Options and Speculative Markets2004-2005Greeks

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

OMS 2004 Greeks |2August 23, 2004

Fundamental determinants of option value

Call value Put Value

Current asset price S

Delta

0 < Delta < 1

- 1 < Delta < 0

Striking price K

Interest rate r Rho

Dividend yield q

Time-to-maturity T Theta ?

Volatility Vega

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Example

BLACK-SCHOLES OPTION PRICING FORMULA A.Farber

Stock price 100 Call PutDividend yield 0.00% Decomposition of valueStriking price 100 Intrinsic val. 0.00 0.00Maturity (days) 365 Time value 4.88 -4.88Interest rate 5.00% Insurance 5.57 10.45Volatility 20.00%

BS partial differential equationTheta -6.41 -1.66

Call Put (r-q)SDelta 3.18 -1.82Price 10.451 5.574 0.5²S²Gamma 3.75 3.75Delta 0.637 -0.363 rf 0.52 0.28Gamma 0.019 0.019Theta (per day) -0.018 -0.005 Put-Call ParityElasticity 6.094 -6.516 Call Value 10.45Vega 0.375 0.375 + PV(Strike) 95.12 105.57Rho 0.532 -0.419 = S * exp(-qT) 100.00

+ Put 5.57 105.57

OMS 2004 Greeks |4August 23, 2004

Delta

• Sensitivity of derivative value to changes in price of underlying asset

Delta = ∂f / ∂S

• As a first approximation : f ~ Delta x S

• In example, for call option : f = 10.451 Delta = 0.637

• If S = +1: f = 0.637 → f ~ 11.088

• If S = 101: f = 11.097 error because of convexity

Binomial model: Delta = (fu – fd) / (uS – dS)

European options:Delta call = e-qT N(d1)Delta put = Delta call - 1

Forward : Delta = + 1Call : 0 < Delta < +1Put : -1 < Delta < 0

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Calculation of delta

0.00

10.00

20.00

30.00

40.00

50.00

60.00

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

Delta

OMS 2004 Greeks |6August 23, 2004

Variation of delta with the stock price for a call

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

OMS 2004 Greeks |7August 23, 2004

Delta and maturity

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

30 days 182 days 365 days

OMS 2004 Greeks |8August 23, 2004

Delta hedging

• Suppose that you have sold 1 call option (you are short 1 call)

• How many shares should you buy to hedge you position?

• The value of your portfolio is:

V = n S – C

• If the stock price changes, the value of your portfolio will also change.

V = n S - C

• You want to compensate any change in the value of the shorted option by a equal change in the value of your stocks.

• For “small” S : C = Delta S V = 0 ↔ n = Delta

OMS 2004 Greeks |9August 23, 2004

Effectiveness of Delta hedging

-1.200

-1.000

-0.800

-0.600

-0.400

-0.200

0.000

90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Stock price

OMS 2004 Greeks |10August 23, 2004

Gamma

• A measure of convexity

Gamma = ∂Delta / ∂S = ∂²f / ∂S²

• Taylor: df = f’S dS + ½ f”SS dS²

• Translated into derivative language: f = Delta S + ½ Gamma S²

• In example, for call : f = 10.451 Delta = 0.637 Gamma = 0.019

• If S = +1: f = 0.637 + ½ 0.019 → f ~ 11.097

• If S = 101: f = 11.097

OMS 2004 Greeks |11August 23, 2004

Variation of Gamma with the stock price

0.000

0.005

0.010

0.015

0.020

0.025

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

OMS 2004 Greeks |12August 23, 2004

Gamma and maturity

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

60 70 80 90 100 110 120 130 140 150

Stock price

90 days 182 days 365 days

OMS 2004 Greeks |13August 23, 2004

Gamma hedging

• Back to previous example.

• We have a delta neutral portfolio:

• Short 1 call option

• Long Delta = 0.637 shares

• The Gamma of this portfolio is equal to the gamma of the call option:

• V = n S – C →∂V²/∂S² = - Gammacall

• To make the position gamma neutral we have to include a traded option with a positive gamma. To keep delta neutrality we have to solve simultaneously 2 equations:

• Delta neutrality

• Gamma neutrality

OMS 2004 Greeks |14August 23, 2004

Theta

• Measure time evolution of asset

Theta = - ∂f / ∂T• (the minus sign means maturity decreases with the passage of time)

• In example, Theta of call option = - 6.41

• Expressed per day: Theta = - 6.41 / 365 = -0.018 (in example)

• Theta = -6.41 / 252 = - 0.025 (as in Hull)

OMS 2004 Greeks |15August 23, 2004

Variation of Theta with the stock price

-0.020

-0.018

-0.016

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

OMS 2004 Greeks |16August 23, 2004

Relation between delta, gamma, theta

• Remember PDE:

f

trS

f

S

f

SS rf

1

2

2

22 2

Theta Delta Gamma