"Frontmatter". In: Analysis of Financial Time Series

244 CONTINUOUS-TIME MODELSChanging the notation x t to P t for the price of an asset, we have a solution for theprice under the assumption that it is a geometric Brownian motion. The price isP t = P 0 exp[(µ − σ 2 /2)t + σw t ]. (6.25)6.9 JUMP DIFFUSION MODELSEmpirical studies have found that the stochastic diffusion model based on Brownianmotion fails to explain some characteristics of asset returns and the prices of theirderivatives (e.g., the “volatility smile” of implied volatilities; see Bakshi, Cao, andChen, 1997, and the references therein). Volatility smile is referred to as the convexfunction between the implied volatility and strike price of an option. Both out-ofthe-moneyand in-the-money options tend to have higher implied volatilities thanat-the-money options especially in the foreign exchange markets. Volatility smileis less pronounced for equity options. The inadequacy of the standard stochasticdiffusion model has led to the developments of alternative continuous-time models.For example, jump diffusion and stochastic volatility models have been proposed inthe literature to overcome the inadequacy; see Merton (1976) and Duffie (1995).Jumps in stock prices are often assumed to follow a probability law. For example,the jumps may follow a Poisson process, which is a continuous-time discrete process.For a given time t, letX t be the number of times a special event occurs during thetime period [0, t].ThenX t is a Poisson process ifPr(X t = m) = λm t mexp(−λt), λ > 0.m!That is, X t follows a Poisson distribution with parameter λt. The parameter λ governsthe occurrence of the special event and is referred to as the rate or intensity ofthe process. A formal definition also requires that X t be a right-continuous homogeneousMarkov process with left-hand limit.In this section, we discuss a simple jump diffusion model proposed by Kou(2000). This simple model enjoys several nice properties. The returns implied by themodel are leptokurtic and asymmetric with respect to zero. In addition, the modelcan reproduce volatility smile and provide analytical formulas for the prices of manyoptions. The model consists of two parts, with the first part being continuous andfollowing a geometric Brownian motion and the second part being a jump process.The occurrences of jump are governed by a Poisson process, and the jump sizefollows a double exponential distribution. Let P t be the price of an asset at time t.The simple jump diffusion model postulates that the price follows the stochasticdifferential equationdP tP t= µdt + σ dw t + d(nt)∑(J i − 1) , (6.26)i=1