(1 − e-kst) . Then the variance function σ(T)² can be calculated from f σ ² ds and it is given by
In the standard Black-Scholes model, σ( T ) reduces to σ 0 V T , which corresponds to the case when σ = σ 0 . Also we have that Y t = e-krt and Y t σ = e-kσ t . By using the definition of Σ( γ ) ( T ) in (2.33), the integration operation gives the following expression:
where we have used the notation ar defined by
Also by using the definition of Σ ( σ )( T ) in (2.35), we can calculate the integration explicitly as
and
The standard notation in the Black-Scholes formula d 1 becomes in the present case
and dշ = d 1 − σ ( T ). Finally, we can simplify considerably the leading term in the discount factor and it is given by

4.2
Numerical Accuracy
We report some results on the numerical accuracy of our formulae by using the CIR interest rates and volatility example. We take ν σ = ν r = 0 and hence λ r (T) = λ σ (T) = 0 for the resulting simplicity. Since we are interested in the case when the covariances between three state variables are not zeros, we give a set of numerical values for the cases of ρ r,ρ σ = −0.5,0.5 . Among many cases of our numerical examples we shall report only one case. It is the case when the initial interest rate is in the downward phase and we set r 0 = 0.11, r = 0.08, k r = 2.0, and ε = 0.1 . For the stochastic process on the volatility function, it is the upward phase case and we set σ 0 = 0.2, σ = 0.3, κσ = 4.0, and δ = 0.1 . Also we take that the time to maturity is assumed to be one year ( T = 1 ). For the futures price, we set S 0 = 100. For the call option case, we set K = 100 and S 0 = 90, 100, 110 to incorporate moneyness of options. Each table in this sub-section corresponds to the numerical value of the approxima- tions up to o(ε,δ) based on the asymptotic expansions in Theorem 3.1 and Theorem 3.3 by ignoring higher order terms. For the call option case, the option value under the original Black-Scholes model has also been given for the comparative purpose. As the benchmark, we also provide the Monte Carlo simulation results in the first row of each table. The number of simulated sample paths is 10,000 and the time interval is 250 . As the discretization method of sample paths, we have adopted the Euler-Maruyama approximation. All results are the mean of 200 simulation trials. Table 1 describes the numerical accuracy of our formula for the futures price. As men- tioned in Section 3, it should be noted that in our analysis the forward price under the stochastic interest and the stochastic volatility differs from the futures price slightly. In addition, the parameters of stochastic volatility have not appeared in our formula which is in the order of o(ε,δ) . We can observe in Table 1 that our formula is numerically very close to the simulation results. For example, when ρ r = ρ σ = −0.5, the pricing bias is 0.006 yen, which is equal to only 0.005 % of the true value. Table 2 describes the numerical example for the European call option for at-the-money case. When κ r = κ σ = 0 and ε = δ = 0, the case corresponds to the original Black- Scholes economy with constant risk-free interest rate and volatility. The Black-Scholes value in this case BS is 13.868. We can observe in Table 2 that our option pricing formula is very close to the simulation results. For example, when ρ r = −0.5 and ρσ = 0.5, the discrepancy between them is 0.013 yen, which is only 0.085%. Table 3 and Table 4 also give the numerical results with the same set of parameter values for the in-the-money and the out-of-the-money cases, respectively.
4.3
Term Structure of Implied Volatilities
In our setting (2.1)-(2.3), the implied volatility σ*( T ) is the same as the implied average volatility σ(T) when ε = δ = 0, where we denote σ(T) 2 = (1/T) f T σζ dt . In the general case, however, by using Theorem 3.3 the implied volatility and the implied average volatility can be defined by the solutions of the nonlinear equation
where k = K/S 0 , d 2 = d 1 − √T σ (T), d 1 = [− log k + T r (T) + T l σ (T) 2 ] /[ σ (T)√T], and we denote the average interest rate r(T) = (1/T) f( rtdt . Here we have interpreted the implied volatility as the volatility calculated from the Black-Scholes formula by using call options market prices. The implied average volatil- ity is a reasonable volatility index if the actual price of the call options in market is equal to the theoretical value in Definition 2.1. Furthermore, by ignoring higher order terms the above equation can be solved with respect to σ *( T ) and it can be written as
Hence it is analytically possible to investigate the various shapes on the implied volatil- ity σ*( T ) or the implied average volatility as the functions of the variables k and T including the well-known phenomena of volatility smile.
