- Email: [email protected]

Option valuation with co-integrated asset prices Jin-Chuan Duana;∗ , Stanley R. Pliskab a Joseph

L. Rotman, School of Management, University of Toronto, 105 St. George Street, Toronto, ON M5S 3E6, Canada b Department of Finance, University of Illinois at Chicago, USA Accepted 4 February 2003

Abstract This paper investigates theoretical and practical aspects of options that are based upon two or more assets which are co-integrated. For this purpose, a new, discrete-time model of asset prices is developed, a model featuring both the co-integration property as well as stochastic volatilities. Using a GARCH, equilibrium-based option pricing approach, it is shown that when volatilities are deterministic the option prices do not depend on the co-integration parameters, except for the mis-speci6cation e7ect as to the manner in which the volatilities are estimated. However, with stochastic volatilities the option prices explicitly depend upon the co-integration parameters. In order to understand these results better, this paper also examines a continuous-time, di7usion limit of the asset price system and empirically studies the co-integration e7ect using spread options based upon the S& P500 and the NASDAQ100. These numerical results suggest that consideration of co-integration can substantially alter the value, delta and vega of a spread option. ? 2003 Elsevier B.V. All rights reserved. JEL classi*cation: G13; C3 Keywords: Co-integration; Option valuation; GARCH; Di7usion limit; Spread options

1. Introduction Many options, such as spread, maximum, minimum, and basket options, are de6ned in terms of two or more underlying price processes. Typically (e.g., see Stulz, 1982; Pearson, 1995) the underlying price system is modeled as a multivariate geometric Brownian motion whose volatility matrix is constant, in which case the option valuation problem is straightforward. However, such a model is unrealistic for certain circumstances. For example, Mbanefo (1997) pointed out that the correlation between ∗

Corresponding author. Tel.: +1-416-946-5653; fax: +1-416-971-3048. E-mail address: [email protected] (J.-C. Duan).

0165-1889/03/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1889(03)00042-3

728

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

pairs of assets in the energy market is highly stochastic. And for two futures prices with the same underlying, the distant price minus the nearby price is bounded above by the cost-of-carry, so by arbitrage considerations this price system is incompatible with multivariate geometric Brownian motion. Thus, it is natural to consider whether more realistic models of price systems will lead to fundamentally new kinds of option formulas, or even to signi6cantly di7erent numerical results. In particular, co-integrated models of 6nancial assets have received considerable attention in recent years, so the purpose of this paper will be to study the valuation of options whose payo7s are based upon two or more asset prices which are co-integrated. The implications of co-integration for option pricing have, to our knowledge, never been explored in the literature. Co-integration refers to two or more non-stationary time series that are driven by one or more common non-stationary time series. If linearly combining several non-stationary time series leads to a stationary time series, these non-stationary time series are said to be co-integrated. Many economic data series are known to exhibit this property. For example, statistical evidence of co-integration has been reported for interest rates by Engle and Granger (1987), Hall et al. (1992), and Alexander and Johnson (1994); for foreign exchange rates by Baillie and Bollerslev (1989), Kroner and Sultan (1993), and Alexander and Johnson (1992, 1994); for futures and spot prices of re6ned petroleum products by Ng and Pirrong (1993a); for futures and spot prices of metals by Ng and Pirrong (1993b); for stock market indices in di7erent countries by Taylor and Tonks (1989) and Alexander and Johnson (1994); and for stock prices within an industry by Cerchi and Havenner (1988). In this paper we introduce, develop, and study the theory of valuation of options based on systems of co-integrated asset prices. The basic model of co-integrated asset prices is presented in Section 2. Since the concept of co-integration is associated with discrete time stochastic processes, this is the approach we take. In particular, we postulate an error correction model with stochastic volatilities that follow a multivariate GARCH process. Since this model involves heteroskedastic errors, existing theoretical results from the literature cannot simply be applied to verify that the asset prices are actually co-integrated. Consequently, in what constitutes one of the major results in this paper, we derive suIcient conditions for the assets in our model to indeed be co-integrated. To better understand our asset pricing model and its implications for option pricing, in Section 3 we present a numerical, two-asset example, with parameters estimated from historical data on the S&P500 and the NASDAQ100. Furthermore, in Section 4 we study a continuous-time, di7usion limit of a two-asset version of our basic, discrete-time model. Not only does this continuous-time process provide some economic intuition and insight about co-integration, but it constitutes a contribution toward the development of general theory for continuous-time, co-integrated processes. In Section 5 we provide our theoretical results for the valuation of options that are based upon two or more co-integrated assets. Since the market corresponding to our discrete-time model is not complete, arbitrage pricing theory cannot be used for option valuation. Instead we take an equilibrium approach and apply the concept of local risk-neutral valuation. The result is a new system of price processes and an

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

729

equilibrium pricing measure under which option values are expressed in the usual fashion as expected discounted payo7s. In what constitutes the most important result of this paper, it will be seen that the usual Black–Scholes results are recovered when volatilities are deterministic; in particular, the parameters associated with the co-integration property will not appear in calculations of option values. However, when co-integration is combined with stochastic volatilities, then the co-integration parameters will explicitly a7ect an option’s value. This last 6nding appears to be counter-intuitive, since co-integration parameters are con6ned to asset appreciation rates and given the fundamental Black–Scholes result that an asset’s expected return will have no e7ect on the value of options based upon this asset. But when volatilities are stochastic then all bets are o7: asset appreciation rates, and thus co-integration parameters, might a7ect the dynamics of the asset(s) under the risk neutral probability measure that is used for equilibrium pricing. We return to this issue at various points in this paper, including in Section 7 where we summarize our principal 6ndings and make some concluding remarks. Meanwhile, in Section 6 we return to our numerical example and demonstrate our theory by examining the theoretical price of a call option on the spread between the S&P500 and NASDAQ100 stock indices, two data series which we showed in Section 3 to be co-integrated. We do this for four versions of our model, corresponding to whether co-integration and/or stochastic volatility is assumed. Then by using Monte Carlo simulation we compute option prices for each of the four versions as well as for a variety of strike prices, maturities, and other parameter values. We also examine option deltas with respect to the change in each of two asset prices and option vegas with respect to the change in each of two conditional volatilities. Our general conclusion is that option prices and the “Greeks” can vary considerably, depending upon whether allowances are made for co-integration and stochastic volatility. 2. Discrete-time co-integrated assets with stochastic volatilities The error-correction model is adopted as the basis for constructing our discrete-time co-integrated asset return dynamics. In light of the well-documented empirical phenomenon of time-varying volatilities, we extend the standard error-correction model to incorporate stochastic volatility. This co-integrated asset return model then serves as the basis for our development of option pricing in later sections. Formally, an n-component vector time series Xt is said to be co-integrated with exactly k co-integrating relations if each component of Xt is integrated of order 1 and there exist exactly k linearly independent non-zero vectors cj ; j = 1; : : : ; k such that cj Xt is stationary for all j’s. Each vector cj is called a co-integrating vector. For more details, readers are referred to Hamilton (1994) for an exposition on co-integration. Under the co-integration set-up, the components of Xt , after being di7erenced once, are stationary, which is a typical feature of the existing models in the option pricing literature. Our analysis di7ers from the existing models in terms of speci6cally addressing the issue related to the dimension of non-stationarity for multivariate asset price processes.

730

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

By Granger’s representation theorem (see Engle and Granger, 1987), the co-integrated vector time series can be expressed as an error correction model. In order to be compatible with the typical 6nancial models in the option pricing literature, we consider a speci6c error correction model, in which changes in asset prices do not directly depend on the past changes in prices, but the unanticipated price changes are allowed to have time-varying volatilities. The time-varying volatility is assumed to follow a multivariate non-linear asymmetric GARCH(1,1) process. The non-linear asymmetric GARCH(1,1) dynamic is the version proposed by Engle and Ng (1993), and the multivariate GARCH speci6cation follows that of Bollerslev (1988). Our theoretical results can be straightforwardly generalized to all other GARCH dynamics and multivariate speci6cations. Let Ft stand for the time-t information set available to economic agents. Its speci6c de6nition is laid down in the assumption below. We assume an error correction dynamic for the n-component asset price time series with k co-integrating relations as follows. Assumption 1. With respect to the data generating probability measure P, the time-t price for the ith asset (i = 1; : : : ; n), denoted by Si; t ; in the n-component asset price vector with k co-integrating relations (k ¡ n) obeys k

1 ij zj; t−1 + i; t i; t ; ln(Si; t ) − ln(Si; t−1 ) = r − i;2 t + i i; t + 2

(1)

i;2 t = i0 + i1 i;2 t−1 + i2 i;2 t−1 (i; t−1 − i )2 ;

(2)

j=1

zj; t = aj + bj t +

n

cij ln(Si; t )

for j = 1; : : : ; k;

(3)

i=1

where (c1j ; c2j ; : : : ; cnj )

for j = 1; : : : k; are k linearly independent vectors with each of the vectors having at least one element equal to 1;

r is the risk-free rate; 1t 1 2t P 21 ∼N 0; .. .. . . nt n1

12

···

1

···

.. .

..

n2

···

i0 ¿ 0; i1 ¿ 0; i2 ¿ 0

.

