Abstract
I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset's price is important for explaining return skewness and strike-price biases in the Black-Scholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems
Many plaudits have been aptly used to describe Black and Scholes' (1973) contribution to option pricing theory. Despite subsequent development of option theory, the original Black-Scholes formula for a European call option remains the most successful and widely used application. This formula is particularly useful because it relates the distribution of spot returns to the cross-sectional properties of option prices. In this article, I generalize the model while retaining this feature.
Although the Black-Scholes formula is often quite successful in explaining stock option prices [Black and Scholes (1972)], it does have known biases [Rubinstein (1985)]. Its performance also is substantially worse on foreign currency options [Melino and Turnbull (1990, 1991), Knoch (1992)]. This is not surprising, since the Black-Scholes model makes the strong assumption that (continuously compounded) stock returns are normally distributed with known mean and variance. Since the Black-Scholes formula does not depend on the mean spot return, it cannot be generalized by allowing the mean to vary. But the variance assumption is somewhat dubious. Motivated by this theoretical consideration, Scott (1987), Hull and White (1987), and Wiggins (1987) have generalized the model to allow stochastic volatility. Melino and Turnbull (1990, 1991) report that this approach is successful in explaining the prices of currency options. These papers have the disadvantage that their models do not have closed-form solutions and require extensive use of numerical techniques to solve two-dimensional partial differential equations. Jarrow and Eisenberg (1991) and Stein and Stein (1991) assume that volatility is uncorrelated with the spot asset and use an average of Black-Scholes formula values over different paths of volatility. But since this approach assumes that volatility is uncorrelated with spot returns, it cannot capture important skewness effects that arise from such correlation. I offer a model of stochastic volatility that is not based on the Black-Scholes formula. It provides a closed-form solution for the price of a European call option when the spot asset is correlated with volatility, and it adapts the model to incorporate stochastic interest rates. Thus, the model can be applied to bond options and currency options.
1. Stochastic Volatility Model
Equations (10), (17), and (18) give the solution for European call options. In general, one cannot eliminate the integrals in Equation (18), even in the Black-Scholes case. However, they can be evaluated in a fraction of a second on a microcomputer by using approximations similar to the standard ones used to evaluate cumulative normal probabilities.3
2. Bond Options, Currency Options, and Other Extensions
One can incorporate stochastic interest rates into the option pricing model, following Merton (1973) and Ingersoll (1990). In this manner, one can apply the model to options on bonds or on foreign currency. This section outlines these generalizations to show the broad applicability of the stochastic volatility model. These generalizations are equivalent to the model of the previous section, except that certain parameters become time-dependent to reflect the changing characteristics of bonds as they approach maturity.
Although the stochastic interest rate models of this section are tractable, they would be more complicated to estimate than the simpler model of the previous section. For short-maturity options on equities, any increase in accuracy would likely be outweighed by the estimation error introduced by implementing a more complicated model. As option maturities extend beyond one year, however, the interest rate effects can become more important [Koch (1992)]. The more complicated models illustrate how the stochastic volatility model can be adapted to a variety of applications. For example, one could value U.S. options by adding on the early exercise approximation of Barone-Adesi and Whalley (1987). The solution technique has other applications, too. See the Appendix for application to Stein and Stein's (1991) model (with correlated volatility) and see Bates (1992) for application to jump-diffusion processes.
3. Effects of the Stochastic Volatility Model Options Prices
The stochastic volatility model can conveniently explain properties of option prices in terms of the underlying distribution of spot returns. Indeed, this is the intuitive interpretation of the solution (10), since |$P_2$| corresponds to the risk-neutralized probability that the option expires in-the-money. To illustrate effects on options prices, we shall use the default parameters in Table 1.5 For comparison, we shall use the Black-Scholes model with a volatility parameter that matches the (square root of the) variance of the spot return over the life of the option.6 This normalization focuses attention on the effects of stochastic volatility on one option relative to another by equalizing “average” option model prices across different spot prices. The correlation parameter |${\rho}$| positively affects the skewness of spot returns. Intuitively, a positive correlation results in high variance when the spot asset rises, and this “spreads” the right tail of the probability density. Conversely, the left tail is associated with low variance and is not spread out. Figure 1 shows how a positive correlation of volatility with the spot return creates a fat right tail and a thin left tail in the distribution of continuously compounded spot returns.7Figure 2 shows that this increases the prices of out-of-the-money options and decreases the prices of in-the-money options relative to the Black-Scholes model with comparable volatility. Intuitively, out-of-the-money call options benefit substantially from a fat right tail and pay little penalty for an increased probability of an average or slightly below average spot return. A negative correlation has completely opposite effects. It decreases the prices of out-of-the-money options relative to in-the-money options.