5
Discussion
5.1
Risk Premium, Complete Market, and Near Completeness
We have interpreted the stochastic processes in (2.1)-(2.4) under the measure Q , which can be different from the probability measure P governing the observable stochastic processes. When we consider the measure P for the observable underlying asset price and the bond prices with the spot interest rate, we could write the stochastic differential equation under P as
where μ ξ ( ·, u ) is the drift function, w r ( ·, u ) is the diffusion function, and B ξ n ( u ) is the Brownian motion for the spot interest rate under P . Also the stochastic differential equation of the volatility function for the underlying asset could be written as
where μ * ( · , u ) is the drift function, w σ ( · , u ) is the diffusion function, B * ω is the Brownian motion for the volatility function which is independent of B ∗ u under P , and c 2 i ( i = 2, 3) are constants with the normalization condition E P [¢22B å t +¢23B å t ] 2 = t . The stochastic differential equation for the underlying asset price under P could be written as
where μ ξ ( ·, ι ) is the drift function, B ξ u is the Brownian motion for the security market under P , which is independent of B i u (i = 2, 3), and c 1i (i = 1, 2, 3) are constants with the normalization condition EP [c 11 B *t + c 12 B *t + c 13 B *t ] 2 = t. We note that the asset price S ( δ ) has only one parameter δ under the probability P in this setting. In financial economics it has been often considered when there exist risk pre- mium functionals associated with the Brownian motions. In this section we denote λ ( ε, δ) ( u ) as the risk premium processes for the Brownian motions B iu ( i = 1,2,3) under the probability measure P. For the simplicity of our discussion on the risk pre- mium functionals we first consider the single factor model in the bond market and let P(c)(t,T) (= P(c)(r){ },t,T)) be the discount bond price with the maturity period T satisfying
where μ ( ξ ) ( t, T ) and ν ( ε ) ( t, T ) are the drift term and the volatility function of the bond price, respectively. By using the standard argument of the no-arbitrage condition the risk premium functional λ ( ε ) ( t ) can be determined by
provided that ι ( ε )( t, T ) > 0 for any ε > 0 and the consistency condition lim ε 10 ι ( ε )( t, T ) = 0. For stochastic volatility let us assume that there exists a traded contingent claim whose price can be written as G ( ε,δ) (= G ( S (δ) , σ (δ) , r (ε ) , t )) at t . Then this price process can be re-written as
where μ ( ε, δ )( t ) and σ ( ε )( t ) are the drift term and the volatility function of the contingent claim price. Together with the stochastic process for S ( δ ) , again by using the standard arguments of the no-arbitrage conditions, the risk premium functional λ ( ε, σ ) ( 1, 2, 3) can be determined by
Then the risk premium functionals can be uniquely given by solving the above equations if we have the non-degeneracy conditions ν ( ε )(t,T ) > 0,σ (ε ,δ )(t) > 0,σ ( δ ) > 0 and (5.63)
for 0 < t ≤ T . Our method here is a kind of spanning problem and market completion by financial contingent claims and this problem has been systematically investigated by Romano and Touzi (1997) in a slightly different formulation. Furthermore, in order to use the change of measures procedure, we need to assume the Novikov condition for the transformation of measures
Then we can take the Brownian Motions under the martingale measure Q , which is equivalent to the original measure P for (5.57)-(5.59) by using the Maruyama-Girsanov transformation with
We note that the expectation in (5.64) is taken with respect to the probability measure P . In this formulation of the risk premium functionals they are dependent on the underlying stochastic processes in a complicated way in the general case.  Because stochastic volatility and the stochastic interest rate become non-stochastic in the limit, we may assume the condition that the risk premium functionals are locally deterministic in the small disturbance asymptotics sense. Assumption III : The risk premium functionals λ ( ε,δ) ( t ) ( i = 1,2,3) are bounded and satisfy (5.63)-(5.64) and as ε ↓ 0 and δ ↓ 0,
This is a restrictive assumption and it is possible to relax this condition, but the fol- lowing analysis will become considerably more complex. By redefining W 1 t = c 11 B 1 t + c12B2t + c13B3t, W2t = c22B2t + c23B3t, and W3t = B3t such that we have the corre- lation structure of (2.4) with c ij ( i ≤ j ), we have the security price process and the interest rate processes with (2.1)-(2.4). The drift condition in the form of (2.1) is the standard formulation under the assumption of the existence of a locally riskless money market. By ignoring the higher order terms, under Assumption III we can write the drift function of r ( ε ) as
