Many have the opinion mathematics is exclusively isolated to esoteric applications with little real-world use. This couldn’t be further from the case. Mathematics is the cornerstone of many industries, none such more than the financial field. In option pricing the mathematical and financial worlds synthesize together to create pricing models to determine the fair price of options. The standard model used in option pricing known as Black-Scholes model, however, makes unrealistic expectations about financial markets that are not replicated in reality. The focus of my report was then to study one approach that adds transaction costs into this Black-Scholes model and observe the effects on option prices.
Options are financial derivatives investors use to protect themselves against movements in stock price. Options involve two parties; an option holder who purchases the option for some premium, entitling them to buy a share at a set price (known as a call) or sell a share (known as a put), with these known as the option’s rights. The option writer receives this option premium but must purchase or sell a share if the rights are exercised by the holder. Options can also be divided based on when these rights can be used, with American options allowing holder to exercise anytime between purchase of option and a pre-determined expiry date. With American options, there are then share prices in which using rights of option before expiry becomes economical for the option holder, known as the optimal exercise boundary.
Pricing of options relies on holding a portfolio of stock and option and adjusting their positions such that this portfolio remains risk-free, with this process known as hedging. It was considered how varying the transaction cost rates for hedging and the re-hedging frequency affects the optimal exercise price and the financial implications that arise from this. The report shows the difference that exists between an idealised mathematical world and one where market imperfections exist. Immediately from relaxing one assumption issues arise with option holder and writer no longer agreeing on the value of an option. Additionally, it was observed as costs of re-hedging increased holders chose to exercise American call option earlier by reducing their optimal exercise price. This is logical as an option holder would choose to avoid the amount of costs incurred from holding this portfolio.
The holder’s optimal exercise boundary is then important as the option writer needs this to be able to price the option from their perspective. Overall, this report was an interesting exploration into the world of option pricing and highlighted the importance of mathematics in the financial world.
Jeremy Sterjovski
University of Wollongong
