Delta hedging is a core risk management technique employed in options trading to reduce the risk associated with the price movements of an underlying asset. Quite simply – the overall delta risk of the options portfolio is neutralized by buying or selling the underlying asset to the tune of the magnitude and direction of that delta. For instance, an ATM call option contract (holding 100 units of the underlying asset) with +0.5 delta would require a sale of 50 units of the underlying to neutralize the risk emanating from a 1% jump in the underlying’s price. The reverse would be true for an ATM put option contract that has -0.5 delta i.e. 50 units of the underlying would have to be bought to neutralize the risk from a 1% up-move in its price.
As a quick recap, Delta of an option is the measure of the change in its price due to a marginal change in the price of the underlying asset. Hence it can also be understood as the fraction of option contract’s notional that one needs to buy/sell of the underlying to neutralise the option’s market price risk.
Delta hedging and its second order effects
As explained above, Delta hedging very simply neutralises the directional market risk on an option in response to a first order sensitivity of the option’s price to a marginal movement in the underlying’s price. But every subsequent move in the price of the underlying changes the delta of the underlying i.e. brings about a second order sensitivity of the option’s price to the change in the price of the underlying. And this brings us to the concept of gamma hedging or the need to constantly rebalance the changing hedge-delta to neutralize the option’s market price risk coming from the price movements in the underlying.
The direction of gamma in the option portfolio importantly governs the cost of hedging market risks – a long/positive gamma portfolio would deliver more delta as the price of the underlying asset goes up and reduces it when the price goes down. Hedging this risk would mean selling the underlying asset when its price increases and buying it when its price declines, which intuitively makes it a positive yielding exercise (adjusting for the portfolio’s theta of course). Negative gamma is just the opposite of the above, and delta hedging such a portfolio comes at a cost.
An example of Delta hedging
Consider a trader seeking a delta-neutral position for a Call option onStock A with ‘at the money’ strike at 100, which means a delta of +0.5 at the time. Assuming a single call option notional amounting to 100 stock units, to neutralize the delta of the long call position onewould need to sell 50 units of the stock. A hedging strategy that neutralizes everyday risk of the long option position depending on the delta as per the closing price of the underlying would show an end of day return/profit composed of that day’s profit on option’s price and the profit on Delta hedge based on the size of previous day’s hedge position.
Returns on this repeated hedging exercise would importantly be a function of the number of times the delta of the option needs to be neutralized i.e. the volatility of the underlying stock, and even if positive gamma is at play one needs to be judicious about the increasing transaction costs of the exercise. Time decay of the option i.e. the loss in its time value as it nears expiry is the cost of holding the long position, hence profits from daily hedging of the delta of the position would need to be adjusted for this time decay to arrive at the more pertinent net returns. In the real marketplace therefore risks of under-hedging or over-hedging are always at play if expectations of lower/higher volatility of the underlying asset relative to what was implied at the inception are proven wrong.
The bottom line
The need for neutralising the delta of an options portfolio is a choice to not run the price risk coming from the movements in the underlying. And the kind of strategy chosen to hedge an option portfolio’s delta would need to consider multiple moving parts as mentioned above.
Ready to learn more with practical examples? Dive deeper into delta and gamma hedging here.
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