1n

2n conditional on Ft−1 ; .. . 1

for i = 1; : : : ; n;

Ft is the -6eld generated by{Si; 0 ; i; 0 ; i; s : s ∈ {0; 1; 2; : : : ; t}; i = 1; : : : ; n}: In order for the above system to rePect the spirit of the error correction model, the processes zj; t , for j = 1; : : : k, must be stationary. The conditions under which they are

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

731

stationary are provided later in Proposition 1. For the moment, let us assume this is indeed the case. The co-integrating vectors under our assumption are (c1j ; c2j ; : : : ; cnj ) for j = 1; : : : ; k. We need to 6x some cij = 1, for every j = 1; : : : ; k, to avoid parameter indeterminacy caused by the fact that co-integrating relations are invariant to linear transformations of co-integrating vectors. Note that we have built a time trend variable into the co-integrating relation to provide a mechanism that can be used to adjust the growth rates in the underlying variables. For example, bj t can be used to neutralize the n potential deterministic trend in i=1 cij ln(Si; t ) so that zj; t can become stationary with mean 0 by adjusting the value of bj ’s (see Proposition 1 later for details). If the bj ’s are forced to take on the value of 0, the means of zj; t ’s are preset to the levels that are completely determined by other parameters. Theoretically, it is preferable to allow bj ’s to be free so that the overall levels of all error correction terms are not unduly constrained. The motivation for the inclusion of a co-integration term like zj; t in the short-run dynamic is based on a belief that many non-stationary time series may form some long-run relationships. Short-run equilibria may deviate from the long-run relationship, but any deviation is stationary and hence a temporary departure. The term “error correction” stems from allowing the short-run dynamic to be corrected by the deviation from the long-run relationship. Parameter i can be interpreted as the risk premium per unit of conditional risk, and ij will be referred to as the ith asset’s co-integration premium for the jth error correction term. In short, our formulation of the conditional expected return is allowed to rePect the conditional risk embodied in the asset return and any temporary deviation from the long-run equilibrium relationship. Our modeling setup can be easily extended, similar to the standard adjustment used in the option pricing literature, to include dividend yields when the underlying assets pay dividends or convenience yields when commodity options are the issue of interest. An alternative way of understanding the conditional mean speci6cation is to view k the term i i; t + j=1 ij zj; t−1 as a time-varying risk premium. Rearranging this term k k as i; t ( i + j=1 ij (zj; t−1 )= i; t ) suggests that i + j=1 ij (zj; t−1 )= i; t represents the total risk premium per unit of risk. According to our model, this unit risk premium is also time-varying, and the time-varying nature is caused by the error correction term. If economic agents are risk-neutral, then the unit risk premium must equal zero. Interestingly (and this can be a source of confusion), this implies that the error correction term vanishes and the (logarithmic) asset prices are not co-integrated under a risk neutral probability measure. This risk premium interpretation can also shed some light on why, say, two asset prices are co-integrated. For example, consider a simple two-asset spread (the 6rst asset price minus the second asset price) as the co-integration relation and assume the mean of the error correction term z1; t is zero. If the mean is not equal to zero, it can always be absorbed into 1 and 2 and causes no loss of generality. The sign of the co-integration premiums (11 and 21 ) can be negative or positive as long as the condition given later in Proposition 1 is satis6ed. In a nutshell, that condition simply requires the relatively cheaper asset to catch up with the more expensive one. This can take place in a number of forms; for example, the more expensive one rises in value but the cheaper one increases even more, or the more expensive one decreases in value but the cheaper one increases. Suppose that both assets have

732

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

positive systematic risks. We argue that this implies 1 ¡ 0 and 2 ¿ 0. If two asset prices are badly out of step, say, z1; t is a large positive number, the systematic risk of the 6rst asset will decrease whereas that of the second asset will increase. This is true because a temporary larger current price for the 6rst asset makes its future return smaller and hence reduces its systematic risk. Consequently, the risk premium of the 6rst asset will decrease, ceteris paribus, which is indeed the result implied by the risk premium formula when 1 ¡ 0. The logic for the second asset is similar and suggests that 2 ¿ 0. If z1; t is negative, a temporary lower current price for the 6rst asset makes its future return larger and thus increases its systematic risk. This results in a higher risk premium for the 6rst asset, and it is consistent with the prediction of the risk premium formula when 1 ¡ 0. It is clear that the similar logic again applies to the second asset. Although we have earlier referred to Granger’s representation theorem to motivate our error-correction representation of the dynamic for asset prices, that theorem does not necessarily lead to our speci6c form of error-correction. Moreover, since Granger’s representation theorem is established under the condition that errors (or innovations) are homoskedastic, it is not technically correct to apply that theorem to our setting, which by assumption has heteroskedastic errors. Can we be certain that the asset prices speci6ed according to Assumption 1 are co-integrated? Or, alternatively, is Assumption 1 self-consistent (i.e., can the endogenous process zj; t be stationary for j = 1; : : : ; k)? The answer is provided in the following proposition. For this we need to de6ne a k × k matrix:

n

ci1 i1 1+ i=1 n ci2 i1 A ≡ i=1 .. . n cik i1 i=1

n

ci1 i2

···

i=1

1+

ci2 i2

···

i=1

i=1

ci1 ik

n ci2 ik i=1 .. . n 1+ cik ik i=1

n

n

n

.. .

..

cik i2

···

.

(4)

i=1

and a matrix norm of A by A = max x=0

|Ax| ; |x|

(5)

where x is an n-dimensional column vector and |x| is its Euclidean norm. Proposition 1. The logarithmic asset prices under Assumption 1 are co-integrated if i;2 t , for i = 1; : : : ; n, are stationary with *nite *rst moments (under measure P), and

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

A ¡ 1. Furthermore, z1; t z2; t −1 EP .. = (I − A) . zk; t n n ci1 − ci1 EP (di; t ) − b1 + r i=1 i=1 n n b + r c − c EP (d ) − 2 i2 i2 i; t i=1 i=1 × n n b +r cik − cik EP (di; t ) − k i=1

i=1

733

n n 1 P 2 P ci1 E ( i; t ) + ci1 i E ( i; t ) 2 i=1 i=1 n n 1 P 2 P ci2 E ( i; t ) + ci2 i E ( i; t ) 2 i=1 i=1 : .. . n n 1 P 2 P cik E ( i; t ) + cik i E ( i; t ) 2 i=1

i=1

(6)

Proof. See appendix. The conditional variance i;2 t is known to be stationary and EP ( i;2 t ) ¡ ∞ under the usual GARCH parameter restriction (see Duan, 1997). The stationarity condition on i;2 t thus presents no diIculty. The condition that the matrix norm of A is less than 1 imposes a parameter constraint. This constraint is easier to understand when there are only two assets (n = 2) and one co-integration relation (k=1). Since one of the c’s must equal 1, we can set c11 =1. The matrix norm condition reduces to |1+11 +c21 21 | ¡ 1. We will verify this condition later in our empirical implementation. It is worth noting that the value of EP (zj; t ) = 0 has nothing to do with aj . This is true because a di7erent value for aj causes ln(Si; t ) to react according to Eq. (1). The combination of values of ln(Si; t ) for i = 1; : : : ; n turns out to o7set aj in equation (3) so that the dynamic of zj; t remains una7ected. This result, however, does not suggest the irrelevance of aj , because its value does a7ect the dynamic of Si; t individually. 3. Empirical example: the S&P500 and NASDAQ100 To illustrate our model we consider a two-asset system in accordance with the speci6cation in Eqs. (1)–(3). In order for this to be a meaningful numerical analysis, we use realistic parameter values obtained from estimations based upon real data. The S&P500 and NASDAQ100 indices are used for this purpose. Here we present and discuss parameter estimates, whereas in a later section we shall build upon this model to value spread options and to compute the associated option deltas and vegas. We would like

734

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754 1300 1100 900 700 500 S&P500

300 NASDAQ100

100 11/23/90 12/28/91 01/31/93 03/07/94 04/11/95 05/15/96 06/19/97 07/24/98

Fig. 1. S&P500 and NASDAQ100 daily closing index values from January 2, 1991 to May 15, 1998.

to emphasize that our principal aim here is simply to illustrate our methods and not, for example, to conduct a comprehensive study of spread options for stock indices or to engage in a debate on whether stock market indices are co-integrated. The parameters of the co-integration model are estimated using the S&P500 and NASDAQ100 daily closing index values from January 2, 1991 to May 15, 1998, a total of 1864 data points. These two series are plotted in Fig. 1. It is evident from this plot that the NASDAQ100 index grows faster than does the S&P500 index. Later, our parameter estimates reveal that the NASDAQ100 index return is more volatile by a factor of roughly about two (measured in terms of standard deviation). Weekly updated three-month US Treasury bill rates are used as the risk-free rates. For the model allowing for co-integration, we need to identify the co-integration relation. Since there are only two series, we only need to consider at most one co-integrating relation. We conduct the following co-integration regression to obtain the co-integrating vector. As argued earlier, a time trend variable is included to take into account the growth in the underlying variables. The model and results are given by ln(S1; t ) = 1:746433 − 0:0000696985t + 0:746413 ln(S2; t ) + ut ; (0:039161) (0:0000062441) (0:007249);

(7)

where S1; t (S2; t ) denotes the S&P500 (NASDAQ100) index value at time t, and the numbers in the parentheses are the OLS standard errors. The Phillips–Ouliaris z test statistic (12 lags) for the co-integration regression in (7) equals −27:6, which implies that the null hypothesis of no co-integration is rejected at the 5% level. The critical value is available from the case 3 section of Table B.8 in Hamilton (1994). In other words, Eq. (7) is not due to the spurious regression e7ect. The co-integration regression gives rise to the following stationary series: zt = −1:746433 + 0:0000696985t + ln(S1; t ) − 0:746413 ln(S2; t ): The constructed series, zt , is needed in the second stage estimation.

