Delta Hedging

Delta of a portfolio of options

The main benefit of delta as a portfolio management tool is that it is additive across individual options – calls and puts, different strikes and different times to expiration. Of course, all options must be for the same underlying.

You can easily calculate the total delta of your position by summing up the deltas of individual options. For example, you have the following portfolio of options:

• you are long 2 ITM calls with a delta of 0.70
• you are short 1 OTM call with a delta of 0.40
• you are long 1 OTM put with a delta of -0.30

The total delta of your position is:

• 2 x 0.70 (2 contracts of long calls)
• less 0.40 (subtract because you are short)
• plus -0.30 (add because you are long the option, but the delta is negative because it is a put)
• = 1.40 – 0.40 – 0.30 = 0.70

You can expect your portfolio’s market value to increase by 70 dollars (as 1 option contract represents 100 shares) for every 1 dollar of the underlying stock’s price increase.

Option delta as number of shares

You can look at delta as a proxy for number of shares. In the example above, your total position has the same directional exposure as 70 shares of the underlying stock. Both positions would appreciate by 70 dollars for every 1 dollar increase per share in the stock price. Both would lose 70 dollars for every 1 dollar decline in the share price.

Delta hedging of an option portfolio using stocks

When you know the total delta of your position and therefore know how many shares it represents in terms of directional exposure, you can easily hedge this directional exposure in case that you want to eliminate it for some reason.

Because all your options together behave as 70 shares of the underlying stock, you can hedge your position by selling 70 shares of the stock short. The resulting directional exposure is zero. By delta hedging your position, you have eliminated the risk resulting from directional moves in the underlying stock’s price.

Your option position may still be exposed to the passage of time, volatility, or interests rates, as these factors can’t be hedged by simply buying or selling the underlying stock.

Delta hedging of a stock position using options

Delta hedging can also be used in the opposite direction – you hedge a position in stocks using options. Let’s say you hold 500 shares in J.P. Morgan stock and for some reason you want to temporarily eliminate the directional exposure. You might be for example going on vacation for a week and you are afraid that your stock will go down while you are away. At the same time, you don’t want to sell your stock for tax reasons.

So you can buy put options to hedge the directional exposure, while keeping your long stock position. You want to buy at the money puts with a delta of -0.50. How many put option contracts do you need to buy? The answer is 10, as you want the total delta of the position to be zero. The delta of your long stock (500 shares) is 5, therefore you need the total delta of your put options to be negative 5.

Delta hedging must be managed continuously

The idea about delta hedging and going on vacation is not entirely perfect. The reason is that an option’s delta itself is not constant and changes with many factors, mainly with the moneyness of the option (and therefore with the underlying stock’s price movement). Delta also changes with passage of time or volatility.

As a result, you must be watching your portfolio continuously and adjust your positions if necessary. In the J.P. Morgan example above, if the stock price declines, the put options you use for hedging will be in the money and their delta will probably decrease from -0.50 to let’s say -0.70. If you hold 10 contracts, you now have an equivalent of 700 shares sold short, far more than the 500 shares of stock you hold (you are overhedged). In this case you will need to adjust your position by selling some (probably 3) of the put option contracts.

On the contrary, the stock going up could make you underhedged, as the options’ delta would go closer to zero, and you would need to buy additional puts.

Delta hedging may be costly

Another problem with delta hedging is that sometimes it comes at a cost, especially when you hedge by buying options, because options tend to lose time value with the passage of time (other factors being constant).

You may come from your vacation and find your J.P. Morgan shares lower and the put options lower as well, because the loss in time value has exceeded the increase resulting from the underlying’s directional movement.

Delta hedging by selling options

If you are now thinking about shorting options instead (that would solve the time decay problem), bear in mind that shorting options makes you short gamma, which simply said means that your profits tend to slow down and your losses tend to accelerate. While such strategy may be suitable in some cases, continuous management and adjustments are even more crucial here. Don’t go on vacation while leaving a short gamma position in the market.