# Calculating Shadow Gamma (Option Greeks)

We use a simple example to illustrate the calculation of Shadow Gamma as describe by Taleb in Dynamic Hedging.

Gamma is the second derivative of the change in option price to a change in the underlying asset price. Alternatively, it is the rate of change in the option delta to a change in the underlying asset price.

Gamma may be calculated using the Black Scholes formula but according to Taleb a more practical way of calculating Gamma is to calculate the actual change in delta that arises when the price of the underlying is changed. Here delta is also taken to be the discrete change in the value of the option for a change in the value of underlying and not delta as determined from the Black Scholes formula.

For example, let us consider a 30-day out of money European call option on NVIDIA stock. From ivolatility.com we have obtained the following data for available options on 14-May-2014:

 symbol exchange date period strike out-of-the-money % call/put Iv delta NVDA NASDAQ 14-May-2014 30 18.1 0 C 0.27 0.49204 NVDA NASDAQ 14-May-2014 30 19.005 5 C 0.2734 0.26021 NVDA NASDAQ 14-May-2014 30 19.91 10 C 0.2987 0.13058 NVDA NASDAQ 14-May-2014 30 20.815 15 C 0.301 0.0516 NVDA NASDAQ 14-May-2014 30 21.72 20 C 0.301 0.01689 NVDA NASDAQ 14-May-2014 30 22.625 25 C 0.301 0.00472 NVDA NASDAQ 14-May-2014 30 23.53 30 C 0.301 0.00114 NVDA NASDAQ 14-May-2014 30 24.435 35 C 0.301 0.00024 NVDA NASDAQ 14-May-2014 30 25.34 40 C 0.301 0.00005 NVDA NASDAQ 14-May-2014 30 26.245 45 C 0.301 0.00001 NVDA NASDAQ 14-May-2014 30 27.15 50 C 0.301 0 NVDA NASDAQ 14-May-2014 30 28.055 55 C 0.301 0 NVDA NASDAQ 14-May-2014 30 28.96 60 C 0.2885 0

Let us consider the 60% 30-day out-of-money call option with strike of 28.96, spot price of 18.1 and annualized volatility of 28.85%. The risk free rate and dividend yield assumed in our calculations are 0.05% and 0% respectively. The value of the option works out to 0.000000003. Now let us assume that the underlying stock price changes to 22.1. The value of the option becomes 0.0003291. Hence an increase in the stock price led to a 0.00033 change in the value of the option, the delta. The following are deltas calculated for changes in the underlying asset price (with respect to the original spot price) for the given option:

 NVIDIA Stock Price Delta 15.6 (0.00000) 16.1 (0.00000) 16.6 (0.00000) 17.1 (0.00000) 17.6 (0.00000) 18.1 – 18.6 0.00000 19.1 0.00000 19.6 0.00000 20.1 0.00000 20.6 0.00001 21.1 0.00003 21.6 0.00011 22.1 0.00033 22.6 0.00088 23.1 0.00218 23.6 0.00498 24.1 0.01058 24.6 0.02105 25.1 0.03935 25.6 0.06946 26.1 0.11624 26.6 0.18519 27.1 0.28201 27.6 0.41198 28.1 0.57942 28.6 0.78721 29.1 1.03647 29.6 1.32654 30.1 1.65513 30.6 2.01866 31.1 2.41271 31.6 2.83249 32.1 3.27323 32.6 3.73050

In this discrete measurement, gamma is calculated twice for each asset price. An up-gamma is the change in the value of delta, given the underlying asset price moves up by an incremental value; down gamma is the change in the value of the delta given that the underlying asset price moves down by an incremental value.

For example, the up-gamma calculated at asset price 25.1 for an increment of 1 unit in the asset price is equal to the delta calculated at asset price 26.1 less the delta calculated at asset price 25.1, i.e. up-gamma (25.1) = delta (26.1)-delta (25.1) = 0.11624 0.03935 = 0.0769.

In like manner, the down-gamma calculated at asset price 25.1 for a decrement of 1 unit in the asset price is equal to the delta calculated at asset price 25.1 less the delta calculated at asset price 24.1, i.e. down-gamma (25.1) = delta (25.1)-delta (24.1) = 0.03935 – 0.01058 = 0.0288.

