Delta hedging example

In order to fully hedge a position taken in the options market, a market maker must calculate the delta of the option position. Delta is a measure of an option’s sensitivity to changes in the price of the underlying asset, and is calculated from option pricing models. It is used to estimate the amount an option price will move given a change in the underlying stock price, all else being equal. The market maker enters other transactions in the physical market and/or the options or warrants market to achieve a delta-neutral position.

The principal behind a delta-neutral hedge is that for a given move in the price of the underlying securities, regardless of the direction of that move, an equal and offsetting move will occur in the other ‘legs’ of the hedge. The aim is that the value of the overall position does not change.

As the price of a stock moves, and as time passes, the delta of an option itself changes. Consequently, delta-neutral, or dynamic hedging, requires constant monitoring and regular adjustments to the position.

Example

Assume a market maker enters into a trade to sell 10 NewsCorp (NWS) call options with a delta of 0.6. While call options have a positive delta (the price of the option rises as the stock price rises), the market maker has a negative delta position, as they have sold the options. To achieve a delta-neutral position, the market maker must buy NWS shares as follows:

Number of shares   =  no.option contracts x 1000 x delta
  = 10 x 1000 x 0.6
  = 6000

Similarly, if the market maker had bought 10 of the same call options they would need to sell 6,000 NCP shares to achieve a fully hedged position.

Assume that a day later the price of NWS shares rises by $0.50. The call options will increase in value, resulting in an (unrealised) loss to the market maker. However, the value of the NWS shares held by the market maker will also have increased. If the hedge is accurate, the overall position should be neutral.

As a result of the increase in the share price, however, the delta of the options will have increased, as the options move further into the money. Assume the delta of the options has increased to 0.65. To maintain a delta neutral hedge, the market maker needs to hold shares as follows:

Number of shares   =  no.option contracts x 1000 x delta
  = 10 x 1000 x 0.65
  = 6500

The market maker should therefore buy 500 more NWS shares.

In this way, option and warrant transactions generate liquidity in the equity market. When an option or warrant trade is executed it often results in a number of trades in the underlying securities, both at the time of the initial trade, and during the remaining life of the derivative product as the market maker actively manages the risk of the position.