(8)

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

735

We obtain four sets of maximum likelihood estimates for three variations of the model: constant volatility, stochastic volatility without co-integration, and stochastic volatility with co-integration. Two sets of parameter estimates for the constant volatility model are presented to examine whether omitting co-integration in the constant volatility model will cause a signi6cant mis-speci6cation bias. It will be shown in a later section that under the constant volatility assumption, the option price does not theoretically depend on co-integration. If co-integration has any e7ect on option prices, it must be due to mis-speci6cation biases on the estimates for other relevant parameters. The parameter values along with their asymptotic standard errors are reported in Table 1. The results for Model I suggest the presence of co-integration. The cointegration premium for the NASDAQ100 index return is signi6cantly positive, i.e., 2 . Examining Model II leads us to conclude the presence of the GARCH e7ect for both data series, i.e., 11 ¿ 0; 12 ¿ 0; 21 ¿ 0; 22 ¿ 0 . The leverage e7ect is also present in two data series because 1 ¿ 0 and 2 ¿ 0. These results taken together motivate the use of Model III, which contains both co-integration and stochastic volatility e7ects. Since 12 ¿ 0, the result suggests that S&P500 and NASDAQ100 indices, in addition to being co-integrated, have correlated return innovations. We now verify the conditions given in Proposition 1 to see whether the estimated system is consistent with co-integration. First, the estimated GARCH parameter values for each asset satisfy the stationary condition that i1 + i2 (1 + i2 ) ¡ 1. As stated earlier, the matrix norm condition for the two-asset case is |1 + 1 + c2 2 | ¡ 1. Given our estimates, namely c2 = −0:746413, 1 = −0:00214, and 2 = 0:022806, it is clear that the matrix norm condition is satis6ed. Thus we do have a two-asset co-integrated price system. 4. A di)usion limit Since a continuous-time framework is commonly used for option pricing, and since continuous-time models are often easier for most 6nancial researchers to comprehend on an intuitive level, it will be interesting and useful for a better understanding to know the di7usion limit of our model. This will also advance the development of suitable continuous-time models of co-integrated assets. For notational simplicity, we restrict our consideration to the case of two assets. We adopt and modify the approach of Nelson (1990) and Duan (1997) to arrive at the following limiting model (see Appendix for details): 1 d ln(S1; t ) = r − 1;2 t + 1 1; t + 1 zt dt + 1; t dB1t ; (9) 2 d 1;2 t = {10 + [11 + 12 (1 + 12 ) − 1] 1;2 t } dt + 12 2(1 + 212 ) 1;2 t dB2t ; (10) 1 2 d ln(S2; t ) = r − 2; t + 2 2; t + 2 zt dt + 2; t dB3t ; (11) 2 d 2;2 t = {20 + [21 + 22 (1 + 22 ) − 1] 2;2 t } dt + 22 2(1 + 222 ) 2;2 t dB4t : (12)

736

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

Table 1 Parameter estimates (standard errors) for three models. Data are S&P500 (data series #1) and NASDAQ100 (data series #2) daily closing index values from January 2, 1991 to May 15, 1998 Model I

Model II

Model III

(Constant volatility, (Constant volatility, (Stochastic volatility, (Stochastic volatility, w/o co-integration) with co-integration) w/o co-integration) with co-integration) Mean parameters

1

2 1

0.075003 (0.023299) 0.071560 (0.023314)

2 Volatility parameters 5:72305 × 10−5 10 (0:09698 × 10−5 ) 11

0.074985 (0.023846) 0.071770 (0.023509) −0:001057 (0.004772) 0.019970 (0.007733)

0.060282 (0.024010) 0.061229 (0.023893)

0.060107 (0.024165) 0.063733 (0.023857) −0:002140 (0.004104) 0.022806 (0.007924)

5:72289 × 10−5 (0:09712 × 10−5 )

0:08052 × 10−5 (0:01554 × 10−5 ) 0.940156 (0.006773) 0.041637 (0.005206) 0.360347 (0.120935)

0:05843 × 10−5 (0:01226 × 10−5 ) 0.951957 (0.005717) 0.036995 (0.004618) 0.244193 (0.123706)

17:92216 × 10−5 (0:48234 × 10−5 )

17:86452 × 10−5 (0:48018 × 10−5 )

0:30952 × 10−5 (0:07704 × 10−5 ) 0.939866 (0.007507) 0.041499 (0.004723) 0.245906 (0.098139)

0:23161 × 10−5 (0:06500 × 10−5 ) 0.950378 (0.006514) 0.037542 (0.004704) 0.037540 (0.093588)

12

0.742089 (0.009869)

0.743597 (0.009867)

0.744265 (0.009499)

0.748906 (0.009368)

Log-likelihood sample size

16008.89 1864

16016.57 1864

16188.61 1864

16197.87 1864

12 1 20 21 22 2

zt = a + bt + ln(S1; t ) + c ln(S2; t );

(13)

where (B1t ; B2t ; B3t ; B4t ) is a four-dimensional standard Brownian motion process (under measure P) with the following correlation structure: Var P (B1t ; B2t ; B3t ; B4t )

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

√ −21

1

√ −21 2(1+22 ) 1 =t √−22 2 2(1+22 )

2(1+212 )

1

√−21 2 2(1+21 )

√−21 2

1

2(1+21 )

√

2

+21 2 (1+212 )(1+222 )

√ −22

2(1+222 )

√−22 2

2(1+22 )

2 √ +22 1 2 2 (1+21 )(1+22 )

√ −22

2(1+222 )

1

737

:

(14)

The limiting system is four-dimensional. Note that zt is only used to simplify notations. The complexity of the system arises from both co-integration and stochastic volatility. In order to better understand the e7ects of co-integration, we switch o7 stochastic volatility for the time being. Setting 11 = 12 = 21 = 22 = 0 gives rise to a much simpler system: 1 (15) d ln(S1; t ) = r − 12 + 1 1 + 1 zt dt + 1 dB1t ; 2 1 d ln(S2; t ) = r − 22 + 2 2 + 2 zt dt + 2 dB3t ; (16) 2 zt = a + bt + ln(S1; t ) + c ln(S2; t ); (17) where i = i0 and CovP (B1t ; B3t ) = t. Quite intuitively, the error correction mechanism acts through the drift coeIcients to keep a weighted average of log prices close to a trend line. For instance, if 1 = 2 = ; 1 = 2 = ; 2 = 0; and −2 ¡ 1 = ¡ 0; then condition (6) implies E(zt ) = −[b + (1 + c)(r − 2 =2 + )]=, which we denote it by %. With the help of stochastic calculus it can then be shown that dzt = −(% − zt ) dt + dB1t + c dB3t :

(18)

Since ¡ 0, it follows that zt is a stationary Ornstein–Uhlenbeck process having long-run mean equal to %. Thus, ln(S1; t ) and ln(S2; t ) are Gaussian random walks with drift, but they are co-integrated because they, through the relationship de6ned by zt , maintain a long-run relative position. Such a limiting result is informative in fact. By the fundamental option pricing theory under di7usion, it is expected that co-integration without stochastic volatility will not a7ect the value of an option. This must be true because co-integration without stochastic volatility only a7ects the system through the drift terms. We will elaborate on this point later.

5. Option pricing under co-integration For option pricing in the discrete-time setting we resort to the use of an equilibrium argument, because the market in our model is inherently incomplete and so arbitrage pricing theory by itself is not suIcient for deriving an operational option valuation model. We adopt the local risk-neutral valuation principle introduced in Duan (1995)

738

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

as a generalization of the equilibrium approach carried out by, for example, Brennan (1979) and Rubinstein (1976). The following assumption provides a description of this valuation principle. Assumption 2. The equilibrium pricing measure Q, which is de6ned over the period from 0 to a 6nite integer T , satis6es the local risk-neutral valuation relationship (LRNVR), that is, (1) Q and P are mutually absolutely continuous; (2) conditioned on Ft−1 , Si; t =Si; t−1 for i = 1; : : : ; n has a multivariate lognormal probability distribution (under Q); (3) EQ (Si; t =Si; t−1 |Ft−1 ) = er for all i; and (4) CovQ (ln(Si; t =Si; t−1 ); ln(Sj; t =Sj; t−1 )|Ft−1 )=CovP (ln(Si; t =Si; t−1 ); ln(Sj; t =Sj; t−1 )|Ft−1 ) for all i and j, almost surely with respect to P. The conditions under which local risk-neutralization holds have been given in Duan (1995). It is referred to as local risk-neutralization because the asset return viewed locally (i.e., one period ahead) under the equilibrium pricing measure Q can be described by a system with a change in all conditional mean returns to the risk-free rate but without altering the conditional variance-covariance structure. In other words, in this suitably constructed economy the economic agents can be viewed as behaving in a risk-neutral fashion. Under Assumptions 1 and 2, the asset price process under the equilibrium pricing measure can be described by the proposition that follows. A formal proof will be omitted since it is a straightforward extension of the ideas in Duan (1995) (i.e., in analogy with Girsanov’s theorem for continuous time processes, just substitute i; t = )i; t + !i; t after choosing the predictable process !i; t in a suitable manner). Proposition 2. With respect to the equilibrium pricing measure Q, the price for the ith asset in the n-component asset price vector obeys ln(Si; t ) − ln(Si; t−1 ) = r − 12 i;2 t + i; t )i; t ; i;2 t = i0 + i1 i;2 t−1 + i2 i;2 t−1 )i; t−1 − i − i − zj; t = aj + bj t +

n

k j=1

cij ln(Si; t )

2

ij

zj; t−2 ; i; t−1

for j = 1; : : : ; k;

i=1

where

)1; t

1

)2; t Q 21 ∼N 0; .. .. . . )n; t n1

12

···

1

···

.. .