The up-gamma and down-gamma changes in delta for a sample set of asset prices are given below:

 NVIDIA Stock Price Delta Up-Gamma Down-Gamma 15.6 (0.00000) 0.0000 0.0000 16.1 (0.00000) 0.0000 0.0000 16.6 (0.00000) 0.0000 0.0000 17.1 (0.00000) 0.0000 0.0000 17.6 (0.00000) 0.0000 0.0000 18.1 – 0.0000 0.0000 18.6 0.00000 0.0000 0.0000 19.1 0.00000 0.0000 0.0000 19.6 0.00000 0.0000 0.0000 20.1 0.00000 0.0000 0.0000 20.6 0.00001 0.0001 0.0000 21.1 0.00003 0.0003 0.0000 21.6 0.00011 0.0008 0.0001 22.1 0.00033 0.0018 0.0003 22.6 0.00088 0.0041 0.0008 23.1 0.00218 0.0084 0.0018 23.6 0.00498 0.0161 0.0041 24.1 0.01058 0.0288 0.0084 24.6 0.02105 0.0484 0.0161 25.1 0.03935 0.0769 0.0288 25.6 0.06946 0.1157 0.0484 26.1 0.11624 0.1658 0.0769 26.6 0.18519 0.2268 0.1157 27.1 0.28201 0.2974 0.1658 27.6 0.41198 0.3752 0.2268 28.1 0.57942 0.4570 0.2974 28.6 0.78721 0.5393 0.3752 29.1 1.03647 0.6187 0.4570 29.6 1.32654 0.6921 0.5393 30.1 1.65513 0.7576 0.6187 30.6 2.01866 0.8138 0.6921 31.1 2.41271 0.8605 0.7576 31.6 2.83249 0.8980 0.8138

The difference between the up-gamma and down-gamma changes is depicted below:

Figure 1 Difference between up gamma and down gamma

We can see that the difference does not remain constant as asset price changes. Hedging exactly for gamma fails to account for the fact that changes in delta are not consistent across asset prices changes.

## Shadow gamma – calculation example

The change in delta is due to a change in the underlying asset price. A change in the asset price suggests a change in volatility (and possibly other price elements as well). However the gamma calculation assumes that volatility remains unchanged. Therefore there needs to be an adjustment in the gamma measure that considers that the underlying asset price volatility has changed. Taleb suggests the calculation of the shadow gamma measure which adjusts the basic gamma measure by calculating the changes in delta taking into account volatility changes as well as price changes.

The revised delta measure used in the calculation of the shadow gamma is the difference between the value of the option considering both a price change and revised volatility level, and the original value of the option at the original spot and volatility level. In our example, we have increase volatility by 5% so that it is now 33.85%. The results for the revised delta are given below for a sample set of asset prices:

 NVIDIA Stock Price Delta Delta – higher vol Delta Difference 15.6 (0.00000) (0.0000) 0.0000 16.1 (0.00000) (0.0000) 0.0000 16.6 (0.00000) (0.0000) 0.0000 17.1 (0.00000) 0.0000 0.0000 17.6 (0.00000) 0.0000 0.0000 18.1 – 0.0000 0.0000 18.6 0.00000 0.0000 0.0000 19.1 0.00000 0.0000 0.0000 19.6 0.00000 0.0000 0.0000 20.1 0.00000 0.0001 0.0000 20.6 0.00001 0.0001 0.0001 21.1 0.00003 0.0004 0.0004 21.6 0.00011 0.0009 0.0008 22.1 0.00033 0.0021 0.0018 22.6 0.00088 0.0045 0.0036 23.1 0.00218 0.0089 0.0067 23.6 0.00498 0.0168 0.0118 24.1 0.01058 0.0301 0.0195 24.6 0.02105 0.0513 0.0303 25.1 0.03935 0.0838 0.0444 25.6 0.06946 0.1312 0.0618 26.1 0.11624 0.1978 0.0816 26.6 0.18519 0.2877 0.1025 27.1 0.28201 0.4050 0.1230 27.6 0.41198 0.5531 0.1411 28.1 0.57942 0.7346 0.1552 28.6 0.78721 0.9512 0.1640 29.1 1.03647 1.2032 0.1667 29.6 1.32654 1.4901 0.1635 30.1 1.65513 1.8101 0.1549 30.6 2.01866 2.1608 0.1421 31.1 2.41271 2.5390 0.1263 31.6 2.83249 2.9416 0.1091 32.1 3.27323 3.3648 0.0916 32.6 3.73050 3.8054 0.0749

