1. Options and Speculative Markets 2004-2005 Greeks Professor André Farber Solvay Business School Université Libre de Bruxelles

2. August 23, 2004 OMS 2004 Greeks |2 Fundamental determinants of option value Call valuePut Value Current asset price S Delta  0 < Delta < 1  - 1 < Delta < 0 Striking price K  Interest rate rRho  Dividend yield q  Time-to-maturity TTheta  ? VolatilityVega 

3. August 23, 2004 OMS 2004 Greeks |3 Example

4. August 23, 2004 OMS 2004 Greeks |4 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 = (f u – f d ) / (uS – dS) European options: Delta call = e -qT N(d 1 ) Delta put = Delta call - 1 Forward : Delta = + 1 Call : 0 < Delta < +1 Put : -1 < Delta < 0

5. August 23, 2004 OMS 2004 Greeks |5 Calculation of delta

6. August 23, 2004 OMS 2004 Greeks |6 Variation of delta with the stock price for a call

7. August 23, 2004 OMS 2004 Greeks |7 Delta and maturity

8. August 23, 2004 OMS 2004 Greeks |8 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

9. August 23, 2004 OMS 2004 Greeks |9 Effectiveness of Delta hedging

10. August 23, 2004 OMS 2004 Greeks |10 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

11. August 23, 2004 OMS 2004 Greeks |11 Variation of Gamma with the stock price

12. August 23, 2004 OMS 2004 Greeks |12 Gamma and maturity

13. August 23, 2004 OMS 2004 Greeks |13 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 = - Gamma call 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

14. August 23, 2004 OMS 2004 Greeks |14 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)

15. August 23, 2004 OMS 2004 Greeks |15 Variation of Theta with the stock price

16. August 23, 2004 OMS 2004 Greeks |16 Relation between delta, gamma, theta Remember PDE: ThetaDelta Gamma