..

n2

···

.

1n

2n conditional on Ft−1 : .. . 1

(19) (20) (21)

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

739

Note that r, i (i = 1; : : : ; n), ij (i = 1; : : : ; n; j = 1; : : : ; k), and ij (i; j = 1; : : : ; n) can all be generalized to predictable stochastic processes without a7ecting the pricing result in Proposition 2. The reason is that the LRNVR is entirely based on a local relationship. Since everything is conditional on Ft−1 , any Ft−1 -measurable random variable behaves locally just like a constant. To gain a better understanding of the co-integration option pricing system, it is convenient to consider two distinct, fundamental cases. In the 6rst case the volatility is deterministic, i.e., i1 = i2 = 0, and then co-integration can only have e7ects through parameter estimation. That is, the volatility parameter estimates will be inPuenced by the error correction speci6cation, but the option pricing model remains as in the standard Black–Scholes (1973) framework. In particular, the co-integration parameters ij and cij will not enter into any formula for the option’s value. For instance, the maximum option has an analytical formula under constant volatility and continuous time that was previously derived by Stulz (1982), and this same formula will be obtained under our discrete time model when i1 =i2 =0: But while the pricing formula is not a7ected by co-integration, the biases in parameter values caused by ignoring co-integration may still be important. The article by Lo and Wang (1995) illustrates the importance of having a correct speci6cation for the mean asset return even if the Black–Scholes option pricing model does not depend on the mean return of the asset. The reason is in a way intuitive. Di7erent speci6cations for the mean will lead to di7erent volatility estimates, and consequently di7erent option prices. As shown in Eq. (1), co-integration alters the mean speci6cation and thus could a7ect the estimates for other parameters that are theoretically important to option pricing. For the other fundamental case of our model, if the volatilities are stochastic, then the co-integration e7ect becomes theoretically important. The volatility dynamic with respect to the risk-neutralized pricing measure is a7ected by the co-integration premium ij . If ij = 0, the pricing result in Proposition 2 reduces to the GARCH option pricing model of Duan (1995) in which only the asset risk premium i enters into pricing system. In summary, our pricing result under stochastic volatility shows that co-integration is both theoretically important and statistically relevant. The general di7usion limit of the model under the risk-neutral pricing measure Q is somewhat complicated. Here we use the constant volatility case to gain better understanding of our theory. Under constant volatility, the di7usion limit corresponding to system (19)–(21) under the risk-neutral pricing measure Q is 1 2 ∗ d ln(S1; t ) = r − 1 dt + 1 dB1t ; (22) 2 1 ∗ d ln(S2; t ) = r − 22 dt + 2 dB3t ; (23) 2 zt = a + bt + ln(S1; t ) + c ln(S2; t );

(24)

∗ ∗ (B1t ; B3t )

where is a two-dimensional standard Brownian motion process (under measure Q). Moreover, ∗ ∗ CovQ (B1t ; B3t ) = CovP (B1t ; B3t ) = t:

(25)

740

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

The two-component asset price system, after discounting by the risk-free rate, is clearly a martingale with respect to measure Q. Of course, (22)–(24) could have been derived directly from (15)–(17) using standard methods, in accordance with the general martingale pricing theory developed by Harrison and Kreps (1979) and Harrison and Pliska (1981). This pricing result is not at all surprising because co-integration only dictates how the drift term is formulated. It is a well-known result of option pricing theory that the drift term under the real-world probability measure is inconsequential as far as option valuation is concerned. In the case of stochastic volatility, this point can be a source of confusion and thus deserves some further discussion. Stochastic volatility might not arise from another source of uncertainty. For example, with the constant elasticity of variance di7usion model, co-integration irrelevancy continues to be true. Again, this should not be a surprise given our knowledge of classical option pricing theory. But the situation becomes di7erent when stochastic volatility arises from another source of uncertainty. The limiting GARCH model given in the preceding section clearly falls in this category. Co-integration a7ects option pricing because it shows up in the volatility dynamics through the measure transformation. In all cases, though, asset prices cannot be co-integrated under a risk neutral pricing measure Q, for otherwise the discounted asset prices would not be martingales.

6. Example continued: S&P500 and NASDAQ100 spread options The pricing dynamic in Proposition 2 can be used to value any claim contingent on the n-component asset price vector. Because the option price is the discounted expected payo7, Monte Carlo simulation or other numerical methods can be used to evaluate the expected value. We illustrate this with the two-asset, S&P500/NASDAQ100 model that was introduced in an earlier section. For derivatives on two assets, the relevant dynamic is actually a Markovian vector system with a dimension equal to four. In particular, (S1; t ; S2; t ; 1;2 t+1 ; 2;2 t+1 ) is a suIcient statistic for the information available at time t. We use spread options to demonstrate the e7ect of co-integration on option valuation. The spread option considered here has the following payout structure: max(S1T −qS2T − X; 0), where X and q are some positive constants (note that a basket option might correspond to the case where q is negative). To obtain standard spread options, we can set q = 1. The time-t value of the spread option can be evaluated by the following expression: SP(S1; t ; S2; t ; 1;2 t+1 ; 2;2 t+1 ; T ) = e−r(T −t) EQ {max(S1; T − qS2; T − X; 0)|Ft }:

(26)

With the co-integration relation in Eq. (8) and the parameter values in Table 1, we are ready to analyze the numerical properties of the option pricing theory under co-integration. We examine the co-integration option pricing model by comparing three scenarios. These scenarios are characterized by three versions of our model:

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

741

(1) Model I: (constant volatility). (2) Model II: (stochastic volatility, without co-integration). (3) Model III: (stochastic volatility, with co-integration). Recall our option pricing result under constant volatility. Because co-integration coef6cients 1 and 2 play no direct role in the pricing system, there is no need to consider co-integration above and beyond the e7ect of the estimation bias due to model mis-speci6cation. As shown in Table 1, the estimates for all relevant option pricing parameters are practically equivalent under constant volatility. We thus only consider the case without co-integration for the constant volatility model. Note that under constant volatility, an analytical formula actually exists. We use Monte Carlo simulation in this case purely for the purpose of coming up with option values in a way comparable to other models. The comparison between Models II and III is a di7erent story, however, because 1 and 2 play a direct role in the option pricing system characterized by equations (19) and (20). Some estimated volatility parameter values are also quite di7erent; for example, 2 . In short, co-integration can make a di7erence because of the di7erence both in the theory itself and in parameter estimates. 6.1. Spread option values Eq. (8) contains a time trend variable to take into account the data position relative to the 6rst observation in the sample. It is therefore more convenient to use an alternative, equivalent form: for s ¿ 0, zt+s = zt + 0:0000696985s + ln

S2; (t+s) S1; (t+s) : − 0:746413 ln S2; t S1; t

(27)

The variable q in the spread formula is set according to q = (1 + w)(S1; t =S2; t ) at the valuation time t. The spread ratio in the payout function can thus be adjusted in relation to the initial spread by altering the parameter w. This way of expressing q also allows us to simplify the presentation by considering the spread option value as a fraction of the S&P500 index value. Speci6cally, we rewrite Eq. (26) as SP(S1; t ; S2; t ; 1;2 t+1 ; 2;2 t+1 ; T ) S1; t S1; T S2; T X = e−r(T −t) EQ max − (1 + w) − ; 0 Ft : S1; t S2; t S1; t

(28)

The normalized strike price (X=S1; t ) and maturity (T ) will be varied to study their e7ects on the spread option price. The speci6c numerical results for spread options are obtained using the parameter estimates in Eq. (8) and Table 1. For all calculations, we assume a zero risk-free rate of interest. All prices under the three models are computed with 200,000 sample paths with common random numbers. We 6rst consider the scenario that S1; t = 1108:73, S2; t = 1249:49, and zt = 0:072165. The values for these two indices are the actual index values on May 15, 1998, which is the last observation

