ABSTRACT
Previous options studies typically assume that the dynamics of the underlying asset price follow a geometric Brownian motion (GBM) when pricing options on stocks, stock indices, currencies or futures. However, there is mounting empirical evidence that the volatility of asset price or return is far from constant. This article, in contrast to studies that use parametric approach for option pricing, employs nonparametric kernel regression to deal with changing volatility and, accordingly, prices options on stock index. Specifically, we first estimate nonparametrically the volatility of asset return in the GBM based on the Nadaraya–Watson (N–W) kernel estimator. Then, based on the N–W estimates for the volatility, we use Monte Carlo simulation to compute option prices under different settings. Finally, we compare the index option prices under our nonparametric model with those under the Black–Scholes model and the Stein–Stein model.
Notes
2 Volatility clustering, first noted by Mandelbrot (Citation1963), refers to a phenomenon in stock returns that ‘large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes.’ Leverage effect, first noted by Black (Citation1976), refers to a phenomenon that volatility of stock tends to increase when its price falls. Leptokurtosis refers to a phenomenon that the distribution of stock returns tends to have fatter tails than the normal distribution.
3 The ARCH and GARCH models are frequently used as a parametric approach for dealing with time-varying volatility of asset returns. See Engle (Citation1982) and Bollerslev, Chou, and Kroner (Citation1992) for using these two models to model asset returns, and Duan (Citation1995) for using the GARCH model to price options.
9 Equation 13 is a little different from the formula used by Stanton (Citation1997). Stanton leaves out the constant factor in his formula, probably for rounding reason. See Equation 55 on page 1991 of Stanton (Citation1997).
10 We did attempt to use an automatic method (i.e., leave-one-out cross-validation) to compute the bandwidth. Interestingly, the bandwidth so obtained is large and, most unexpectedly, leads to some absurd results for our option prices. Hence, we use Equation 13 for computing the bandwidth. As a case in point, Lo, Mamaysky, and Wang (Citation2000) encounter a similar situation as ours. When they use the cross-validation (CV) method, they obtain a bandwidth that is too large. To ‘fix’ the problem, they adjust their bandwidth by a factor of 0.3. That is, they use a bandwidth of 0.3 × h*, where h* minimizes CV(h). Lo, Mamaysky, and Wang (Citation2000, p. 1714) admit that ‘this is an ad hoc approach.’
11 Their studies involve estimating the volatility of the short rate; our study involves estimating the volatility in Equation 2.
12 In this study, we price index options starting from 2 January 2013, the first day of trading in 2013. Hence, the data over the year 2012 are more relevant to computing the bandwidth.
13 The estimation is done using daily log returns from the S&P 500 index over the year 2012.
14 The exact value of d is not important to our study. Our purpose is to investigate the difference in index option prices between our nonparametric model and the Black–Scholes parametric model.
15 The risk-neutral probability measure, not the original probability measure, is the relevant one for pricing options. This insight was first pointed out by Cox and Ross (Citation1976) and later theoretically developed by Harrison and Kreps (Citation1979).
17 We divide the life of this 12-month option into 252 (not 250) time intervals so that the life of other shorter maturity options is divisible. Hence, 1-month options, 3-month options, 6-month options and 9-month options will have, respectively, 21, 63, 126 and 189 time intervals.
Black, F. 1975a. “Forecasting Variance of Stock Prices for Options Trading and Other Purposes.” In Seminar on the Analysis of Security Prices. Chicago, IL: University of Chicago. Blattberg, R. C., and N. J. Gonedes. 1974. “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices.” The Journal of Business 47: 244–280. doi:10.1086/jb.1974.47.issue-2. Castanias, R. P. 1979. “Macroinformation and the Variability of Stock Market Prices.” Journal of Finance 34: 439–450. Christie, A. A. 1982. “The Stochastic Behavior of Common Stock Variances Value, Leverage and Interest Rate Effects.” Journal of Financial Economics 10: 407–432. doi:10.1016/0304-405X(82)90018-6. Latane, H. A., and R. J. Rendleman. 1976. “Standard Deviations of Stock Price Ratios Implied in Option Prices.” Journal of Finance 31: 369–381. doi:10.1111/j.1540-6261.1976.tb01892.x. MacBeth, J., and L. Merville. 1979. “An Empirical Examination of the Black-Scholes Call Option Pricing Formula.” Journal of Finance 34: 1173–1186. Oldfield, G. S., R. J. Rolgalski, and R. A. Jarrow. 