and in the present case A r ( t ) of (2.18) should be the solution of the stochastic integral equation,
Then it can be solved as
Similarly, we can write the drift function of σ ( o )
and we have the corresponding equation for (2.23) as
Also in the present case we have
and
In this way we can obtain the corresponding expressions in Sections 2 and 3 under the probability measure Q . The correlation coefficients among the underlying Brownian motions are invariant with respect to the change of measures that we have discussed. In order to use our derived formulae in practice, however, we have to estimate the appeared values of the parameters. We can ignore the extra terms discussed in this subsection if the risk premum functions are small and in the order of op (1). We should mention that for the underlying asset prices and their derivatives there have been different approaches to determine the prices of contingent claims at actual
20
financial markets, i.e., the (economic) general equilibrium prices, which have been ex- tensively investigated. (See Cox, Ingersoll, Ross (1985a,b) for instance.) In the general equilibrium framework, the risk functional λ (ε, δ) (t) (i = 1, 2, 3) in (5.62) should be de- termined as the function of individual preferences. Then the resulting relation between the drift functions of the underlying asset prices and the risk premium functionals could be in the form of (5.62). When ε ↓ 0 and δ ↓ 0, the limiting case is the complete market for the underlying asset S ( δ ) and its derivatives in any case because lim ε→ o r ( ε ) = r t and lim δ→ o σ ( δ ) = σ t in the sense of probability. This is the case regardless whether there exists any additional bond markets or contingent claim markets depending on the interest rates and volatility of underlying asset price. However, when δ > 0, the market is incomplete in the case when there does not exist an additional financial market on the volatility of the underlying asset. Thus, in this case, the security market with the price processes developed in Section 2 is incomplete, but not far from the complete market as the limiting case. We can call the situation which we have been considering the near complete market in this respect.
The HJM Case
5.2
In our formulation of the spot interest rate process, we have assumed (2.3) or (5.57), which is a Markovian-type continuous process. There has been another type of term structure of interest rate processes originally developed by Heath, Jarrow, and Morton (1992). In the HJM framework the instantaneous spot interest rate is not necessarily a Markovian continuous process. Let also the price of discount bond P(c)(s,t) be continuously differentiable with respect to t and P(c)(s,t) > 0 for 0 ≤ s ≤ t ≤ T and 0 < ε ≤ 1 . The instantaneous forward rate at s for future date t (0 ≤ s ≤ t ≤ T) is defined by
In the HJM term structure model of interest rates we assume that the bond market is complete and under the equivalent martingale measure, say Q*, a family of forward rate processes f(ε)(s,t) follow the stochastic integral equations
where f(0,t) (= f(0)(0,t)) are assumed to be observable and the non-random ini- tial forward rates, W 3 , i ( v ) ( i = 1, . . . , n ) are n independent Brownian motions, and ωƒ,i(ƒ(ε)(v,t),v,t) (i = 1, . . .,n) are the diffusion functions. This formulation has been adopted by Kunitomo and Takahashi (1995, 2001), who have investigated the valu- ation problems of interest rate-based contingent claims in detail. When f(ε)(s,t) is continuous at s = t for 0 ≤ s ≤ t ≤ T , the spot interest rate at t can be defined by
Kunitomo and Takahashi (1995, 1998, 2001) have derived the asymptotic expansion of the discount factor when ε ↓ 0 under a set of assumptions on the boundedness and smoothness of volatility functions. The equation (3.21) of Kunitomo and Takahashi (2001) implies
Then it is straightforward to modify our analysis developed in Sections 2 and 3. The essential point is to replace the formula for Σ(r) by
where ρf.i are the correlation coefficients between dW 1 t and dW 3 s ( t ) ( i = 1, . . ,n ). Then it is possible to obtain the corresponding results on the futures, forward, and options prices to Theorem 3.1, Theorem 3.2, Theorem 3.3, and Theorem 3.4 when the forward interest rate processes follow the stochastic integral equation given by (5.73) and the security returns exhibit stochastic volatility.
