Options Delta Hedging with Example

What is Hedging?

Hedging is a term used in finance to describe the process of eliminating (or minimizing at best) the risk of a position. Typically, the risk referred to is the directional, or price risk, and the hedge is accomplished by taking the opposite view/position in a similar asset (or same asset traded elsewhere).

For example, take Vodafone stock. This is traded on the London Stock Exchange in GBP as Vodafone Group PLC (VOD.L) and also traded on America's NASDAQ in USD as Vodafone Group Public Limited Company (VOD). If you were long VOD in the US you could hedge your exposure to the company's stock price risk by shorting the same notional value of VOD.L in GBP. Your price risk would be reduced but you would now have exposure to currency and dividend risk.

The same concept applies to options when hedging the option delta; you remove the delta (position/price) risk by buying/selling the underlying instrument (stock, future etc).

Hedging Options Using Option Delta

As mentioned, Option Delta represents the relative price movement that an option will experience given a one point move in the underlying. The delta therefore results in a sort of proxy for the underlying stock; that is, the number itself tells you the proportion of equivalent shares you are long/short.

Example: 10 call options on MSFT, where the option has a delta of 0.25, means you have effectively 250 shares in MSFT (10 * 0.25 * 100). Delta hedging this option position with shares means you would sell 250 MSFT stock to offset the 250 "deltas" of call options.

Example 2: 50 put options on AAPL, where the option delta is 0.85, means your effective position in the stock is short 4,250 shares (50 * -0.85 * 100). As you are short deltas, your hedge would involve buying 4,250 shares of AAPL stock.

The Gamma and the Delta

The tricky part when using the delta of an option to determine the hedge volume is that the actual delta value is always changing. You might sell 500 shares to hedge a delta value 0.50 but after your hedge, the market moves and the delta of the option now becomes 0.60. Now, your total position delta has increased to 100 meaning you will need to sell another 100 shares to square off the delta to zero.

This changing of the delta can be measured and estimated by an other option Greek called option gamma.

Let's go through an example showing the gamma effect on delta and the resulting hedging that is needed.

1) Buy 10 call options on ABC with a delta of 0.544. Position delta = 544 (10 * 0.544 * 100)

Contract	Delta	Position	Pos Delta
Call Option	0.544	10	544
ABC Stock	1	0	0
		Total Delta	544

2) You sell 544 shares of ABC (delta 1 of course). Position delta = 0 (544 option delta minus 544 shares of 1 delta)

Contract	Delta	Position	Pos Delta
Call Option	0.59	10	590
ABC Stock	1	-544	-544
		Total Delta	46

3) ABC shares increase by 1%. The delta of the call option is now 0.59. Your long 10 calls is now worth 590 deltas. Your short ABC stock is worth - 544 deltas. Net position delta = 46

Contract	Delta	Position	Pos Delta
Call Option	0.544	10	544
ABC Stock	1	-544	-544
		Total Delta	0

4) You sell an additional 46 ABC shares. Total deltas of the call position = 590. Total deltas of ABC -590. Total position delta = 0

Contract	Delta	Position	Pos Delta
Call Option	0.59	10	590
ABC Stock	1	-590	-590
		Total Delta	0

5) ABC shares trade lower by 0.50%. Call option delta now 0.57 so your position of 10 calls is 570. You are short 590 shares of ABC. Total position delta -20

Contract	Delta	Position	Pos Delta
Call Option	0.57	10	570
ABC Stock	1	-590	-590
		Total Delta	-20

6) Now you have a net short delta, which means you will have to buy back 20 shares to square off the delta. Option position delta 570, ABC share delta -570 net = 0

Contract	Delta	Position	Pos Delta
Call Option	0.57	10	570
ABC Stock	1	-570	-570
		Total Delta	0

Here are the transactions for the above scenario;

Buy 10 calls @ 4.13
Sell 544 stock @ 100
Sell 46 stock @ 101 (calls at 4.69)
Buy 20 stock @ 100.50 (calls at 4.40)

Note: the option position in this example didn't change after the first trade; the price fluctuations of the underlying stock resulted in the changing delta of the option. As the delta changes, the total position also changes resulting in the need to re-hedge.

Factors Affecting Delta

In the example above the only input to the delta that changed was the underlying price. There are, however, keep in mind that there are four variables that will effect the delta of an option and hence change your hedge position; underlying price, volatility, time and interest rates.

Increasing underlying price, vol and interest rates all increase the value of delta. I.e call delta will move from 0.50 to 0.60 and put delta from -0.40 to -0.30.

The effect of Time value depends on the options moneyness. If you;ve read the section on Option Charm you'll be aware that for in-the-money call options a decrease in time will increase the value of the option's delta; with less time to expiration, the already in-the-money option becomes even more likely to remain in-the-money. Conversely, a call option already out-of-the-money becomes less certain with each day that passes (all other factors staying the same). So, for ITM call options decreasing time increases delta (your position becomes "longer") while your OTM call becomes shorter.

For an ITM put option the delta is already negative, so as expiration approaches and the option becomes more likely to expire ITM the delta moves closer to -1.0 i.e. becoming shorter.

Delta Effect due to Time

Reducing the Time to Expiration

	Option Delta
Time	ITM Call	ITM Put	OTM Call	OTM Put
Today	0.89	-0.82	0.18	-0.11
Less Time	0.90	-0.84	0.16	-0.10
Change	0.01	-0.02	-0.02	0.01
Result	Longer	Shorter	Shorter	Longer

Increasing the Time to Expiration

	Option Delta
Time	ITM Call	ITM Put	OTM Call	OTM Put
Today	0.89	-0.82	0.18	-0.11
More Time	0.86	-0.77	0.23	-0.14
Change	-0.03	0.05	0.05	-0.03
Result	Shorter	Longer	Longer	Shorter

How Often to Hedge?

As the stock price fluctuates, the implied volatility changing and the time to expiration ticks away, the delta is constantly changing, which raises the important question of how often and on what criteria do you hedge?

In theory, the idea is that you should hedge whenever the position delta is non-zero, so you transact with the underlying to bring the delta back to zero. However, the transaction costs involved in this level of trading prohibit its use in practice.

Dynamic Hedging: Managing Vanilla and Exotic Options