1. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.1 The Greek Letters Chapter 15 Pages 328 - 343

2. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.2 Delta (See Figure 15.2, page 329) Delta (  ) is the rate of change of the option price with respect to the underlying Option price A B Slope =  Stock price

3. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.3 Delta Hedging This involves maintaining a delta neutral portfolio The delta of a European call on a non- dividend-paying stock is N (d 1 ) The delta of a European put on the stock is [N (d 1 ) – 1]

4. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.4 Delta Hedging continued The hedge position must be frequently rebalanced Delta hedging a written option involves a “buy high, sell low” trading rule See Tables 15.2 (page 332) and 15.3 (page 333) for examples of delta hedging

5. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.5 Theta Theta (  ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time See Figure 15.5 for the variation of  with respect to the stock price for a European call

6. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.6 Gamma Gamma (  ) is the rate of change of delta (  ) with respect to the price of the underlying asset See Figure 15.9 for the variation of  with respect to the stock price for a call or put option

7. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.7 Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 15.7, page 337) S C Stock price S′S′ Call price C′C′ C′′

8. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.8 Interpretation of Gamma For a delta neutral portfolio,     t + ½  S 2  SS Negative Gamma  SS Positive Gamma

9. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.9 Relationship Among Delta, Gamma, and Theta For a portfolio of derivatives on a non- dividend-paying stock paying

10. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.10 Vega Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility See Figure 15.11 for the variation of with respect to the stock price for a call or put option

11. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.11 Managing Delta, Gamma, & Vega Delta, , can be changed by taking a position in the underlying asset To adjust gamma,  and vega,  it is necessary to take a position in an option or other derivative

12. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.12 Rho Rho is the rate of change of the value of a derivative with respect to the interest rate

13. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.13 Hedging in Practice Traders usually ensure that their portfolios are delta-neutral at least once a day Whenever the opportunity arises, they improve gamma and vega As portfolio becomes larger hedging becomes less expensive

14. Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 15.14 Suggested Study Problems / Guide Quiz Questions 15.2,15.3,15.4,15.5,15.7 Greek Definitions on page 540 Ability to use tables on pages 544 & 545 Know: Delta / Theta / Gamma / Vega Delta – Bottom pg 330 = N (d1) Theta – Top pg 336 Gamma – Top pg 340 Vega – Bottom Page 342