Non-Gaussian Price Dynamics and Implications for Option Pricing

It is well known that the probability distribution of stock returns is non-Gaussian. The tails of the distribution are too “fat,” meaning that extreme price movements, such as stock market crashes, occur more often than predicted given a Gaussian model. Numerous studies have attempted to characterize and explain the fat-tailed property of returns. This is because understanding the probability of extreme price movements is important for risk management and option pricing. In spite of this work, there is still no accepted theoretical explanation. In this chapter, we use a large collection of data from three different stock markets to show that slow fluctuations in the volatility (i.e., the size of return increments), coupled with a Gaussian random process, produce the non-Gaussian and stable shape of the return distribution. Furthermore, because the statistical features of volatility are similar across stocks, we show that their return distributions collapse onto one universal curve. Volatility fluctuations influence the pricing of derivative instruments, and we discuss the implications of our findings for the pricing of options.