Option pricing and hedging in illiquid markets in presence of jump clustering

The topic of this paper is the pricing and hedging of options on small capitalization stocks. Such stocks tend to exhibit two features. The first is the presence of motionless periods in their prices. It is a consequence of a lack of liquidity, since these stocks are not heavily traded. The second is the occurrence of clustered sudden moves in the price at the times the stock is traded. The model we propose is therefore a self-exciting Hawkes jump-diffusion process that is time-changed by the inverse of an alpha-stable subordinator, a process that exhibits motionless periods. This article is divided into two parts. In the first part, we prove that, when adding some information to the inverse alpha-stable subordinator, we obtain a Markov process. This result allows us to obtain the dynamic framework we need to establish a hedging strategy. In the second part, we deal with the pricing and hedging of options. To this end, we derive a fractional partial differential equation (FPDE) for the Fourier transform of the log asset price. We introduce a finite difference method to solve this FPDE. Prices of options can be obtained by a numerical inversion of the Fourier transform with a fast-Fourier transform algorithm. Changes of measures are then discussed, as well as an optimal quadratic hedging strategy for options. Finally, the last section of this paper presents some numerical experiments.