Dynamic Hedging Strategies

To hedge its written call, the issuing firm decides to buy shares of the underlying stock or portfolio. The number of shares purchased at time t will depend on the price of the underlying stock at t and on the amount of time remaining until the expiration of the call. Another way of viewing this is that the amount of stock held against the call position depends on the probability that the option will be exercised. To hedge its anticipated receipt of German marks, the firm might enter into a forward contract. This would be a static hedge position. On the other hand, it might buy a put option on Deutschmarks (DM), which would give it the right to sell DM for a given dollar price; this is still a static hedge position, though somewhat more sophisticated. A third possibility would be to dynamically simulate a put on the DM position, replicating it by holding amounts of dollars and DM (these amounts changing with market conditions). In this article we consider some simple dynamic hedging strategies and show how to simulate them in Mathematica. Our discussion will lead us naturally to the consideration of Value-at-Risk (VaR), a measure of possible future losses which has become increasingly popular. 1. A SIMPLE EXAMPLE To start off, consider the following example, which we have adapted from Hull (1997): A financial institution has sold a European call option for $300,000. The call is written on 100,000 shares of a non-dividend paying stock with the following parameters: