Affine Heston model style with self-exciting jumps and long memory

Classic diffusion processes fail to explain asset return volatility. Many empirical findings on asset return time series, such as heavy tails, skewness and volatility clustering, suggest decomposing the volatility of an asset's return into two components, one caused by a Brownian motion and another by a jump process. We analyze the sensitivity of European call options to memory and self-excitation parameters, underlying price, volatility and jump risks. We expand Heston's stochastic volatility model by adding to the instantaneous asset prices, a jump component driven by a Hawkes process with a kernel function or memory kernel that is a Fourier transform of a probability measure. This kernel function defines the memory of the asset price process. For instance, if it is fast decreasing, the contagion effect between asset price jumps is limited in time. Otherwise, the processes remember the history of asset price jumps for a long period. To investigate the impact of different rates of decay or types of memory, we consider four probability measures: Laplace, Gaussian, Logistic and Cauchy. Unlike Hawkes processes with exponential kernels, the Markov property is lost but stationarity is preserved; this ensures that the unconditional expected arrival rate of the jump does not explode. In the absence of the Markov property, we use the Fourier transform representation to derive a closed form expression of a European call option price based on characteristic functions. A numerical illustration shows that our extension of the Heston model achieves a better fit of the Euro Stoxx 50 option data than the standard version.