742

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

of our data sample. The value for zt is computed using Eq. (8) by setting t = 1864, the sequence number of that data point. For Model I, the volatilities are available from Table 1; that is, the annualized (252 days) standard deviations for the two assets are 12% and 21.25%, respectively. If we consider stochastic volatility, the situation is di7erent. For both Models II and III we need to specify the two conditional volatilities, i.e., 1; t+1 and 2; t+1 . Although these values are available as a by-product of estimation, varying them allows us to gain a better understanding of the model. We thus consider two cases: the stationary level (referred to as average volatility) and 20% above the stationary level (referred to as high volatility). The stationary levels under measure P can be computed using the following formula: i0 Vi = : (29) 1 − i1 − i2 (1 + i2 ) Two sets of stationary volatilities are computed, one for Model II and the other for Model III, using the parameter values reported in Table 1. For Model II, V1 = 0:007931 (12.59% if annualized), V2 = 0:013028 (20.68% if annualized). For Model III, we have V1 = 0:008129 (12.90% if annualized), V2 = 0:013877 (22.03% if annualized). Table 2 corresponds to the situation that w = 0 or q = (S1; t =S2; t ) = 1108:73=1249:49 = 0:8873. For Table 3, we set w = 0:1 or q = 1:1 × (S1; t =S2; t ) = 1:1 × 0:8873 = 0:9760. We also consider two maturities: 3 months and 1 year, and the results are reported in two panels of each table. For ease of comprehending the general pricing results, we have plotted the second panel of Table 3 and presented the option values along with two-standard deviation con6dence bands in Fig. 2. Since Fig. 2 pretty much summarizes the option pricing behavior reported in Tables 2 and 3, we will focus on Fig. 2 and refer readers to Tables 2 and 3 for speci6c values. In order to gain a better understanding of the role played by the error correction term, we also consider a numerical scenario for which we keep the same value for S1; t , i.e., 1108:73, but change the value for zt to a negative number. Speci6cally, we set zt =−0:05. This combination implies, by equation (8), that S2; t =1471:6776. The spread option value, stated as a fraction of the S&P500 index value, will not be a7ected by this change if the valuation model is Model I or II. This is true because the pricing dynamics (with respect to measure Q) for S1; T =S1; t and S2; T =S2; t do not depend on zt under these two models. This change does a7ect Model III, however. Our results corresponding to negative zt are organized under the heading of Model III’ to rePect that they are based on Model III but use a di7erent value of zt . It is clear from Fig. 2 that stochastic volatility increases spread option values because Model I yields the lowest values. Comparing Models II, III and III’, we conclude that co-integration a7ects option values and depends on the value of zt . Consideration of co-integration under stochastic volatility has a tendency to push down in a signi6cant fashion spread option values if the error correction term switches from positive to negative, i.e., moving from Model III to Model III’. The valuation system is too complex to give an intuitive reason why the spread option value moves in such a direction. We know, however, that changing the value of zt a7ects the volatilities 1; t and 2; t in opposite way under the risk-neutral pricing measure Q (see Eq. (20)). This is true

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

743

Table 2 Spread option values (reported as fractions of the S&P500 index value) and Monte Carlo errors (in parentheses) for three models using 200,000 sample paths. Strike price ratio stands for the strike price divided by the S&P500 index value. Model III’ is the same as Model III except the error correction term has been changed from 0.072165 to −0:05. Parameter values are taken from Table 1. In the case of Model I, the parameter estimates are under no co-integration. The spread formula uses q = S1t =S2t . “Avg. vol.” stands for using the stationary standard deviations as the conditional volatilities, whereas “High vol.” corresponds to using the conditional volatilities that are 20% above their respective stationary levels Model I

Model II

Model III

Model III

Avg. vol.

High vol.

Avg. vol.

High vol.

Avg. vol.

High vol.

0.029460 (0.000091) 0.028941 (0.000090) 0.028429 (0.000089) 0.026923 (0.000087) 0.024520 (0.000083) 0.020122 (0.000075) 0.010135 (0.000052) 0.004330 (0.000033) 0.002206 (0.000022)

0.030535 (0.000098) 0.030028 (0.000098) 0.029526 (0.000097) 0.028054 (0.000095) 0.025706 (0.000091) 0.021408 (0.000083) 0.011549 (0.000061) 0.005591 (0.000042) 0.003240 (0.000031)

0.034437 (0.000110) 0.033928 (0.000110) 0.033423 (0.000109) 0.031939 (0.000107) 0.029557 (0.000103) 0.025150 (0.000095) 0.014666 (0.000072) 0.007846 (0.000052) 0.004915 (0.000041)

0.030530 (0.000098) 0.030022 (0.000098) 0.029519 (0.000097) 0.028044 (0.000095) 0.025693 (0.000091) 0.021387 (0.000083) 0.011530 (0.000061) 0.005588 (0.000042) 0.003245 (0.000031)

0.034736 (0.000111) 0.034224 (0.000110) 0.033717 (0.000110) 0.032226 (0.000107) 0.029833 (0.000103) 0.025401 (0.000096) 0.014849 (0.000073) 0.007972 (0.000053) 0.005011 (0.000041)

0.029788 (0.000094) 0.029270 (0.000093) 0.028758 (0.000092) 0.027254 (0.000090) 0.024859 (0.000086) 0.020483 (0.000078) 0.010562 (0.000056) 0.004783 (0.000037) 0.002615 (0.000026)

0.034025 (0.000106) 0.033505 (0.000106) 0.032990 (0.000105) 0.031473 (0.000103) 0.029042 (0.000099) 0.024543 (0.000091) 0.013906 (0.000068) 0.007128 (0.000048) 0.004301 (0.000037)

0.058813 (0.000172) 0.058273 (0.000172) 0.057736 (0.000171) 0.056141 (0.000168) 0.053536 (0.000164) 0.048530 (0.000157) 0.035215 (0.000133) 0.024458 (0.000109) 0.018655 (0.000094)

0.062104 (0.000189) 0.061576 (0.000189) 0.061050 (0.000188) 0.059490 (0.000185) 0.056941 (0.000182) 0.052041 (0.000174) 0.038941 (0.000150) 0.028217 (0.000127) 0.022324 (0.000112)

0.065203 (0.000198) 0.064674 (0.000197) 0.064147 (0.000197) 0.062582 (0.000194) 0.060024 (0.000190) 0.055093 (0.000183) 0.041825 (0.000159) 0.030832 (0.000136) 0.024718 (0.000121)

0.063542 (0.000202) 0.063025 (0.000202) 0.062510 (0.000201) 0.060982 (0.000199) 0.058484 (0.000195) 0.053688 (0.000187) 0.040884 (0.000164) 0.030393 (0.000142) 0.024594 (0.000127)

0.067304 (0.000214) 0.066785 (0.000213) 0.066268 (0.000212) 0.064732 (0.000210) 0.062222 (0.000206) 0.057388 (0.000198) 0.044384 (0.000175) 0.033591 (0.000153) 0.027549 (0.000138)

0.059767 (0.000183) 0.059232 (0.000182) 0.058700 (0.000181) 0.057119 (0.000179) 0.054540 (0.000175) 0.049597 (0.000167) 0.036510 (0.000144) 0.025991 (0.000121) 0.020316 (0.000107)

0.063633 (0.000194) 0.063097 (0.000193) 0.062563 (0.000193) 0.060978 (0.000190) 0.058389 (0.000187) 0.053413 (0.000179) 0.040115 (0.000156) 0.029256 (0.000133) 0.023297 (0.000118)

Strike price ratio 0.000 0.001 0.002 0.005 0.010 0.020 0.050 0.080 0.100 0.000 0.001 0.002 0.005 0.010 0.020 0.050 0.080 0.100

744

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

Table 3 Spread option values (reported as fractions of the S&P500 index value) and Monte Carlo errors (in parentheses) for three models using 200,000 sample paths. Strike price ratio stands for the strike price divided by the S&P500 index value. Model III’ is the same as Model III except the error correction term has been changed from 0.072165 to −0:05. Parameter values are taken from Table 1. In the case of Model I, the parameter estimates are under no co-integration. The spread formula uses q = 1:1 × (S1t =S2t ). “Avg. vol.” stands for using the stationary standard deviations as the conditional volatilities, whereas “High vol.” corresponds to using the conditional volatilities that are 20% above their respective stationary levels Model I

Model II

Model III

Model III

Avg. vol.

High vol.

Avg. vol.

High vol.

Avg. vol.

High vol.

