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.

著名的布莱克-斯科尔斯模型以其优雅的欧式期权定价公式而闻名。然而，与许多其他模型一样，Black-Scholes 模型并不完美，这主要是因为模型中的假设是理想化的，并非所有的经验都有效。假设之一是市场没有交易成本，这在真实市场中是不满足的。利兰 [H. Leland，期权定价和复制交易成本，J. Financ。40 (1985), pp. 1283–1301] 通过纳入交易成本，开创了一种改进的欧式期权复制策略。在本文中，我们进一步考虑了带有交易成本的随机利率和随机波动率模型下的欧式期权定价问题，并从该模型推导出了非线性偏微分方程（PDE）。然后，