1977. “An Autoregressive Jump Process for Common Stock Returns.” Journal of Financial Economics 5: 389–418. doi:10.1016/0304-405X(77)90045-9. Schmalensee, R., and R. R. Trippi. 1978. “Common Stock Volatility Expectations Implied by Option Premia.” The Journal of Finance 33: 129–147. doi:10.1111/j.1540-6261.1978.tb03394.x. Mandelbrot, B. 1963. “The Variation of Certain Speculative Prices.” Journal of Business 36: 394–419. Black, F. 1976. “Studies of Stock Price Volatility Changes.” In Proceedings of the 1976 Meetings of the Business & Economics Statistics Section, 177–181. Washington, DC: American Statistical Association. Engle, R. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50: 987–1008. doi:10.2307/1912773. Bollerslev, T., R. Y. Chou, and K. F. Kroner. 1992. “ARCH Modeling in Finance.” Journal of Econometrics 52: 5–59. doi:10.1016/0304-4076(92)90064-X. Duan, J.-C. 1995. “The GARCH Option Pricing Model.” Mathematical Finance 5: 13–32. doi:10.1111/mafi.1995.5.issue-1. Fan, J., and Q. Yao. 2003. Nonlinear Time Series: Nonparametric and Parametric Methods. New York: Springer-Verlag. Hardle, W. 1990. Applied Nonparametric Regression. Cambridge: Cambridge University Press. Hardle, W. 1991. Smoothing Techniques: With Implementation in S. New York: Springer-Verlag. Pagan, A., and A. Ullah. 1999. Nonparametric Econometrics. Cambridge: Cambridge University Press. Scott, D. W. 1992. Multivariate Density Estimation: Theory, Practice, and Visualization. New York: John Wiley & Sons. Silverman, B. W. 1985. “Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting.” Journal of the Royal Statistical Society Series B 47: 1–52. Silverman, B. W. 1986. Density Estimation for Statistics and Data Analysis. London: Chapman and Hall. Black, F. 1975b. “Fact and Fantasy in the Use of Options.” Financial Analysts Journal 31: 36–41 and 61–72. doi:10.2469/faj.v31.n4.36. MacBeth, J., and L. Merville. 1979. “An Empirical Examination of the Black-Scholes Call Option Pricing Formula.” Journal of Finance 34: 1173–1186. Gultekin, B., R. Rogalski, and S. Tinic. 1982. “Option Pricing Model Estimates: Some Empirical Results.” Financial Management 11: 58–69. doi:10.2307/3665506. Whaley, R. 1982. “Valuation of American Call Options on Dividend-Paying Stocks.” Journal of Financial Economics 10: 29–58. doi:10.1016/0304-405X(82)90029-0. Ait-Sahalia, Y. 1996. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies 9: 385–426. doi:10.1093/rfs/9.2.385. Fan, J., and Q. Yao. 2003. Nonlinear Time Series: Nonparametric and Parametric Methods. New York: Springer-Verlag. Pagan, A., and A. Ullah. 1999. Nonparametric Econometrics. Cambridge: Cambridge University Press. Scott, D. W. 1992. Multivariate Density Estimation: Theory, Practice, and Visualization. New York: John Wiley & Sons. Silverman, B. W. 1986. Density Estimation for Statistics and Data Analysis. London: Chapman and Hall. Stanton, R. A. 1997. “A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk.” The Journal of Finance 52: 1973–2002. doi:10.1111/j.1540-6261.1997.tb02748.x. Hardle, W. 1991. Smoothing Techniques: With Implementation in S. New York: Springer-Verlag. Janssen, P., J. S. Marron, N. Veraverbeke, and W. Sarle. 1995. “Scale Measures for Bandwidth Selection.” Journal of Nonparametric Statistics 5: 359–380. doi:10.1080/10485259508832654. Jones, M. C., J. S. Marron, and S. J. Sheather. 1996. “A Brief Survey of Bandwidth Selection for Density Estimation.” Journal of the American Statistical Association 91: 401–407. doi:10.1080/01621459.1996.10476701. Marron, J. S. 1988. “Automatic Smoothing Parameter Selection: A Survey.” Empirical Economics 13: 187–208. doi:10.1007/BF01972448. Silverman, B. W. 1985. “Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting.” Journal of the Royal Statistical Society Series B 47: 1–52. Wand, M. P., and M. C. Jones. 1995. Kernel Smoothing. London: Chapman and Hall. Stanton, R. A. 1997. “A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk.” The Journal of Finance 52: 1973–2002. doi:10.1111/j.1540-6261.1997.tb02748.x. Stanton, R. A. 1997. “A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk.” The Journal of Finance 52: 1973–2002. doi:10.1111/j.1540-6261.1997.tb02748.x. Lo, A. W., H. Mamaysky, and J. Wang. 2000. “Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation.” Journal of Finance 55: 1705–1765. Lo, A. W., H. Mamaysky, and J. Wang. 2000. “Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation.” Journal of Finance 55: 1705–1765. Cox, J. C., and S. A. Ross. 1976. “The Valuation of Options for Alternative Stochastic Processes.” Journal of Financial Economics 3: 145–166. Harrsion, J. M., and D. M. Kreps. 1979. “Martingales and Arbitrage in Multiperiod Security Markets.” Journal of Economic Theory 20: 381–408. doi:10.1016/0022-0531(79)90043-7.