A nonparametric kernel regression approach for pricing options on stock market index

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.

KEYWORDS:

• Geometric Brownian motion
• stock index option
• nonparametric kernel regression
• Nadaraya–Watson kernel estimator
• Black–Scholes model
• Stein–Stein model

JEL CLASSIFICATION:

• C14
• C15
• G13

Notes

¹ See Black (1975a), Blattberg and Gonedes (1974), Castanias (1979), Christie (1982), Latane and Rendleman (1976), MacBeth and Merville (1979), Oldfield, Rolgalski, and Jarrow (1977) and Schmalensee and Trippi (1978).

² Volatility clustering, first noted by Mandelbrot (1963), 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 (1976), 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 (1982) and Bollerslev, Chou, and Kroner (1992) for using these two models to model asset returns, and Duan (1995) for using the GARCH model to price options.

⁴ See Fan and Yao (2003), Hardle (1990, 1991), Pagan and Ullah (1999), Scott (1992) and Silverman (1985, 1986).

⁵ See Black (1975b), MacBeth and Merville (1979), Gultekin, Rogalski, and Tinic (1982) and Whaley (1982).

⁶ See Ait-Sahalia (1996), Fan and Yao (2003), Pagan and Ullah (1999), Scott (1992), Silverman (1986) and Stanton (1997).

⁷ Note that, in the context of this study, the above $x_t$ and $y_t$ become $r_t$ and ${\sigma }_t$, respectively.

⁸ See Hardle (1991), Janssen et al. (1995), Jones, Marron, and Sheather (1996), Marron (1988), Silverman (1985) and Wand and Jones (1995).

⁹ Equation 13 is a little different from the formula used by Stanton (1997). Stanton leaves out the constant factor in his formula, probably for rounding reason. See Equation 55 on page 1991 of Stanton (1997).

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 (2000) 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 (2000, 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 (1976) and later theoretically developed by Harrison and Kreps (1979).

¹⁶ We use $r_t$ to represent daily log return of the index and $r_f$ to represent the risk-free interest rate.

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.