Evaluation of lookback options in the Heston model using the finite difference method in Python
Creators: name: Schroeter, Moritz Leon
Publisher: name: ZHAW Zürcher Hochschule für Angewandte Wissenschaften
Year: 2023
Subjects: 332: Finanzwirtschaft
510: Mathematik

description: Financial markets consist not only of exchanges for publicly traded assets but also comprise decisive private financial transactions with various participants. The fair value of some assets can be determined directly by the market through supply and demand or by simple calculations. On the other hand, the complexity of price determination is significantly exacerbated for the issuance of over-the-counter products without liquid markets, such as lookback options. The value of these options is path-dependent and considers the future movement of the underlying asset’s price. Thus, the option price depends on future uncertainty but must be determined at the time of emission. Although an explicit analytical solution for the evaluation is known, the calculation of this solution is highly complex and unwarranted in terms of practicality and efficiency. Thus, the option price must be approximated by numerical-algorithmic processes in a stochastic volatility model such as the Heston model. Despite notable advancements in recent research, the study of lookback options in the Heston model exhibits a limited body of literature and a scarcity of practical implementations.
Hence, this thesis lays the foundation for the evaluation of lookback options by expounding the functionality and payoff of options, the Black-Scholes model with its deficiencies, and the Heston model as an extension. Furthermore, the finite difference approach is introduced as a numerical approximation method for one and two-dimensional problems. First, the numerical method is converted into matrix notation for further application and fundamental properties such as multiple, combinable boundary conditions are covered. Subsequently, an analytical partial differential equation for the pricing of lookback options is transformed from four to three dimensions with a new domain satisfying the requirement of the finite difference method, and a modified payoff function is defined. The implementation of the evaluation entails the construction of univariate functions and the development of the finite difference model in the software Python. Furthermore, the option value is interpolated, and node points at the lower boundary are extrapolated to receive the price of floating lookback options. The price for fixed lookback options is obtained by converting floating lookback option prices through a specific put-call parity. Moreover, the processing time is significantly reduced by applying the Craig-Sneyd scheme with specialised algorithms, and a graphic user interface is developed for enhanced usability.
The developed model is verified on the basis of a Monte Carlo simulation and exhibits accurate results up to the second decimal place with expeditious discretisation processes. In addition, this thesis proposes amended option prices for an erroneous study conducted in 2019, thereby contributing to the field of quantitative finance. The codes for the developed model, graphical user interface, and complex graphics are disclosed in the appendix for extended transparency and further research.