Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

Chapter 4Change of Measure

In finance, derivative instruments such as options, swaps or futures can be used for both hedging and speculation purposes. In a hedging scenario, traders can reduce their risk exposure by buying and selling derivatives against fluctuations in the movement of underlying risky asset prices such as stocks and commodities. Conversely, in a speculation scenario, traders can also use derivatives to profit in the future direction of underlying prices. For example, if a trader expects an asset price to rise in the future, then he/she can sell put options (i.e., the purchaser of the put options pays an initial premium to the seller and has the right but not the obligation to sell the shares back to the seller at an agreed price should the share price drop below it at the option expiry date). Given the purchaser of the put option is unlikely to exercise the option, the seller would be most likely to profit from the premium paid by the purchaser. From the point of view of trading such contracts, we would like to price contingent claims (or payoffs of derivative securities such as options) in such a way that there is no arbitrage opportunity (or no risk-free profits). By doing so we will ensure that even though two traders may differ in their estimate of the stock price direction, yet they will still agree on the price of the derivative security. In order to accomplish this we can rely on Girsanov's theorem, which tells us how a stochastic process can have a ...