5.3
Concluding Remarks
We have investigated the value of contingent claims when both the interest rate and volatility are stochastic. We have found that the European options value can be de- composed into the original Black-Scholes option price and adjustment terms, which reflect the effects of the randomness of interest rates and volatility in the underlying stochastic processes. Similarly, the futures price can be decomposed into the futures price under constant interest rate and some adjustment terms. We also have illustrated numerical examples and examined the accuracy of our formulae. Since our formulae are relatively simple and quite accurate, they shall be useful for practical applications. The framework we have developed includes many situations as special cases and we have the standard Black-Scholes economy as the limiting case as both ε ↓ 0 and δ ↓ 0. This makes it possible to examine the effects of stochastic volatility and stochastic interest rate in a unified way. As we have illustrated by examples of futures and options in Section 3, our method can be also applicable to analyze other European- type contingent claims. Since the asymptotic expansion method we have developed has a solid mathematical basis called the Malliavin-Watanabe Calculus or the Watanabe-Yoshida theory on the Malliavin Calculus in stochastic analysis, the formulae in this paper are not ad-hoc approximations and they have a rigorous mathematical basis.  Also they are numerically accurate even for practical applications as we have illustrated in Section 4. In addition to these aspects we should mention that we have derived the asymptotic expansions for the futures price and options price in the order of ρ ( ε, δ ), but it is certainly possible to obtain the approximations up to any higher order terms by using the asymptotic expansion approach. However, the calculations we need will become more demanding than we have derived in Section 2. Finally, we should mention that the theoretical results reported in this paper can be utilized in empirical studies of actual devivatives markets. Kim (2002), for instance, has shown some preliminary results, but they are quite involved and the full details shall be reported on another occasion.
6
Mathematical Appendix
In this appendix we give the detailed proofs of Lemmas and Theorems stated in the previous sections.
6.1
Proof of Lemma 2.1
[i] Let X { c, δ } = log( S { c, δ } ) and X t = log( S t ) . By using (2.1) and (2.25), we represent the difference of two stochastic processes as
Then by using Assumption II, for any 0 ≤ t ≤ T there exists a positive constant M 1 such that
exist a positive constant M2 such that
By using the Borel-Cantelli Lemma and the standard argument in stochastic analysis (see Chapter IV of Ikeda and Watanabe (1989)), we have the first part of our results under Assumptions I and II as ε ↓ 0. [ii] We denote 3 × 1 vector h = ( h i ) ∈ H be the square-integrable continuous functions in [0, T ]. By differentiating (2.3) and (2.2) in the direction of h , we have the stochastic differential equations

23

Then we can represent the solution of these equations as
and (6.85)
where Y ( δ ) ( t ) and Y r ( ε ) ( t ) are the solutions of
tion as
where we denote g(c,s)(s) = σξδ)S(c,s)h,s, g(c,s)(s) = (Dhσ(s)S(c,s), and g(c,s)(s) = (Dhr's) See, δ) By applying the Fubini-type theorem to (6.82) and (6.83), we can further represent
equation as
where Y ( ε, δ) ( t ) is the solution of the stochastic differential equation (6.92) Hence we can represent the Malliavin-covariance of S ( ε,δ) as
We notice that as ε ↓ 0 and δ ↓ 0, σ (δ) → σ t, r (ε) → r t, and Y (ε ,δ) ( t) → S t by using similar but tedious standard arguments. Then we have the convergence results such that as ε ↓ 0 and δ ↓ 0 g ( ε ,δ ) (t) → S tσ th 1s and g (ε ,δ ) (t) → 0 (i = 2,3). Hence we have the desired result.  (Q.E.D.)
