#### Year

2021

#### Degree Name

Doctor of Philosophy

#### Department

School of Mathematics and Applied Statistics

#### Abstract

Option pricing has become a key problem studied in academia as well as in finance industry ever since the publication of the seminal papers by Black and Scholes (1973) and Merton (1973). The Black-Scholes model has laid a solid foundation for the rapid development of various option pricing theories in the next half a century. However, the Black-Scholes model imposes some unrealistic assumptions in order to achieve analytical tractability, among which, the assumption of no transaction costs when trading stocks contradicts the fact there would always be costs associated with transactions of stocks in real markets. Although significant development has been made in studying the effects of transaction costs on option pricing in recent years, there are still gaps to fill in the literature.

In general, there are two different approaches to tackle the problem of pricing options with transactions costs: the hedging strategies and utility indifference pricing. Each of these two approaches has its own merits for pricing options under incomplete markets. The former method is easy to implement, but does not take investors’ preferences into consideration, the latter is very computational intensive. This thesis aims to provide a comprehensive study of option pricing with transaction costs under both the hedging strategy and the utility maximization theory, where the effects of transaction costs and stochastic volatility on option prices are analyzed, with the emphasis on American option prices and their optimal exercise boundaries.

The thesis is composed of seven chapters, with Chapter 1 being the introduction, Chapter 2 providing a review of preliminary knowledge which are necessary for our works in later chapters, and Chapter 3 presents a pricing model for European options with transaction costs under Heston-type stochastic volatility. This approach is formulated using the hedging strategy with some approximation in order to simplify the calculation of the expected transaction costs in hedging. This new approach is different from the existing literature (Mariani and SenGupta 2012) in two different aspects: Heston volatility is used and the option price does not depend on another option. The solution of the non-linear partial differential equation is obtained by a finite-difference scheme, proving a fair price range.

Then we focus on the hedging strategy and utility indifference method for pricing options with transaction costs under constant volatility in Chapter 4-6. Our ultimate goal is to study the American option pricing problem with transaction costs via utility indifference approach. Due to the nonlinearity resulted from the early exercise right of an American option, pricing American options via utility indifference approach raised two key issues: the optimal exercise boundary which needs to be solved as part of the solution and the heavy computational need for the none-linear problem. To solve such a complicated option pricing problem, we start with dealing two fundamental problems as the base of our ultimate goal. Firstly, we prove that the utility indifference approach is equivalent to the hedging strategy for the American option pricing problem in a complete market in Chapter 4. This problem is not well addressed due to the nonlinearity of the problem resulted from the optimal exercise boundary. A numerical study is conducted to deal with such an important problem. Then, in Chapter 5, we derive a new pricing approach for European options with transaction costs, where the ideas of hedging strategy and utility indifference approach are combined to achieve a balance of efficiency and accuracy. Our utility indifference approach reduces the dimension of the portfolio problem without options, thus, achieves better efficiency than the standard utility approach. In Chapter 6, we price American options with transaction costs via these two approaches. Since the impact of transaction costs on the American option price particularly on the most important feature of American options, is much less investigated, we provide a supplement study in this area by analyzing the effect of transaction costs to the optimal exercise price of an American option in addition to the option price itself through a utility-based approach. With a computationally efficient numerical scheme, we are able to demonstrate clearly how the optimal exercise price should be calculated and consequently how the option prices for the buyer and writer as well as the early exercise decision are affected by the inclusion of transaction cost.

The comparison presented in the thesis proves that utility indifference methods compared to hedging strategies produce more realistic option prices in the presence of transaction costs. The utility indifference approach proposed in this study could provide a useful, computational efficient way for pricing options with transaction costs.

#### Recommended Citation

Yan, Dong, A comprehensive study of option pricing with transaction costs, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2021. https://ro.uow.edu.au/theses1/1136

**Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.**