## Year

2018

## Degree Name

Doctor of Philosophy

## Department

School of Mathematics and Applied Statistics

## Abstract

The Financial Crisis 2007-2009 is considered as the worst one since the Great Depression of the 1930s. During this financial crisis, most regulatory authorities around the world imposed restrictions or bans on short selling to reduce the volatility of financial market and limit the negative impacts of a downturn market (Beber & Pagano 2013). Such interventions were implemented to restore the orderly functioning of financial markets and limit drops in stock price. However, these regulations also resulted in some new problems, one of which is how to price options in a market with short selling restrictions or bans being imposed. The motivation of this Ph.D thesis is to study the effects of short selling restrictions or bans on option pricing.

The thesis is divided into two parts, the first one of which is consist of Chapter 3 and Chapter 4, where option pricing is explored under a new hard-to-borrow stock model. Such a model was proposed by Avellaneda & Lipkin (2009) to characterize the price-evolution of stocks subject to short selling restrictions. Although an approximate semi-explicit pricing formula has been produced for European call option, its derivation requires an independence assumption, which has limited its application to more general cases. In Chapter 3, we propose a new partial differential equation (PDE) approach to price European call options under the hard-to-borrow stock model and then an alternative direction implicit (ADI) scheme is applied to solve it numerically. This new PDE approach has also laid a solid foundation for the study on option pricing of American-style options. In Chapter 4, we extend the PDE approach to the American case and reformulate it as a linear complementarity problem, which is numerically solved with the Lagrange multiplier approach. A significantly important conclusion is that it may be optimal to exercise an American call option before expiration even though the underlying stock pays no dividends. Such a conclusion supports the recent work by Jensen & Pedersen (2016) and overturns a classic result by Merton (1973).

The second part of this thesis is about option pricing with short selling bans being imposed. Recently, Guo & Zhu (2017) proposed a new equal-risk pricing approach to study the effects of short selling bans on option pricing. Their analysis method appears to be but not the same as the existing utility indifference pricing methods. Only when the payoff function is monotonic, can an analytical pricing formula be produced. However, it is still difficult to apply equal-risk pricing approach to the case where the payoff function is non-monotonic. We intend to expand its application by establishing a PDE framework. Since Hamilton-Jacobi-Bellman (HJB) equation would be involved, we first explore different solution approaches to the HJB equation in Chapter 5, Chapter 6 and Chapter 7 as preliminaries before taking on the tough challenge of establishing the PDE framework for equal-risk pricing approach.

In Chapter 5, we successfully apply the homotpy analysis method to decompose the highly nonlinear HJB equation into an infinite series of linear PDEs and finally derive an exact and explicit solution for the HJB equation subject to general utility functions for the first time. In Chapter 6, a closed-form analytical solution for the Merton problem defined on a finite horizon with exponential utility function is obtained through two different methods without any one of the following assumptions: (1) the utility function belongs to the constant relative risk aversion (CRRA) class; (2) the utility function is defined over terminal wealth only and consumption is not allowed; (3) the investment horizon is infinite. In Chapter 7, a monotone numerical scheme method is presented to solve the HJB equation with general utility functions. Such a scheme is proved to be convergent through demonstrating its stability, consistency and monotonicity.

After proposing these three solution approaches to the HJB equation, we finally establish a PDE framework for equal-risk pricing approach in Chapter 8 and successfully solve the HJB equation analytically and numerically. When the payoff function is monotonic, analytical pric- ing formula is derived from our PDE framework, which matches perfectly with the pricing formula derived by Guo & Zhu (2017). When the payoff function is non-monotonic, such as a butter y spread option, equal-risk price is also produced through solving the PDE system numerically, which is absent in Guo & Zhu (2017). Consequently, our PDE framework has really expanded the range of application of equal-risk pricing approach so that effects of short selling bans are discussed in more general cases.

## Recommended Citation

Ma, Guiyuan, Option pricing with short selling restrictions or bans being imposed, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2018. https://ro.uow.edu.au/theses1/283

## FoR codes (2008)

010205 Financial Mathematics, 010302 Numerical Solution of Differential and Integral Equations

**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.**