Derivatives

CHAPTER 27Delta Hedging

Aims

• To demonstrate simple delta hedging of calls and puts and the need for rebalancing.
• To show how dynamic delta hedging is used to protect the value of an options portfolio from (small) changes in the price of the underlying asset.
• To show that after dynamic delta hedging, you can close out your positions at any time and you will have neither gained nor lost money.

27.1 DELTA

A portfolio can be constructed so that any gain or loss from a small change in the option price (over a short interval of time) is offset by changes in the price of the underlying asset – this is delta hedging. Delta hedging applies to all kinds of options, where the underlying can be a stock price, stock index, commodity price, exchange rate, futures price, or an interest rate. The delta of a long call (which is written on one stock) is defined as:

Hence:

The delta of a long call is positive because a rise in the stock price leads to a rise in the call premium (Figure 27.1). Consider delta hedging a position in European call options. The value V of a portfolio consisting of