0.003569 (0.000031) 0.003462 (0.000030) 0.003358 (0.000030) 0.003060 (0.000028) 0.002612 (0.000026) 0.001873 (0.000021) 0.000607 (0.000011) 0.000159 (0.000005) 0.000057 (0.000003)

0.004889 (0.000041) 0.004772 (0.000040) 0.004657 (0.000040) 0.004325 (0.000038) 0.003816 (0.000036) 0.002948 (0.000031) 0.001281 (0.000020) 0.000516 (0.000012) 0.000268 (0.000009)

0.007086 (0.000052) 0.006940 (0.000052) 0.006796 (0.000051) 0.006378 (0.000049) 0.005728 (0.000046) 0.004592 (0.000041) 0.002252 (0.000028) 0.001030 (0.000019) 0.000590 (0.000014)

0.004895 (0.000041) 0.004778 (0.000040) 0.004663 (0.000040) 0.004332 (0.000038) 0.003825 (0.000036) 0.002960 (0.000031) 0.001296 (0.000020) 0.000527 (0.000012) 0.000276 (0.000009)

0.007209 (0.000053) 0.007060 (0.000052) 0.006914 (0.000052) 0.006492 (0.000050) 0.005836 (0.000047) 0.004685 (0.000042) 0.002313 (0.000029) 0.001065 (0.000019) 0.000614 (0.000014)

0.004024 (0.000035) 0.003916 (0.000035) 0.003810 (0.000034) 0.003508 (0.000033) 0.003049 (0.000030) 0.002282 (0.000026) 0.000889 (0.000016) 0.000313 (0.000009) 0.000146 (0.000006)

0.006260 (0.000047) 0.006119 (0.000046) 0.005981 (0.000046) 0.005582 (0.000044) 0.004963 (0.000041) 0.003896 (0.000036) 0.001776 (0.000024) 0.000744 (0.000015) 0.000397 (0.000011)

0.024154 (0.000113) 0.023860 (0.000112) 0.023569 (0.000111) 0.022710 (0.000109) 0.021326 (0.000105) 0.018733 (0.000098) 0.012298 (0.000077) 0.007649 (0.000059) 0.005387 (0.000048)

0.028214 (0.000133) 0.027917 (0.000132) 0.027622 (0.000131) 0.026752 (0.000129) 0.025346 (0.000125) 0.022695 (0.000118) 0.015937 (0.000098) 0.010817 (0.000079) 0.008192 (0.000068)

0.030978 (0.000142) 0.030670 (0.000141) 0.030364 (0.000141) 0.029460 (0.000138) 0.027995 (0.000135) 0.025223 (0.000127) 0.018086 (0.000106) 0.012567 (0.000087) 0.009688 (0.000076)

0.030835 (0.000150) 0.030543 (0.000149) 0.030254 (0.000149) 0.029397 (0.000146) 0.028009 (0.000143) 0.025385 (0.000136) 0.018634 (0.000116) 0.013402 (0.000097) 0.010637 (0.000086)

0.034223 (0.000162) 0.033918 (0.000161) 0.033616 (0.000161) 0.032720 (0.000158) 0.031268 (0.000155) 0.028512 (0.000148) 0.021342 (0.000127) 0.015677 (0.000108) 0.012634 (0.000096)

0.025670 (0.000126) 0.025384 (0.000125) 0.025101 (0.000125) 0.024265 (0.000123) 0.022918 (0.000119) 0.020393 (0.000112) 0.014084 (0.000092) 0.009431 (0.000075) 0.007101 (0.000064)

0.029082 (0.000138) 0.028782 (0.000138) 0.028484 (0.000137) 0.027604 (0.000135) 0.026183 (0.000131) 0.023504 (0.000124) 0.016707 (0.000104) 0.011556 (0.000085) 0.008907 (0.000074)

Strike price ratio 0.000 0.001 0.002 0.005 0.010 0.020 0.050 0.080 0.100 0.000 0.001 0.002 0.005 0.010 0.020 0.050 0.080 0.100

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

745

0.035

0.030

Model III

0.025 Model II

0.020

0.015 Model III'

0.010 Model I

0.005 0.000 0.00

0.02

0.04

0.06

0.08

0.10

Fig. 2. Spread option values of one-year maturity (reported as fractions of the S&P500 index value) along with two-standard deviation con6dence bands (based on 200,000 sample paths). The horizontal axis is the strike price ratio (the strike price divided by the S&P500 index value). The spread formula uses q = 1:1 × (S1t =S2t ) and the option values are based on the assumption that the conditional volatilities equal their respective stationary volatilities.

because the parameter values for 1 and 2 , as reported in Table 1, are exactly opposite in sign. Comparing to Model II, we learn that co-integration adds additional Pexibility in adjusting spread option values depending on how far the two underlying assets are o7 their long-run relationship. This suggests that one should pay attention to the co-integration feature when derivatives on multiple underlying assets are considered. 6.2. Spread option deltas and vegas To analyze how the spread option value reacts to the change in one of its determining factors, we need to be speci6c about which factors are stochastic and which are not. In accordance with the parametric model, only stochastic variables are going to change over time and thus constitute legitimate sources of risk in the strict sense of parametric models. As stated earlier in Eq. (26), the spread option value is a function of four stochastic variables—two asset prices and two conditional standard deviations. We thus consider the partial derivatives of the spread option value with respect to each one of these four variables. Note that our interpretation e7ectively views deltas and vegas as the four pillars of the 6rst-order Taylor expansion of the spread option valuation function. To compute the delta with respect to, say, the 6rst asset, we must make a corresponding adjustment to zt . Since zt is a combination of two asset prices, a change in

746

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

Table 4 Spread option deltas (reported as fractions of the S&P500 index value) for two models. Strike price ratio stands for the strike price divided by the S&P500 index value. Model III’ is the same as Model III except the error correction term has been changed from 0.072165 to −0:05. Parameter values are taken from Table 1. The spread formula uses q = S1t =S2t . For both models, the initial conditional standard deviations are set at their respective stationary levels Model II 1st asset

Model III

Model III 2nd asset

1st asset

2nd asset

1st asset

2nd asset

Maturity = 3 months 0.000 0.000455 0.001 0.000450 0.002 0.000445 0.005 0.000431 0.010 0.000407 0.020 0.000359 0.050 0.000229 0.080 0.000129 0.100 0.000082

−0:000379 −0:000375 −0:000370 −0:000358 −0:000337 −0:000295 −0:000184 −0:000100 −0:000062

0.011991 0.012063 0.012127 0.012303 0.012594 0.012939 0.012375 0.009706 0.007498

−0:008990 −0:009043 −0:009090 −0:009220 −0:009433 −0:009685 −0:009250 −0:007249 −0:005598

0.001151 0.001230 0.001310 0.001552 0.001938 0.002658 0.004010 0.003864 0.003169

−0:000842 −0:000901 −0:000960 −0:001141 −0:001429 −0:001967 −0:002979 −0:002874 −0:002358

Maturity = 1 year 0.000 0.001 0.002 0.005 0.010 0.020 0.050 0.080 0.100

−0:000365 −0:000363 −0:000361 −0:000355 −0:000345 −0:000324 −0:000263 −0:000207 −0:000172

0.056999 0.057147 0.057283 0.057557 0.058041 0.058825 0.059327 0.057121 0.054368

−0:042558 −0:042668 −0:042769 −0:042973 −0:043333 −0:043915 −0:044282 −0:042630 −0:040573

0.006320 0.006481 0.006607 0.007114 0.007925 0.009477 0.013436 0.016042 0.016804

−0:004680 −0:004800 −0:004894 −0:005272 −0:005878 −0:007037 −0:009993 −0:011941 −0:012513

Strike price ratio

0.000468 0.000465 0.000463 0.000456 0.000444 0.000421 0.000350 0.000281 0.000238

the value of the 6rst asset while keeping the second one constant necessarily forces a change in zt . There is, however, no such complication when we compute vegas. We compute deltas and vegas by numerical derivatives, which are calculated by repeating Monte Carlo simulation with two di7erent values for the variable in question. Take the delta for the 6rst asset as an example. We 6rst compute the spread option value using a 6xed set of values for the four variables. We then add a small value, 10−8 , to the 6rst asset price and recompute the spread option price. As a result, the numerical derivative can be computed. Common random numbers are used in these Monte Carlo calculations to avoid discontinuity. We use 200,000 sample paths in computing every Monte Carlo price. Models II–III are used to compute deltas and vegas for spread option contracts with various strike price ratios and two maturities. The results are reported in Tables 4 and 5. All deltas and vegas are reported as fractions of the S&P500 index value. For example, in Table 4 the delta for the 6rst asset for a 3-month maturity and a strike price ratio of 0.005 under Model II is reported to be 0:000431. Its actual value is

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

747

Table 5 Spread option vegas (reported as fractions of the S&P500 index value) for two models. Strike price ratio stands for the strike price divided by the S&P500 index value. Model III’ is the same as Model III except the error correction term has been changed from 0.072165 to −0:05. Parameter values are taken from Table 1. The spread formula uses q = S1t =S2t . For both models, the initial conditional standard deviations are set at their respective stationary levels Model II 1st asset

Model III

Model III 2nd asset

1st asset

2nd asset

1st asset

2nd asset

Maturity = 3 months 0.000 −0:476962 0.001 −0:476500 0.002 −0:476298 0.005 −0:472507 0.010 −0:467795 0.020 −0:451798 0.050 −0:364486 0.080 −0:247419 0.100 −0:180153

1.635696 1.634539 1.633482 1.626938 1.611809 1.560880 1.272254 0.882143 0.635726

−0:530475 −0:528974 −0:527878 −0:523961 −0:518414 −0:498952 −0:401510 −0:272270 −0:197820

1.784047 1.781748 1.779470 1.771578 1.752543 1.691527 1.367445 0.939803 0.673419

−0:474945 −0:471728 −0:470665 −0:465947 −0:451522 −0:421187 −0:307941 −0:187130 −0:120262

1.763303 1.760825 1.759276 1.751331 1.729531 1.664465 1.318925 0.870324 0.594515

Maturity = 1 year 0.000 0.001 0.002 0.005 0.010 0.020 0.050 0.080 0.100

1.276863 1.275610 1.274472 1.270441 1.263806 1.245580 1.157386 1.028011 0.929933

−0:527999 −0:528464 −0:527699 −0:523899 −0:515923 −0:499238 −0:439100 −0:366190 −0:315910

1.578630 1.577553 1.576738 1.572689 1.563616 1.538878 1.435390 1.288205 1.173703

−0:418646 −0:416795 −0:414445 −0:410400 −0:401662 −0:376334 −0:296113 −0:216402 −0:161678