Proof of Lemma 2.2
6.2
By using the stochastic expansions, we shall focus on the stochastic approximation based on X 1 t as the leading term with respect to ε and δ in (2.29). By the Fubini-type theorem 8  for stochastic integrals, we notice that
Then by using the Gaussian property of W t , we have the representation dW 2t = ρσdW1t + dW žt and dW žt = ρrdW1t + dW žt , where we can construct the random variables W ž t and W ž t being independent of W 1 t under Q . The conditional expectation given X 1 t as the leading term is determined by
where we denote
Then from (2.21) we confirm the relations (2.33) and (2.34). By the same token, the second conditional expectation given X 1 t as the leading term is given by
Also by using the independent Gaussian random variables, we have the expression that the third conditional expectation given X 1 t is determined by

25
We note that the second equation in (6.98) can be obtained by applying Lemma 6.1 in Kunitomo and Takahashi (2001) ( or Lemma 6.4 of Kunitomo and Takahashi (2003b) for the details ). From the above relatively tedious expositions, we sum up the expressions of three random variables as
he random variables being uncorrelated with X 1 t and we where zit (i = 1,2,3) are have the relations E Ω  zit = x ] = 0 (i = 1, 2, 3) and EQ[ z 2 ] < +∞ by our constructions. By substituting (6.99)-(6.101) into (2.29), we have the result. We need the order calculations and their justifications as we have illustrated for r ( ε ) . It is straightfor- ward to do them as we have indicated at the end of Section 2 and we have omitted them. (Q.E.D.)
6.3
Proof of Theorem 3.3
Our derivation of the formula in Theorem 3.3 consists of three steps. The mathematical justifications of formal calculations are given. [i] Since the payoff function in the valuation problem of option contracts is not differ- entiable in the usual sense, we need to evaluate some expectations with the indicator function. First we prepare a useful lemma. Lemma 6.1 : Under the assumptions, the probability of the event { S { ε,σ } K > 0} can be expressed as
where Rg is the remainder terms of the order o(ε,δ), CT2 and
Proof of Lemma 6.1 : By using Lemma 2.2 for the event { S ( ε,δ ) − K ≥ 0} , the probability we need to evaluate is given by
where R 5 is the remainder term of the order o n ( ε, δ ) in (2.31). When Σ ( 2 ( T ) ≠ 0 9 , we can evaluate
by using the Gaussianity of the random variable X 1 T and the Tchebichev inequality. We note that when Σ ( σ ) ( T ) = 0 we can take ε and δ sufficiently small such that the first parenthesis of left-hand side of (6.105) is positive. Hence by using (6.105) we can evaluate the conditional probability as
where we have used the notations in the present Lemma and the remainder term R 9 . Finally we notice that the remainder term R 9 is R 5 times some polynomial function of X 1 T which is the Gaussian random variable. Because we have E [ R 2 ] < ∞ and R5 = op(ε, δ), we have (6.102).   (Q.E.D.) [ii] The next step is to evaluate the expectation operators. We define Λ 1 = EQ[ZoI(S(e,s) – K)], Λ 5 = EQ[Zf] I(S(e,s) – K)], and Λ 5 = EQ[Zf] I(S(e,s) – K)], where I( · ) is the indicator function of [0, +∞) . By recalling the definition of Z(c, o) and using the inequality (6.102), Λ 1 is represented as
where φ σ ( τ ) 2 ( x ) is the density function of the normal random variable X 1 T = x with zero mean and the variance σ(T) 2 . By transforming the random variable from x to
27

