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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:

symbolexchangedateperiodstrikeout-of-the-money %call/putIvdelta
NVDANASDAQ14-May-20143018.10C0.270.49204
NVDANASDAQ14-May-20143019.0055C0.27340.26021
NVDANASDAQ14-May-20143019.9110C0.29870.13058
NVDANASDAQ14-May-20143020.81515C0.3010.0516
NVDANASDAQ14-May-20143021.7220C0.3010.01689
NVDANASDAQ14-May-20143022.62525C0.3010.00472
NVDANASDAQ14-May-20143023.5330C0.3010.00114
NVDANASDAQ14-May-20143024.43535C0.3010.00024
NVDANASDAQ14-May-20143025.3440C0.3010.00005
NVDANASDAQ14-May-20143026.24545C0.3010.00001
NVDANASDAQ14-May-20143027.1550C0.3010
NVDANASDAQ14-May-20143028.05555C0.3010
NVDANASDAQ14-May-20143028.9660C0.28850

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 PriceDelta
15.6(0.00000)
16.1(0.00000)
16.6(0.00000)
17.1(0.00000)
17.6(0.00000)
18.1
18.60.00000
19.10.00000
19.60.00000
20.10.00000
20.60.00001
21.10.00003
21.60.00011
22.10.00033
22.60.00088
23.10.00218
23.60.00498
24.10.01058
24.60.02105
25.10.03935
25.60.06946
26.10.11624
26.60.18519
27.10.28201
27.60.41198
28.10.57942
28.60.78721
29.11.03647
29.61.32654
30.11.65513
30.62.01866
31.12.41271
31.62.83249
32.13.27323
32.63.73050

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 PriceDeltaUp-GammaDown-Gamma
15.6(0.00000)0.00000.0000
16.1(0.00000)0.00000.0000
16.6(0.00000)0.00000.0000
17.1(0.00000)0.00000.0000
17.6(0.00000)0.00000.0000
18.10.00000.0000
18.60.000000.00000.0000
19.10.000000.00000.0000
19.60.000000.00000.0000
20.10.000000.00000.0000
20.60.000010.00010.0000
21.10.000030.00030.0000
21.60.000110.00080.0001
22.10.000330.00180.0003
22.60.000880.00410.0008
23.10.002180.00840.0018
23.60.004980.01610.0041
24.10.010580.02880.0084
24.60.021050.04840.0161
25.10.039350.07690.0288
25.60.069460.11570.0484
26.10.116240.16580.0769
26.60.185190.22680.1157
27.10.282010.29740.1658
27.60.411980.37520.2268
28.10.579420.45700.2974
28.60.787210.53930.3752
29.11.036470.61870.4570
29.61.326540.69210.5393
30.11.655130.75760.6187
30.62.018660.81380.6921
31.12.412710.86050.7576
31.62.832490.89800.8138

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

Calculating Shadow Gamma - Taleb's approach for the second order option Greek.

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 PriceDeltaDelta – higher volDelta 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)0.00000.0000
17.6(0.00000)0.00000.0000
18.10.00000.0000
18.60.000000.00000.0000
19.10.000000.00000.0000
19.60.000000.00000.0000
20.10.000000.00010.0000
20.60.000010.00010.0001
21.10.000030.00040.0004
21.60.000110.00090.0008
22.10.000330.00210.0018
22.60.000880.00450.0036
23.10.002180.00890.0067
23.60.004980.01680.0118
24.10.010580.03010.0195
24.60.021050.05130.0303
25.10.039350.08380.0444
25.60.069460.13120.0618
26.10.116240.19780.0816
26.60.185190.28770.1025
27.10.282010.40500.1230
27.60.411980.55310.1411
28.10.579420.73460.1552
28.60.787210.95120.1640
29.11.036471.20320.1667
29.61.326541.49010.1635
30.11.655131.81010.1549
30.62.018662.16080.1421
31.12.412712.53900.1263
31.62.832492.94160.1091
32.13.273233.36480.0916
32.63.730503.80540.0749
Calculating Shadow Gamma - Taleb's approach for the second order option Greek.

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
15.60.00000.00000.0000(0.0000)
16.10.00000.00000.0000(0.0000)
16.60.00000.00000.0000(0.0000)
17.10.00000.00000.0000(0.0000)
17.60.00000.00000.0000(0.0000)
18.10.00000.00000.0000(0.0000)
18.60.00000.00000.0000(0.0000)
19.10.00000.00000.0001(0.0000)
19.60.00000.00000.0001(0.0000)
20.10.00000.00000.0004(0.0000)
20.60.00010.00000.0009(0.0000)
21.10.00030.00000.0021(0.0000)
21.60.00080.00010.0043(0.0000)
22.10.00180.00030.0086(0.0001)
22.60.00410.00080.0159(0.0000)
23.10.00840.00180.02790.0001
23.60.01610.00410.04630.0005
24.10.02880.00840.07320.0017
24.60.04840.01610.11020.0043
25.10.07690.02880.15850.0093
25.60.11570.04840.21830.0181
26.10.16580.07690.28880.0324
26.60.22680.11570.36790.0539
27.10.29740.16580.45260.0842
27.60.37520.22680.53920.1242
28.10.45700.29740.62380.1744
28.60.53930.37520.70290.2341
29.10.61870.45700.77360.3018
29.60.69210.53930.83420.3754
30.10.75760.61870.88390.4519
30.60.81380.69210.92290.5286
31.10.86050.75760.95210.6026
31.60.89800.81380.97290.6717
Calculating Shadow Gamma - Taleb's approach for the second order option Greek.

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