Delta Hedging

Delta Hedging

Delta hedging is a hedging strategy for managing the risk of holding options. It entails buying or selling an amount of the underlying asset that corresponds to the delta of the portfolio. By adjusting the amount bought or sold on new positions, the portfolio delta can be made to sum to zero, and the portfolio is then delta neutral.

The Nobel Prize was awarded to Merton and Scholes in 1997 for applying risk neutrality to option valuation and overcoming a problem that researchers had not managed to find a workable approach to discounting payoffs from option contracts. The Nobel Committee explained the rationale for awarding the prize in the following way:

"This year's laureates resolved these problems by recognizing that it is not necessary to use any risk premium when valuing an option. This does not mean that the risk premium disappears, but that it is already incorporated in the stock price. In 1973 Fischer Black and Myron S. Scholes published the famous option pricing formula that now bears their name (Black and Scholes (1973)). They worked in close cooperation with Robert C. Merton, who, that same year, published an article which also included the formula and various extensions (Merton (1973)).

The idea behind the new method developed by Black, Merton and Scholes can be explained in the following simplified way. Consider a so-called European call option that gives the right to buy a certain share at a strike price of $100 in three months. (A European option gives the right to buy or sell only at a certain date, whereas a so-called American option gives the same right at any point in time up to a certain date.) Clearly, the value of this call option depends on the current share price; the higher the share price today the greater the probability that it will exceed $100 in three months, in which case it will pay to exercise the option. A formula for option valuation should thus determine exactly how the value of the option depends on the current share price. How much the value of the option is altered by a change in the current share price is called the “delta” of the option.

Assume that the value of the option increases by $1 when the current share price goes up $2 and decreases by $1 when the stock goes down $2 (i.e. delta is equal to one half). Assume also that an investor holds a portfolio of the underlying stock and wants to hedge against the risk of changes in the share price. He can then, in fact, construct a risk-free portfolio by selling (writing) twice as many options as the number of shares he owns. For reasonably small increases in the share price, the profit the investor makes on the shares will be the same as the loss he incurs on the options, and vice versa for decreases in the share price. As the portfolio thus constructed is risk free, it must yield exactly the same return as a risk-free three-month treasury bill. If it did not, arbitrage trading would begin to eliminate the possibility of making risk-free profits.

As the share price is altered over time and as the time to maturity draws nearer, the delta of the option changes. In order to maintain a risk-free stock-option portfolio, the investor has to change its composition. Black, Merton and Scholes assumed that such trading can take place continuously without any transaction costs (transaction costs were later introduced by others). The condition that the return on a risk-free stock-option portfolio yields the risk-free rate, at each point in time, implies a partial differential equation, the solution of which is the Black-Scholes formula for a call option."

We set out the following videos to explain how a simplified discrete time re-balancing of the portfolio options combined with stocks from from the underlying security can be implemented. The effectiveness of this strategy is dependent on the extent to Black Scholes model assumptions can be assured and also on the extent or frequency to which re-balancing can be carried out. It may not be practicable to re-balance at high frequency and this is a concern when the gamma of the option is high. Please feel free to download spreadsheet used.

Useful Links

Link to VBA code for Delta. Link to explanation of Stock Price simulation and Geometric Brownian Motion.