A closed-form solution for pricing European-style options under the Heston model with credit and liquidity risks

Credit risks are one type of hazardous financial risks, which results in the necessity of considering vulnerable options. Two involved assets, corresponding to underlying and option seller's ones, both follow Heston stochastic volatility with different parameters, and their prices are discounted via a stochastic factor relying on stochastic market-wide liquidity. We then develop a general formula after the establishment of a risk-neutral measure, and it can be computed analytically since we are able to derive a closed-form joint characteristic function. Liquidity and volatility risks are shown through numerical experiments to significantly impact vulnerable option prices.