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Calculating Shadow Gamma – Taleb’s approach for the second order option Greek.

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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