Conditional probability density of the continuously compounded spot return over a six-month horizon
Spot asset dynamics are |$dS(t) = \mu S\,dt + \sqrt {\upsilon (t)} S\,dz_1 (t),$| where |$d\upsilon (t) = \kappa ^{\mathbf {*}} [\theta ^{\mathbf {*}} - \upsilon (t)]dt + \sigma \sqrt {\upsilon (t)} dz_2 (t).$| Except for the correlation |${\rho}$| between |$z_1$| and |$z_2$| shown, parameter values are shown in Table 1. For comparison, the probability densities are normalized to have zero mean and unit variance.
Option prices from the stochastic volatility model minus Black-Scholes values with equal volatility to option maturity
Except for the correlation |${\rho}$| between |$z_1$| and |$z_2$| shown, parameter values are shown in Table 1. When |${\rho = -5}.$|5 and |${\rho =.5},$| respectively, the Black-Scholes volatilities are 7.10 percent and 7.04 percent, and at-the-money option values are $2.83 and $2.81.
Default parameters for simulation of option prices
$\begin{array} {l} {dS(t) = \mu S\,dt + \sqrt {\upsilon (t)S} \,dz_1 (t),} & {(10)} \cr {d\upsilon (t) = \kappa ^{\mathbf {*}} [\theta ^{\mathbf {*}} - \upsilon (t)]dt + \sigma \sqrt {\upsilon (t)} dz_2 (t).} & {(30)}\end{array}$ | |
|---|---|
| Parameter | Value |
| Mean reversion | |$\kappa ^{\mathbf {*}}$| = 2 |
| Long-run variance | |$\theta ^{\mathbf {*}}$| = .01 |
| Current variance | |${\upsilon}$||${(t)=.01}$| |
| Correlation of |$z_1 (t)$| and |$z_2 (t)$| | |${\rho =0.}$| |
| Volatility of volatility parameter | |${\sigma =.1}$| |
| Option maturity | .5 year |
| Interest rate | |${r=0}$| |
| Strike price | |${K=100}$| |
$\begin{array} {l} {dS(t) = \mu S\,dt + \sqrt {\upsilon (t)S} \,dz_1 (t),} & {(10)} \cr {d\upsilon (t) = \kappa ^{\mathbf {*}} [\theta ^{\mathbf {*}} - \upsilon (t)]dt + \sigma \sqrt {\upsilon (t)} dz_2 (t).} & {(30)}\end{array}$ | |
|---|---|
| Parameter | Value |
| Mean reversion | |$\kappa ^{\mathbf {*}}$| = 2 |
| Long-run variance | |$\theta ^{\mathbf {*}}$| = .01 |
| Current variance | |${\upsilon}$||${(t)=.01}$| |
| Correlation of |$z_1 (t)$| and |$z_2 (t)$| | |${\rho =0.}$| |
| Volatility of volatility parameter | |${\sigma =.1}$| |
| Option maturity | .5 year |
| Interest rate | |${r=0}$| |
| Strike price | |${K=100}$| |
Default parameters for simulation of option prices
$\begin{array} {l} {dS(t) = \mu S\,dt + \sqrt {\upsilon (t)S} \,dz_1 (t),} & {(10)} \cr {d\upsilon (t) = \kappa ^{\mathbf {*}} [\theta ^{\mathbf {*}} - \upsilon (t)]dt + \sigma \sqrt {\upsilon (t)} dz_2 (t).} & {(30)}\end{array}$ | |
|---|---|
| Parameter | Value |
| Mean reversion | |$\kappa ^{\mathbf {*}}$| = 2 |
| Long-run variance | |$\theta ^{\mathbf {*}}$| = .01 |
| Current variance | |${\upsilon}$||${(t)=.01}$| |
| Correlation of |$z_1 (t)$| and |$z_2 (t)$| | |${\rho =0.}$| |
| Volatility of volatility parameter | |${\sigma =.1}$| |
| Option maturity | .5 year |
| Interest rate | |${r=0}$| |
| Strike price | |${K=100}$| |
$\begin{array} {l} {dS(t) = \mu S\,dt + \sqrt {\upsilon (t)S} \,dz_1 (t),} & {(10)} \cr {d\upsilon (t) = \kappa ^{\mathbf {*}} [\theta ^{\mathbf {*}} - \upsilon (t)]dt + \sigma \sqrt {\upsilon (t)} dz_2 (t).} & {(30)}\end{array}$ | |
|---|---|
| Parameter | Value |
| Mean reversion | |$\kappa ^{\mathbf {*}}$| = 2 |
| Long-run variance | |$\theta ^{\mathbf {*}}$| = .01 |
| Current variance | |${\upsilon}$||${(t)=.01}$| |
| Correlation of |$z_1 (t)$| and |$z_2 (t)$| | |${\rho =0.}$| |
| Volatility of volatility parameter | |${\sigma =.1}$| |
| Option maturity | .5 year |
| Interest rate | |${r=0}$| |
| Strike price | |${K=100}$| |
The parameter |${\sigma}$| controls the volatility of volatility. When |${\sigma}$| is zero, the volatility is deterministic, and continuously compounded spot returns have a normal distribution. Otherwise, |${\sigma}$| increases the kurtosis of spot returns. Figure 3 shows how this creates two fat tails in the distribution of spot returns. As Figure 4 shows, this has the effect of raising far-in-the-money and far-out-of-the-money option prices and lowering near-the-money prices. Note, however, that there is little effect on skewness or on the overall pricing of in-the-money options relative to out-of-the-money options.
