Delta Hedging: Calculations, Adjustments, Long vs. Short Options

This page explains delta hedging using stocks or options, the difference between hedging by buying vs. selling options, and implications on hedging cost, gamma exposure, ongoing management and adjustments.

Calculating total delta of option portfolio

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 expirations, as long as all the options are on the same underlying asset.

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:

• Long 2 ITM calls with a delta of 0.70
• Short 1 OTM call with a delta of 0.40
• Long 1 OTM put with a delta of -0.30

Total delta of your position is:

• 2 x 0.70 (2 contracts of long calls)
• minus 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

Total delta of 0.70 means the portfolio value is expected to increase by approximately 70 dollars for every 1 dollar of the underlying stock's price increase (this applies to US stock options, where one option contract represents 100 shares).

Option delta as number of shares

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

One unit of delta is like being long 100 shares (for US stock options, with 1 option contract = 100 shares).

Delta hedging using stocks

When you know total delta of your position and therefore know how many shares it represents in terms of directional exposure, you can hedge this directional exposure to eliminate it (if you want to avoid the directional exposure, but don't want to close the position for some reason).

Because all the options together behave as 70 shares of the underlying stock, you can hedge your position by short selling 70 shares of the stock. 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 using options

Delta hedging can also be used in the opposite direction – hedge a stock position 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. For instance, you may be 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.

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 contracts, because you want 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.

The idea about delta hedging and going on vacation is not perfect. The reason is that option delta itself is not constant and changes with many factors, mainly with the moneyness of the option (and therefore with the underlying stock 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 of the put option contracts. You should sell 3 put contracts and keep 7 contracts. Total delta of the puts will be 7 x -0.70 = -4.90, which is closest you can get to hedge your +5 long stock delta.

On the contrary, the stock going up can make you underhedged, as the put option delta goes closer to zero. In this case you need to buy additional put contracts.

The cost of delta hedging

Delta heding always involves transaction costs such as commissions and bid-ask spreads, which depend mainly on how often adjustments are made and how liquid the options are.

When hedging is done by buying options, there can be another, even greater cost in time decay, as options tend to lose time value with passage of time (other factors being constant). This depends mainly on the strikes and expirations used.

At the money options with short time to expiration have the greatest time value (on a relative basis) and therefore can be expensive as hedging tool (at the same time, they also tend to be the most liquid and most effective).

Delta hedging by selling options

This solves the time decay problem. The short call position will actually gain with passing time if the underlying stock does not move. And that is a big if.

If the stock prices increases, your long stock position gains. Your short call position loses (the call options' value increases but you are short), initially by the same amount as your stock gains. So far the hedge works as intended.

However, if the stock price keeps rising, your long stock position keeps gaining value at the same rate, but the losses in your short call position accelerate, making you overhedged.

Shorting options makes you short gamma, which means delta always changes in the direction that is unfavorable to the position. As a result, your profits slow down and your losses accelerate in large underlying price moves. Such position must be continuously monitored and adjusted.

In other words, don't go on vacation while leaving a short gamma position in the market.

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