Black-Scholes Approximations of Call Option Prices With Stochastic Volatilities: A Note
提出用分段线性加权函数处理布莱克-舒尔斯隐含波动率，结合该模型近似随机波动率下的欧式看涨期权价格。敏感性分析表明，该简单模型能以较少计算量接近模拟价格，对需要快速定价的从业者有用。
A piecewise linear weighting function for Black-Scholes implied volatilities is used in conjunction with the Black-Scholes call pricing model to approximate stochastic volatility European call prices. A sensitivity analysis is conducted to compare simulated stochastic volatility call prices to the Black-Scholes prices calculated with the weighted implied vo? latilities. The analysis indicates that a simple model can provide close approximations to the simulated prices with far less computational effort. I. Introduction The volatility of the underlying asset is one of the more difficult parameters to estimate in any option pricing model. The stochastic nature of the volatility of most financial assets is responsible for much of the difficulty. While the early tests of the Black-Scholes (1973) option pricing model rely upon historical prices for volatility estimates, Latane and Rendleman's (1976) implied volatility technique has become the standard method of estimation. Unfortunately, as Beckers (1981) stresses, although implied volatilities calculated from a stationary vari? ance model may be of practical use, it is inconsistent to use the Black-Scholes model with a constant variance to obtain estimates of nonstationary variances.l Recent work conducted by Hull and White (1987) and Johnson and Shanno (1987) has been directed at solving the problem of pricing European calls on assets with stochastic volatilities. While Hull and White provide a series solution for the case in which the variance and stock price are uncorrelated, convergence is slow unless the variance of the assets' volatility is relatively small. For more general cases, both Hull and White and Johnson and Shanno rely upon Monte Carlo simulations to estimate option prices. Although the application of simulation techniques to option pricing, first explored by Boyle (1977), is often a useful method of estimating option prices, the Monte Carlo method tends to be expensive and is often too time consuming for real-time applications. This study examines a simple method of approximat