1.549379 1.548162 1.547036 1.542268 1.534497 1.506765 1.385864 1.218664 1.086281

Strike price ratio

−0:408495 −0:408181 −0:405810 −0:400233 −0:394418 −0:382314 −0:329832 −0:269876 −0:229683

0:000431 × 1108:73 = 0:4779 because the S&P500 index value equals 1108.73. It is fairly clear from Table 4 that the deltas under Model II are, after being restored back to their natural values, similar in magnitude to the standard delta for the European option on one asset only. The situation is very di7erent when we move to Model III where deltas are easily thirty or forty times the deltas under Model II. For Model III’ the deltas are easily three or four times those under Model II. The reason for these large di7erences is related to our earlier description about how deltas should be calculated under Models III and III’. When we change, say, the price of the 6rst asset, the value of the error correction term is also a7ected. In other words, incorporation of co-integration makes the spread option value much more sensitive to a change in the asset price, and the sensitivity is also highly dependent upon the value of the error correction term. The vega considered here is the partial derivative of the spread option price with respect to a change in the conditional standard deviation for each of the two assets. The results for spread option vegas are presented in Table 5. It is interesting to note that

748

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

the vega of the 6rst asset is negative, but for the second asset it is positive regardless which model is used. This result may appear puzzling at 6rst glance, but actually it is somewhat intuitive. To see this, consider the variance of the spread between two asset prices and analyze its derivative with respect to each of the two individual conditional standard deviations. Let v12 and v22 be the variances of S1; T =S1; t and S2; T =S2; t , respectively, and / be the correlation coeIcient. We have the following simple result: S1; T S2; T 0(v1 ; v2 ) = Var = v12 + (1 + w)2 v22 − 2(1 + w)v1 v2 /; − (1 + w) S1; t S2; t v2 @0(v1 ; v2 ) = 2v1 − 2(1 + w)v2 / = 2v1 1 − (1 + w) / ; @v1 v1 @0(v1 ; v2 ) v2 = 2(1 + w)2 v2 − 2(1 + w)v1 / = 2v1 (1 + w) (1 + w) − / : @v2 v1 ∼ ∼ Our earlier parameter estimates suggest that v2 =v1 = 2 and / = 0:75. Since the results in Table 5 are obtained by setting w = 0, it is clear that @0(v1 ; v2 )[email protected] ¡ 0 and @0(v1 ; v2 )[email protected] ¿ 0, which is a result in agreement with the signs reported in Table 5. Although there are noticeable di7erences in vegas across models, the magnitude of the variation is not as pronounced as that for the deltas. In summary, consideration of co-integration can have major implications for hedging derivative contracts. 7. Conclusion In this paper we have developed an option pricing theory for the setting in which several assets have co-integrated prices subject to time-varying volatility. Indeed, our paper seems to be the 6rst to produce explicit values for options on assets whose appreciation rates are connected by co-integration. Since the literature on co-integration lives in a discrete-time setting, it was natural for us to utilize the multivariate GARCH modeling approach. Wanting to develop a co-integration model featuring heteroskedastic errors, our 6rst main contribution (Proposition 1) is a set of suIcient conditions for the assets in our model to indeed be co-integrated. Our resulting model is, of course, one for which markets are incomplete, meaning there is not necessarily a single arbitrage-free price for an option (because there are in6nitely many risk neutral probability measures). To resolve the valuation problem in our setting, it was natural to employ the local risk-neutralization technique developed by Duan (1995) for the GARCH model, which relies on the competitive equilibrium concept to yield a unique option price. As can be seen in this paper, setting up a proper co-integrated GARCH system and then working out the characterizations of option prices were complicated matters and not just simple applications of existing theory. Hence our option pricing results for systems of co-integrated assets constitute the second major contribution of this paper. In particular, we show that in a discrete time context the terms referred to as co-integration premiums enter the pricing dynamics if the underlying assets’ volatilities are stochastic. In this case the impact of incorporating co-integration goes beyond the mis-speci6cation induced parameter bias

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

749

that was articulated by Lo and Wang (1995) in the case of the Black–Scholes option pricing model. Thus the speci6cs of the co-integration mechanism are important not only for correct estimates of volatility parameters, but also for correct speci6cations of option pricing formulas. The contributions of our paper do not stop here. We provide for the 6rst time a diffusion speci6cation of co-integrated systems. It was not clear to us, nor to many other specialists whom we know, how a co-integrated di7usion system should be formulated. We managed to produce such a system by using a discrete-time error correction model and then deducing its weak limit. The result initially surprised us, but later we discovered that it was rather intuitive once the issue is viewed from a proper angle. We did not develop option pricing theory for continuous-time systems of co-integrated assets except for the case of constant volatility di7usion. Thus a few remarks are in order. The di7usion model (15)–(17) constitutes a complete market model and, as seen by (22)–(24), the prices of options will not depend on the co-integration mechanism. It is clear that a similar outcome will apply to some other complete market, co-integrated di7usion models. However, this conclusion does not necessarily apply to all co-integrated, complete market models. Kallsen and Taqqu (1998) and Duan (2001) provided continuous- and discrete-time, respectively, GARCH option pricing models in which the markets are complete and option values are uniquely determined by arbitrage. Interestingly, their models yield a theoretical pricing conclusion identical to that of Duan (1995), suggesting that option prices are functions of the underlying asset’s risk premium even if the market is complete. Analogously, a complete market, co-integrated continuous-time (or discrete-time) model with stochastic volatility, if constructed using the idea from Kallsen and Taqqu (1998) (or Duan (2001)), will have a pricing implication identical to the one developed in our paper; that is, co-integration mechanism will impact option prices beyond the mere parameter estimation e7ect. Acknowledgements The authors thank Jae Young Kim for his valuable inputs. Duan acknowledges support received as the Manulife Chair in Financial Services and research funding from both the Social Sciences and Humanities Research Council of Canada and the Natural Sciences and Engineering Research Council of Canada. Pliska is grateful for support by NSF Grant DMS-9971424 as well as by the Center for Financial Engineering in the Institute for Economic Research at Kyoto University. Appendix A. Proof of Proposition 1. Since each i;2 t is stationary, it is clear that ln(Si; t ) is integrated of order 1 if zj; t for all j’s are stationary. If we can show that zj; t for all j’s are indeed stationary under the matrix norm condition, then we have two results: (1) the logarithmic asset prices are individually integrated of order 1, and (2) some linear combinations of these logarithmic prices are stationary. Together, they imply

750

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

co-integration. Our task is thus to show zj; t for all j’s are indeed stationary. First, we use Eq. (1) to obtain n n cij ln(Si; t ) − cij ln(Si; t−1 ) i=1

i=1

=r

n

cij −

i=1

n

n

k

n

n

i=1

i=1

l=1

i=1

i=1

1 cij i;2 t + cij i i; t + zl; t−1 cij il + cij i; t i; t : 2

The above equation in turn implies that zj; t − zj; t−1 − bj =r

n

cij −

i=1

or

zj; t = bj + r

n

n

n

k

n

n

i=1

i=1

l=1

i=1

i=1

1 cij i;2 t + cij i i; t + zl; t−1 cij il + cij i; t i; t 2 n

cij −

i=1

+ zj; t−1

1+

n

1 cij i;2 t + cij i i; t 2 i=1

n

i=1

cij ij

i=1

+

k

zl; t−1

n i=1

l=j

cij il +

n

cij i; t i; t :

i=1

Expressing the system in a vector form of zj; t for all j’s yields the following: Zt = AZt−1 + Ut ; where Zt ≡ [z1; t ; z2; t ; : : : ; zk; t ] and

n n n 1 2 ci1 − ci1 i; t + ci1 i i; t + ci1 i; t i; t b1 + r 2 i=1 i=1 i=1 i=1 n n n n 1 b +r ci2 − ci2 i;2 t + ci2 i i; t + ci2 i; t i; t 2 2 i=1 i=1 i=1 i=1 Ut ≡ : .. . n n n n 1 b +r cik − cik i;2 t + cik i i; t + cik i; t i; t k 2 n

i=1

i=1

i=1

i=1

The above multivariate system can be solved recursively to yield n−1 Zt = An Zt−n + Ai Ut−i for n ¿ 1: i=0

Suppose that 1) An Y , for any k × 1 random vector Y , converges to the zero vector n−1 almost surely (measure P) as n tends to in6nity, and 2) i=0 Ai Ut−i converges to a

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

751

∞ random vector almost surely (measure P) as n tends to in6nity. Then, Zt = i=0 Ai Ut−i is the unique (almost surely under measure P) solution to the recursive system. It is clear that such a Zt is stationary because any shift in time index does not alter the distribution. It remains to show that the two assertions are indeed true. It is clear that Ut forms a stationary sequence of vectors by our stationarity assumption on i;2 t . Moreover, every element of Ut has a 6nite 6rst moment by the assumption of 6nite 6rst moments for i;2 t . (Note that EP ( i;2 t ) ¡ ∞ implies EP ( i; t ) ¡ ∞.) First, n→∞

|An Y | 6 An × |Y | 6 A n × |Y | → 0 almost surely (measure P) because A ¡ 1: The 6rst assertion is thus true. To prove the second assertion, we claim that lim supi→∞ 1=i ln|Ut−i | 6 0 almost surely (measure P). If not, lim supi→∞ ln |Ut−i | = ∞ for a set of states with a positive probability, which contradicts the fact that all elements of Ut−i have 6nite 6rst moments. This leads us to the following almost sure relationship: 1 1 ln|Ai Ut−i | 6 lim sup ln( A i |Ut−i |) i i→∞ i→∞ i 1 = lim sup ln A + ln|Ut−i | ¡ 0: i i→∞

lim sup

The strict inequality is true because A ¡ 1. The above result in turn implies lim sup |Ai Ut−i |1=i ¡ 1 almost surely (measure P): i→∞

