Options and Speculative Markets 2005 2006 Greeks Professor

Options and Speculative Markets 2005 -2006 Greeks Professor André Farber Solvay Business School Université Libre de Bruxelles OMS 08 Greeks

Fundamental determinants of option value Current asset price S Delta Striking price K Interest rate r Rho Dividend yield q Call value Put Value 0 < Delta < 1 - 1 < Delta < 0 Time-to-maturity T Theta ? Volatility Vega 16 February 2022 OMS 08 Greeks 2

Example 16 February 2022 OMS 08 Greeks 3

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 Forward : Delta = + 1 Call : 0 < Delta < +1 Put : -1 < Delta < 0 16 February 2022 Binomial model: Delta = (fu – fd) / (u. S – d. S) European options: Delta call = e-q. T N(d 1) Delta put = Delta call - 1 OMS 08 Greeks 4

Calculation of delta 16 February 2022 OMS 08 Greeks 5

Variation of delta with the stock price for a call 16 February 2022 OMS 08 Greeks 6

Delta and maturity 16 February 2022 OMS 08 Greeks 7

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 16 February 2022 OMS 08 Greeks 8

Effectiveness of Delta hedging 16 February 2022 OMS 08 Greeks 9

Gamma • A measure of convexity Gamma = ∂Delta / ∂S = ∂²f / ∂S² • Taylor: df = f’S d. S + ½ f”SS d. S² • 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 16 February 2022 OMS 08 Greeks 10

Variation of Gamma with the stock price 16 February 2022 OMS 08 Greeks 11

Gamma and maturity 16 February 2022 OMS 08 Greeks 12

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 16 February 2022 OMS 08 Greeks 13

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) 16 February 2022 OMS 08 Greeks 14

Variation of Theta with the stock price 16 February 2022 OMS 08 Greeks 15

Relation between delta, gamma, theta • Remember PDE: Theta 16 February 2022 Delta Gamma OMS 08 Greeks 16