How to hedge the option delta

Delta hedging is a widely used technique for managing option portfolios, regularly used by market makers. Before considering delta hedging, it is better to review what delta is. As discussed last week, delta is the measure of how the value of an option changes with respect to changes in the value of the underlying contract.

So a delta of 0.6 for a call option means the value of the option increases or decreases by 60 paise for every Re 1 increase and decrease in the underlying.

A call option does not move in the opposite direction of the underlying, hence the delta cannot be lower than zero (negative) and thus it ranges from 0 to 1. Value of the put option move in the opposite direction of the underlying and therefore the delta for put option ranges from 0 to -1. Deep in the money call/put options have delta at or close to 1/-1, whereas far out of the money call/put options have delta close to zero and at the money options have delta of 0.5/-0.5.

Now a portfolio is delta neutral if the total delta of the portfolio is zero, and thus the portfolio position is unbiased in terms of direction of any price movement of the underlying. A negative delta can be considered equivalent to being short in the underlying market, hence exposed to the same directional price risk. Conversely, a positive delta can be considered equivalent to being long in the underlying market. Now in order to establish a neutral or unbiased hedge, for options purchased one must also sell an appropriate number of underlying contracts. Now the appropriate number of contracts is determined by delta or hedge ratio and this theory is the basis for delta hedging. Now delta of the option changes as market conditions changes and thus adjustments need to be made to the portfolio to keep it delta neutral.

Suppose stock price is Rs 100 and the option price is Rs 10 and delta of the option contract is 0.5. Now the market maker has sold 10 call option contracts, i.e. option to buy 1000 shares. The market maker’s position can be hedged by buying 0.5*1000=500 shares. As we can see the position has been completely hedged, for example if the stock price goes up by Rs 1 then the market maker will gain Rs 500 on its shares and loose Rs 500 on the options sold. Now suppose the delta increases from 0.5 to 0.6, then an extra .1 per cent of 1000 shares = 10 shares would then have to be purchased to maintain the hedge.

The above delta hedging scheme is an example of dynamic hedging scheme

> shaurya.mishra@thehindu.co.in

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