The Mathematics of Financial Models: Solving Real-World Problems with Quantitative Methods

CHAPTER 7 The Effectiveness of Hedging Strategies

In Chapters 3, 4, and 5, I presented examples of option pricing and showed how one can use models to analyze and value various customized options. In this chapter, I discuss how sellers of these options (or any guarantees for that matter) can analyze the effectiveness and efficiency of a hedging strategy to understand and mitigate the risks arising from the initiation of these option transactions.

Black-Scholes developed the famous option-pricing model in 1973 to value European-style vanilla options on non-dividend paying stocks. One assumption used by the authors to arrive at their final result was that the hedger had the ability to construct a riskless portfolio at the time the option is sold and maintain this riskless portfolio continuously and dynamically throughout the life of the option as market conditions changed and the option life diminishes. Applying Ito's lemma to the resulting diffusion equations in continuous time, the authors were able to arrive at equations (3.4a) and (3.4b). As mentioned in Chapter 3, each of the assumptions used by Black-Scholes has since been violated by many subsequent publications in the spirit of building more realistic models. The assumption tampering, fuelled by the cheapness of fast computing power, has resulted in the use of simulations as a tool to value options and analyze risk-management strategies associated with the sale of these options in discrete time.

Starting with an example ...