Pricing European options with a log Student’s t-distribution: A Gosset formula

Abstract

The distribution of the returns for a stock are not well described by a normal probability density function (pdf). Student's t-distributions, which have fat tails, are known to fit the distributions of the returns. We present pricing of European call or put options using a log Student's t-distribution, which we call a Gosset approach in honour of W.S. Gosset, the author behind the nom de plume Student. The approach that we present can be used to price European options using other distributions and yields the Black-Scholes formula for returns described by a normal pdf.

Extended Model of Stock Price Behaviour

“…This model is realistic in that it allows for a finite supply of money to be saturated by the net rate of transactions and does not require truncation or capping [8] [9] [13] [14], or modification of the distribution of returns [10] [11] [12] to avoid very large and unobserved stock prices. This homogeneously saturated model, which borrows from laser physics [15], will have widely spread repercussions, most importantly for the Black-Scholes formula for option pricing, which will be investigated in future work.…”

“…It has been known for some time, however, that this assumption is not supported by actual stock prices (e.g., [5] [6] [7]). For example, daily returns of the DJIA and the S&P 500 indices are described by a fat-tailed distribution [3] [8]. Prices predicted by the standard model, using a normal distribution as the noise driving term, will therefore be inaccurate and simply substituting a fat-tailed distribution will permit the infinite prices mentioned above.…”

Option Pricing With Heavy-Tailed Distributions of Logarithmic Returns

“…Power-law-tailed distributions of returns are utilized as a starting point in the last group of models. Examples of such alternatives are given in Matacz (2000), Borland (2002b), Borland & Bouchaud (2004), Moriconi (2007) and Cassidy et al (2010). Another feature that favors the Gaussian distribution over others for being used as a building block in the modeling of logarithmic returns is that it is stable, i.e.…”

“…In one non-Gaussian option valuation approach by Borland (2002a,b), the Student's t-distribution was obtained as a result of a correlated stochastic process. In another proposal by Cassidy et al (2010), as a basis was considered the theoretical result that a mixture of Gaussians with stochastic volatility, such that the inverse of the volatility is chi-squared distributed, follows the Student's t-distribution. Empirical verification of that fact by using returns of real data has provided justification for the same authors to apply it for options pricing.…”

“…The resulting models for pricing options offer determination of the values for a whole range of strikes and maturities by using a single parameter. In the second contribution, Cassidy et al (2010) and Cassidy et al (2013) apply truncated Student's t-distribution directly, and provide an analysis for the position of the truncation point. Our valuation framework is primarily driven by the empirical observations rather than aiming to provide a strong analytical theory.…”