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The celebrated Black–Scholes model is well known for its elegant pricing formula for European options. However, like many other models, the Black–Scholes model is not perfect, which is largely due to the fact that assumptions in the model are idealized and not all empirically valid. One of the assumptions is that the market does not have transaction costs, which is not satisfied in a real market. Leland [H. Leland, Option pricing and replication with transactions costs, J. Financ. 40 (1985), pp. 1283–1301] pioneers a modified replicating strategy for European options by incorporating transaction costs. In this paper, we further consider the problem of pricing European options under a stochastic interest rate and stochastic volatility model with transaction costs, and derive a nonlinear partial differential equation (PDE) from this model. Then, we apply the finite-difference scheme to solve this PDE and conduct numerical experiments.

Valuation of European options with stochastic interest rates and transaction costs