Delta and gamma hedging

By roma, 14 August, 2020

Trader can buy or sell the underlying asset in order to hedge delta of an option position. Now we will analyze the principle of delta hedging in more detail and show an example of gamma hedging.

 

Example 

Let's say that the trader sold 500 call options on the Enron stock.

Spot price of underlying asset = $500
Strike = $505
Time to Expiration = 1 year
Interest rate = 4.25%
Dividend yield = 2.54%
Implied volatility = 18%

Implied volatility is obtained from the option price. Using the above values, the Black-Scholes model produces a call option value of $36.60. Considering that the trader sells options for 500 shares, the value of all call options is $18,300. The delta obtained from the Black-Scholes model is 0.552 or 55.2%. Consequently, since the value of all call options is $18,300:
Delta position = -500 x 0.552 = -276 shares

The delta position is negative because the trader sold call options; if the stock price goes up, the option price will go up. In order to hedge the delta, trader decides to buy 276 shares. If the Enron share price fluctuates slightly, the profit from the option position will be balanced by losses from the position in stock and vice versa.

 

Big jump in stock price

What happens if the share price rises significantly? A 10% increase in the share price would raise the option value to $68.44, according to the Black-Scholes model:
Payoff from call options = $18,300 - $34,220 = -$15,920
Income from hedging = 276 shares x $50 = $13,800
Net income = $13,800 - $15,920 = -$2,120

The trader's loss would be $2,120 due to the impact of the gamma on the value of delta. The value of the call's delta is only useful for small deviations in the price of the underlying asset. From the example, it can be seen that when the share price rose by 10% a short option position generated greater losses than long stock position. However, due to the gamma the delta increased to 74.5% after the stock price movement. Initially, the delta of the short option position was 55.2%. A 10% increase in the share price implies the following loss:
Expected loss from delta 0.552 = - 500 x 0.552 x $50 = - $13.800

However, the actual loss is $15,920. The difference between the actual and estimated loss is $2,120, which is explained by gamma or convexity of the option price.

Call price

 

Delta

Gamma

Position

 

 

Short 500 call options

–276 shares

–2,2 shares

Hedge

 

 

Long 276 shares

276 shares

0 shares

Total

0 shares

–2,2 shares

 

Delta-Gamma balancing

Sometimes traders make small adjustments to the delta so that the delta more closely matches the forecasted changes in the value of the option position in case of high fluctuations in the price of the underlying asset. Such adjustments are called delta-gamma balancing. The gamma of a call option in the previous example approximately corresponds to the value of 0.0044. The influence of the gamma on the income from the option position is calculated using the Taylor series:
(502/2) x 0.0044 = $5.5 per option

This means that the additional loss due to gamma, if the share price rises by $50, is approximately $5.5. For 500 options the losses will increase by $2750 if the share price moves by $50. The delta-gamma balancing is aimed at correcting the hedge and takes into account the convexity of the option price chart with respect to price of the underlying asset. 

 

Gamma Hedging

Buying or selling an underlying asset will help to manage delta of option position, but will not cover the gamma (Read How to trade gamma competently?). The value of an option is not linearly linked to the price movement of the underlying asset. Fixed number of shares bought or sold to hedge the delta react linearly to changes in the price of the underlying asset. In the previous example, the trader sold 500 call options. An alternative hedging method is to purchase call options on the same asset, but with a earlier expiry date. Such options will have a positive delta as well as a positive gamma.

Gamma and expiration

Let's assume that the trader in our example decides to hedge using options with earlier expiration date and buys 9-month calls with the following parameters:
Strike = $500
Implied volatility = 18%
Expiration = 9 months

With a spot share price of $500, the value of the option is $33.51 according to the Black-Scholes model. The delta and gamma are 0.5636 and 0.00505, respectively. The gamma of a 1-year call option with a strike of $505, which the trader sold, is 0.0044. This means that the trader needs to buy a smaller number of short-term options with a strike of $500 to neutralize the gamma risk. The required number of 9-month options with a strike of $500 to buy can be calculated as follows:
500 x (0,00440/0,00505) = 435,64

By buying 436 options, we will reduce the gamma risk, but the delta risk will still be present. The delta of options sold with a strike of $505 is minus 276 shares (- 500 x 0.552), and the delta of options purchased with a strike of $500 is 246 shares (436 x 0.564). To hedge the remaining delta, the trader must buy
276 - 246 = 30 shares.

 

Delta

Gamma

Position

 

 

Short 500 call options

–276 shares

–2,2 shares

Hedge

 

 

Long 436 call options

246 shares

2,2 shares

Long 30 shares

30 shares

0

Total

0 shares

0 shares

 

Remaining risks after gamma hedge

The combination of 436 short-term options purchased and 30 shares purchased will help to effectively reduce the delta and gamma risks from initial short position in 500 calls if the price of the underlying asset fluctuates moderately. However, such a double hedge is not able to cover all possible risks of the option portfolio. For example, there will remain vega risk.

The vega of options sold equals 1.98, i.e. an increase in implied volatility by 1% would result in an option appreciation of $1.98, or 500 x $1.98 = $990 for all options. However, the vega of purchased calls is 1.71 for one option, or 436 x 1.71 = $745.56. Consequently, if the implied volatility grows by 1%, the options purchased will rise by $745.56.

Thus, the net loss from the increase in the volatility by 1% will be as follows
$745,56 –$990 = –$144,44

 

Delta

Gamma

Vega

Position

 

 

 

Short 500 call options

–276 shares

–2,2 shares

–$990

Hedge

 

 

 

Long 436 call options

246 shares

2,2 shares

$745,56

Long 30 shares

30 shares

0 shares

$0

Total

0 shares

0 shares

–$144,44

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