# Chapter 17: Delta Hedging

6 Mins 25 Feb 2022 0 COMMENT

Abhinav’s boss now asks him to think of an alternative strategy that can be used to hedge short positions in the Options of ABC Ltd, and he suggests Delta Hedging. Let’s take a look at the features of this Options strategy.

## What is Delta Hedging

Delta Hedging allows you to hedge the downside risk of short Option positions. For example, a short Call will start making losses as the underlying increases. However, if you have a long position in the underlying, it will produce an offsetting gain to hedge the short Call position.

Similarly, a short Put will start reflecting losses as the underlying decreases. However, if you have a short position in the underlying, that will produce an offsetting gain to hedge the short Put position.

But we know that the price movement of the underlying and Options are not 1 for 1, and the Greek Delta determines that ratio. Delta calculates the extent to which the Option premium would change because of a minor change in the underlying price.

Delta = Change in the price of an Option with respect to the change in the unit price of an underlying

Delta ranges from 0 to 1, where deep OTM Options have values closer to 0 and deep ITM Options have a value closer to 1. This means that movement in deep ITM Options value is closely equal to movement in the underlying. ATM Options have a value closer to 0.5. The actual value of Delta can be in the range of – 1 to + 1, but here, we are talking about the absolute value of Delta.

The actual value of Delta will be in the following ranges for various Option positions:

A long Call or short Put = between 0 to + 1

A short Call or long Put = between 0 to – 1

Let’s understand this with an example. Suppose you short Puts on 100,000 shares of ABC Ltd. and the Put Delta is 0.6 of a particular strike. You lose approximately Rs. 60,000 if the stock price decreases by Re. 1. So, if you shorted 60,000 shares, you will be hedged as the 60,000 short position produces a gain of Rs. 60,000 for a Re. 1 decrease in the stock price.

Similarly, if you short Calls on 100,000 shares of ABC Ltd., the Call Delta is – 0.4. You lose approximately Rs. 40,000 if the stock price increases by Re. 1. So, if you buy 40,000 shares, you will be hedged, as the 40,000 long position produces a gain of Rs. 40,000 for a Re. 1 increase in the stock price.

So, a hedge position in the stock will be = – (Delta * number of Options)

• For the 1,00,000 short Puts, hedge = – 0.6 * 100,000 = – 60,000 i.e., short 60,000 shares
• For the 1,00,000 short Calls, hedge = – (– 0.4) * 100,000 = 40,000 i.e., buy 40,000 shares

In a nutshell, if you have a short Call Option, you should buy stock equivalent to the Delta multiplied by the Option quantity. Similarly, if you have a short Put Option, you should short stock equivalent to the Delta multiplied by Option quantity.

## In reality

It seems perfect, but in reality, things are not as simple as it appears to be. This is because Delta changes over time and with changing market conditions. This issue can be addressed by continuously rebalancing the hedge position, also called dynamic Delta Hedging. You need to recompute the number of shares to hedge and if more shares are required, buy more or sell if fewer shares are needed.

There are a few other factors that also need to be considered.

• If the Option is deep OTM, the Delta will be closer to zero and stable. The Option will have little value and a few shares are required to hedge; the number needed for hedge will not change significantly and the hedge will work well.
• If the Option is deep ITM, the Delta will be closer to one and stable. The Option will have more value; the number of shares needed for hedging will not change significantly and the hedge will work well.
• But if the Option is closer to ATM, the Delta can fluctuate quickly between zero and one, and Delta Hedging becomes very difficult.

In general, if the underlying changes more, the Delta could have changed more, a more significant rebalancing would be needed. It is unfortunate that for both short Call and Put Options, after the share price moves, the Delta will be shifted in a way that you must buy shares at a higher price and sell shares at a lower price and the initial hedge position will not be compensated for, entirely.

## Delta Hedging example

Suppose you are short on 1000 ATM Call Options of the strike price of Rs.50. The Delta of this Call is 0.5. To hedge this position, you bought 0.5 * 1000 = 500 shares.

### Case 1: If the stock price reduces to Rs. 49

In this case, the underlying stock position value decreases by 500 * 1= – Rs. 500.

The change in a Call Option value = Delta * change in the underlying price

Change in a Call Option premium = 0.5 * (– 1) = – Rs. 0.5

Total change in the value of 1000 Call Options = 1000 * (– 0.5) = – Rs. 500

As this is a short Option position, it will lead to a profit of Rs. 500.

Thus, the net change in the portfolio value is = 0. This is called Delta Neutral Strategy.

Now the Delta at this point becomes, let’s say, 0.49. Thus, to remain Delta neutral, the new hedge requires you to have 0.49 * 1000 = 490 shares. It means you need to sell 500 – 490 = 10 shares at a current price of Rs. 49 and this price is lower than your initial purchase price.

### Case 2: If the stock price increases to Rs. 51

In this case, the underlying stock position value increases by 500 * 1= Rs. 500.

The change in a Call Option value = Delta * Change in the underlying price

Change in a Call Option premium = 0.5 * 1 = Rs. 0.5

Total change in the value of 1000 Call Options = 1000 * 0.5 = Rs. 500

As this is a short Option position, it will lead to loss of Rs. 500.

Thus, the net change in the portfolio value is = 0

Now the Delta at this point becomes, let’s say, 0.51. Therefore, to remain Delta neutral, the new hedge requires you to have 0.51 * 1000 = 510 shares. It means you need to purchase 510 –500 = 10 shares at a current price of Rs. 51 and this price is higher than your initial purchase price.

### Case 3: If the stock price reduces to Rs. 48

In this case, the underlying stock position value decreases by 500 * 2= – Rs. 1000.

The change in a Call Option value = Delta * change in the underlying price

Change in a Call Option premium = 0.5 * (– 2) = – Re. 1

Total change in the value of 1000 Call Options = 1000 * (– 1) = – Rs. 1000

As this is a short Option position, it will lead to a profit of Rs. 1000.

Thus, the net change in the portfolio value is = 0

Now the Delta at this point becomes, let’s say, 0.48. Therefore, to remain Delta neutral, the new hedge requires you to have 0.48 * 1000 = 480 shares. It means you need to sell 500 – 480 = 20 shares at a current price of Rs. 48 and this price is lower than your initial purchase price.

### Case 4: If the stock price increases to Rs. 52

In this case, the underlying stock position value increases by 500 * 2= Rs. 1000.

The change in a Call Option value = Delta * change in the underlying price

Change in a Call Option premium = 0.5 * 2 = Re. 1

Total change in the value of 1000 Call Options = 1000 * 1 = Rs. 1000

As this is a short Option position, it will lead to a loss of Rs. 1000.

Thus, the net change in the portfolio value is = 0

Now the Delta at this point becomes, let’s say, 0.52. Therefore, to remain Delta neutral, the new hedge requires you to have 0.52 * 1000 = 520 shares. It means you need to purchase 520 –500 = 20 shares at a current price of Rs. 52 and this price is higher than your initial purchase price.

If you increase the rebalancing frequency, the transaction cost will increase significantly.

The other significant impact on Option pricing is of volatility. Volatility increases the Option price, but does not affect the current share price much. An increase in volatility makes both Calls and Puts more valuable. That increase in value will be a loss for you as you have shorted the Options, and this is not offset by an immediate change in the value of shares used for the hedge. There may be an immediate loss on the hedged position.