Options and Speculative Markets 2005-2006 Greeks Professor André Farber Solvay Business School Université Libre de Bruxelles

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 OMS 08 Greeks

Example OMS 08 Greeks

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 = + 1 Call : 0 < Delta < +1 Put : -1 < Delta < 0 OMS 08 Greeks

Calculation of delta OMS 08 Greeks

Variation of delta with the stock price for a call OMS 08 Greeks

Delta and maturity OMS 08 Greeks

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 08 Greeks

Effectiveness of Delta hedging This diagram illustrate the effectiveness of delta hedging. The initial stock price is S = 100 You are are short on 1 call option. The value of this option is 10.451 To hedge you position, you buy Delta = 0.637 shares. Suppose that the stock price suddenly drops to 95. The value of your short call option drops to 7.511. Since you are short, you gain (10.451 – 7.511) = 2.940 on your short call position. On the other hand, you are long on 0.637 shares. As the price change is S = -5, you loose (-5)(0.637) = -3.184 on you share. The net result in a change in the value of you portfolio: V = -3.184 + 2.940 = -0.244 OMS 08 Greeks

Gamma = ∂Delta / ∂S = ∂²f / ∂S² 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 08 Greeks

Variation of Gamma with the stock price OMS 08 Greeks

Gamma and maturity OMS 08 Greeks

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 08 Greeks

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 08 Greeks

Variation of Theta with the stock price OMS 08 Greeks

Relation between delta, gamma, theta Remember PDE: Theta Delta Gamma OMS 08 Greeks