Jump Diffusion Model: An Application to the Japanese Stock Market

The Black-Scholes(BS) model has been widely and successfully used to model the return of asset and to price financial options. Despite of its success the basic assumptions of this model, that is, Brownian motion and normal distribution are not always supported by empirical studies. Those studies showed the two empirical phenomena: (1) the asymmetric leptokurtic features, (2) the volatility smile. The first means that the return distribution is skewed to the left and has a higher peak and two heavier tails than those of normal distribution, and the second means that if the BS model is correct, then the implied volatility should be flat. But the graph of the observed implied volatility curve often looks like the smile of the Cheshire cat. One of the causes for such phenomena is jumps in assets price processes. Figure 1 shows time series of 1 minute tick data of Yen/$ exchange rate between 15:00 and 24:00 on 21st of July 2005 when Chinese Yuan was revaluated. It seems obvious that there was a jump at the time of the revaluation. Many models were proposed to explain the two empirical phenomena. For example popular ones are normal jump diffusion model(Merton(1976)), stochastic volatility models(Heston(1993)), ARCHGARCH models(Duan(1993)), etc. For other models see references in Kou(2002)). Among others we focus on a double exponential jump diffusion model proposed by Kou(2002) in this paper. Kou’s model is very simple and has rich theoretical implication as described below: The logarithm of the asset price is assumed to follow a Brownian motion plus a compound Poisson process with jump sizes double exponentially distributed. This model has the following advantages: (1) it can explain the two empirical phenomena, that is asymmetric leptokurtic feature and the volatility smile, (2) it leads to analytical solutions to many option-pricing problems. Despite of these advantages there are not many empirical stud110.5 111.0 111.5 112.0 112.5