Conditional probability density of the continuously compounded spot return over a six-month horizon
Spot-asset dynamics are |$dS(t) = \mu S\,dt + \sqrt {\upsilon (t)} S\,dz_1 (t),$| where |$d\upsilon (t) = \kappa ^{\mathbf {*}} [\theta ^{\mathbf {*}} - \upsilon (t)]dt + \sigma \sqrt {\upsilon (t)} dz_2 (t).$| Except for the volatility of volatility parameter |${\sigma}$| shown, parameter values are shown in Table 1. For comparison, the probability densities are normalized to have zero mean and unit variance.
Option prices from the stochastic volatility model minus Black-Scholes values with equal volatility to option maturity
Except for the volatility of volatility parameter |${\sigma}$| shown, parameter values are shown in Table 1. In both curves, the Black-Scholes volatility is 7.07 percent and the at-the-money option value is $2.82.
These simulations show that the stochastic volatility model can produce a rich variety of pricing effects compared with the Black-Scholes model. The effects just illustrated assumed that variance was at its long-run mean, |$\theta ^{\mathbf {*}}.$| In practice, the stochastic variance will drift above and below this level, but the basic conclusions should not change. An important insight from the analysis is the distinction between the effects of stochastic volatility per se and the effects of correlation of volatility with the spot return. If volatility is uncorrected with the spot return, then increasing the volatility of volatility |${(\sigma)}$| increases the kurtosis of spot returns, not the skewness. In this case, random volatility is associated with increases in the prices of far-from-the-money options relative to near-the-money options. In contrast, the correlation of volatility with the spot return produces skewness. And positive skewness is associated with increases in the prices of out-of-the-money options relative to in-the-money options. Therefore, it is essential to choose properly the correlation of volatility with spot returns as well as the volatility of volatility.
4. Conclusions
I present a closed-form solution for options on assets with stochastic volatility. The model is versatile enough to describe stock options, bond options, and currency options. As the figures illustrate, the model can impart almost any type of bias to option prices. In particular, it links these biases to the dynamics of the spot price and the distribution of spot returns. Conceptually, one can characterize the option models in terms of the first four moments of the spot return (under the risk-neutral probabilities). The Black-Scholes (1973) model shows that the mean spot return does not affect option prices at all, while variance has a substantial effect. Therefore, the pricing analysis of this article controls for the variance when comparing option models with different skewness and kurtosis. The Black-Scholes formula produces option prices virtually identical to the stochastic volatility models for at-the-money options. One could interpret this as saying that the Black-Scholes model performs quite well. Alternatively, all option models with the same volatility are equivalent for at-the-money options. Since options are usually traded near-the-money, this explains some of the empirical support for the Black-Scholes model. Correlation between volatility and the spot price is necessary to generate skewness. Skewness in the distribution of spot returns affects the pricing of in-the-money options relative to-out-of-the money options. Without this correlation, stochastic volatility only changes the kurtosis. Kurtosis affects the pricing of near-the-money versus far-from-the-money options.
With proper choice of parameters, the stochastic volatility model appears to be a very flexible and promising description of option prices. It presents a number of testable restrictions, since it relates option pricing biases to the dynamics of spot prices and the distribution of spot returns. Knoch (1992) has successfully used the model to explain currency option prices. The model may eventually explain other option phenomena. For example, Rubinstein (1985) found option biases that changed through time. There is also some evidence that implied volatilities from options prices do not seem properly related to future volatility. The model makes it feasible to examine these puzzles and to investigate other features of option pricing. Finally, the solution technique itself can be applied to other problems and is not limited to stochastic volatility or diffusion problems.
Appendix: Derivation of the Characteristic Functions
This is analogous to extracting an implied volatility parameter in the Black-Scholes model.
Note that characteristic functions always exist; Kendall and Stuart (1977) establish that the integral converges.
Note that when evaluating multiple options with different strike options, one need not recompute the characteristic functions when evaluating the integral in Equation (18).
This occurs for exactly the same reason that the Black-Scholes formula does not depend on the mean stock return. See Heston (1992) for a theoretical analysis that explains when parameters drop out of option prices.
These parameters roughly correspond to Knoch's (1992) estimates with yen and deutsche mark currency options, assuming no risk premium associated with volatility. However, the mean-reversion parameter is chosen to be more reasonable.
This variance can be determined by using the characteristic function.
This illustration is motivated by Jarrow and Rudd (1982) and Hull (1989).
References
Author notes
I thank Hans Knoch for computational assistance. I am grateful for the suggestions of Hyeng Keun (the referee) and for comments by participants at a 1992 National Bureau of Economic Research seminar and the Queen's University 1992 Derivative Securities Symposium. Any remaining errors are my responsibility.