By construction the random variables z i T ( i = 1,2,3) are uncorrelated to the random variable X 1 T so that they are uncorrelated with y 1 and y 2 . Then by taking the expec- tations with respect to y 1 and y 2 and using the distribution function of the standard normal random variables Φ[·], we have
where C(€, δ) = [−єCT2 − δCT2 + єz1т + δ( z3т − z2т )]/σ(T ). By using the notation d1 = σ(T) − C 0 /σ(T),d2 = d1 − σ(T), and rearranging each terms, we have
Next, we try to evaluate the second term Λ 2 . By recalling the definition of Z 2 and using the inequality (6.103), it can be written by
By tranforming from x to y 1 = [ x − σ ( T ) 2 ] / σ ( T ) and using the Gaussian property such as ∂φ(y1)/∂y1 = −y1φ(y1) and ∂(y1φ(y1))/∂y1 = (1 − y2)φ(y1), we can express λ 2 as
where

28
Then by expanding the Gaussian density function and using the notation d 1 = σ ( T ) − σ(T), we have
where C(ε) = [−ε(C 12 − z 1τ)]/σ(T)  and C(δ) = [−δ(C 12 − ( z 3τ − z 2τ))]/σ(T) . By ignoring all the terms involving ε and δ because we are concerned with the order of o(ε, δ), we have
where R 10 is the remainder term of the order o(ε,δ) . Now we shall evaluate the term Λ ε . By using the definition of Z ( ε, δ ) and the inequality (6.102), we can express Λξ as
where R 11 is the remaining term of the order o(ε,δ) . [iii] : As the third step, by taking the conditional expectations and ignoring higher order terms, the first term Λ 1 is given by
and the density function of the standard normal random variable, respectively. Then by collecting three terms and ignoring higher order terms we have the expression as
and
After simple manipulations of three terms, we can derive the formula in Theorem 3.3. For the sake of completeness, we give a key lemma which makes our final results in compact forms. The proof is a result of direct calculations. Lemma 6.2 : With the notations we have used in this section and Section 3 we have the equality

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33
Table 1:  Futures Price under Downward Stochastic Interest Rate and Upward Volatility We set r 0 = 0.11 > ñ = 0.08, ñ r = 2.0, c = 0.1 for the interest rate process and σ 0 = 0.20 < δ = 0.30, κσ = 4.0, δ = 0.1 for the volatility process. Furthermore, we assume S o = 100 and T = 1.0. F o denotes the futures price at initial time and fo represents the forward price and Fo = Fc + [FD − FC] + c Fr +
Table 2: European Stock Call Option Value under Downward Stochastic Interest Rate and Upward Volatility: At-the-money Case We set r 0 = 0.11 > ñ = 0.08, ñ r = 2.0, c = 0.1 for the interest rate process and σ 0 = 0.20 < δ = 0.30, κ σ = 4.0, δ = 0.1 for the volatility process. Furthermore, we assume S 0 = K = 100 and T = 1.0. BS is the original Black-Scholes value and BS i ( i = D,r,σ) are the same as in (3.51). Vo stands for the option value under the stochastic volatility and interest rate BS + [ BSD − BS ] + cBS, + δBS,. SRvc is the option value under stochastic interest rate and constant volatility BS + [ BSD − BS ] + cBSr when σ = σ 0 and δ = 0. SVRC is the option value under constant interest rate and stochastic volatility BS + [BSD − BS] + δBSs when r = ro and ε = 0. SRVD is the option value under the stochastic interest rate and deterministic volatility BS + [ BSD − BS ] + cBSr. SVrD is the option value under the deterministic interest rate and stochastic volatility BS + [ BSD − BS ] + cBSg. Simul. represents the Monte Carlo simulation result for V 0 and Diff. is the difference between simulation result and V 0 .