Intraday Stock Return Distribution for Black-scholes Option Pricing

The availability of intraday stock/index return in the web facilitates the improvement of return volatility estimation over the traditional method that is based on inter-day return data. Truncated Levy process distribution is used to extract the intraday return distribution parameters. The calibration to the volatility for Black-Scholes option pricing is studied using the data from Levy-Gaussian convergence in physical diffusive systems as well as using the empirical implied volatility values. It appears that intraday return distribution parameters give short-term call option prices closer to the market values. Robustness is investigated using bootstrap data. The resulting volatility variability is studied in contrast to the observed intraday fluctuation of the implied volatility. Application to investment strategy is also discussed. Introduction The volatility parameter in the Black-Scholes (BS) Option pricing model has been a subject of intense research ever since Black and Scholes introduced it in 1973 . The original volatility of the underlying asset was estimated using end of day prices. The return follows a Gaussian distribution with its standard deviation as the volatility value. As web information emerges in the nineties, volatility estimation based on 5minute interval prices also becomes a reality. The CBOE implied volatility VIX data is a good example. Similar advances in research have also been reported throughout the open literature and financial engineering has emerged as a new discipline in the last ten years. The 5-minute interval S&P500 index futures dataset provides solid evidence that the stock return deviates much from a Gaussian distribution. The dataset spans several years from 1991-1995 with some 75,000 data points. With success in describing the recent advances of physical science diffusion systems, Levy distribution has also become a favorite tool for the analysis of the short time interval stock dataset. This paper addresses the application of truncated Levy distribution to intraday dataset with a few hundreds data points. The distribution width is calibrated to the volatility value of the B-S model. Computer generated bootstrap data is also used to study robustness. The QQQ stock and OEX index are used as application examples. Project Design Louis Bachelier, a student of Henri Poincare, published his thesis “Theory of Speculation” within Gaussian diffusion framework in 1900. Since then physical science diffusion reveals many examples of Non-Gaussian diffusion in recent studies of turbulent phenomena. Transport variables such as temperature fluctuation in turbulent cell study, force distribution in plastic materials, light and microwave fluctuation in turbid media obey a characteristic function proportional to exp((constant) |variable| k ) for 0 < k < 2, which is commonly referred to as Levy distribution. The case k = 2 is the usual Gaussian probability density function after performing the Fourier transform. For both stock return and turbulent transport, the strength of Levy distribution lies in its ability to account for statistical outliers. Outliers are exactly what the stock market has and the return is seldom Gaussian especially in a short term period. Recently, Matacz published the use of the geometric truncated Levy process for the stock price and a hedging strategy minimizing some appropriate measure of risk because the risk-free B-S delta strategy is not available. The report uses the k = 1.2 Levy distribution and it also shows the theoretical conditions that multiple truncated Levy processes converge to an overall Gaussian process, which has already been observed experimentally in the physical science such as photon diffusion with small absorption. Wilmott presents the trinomial random walk model and deduces that the stock price probability satisfies the Kolmogorov equation with Taylor series approximation. This means that the B-S option value is equivalent to the present value of the expected payoff at expiry under a risk neutral random walk for the underlying stock. If S is the stock price, r is the risk-free date, σ is the volatility, dX is a Brownian/Wiener process, then dS = rSdt + σ Joint Statistical Meetings Business & Economic Statistics Section