Valuation of vulnerable options with stochastic corporate liabilities in a mixed fractional Brownian motion environment

In this paper, we deal with the problem of European vulnerable option pricing under the mixed fractional Brownian motion with stochastic corporate liabilities and jumps. Assume that the dynamics of all assets including the underlying asset, the counterparty’s assets and corporate liabilities are given by three interrelated jump-diffusion processes. Jumps are divided into a systematic component which affects the prices of all assets and an idiosyncratic component affecting each asset price. Under this framework, the pricing formulae for vulnerable options are derived using an actuarial approach. Further, we discuss some properties of the pricing formulae, which reveal that our model contains not only the traditional pure diffusion processes such as Black–Scholes model (J Polit Econ 81:637–659, 1973) and Klein model (J Bank Finance 20:1211–1229, 1996), but also the jump-diffusion processes such as Merton jump model (J Financial Econ 3:125–144, 1976), Tian et al. model (J Futures Mark 34:957–979, 2014) and Zhou et al. model (Acta Math Appl Sin-E 35:305–318, 2019). The numerical results show that the randomness of the corporate liabilities may reduce the expected payoff of the option holder at maturity time and also increase the likelihood of default, thus it reduces the vulnerable option price.