Figure 2 The modified delta approach

The shadow gamma is calculated as follows for a given stock price point, B, such that A<B<C where A is a stock price less than B and C is a stock price greater than B:

Shadow up –gamma (B) = (Delta (C, revised vol) – Delta (B, original vol))/(C-B), where underlying asset price is assumed to go up to C from B

Shadow down-gamma (B) = (Delta (B, original vol) – Delta (A, revised vol))/(B-A), where underlying asset price is assumed to go down to A from B

For example, the up-gamma calculated at asset price 25.1 for an increment of 1 unit in the asset price is equal to the delta using the revised volatility calculated at asset price 26.1 less the delta using the original volatility calculated at asset price 25.1, i.e. shadow up-gamma (25.1) = [delta_higher vol (26.1)-delta (25.1)]/(26.1-25.1) = 0.1978 – 0.03935 = 0.1585.

In like manner, the down-gamma calculated at asset price 25.1 for a decrement of 1 unit in the asset price is equal to the delta using the original volatility calculated at asset price 25.1 less the delta using the revised volatility calculated at asset price 24.1, i.e. down-gamma (25.1) = [delta (25.1)-delta_higher vol (24.1)]/(25.1-24.1) = 0.03935 –0.0301 = 0.0093.

The gammas and shadow gammas for a sample set of asset prices are given below:

 NVIDIA Stock Price Up-Gamma Down-Gamma Shadow up gamma Shadow down gamma 15.6 0.0000 0.0000 0.0000 (0.0000) 16.1 0.0000 0.0000 0.0000 (0.0000) 16.6 0.0000 0.0000 0.0000 (0.0000) 17.1 0.0000 0.0000 0.0000 (0.0000) 17.6 0.0000 0.0000 0.0000 (0.0000) 18.1 0.0000 0.0000 0.0000 (0.0000) 18.6 0.0000 0.0000 0.0000 (0.0000) 19.1 0.0000 0.0000 0.0001 (0.0000) 19.6 0.0000 0.0000 0.0001 (0.0000) 20.1 0.0000 0.0000 0.0004 (0.0000) 20.6 0.0001 0.0000 0.0009 (0.0000) 21.1 0.0003 0.0000 0.0021 (0.0000) 21.6 0.0008 0.0001 0.0043 (0.0000) 22.1 0.0018 0.0003 0.0086 (0.0001) 22.6 0.0041 0.0008 0.0159 (0.0000) 23.1 0.0084 0.0018 0.0279 0.0001 23.6 0.0161 0.0041 0.0463 0.0005 24.1 0.0288 0.0084 0.0732 0.0017 24.6 0.0484 0.0161 0.1102 0.0043 25.1 0.0769 0.0288 0.1585 0.0093 25.6 0.1157 0.0484 0.2183 0.0181 26.1 0.1658 0.0769 0.2888 0.0324 26.6 0.2268 0.1157 0.3679 0.0539 27.1 0.2974 0.1658 0.4526 0.0842 27.6 0.3752 0.2268 0.5392 0.1242 28.1 0.4570 0.2974 0.6238 0.1744 28.6 0.5393 0.3752 0.7029 0.2341 29.1 0.6187 0.4570 0.7736 0.3018 29.6 0.6921 0.5393 0.8342 0.3754 30.1 0.7576 0.6187 0.8839 0.4519 30.6 0.8138 0.6921 0.9229 0.5286 31.1 0.8605 0.7576 0.9521 0.6026 31.6 0.8980 0.8138 0.9729 0.6717

Figure 3 The Shadow Gamma plot

The up-gamma at a stock price of 28.6 forecasts a change in delta of 0.5393 if underlying price increases by 1. The shadow up-gamma however suggests that the forecast of the change in delta taking into account volatility is 0.9512. An exact gamma hedge that does not factor in volatility means that there could be a larger portion of the position that is un-hedged and exposed to risk than expected when volatility levels change.

References:

“Dynamic Hedging – Managing Vanilla and Exotic Options” – Nassim Taleb, John Wiley & Sons, Inc. 1996

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