Abstract

This paper investigates the pricing of European options with transaction costs under the Scott stochastic volatility model. Based on the Delta hedging strategy, a nonlinear partial differential equation for option pricing incorporating transaction costs is established. To solve this equation, the finite difference method is employed for discretization, leading to the construction of an explicit Euler numerical scheme. Subsequently, conditions for the non-negativity of the numerical method are studied and rigorously proven using mathematical induction. Finally, numerical experiments validate the effectiveness of the theoretical findings: when the conditions of the lemma are satisfied, the numerical solutions remain non-negative.

Keywords

Scott model, Transaction costs, Positivity analysis, Explicit Euler method, Partial differential equation.

References

[1] Guy Barles, and Halil Mete Soner, “Option Pricing with Transaction Costs and a Non-Linear Black-Scholes Equation,” Finance and Stochastics, vol. 2, pp. 369-397, 1998.
[CrossRef] [Google Scholar] [Publisher Link]

[2] Hayne E. Leland, “Option Pricing and Replication with Transactions Costs,” The Journal of Finance, vol. 40, no. 5, pp. 1283-1301, 1985.
[CrossRef] [Google Scholar] [Publisher Link]

[3] Xiaoping Lu, Song-Ping Zhu, and Dong Yan, “Nonlinear PDE Model for European Options with Transaction Costs Under Heston Stochastic Volatility,” Communications in Nonlinear Science and Numerical Simulation, vol. 103, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[4] Jianguo Tan, and Jiling Cao, “Nonlinear PDE Model for Pricing European Options with Transaction Costs Under the 3/2 Non-Affine Stochastic Volatility Model,” Computers and Mathematics with Applications, vol. 196, pp. 246-262, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Elham Mashayekhi, Javad Damirchi, and Ahmad Reza Yazdanian, “Alternating Direction Implicit Method for Approximation Solution of the HCIR Model, Including Transaction Costs in a Jump-Diffusion Model,” Computational Methods for Differential Equations, vol. 13, no. 1, pp. 339-356, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[6] Steven L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” The Review of Financial Studies, vol. 6, no. 2, pp. 327-343, 1993.
[CrossRef] [Google Scholar] [Publisher Link]

[7] Louis O. Scott, “Option Pricing When the Variance Changes Randomly: Theory, Estimation, and an Application,” The Journal of Financial and Quantitative Analysis, vol. 22, no. 4, pp. 419-438, 1987.
[CrossRef] [Google Scholar] [Publisher Link]

[8] Rainer Schöbel, and Jianwei Zhu, “Stochastic Volatility with an Ornstei-Uhlenbeck Process: An Extension,” European Finance Review, vol. 3, no. 1, pp. 23-46, 1999.
[CrossRef] [Google Scholar] [Publisher Link]

[9] Freddy H. Marín-Sánchez et al., “Valuation of European Call Options for the Scott’s Stochastic Volatility Model: An Explicit Finite Difference Scheme,” Mathematics and Computers in Simulation, vol. 236, pp. 411-425, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[10]  Ionut¸ Florescu, Maria C. Mariani, and Indranil Sengupta, “Option Pricing with Transaction Costs and Stochastic Volatility,” Electronic Journal of Differential Equations, vol. 165, pp. 1-19, 2014.
[Google Scholar] [Publisher Link]