By the standard root test for convergence, we have the result that converges almost surely (measure P). We now compute ∞ ∞ P i P E (Zt ) = A E (Ut−i ) = Ai EP (Ut ) = (I − A)−1 EP (Ut ): i=0

where

i=0

n n 1 P 2 P ci1 − ci1 E ( i; t ) + ci1 i E ( i; t ) b1 + r 2 i=1 i=1 i=1 n n n 1 b +r P 2 P ci2 − ci2 E ( i; t ) + ci2 i E ( i; t ) 2 2 i=1 i=1 i=1 EP (Ut ) = : .. . n n n 1 b +r P 2 P cik − cik E ( i; t ) + cik i E ( i; t ) k 2 n

i=1

The proof is thus complete.

i=1

i=1

n−1 i=0

Ai Ut−i

752

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

Di?usion Limit We adopt and modify the approach of Nelson (1990) and Duan (1997) by considering the following approximating system (two assets): for k = 1; : : : ; mT; √ 1 (m)2 (m) (m) (m) (m) ln(S1; ks ) − ln(S1; (k−1)s ) = r − 1; ks + 1 1; ks + 1 z(k−1)s s + 1;(m)ks 1k s 2 (m)2 (m)2 2 1;(m)2 (k+1)s − 1; ks = 10 s + 1; ks [11 + 12 (1 + 1 ) − 1]s

√ 2 2 + 12 1;(m)2 ks [(1k − 1 ) − (1 + 1 )] s √ 1 (m)2 (m) (m) (m) (m) ln(S2;(m) + z ) − ln(S ) = r − +

2 (k−1)s s + 2; ks 2k s 2 2; ks 2; (k−1)s 2; ks ks 2 (m)2 (m)2 2 2;(m)2 (k+1)s − 2; ks = 20 s + 2; ks [21 + 22 (1 + 2 ) − 1]s

√ 2 2 + 22 2;(m)2 ks [(2k − 2 ) − (1 + 2 )] s (m) (m) zks = a + bks + ln(S1;(m) ks ) + c ln(S2; ks );

where s = 1=m and the time interval [0; T ] is divided into mT subintervals of length s and where (1k ; 2k ), k = 1; 2; : : : ; is a sequence of i:i:d: standard normal random vectors with Cov(1k ; 2k )=. A weak convergence argument similar to Nelson (1990) or Duan (1997) gives rise to a di7usion limit as m goes to in6nity: 1 2 d ln(S1; t ) = r − 1; t + 1 1; t + 1 zt dt + 1; t dB1t 2 d 1;2 t = {10 + [11 + 12 (1 + 12 ) − 1] 1;2 t } dt + 12 2(1 + 212 ) 1;2 t dB2t 1 2 d ln(S2; t ) = r − 2; t + 2 2; t + 2 zt dt + 2; t dB3t 2 2 d 2; t = {20 + [21 + 22 (1 + 22 ) − 1] 2;2 t } dt + 22 2(1 + 222 ) 2;2 t dB4t zt = a + bt + ln(S1; t ) + c ln(S2; t ); where (B1t ; B2t ; B3t ; B4t ) is a four-dimensional standard Brownian motion process (under measure P) with the following correlation structure: Var P (B1t ; B2t ; B3t ; B4t ) √ −21 2 1 2(1+21 ) √ −21 1 2(1+22 ) 1 =t √−21 2 2(1+21 ) 2 √−22 2 √ +22 1 2 2(1+22 )

(1+21 )(1+222 )

√−22 2

√−21 2

2(1+21 )

1 √ −22

2(1+222 )

2(1+22 )

√

2

+21 2 (1+212 )(1+222 )

√ −22

2(1+222 )

1

:

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

753

The above result is derived by considering the contemporaneous relationship among four random variables: (1k − 1 )2 − (1 + 12 ) (2k − 2 )2 − (1 + 22 ) 1k ; : ; 2k ; Var[(2k − 2 )2 ] Var[(1k − 1 )2 ] Since all four random variables have mean 0 and variance 1, we can invoke Donsker’s Theorem (see Either and Kurtz, 1986, Theorem 1.2(c), p. 278) to show that √ (1k − 1 )2 − (1 + 12 ) (2k − 2 )2 − (1 + 22 ) ; 2k ; s 1k ; Var[(2k − 2 )2 ] Var[(1k − 1 )2 ] converges weakly to a four-dimensional Brownian increment with a particular correlation structure. The correlation structure depends on the following quantities: Var P (1k ) = 1; Var P [(1k − 1 )2 ] = 2(1 + 212 ); Var P (2k ) = 1; Var P [(2k − 2 )2 ] = 2(1 + 222 ); CovP [1k ; (1k − 1 )2 ] = −21 ; CovP (1k ; 2k ) = ; CovP [1k ; (2k − 2 )2 ] = −22 ; CovP [(1k − 1 )2 ; 2k ] = −21 ; CovP [(1k − 1 )2 ; (2k − 2 )2 ] = 22 + 41 2 ; CovP [2k ; (2k − 2 )2 ] = −22 : The limiting model for Si; t and i;2 t (for i = 1; 2) then follows. References Alexander, C., Johnson, A., 1992. Are foreign exchange markets really eIcient? Economics Letters 40, 449–453. Alexander, C., Johnson, A., 1994. Dynamic links. RISK 7 (2), 57–60. Baillie, R., Bollerslev, T., 1989. Common stochastic trends in a system of exchange rates. Journal of Finance 44, 137–151. Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659. Bollerslev, T., 1988. On the correlation structure on the generalized autoregressive conditional heteroskedastic process. Journal of Time Series Analysis 9, 121–131. Brennan, M., 1979. The pricing of contingent claims in discrete time models. Journal of Finance 34, 53–68.

754

J.-C. Duan, S.R. Pliska / Journal of Economic Dynamics & Control 28 (2004) 727 – 754

Cerchi, M., Havenner, A., 1988. Cointegration and stock prices: the random walk on Wall Street revisited. Journal of Economic Dynamics and Control 12, 333–346. Duan, J.-C., 1995. The GARCH option pricing model. Mathematical Finance 5, 13–32. Duan, J.-C., 1997. Augmented GARCH(p; q) process and its di7usion limit. Journal of Econometrics 79, 97–127. Duan, J.-C., 2001. Risk premium and pricing of derivatives in complete markets. Unpublished manuscript, University of Toronto. Either, S., Kurtz, T., 1986. Markov Processes—Characterization and Convergence. Wiley, New York. Engle, R., Granger, C., 1987. Co-integration and error correction: representation, estimation and testing. Econometrica 55, 251–276. Engle, R., Ng, V., 1993. Measuring and testing the impact of news on volatility. Journal of Finance 48, 1749–1778. Hall, A., Anderson, H., Granger, C., 1992. A cointegration analysis of treasury bill yields. Review of Economics and Statistics 74, 116–126. Hamilton, J., 1994. Time Series Analysis. Princeton University Press, Princeton, NJ. Harrison, M., Kreps, D., 1979. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20, 381–408. Harrison, M., Pliska, S., 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Applications 11, 215–260. Kallsen, J., Taqqu, M., 1998. Option pricing in ARCH-type models. Mathematical Finance 8, 13–26. Kroner, K., Sultan, J., 1993. Time varying distributions and dynamic hedging with foreign currency futures. Journal of Financial and Quantitative Analysis 28, 535–551. Lo, A., Wang, J., 1995. Implementing option pricing models when asset returns are predictable. Journal of Finance 50, 87–129. Mbanefo, A., 1997. Co-movement term structure and the valuation of energy spread options. In: Dempster, M.A.H., Pliska, S.R. (Eds.), Mathematics of Derivative Securities. Cambridge University Press, Cambridge, pp. 88–102. Nelson, D., 1990. ARCH models as di7usion approximations. Journal of Econometrics 45, 7–38. Ng, V., Pirrong, C., 1993a. Price dynamics in physical commodity spot and futures markets: spreads, spillovers, volatility, and convergence in re6ned petroleum products. Unpublished manuscript, University of Michigan. Ng, V., Pirrong, C.,1993b. Fundamentals and volatility: storage, spreads, and the dynamics of metals prices. Unpublished manuscript, University of Michigan. Pearson, N., 1995. An eIcient approach for pricing spread options. Journal of Derivatives, 76 –91. Rubinstein, M., 1976. The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics and Management Science 7, 407–425. Stulz, R., 1982. Options on the minimum or maximum of two risky assets: analysis and applications. Journal of Financial Economics 10, 161–186. Taylor, M., Tonks, I., 1989. The internationalisation of stock markets and the abolition of U.K. exchange control. Review of Economics and Statistics 71, 332–336.