1. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 The Greek Letters Chapter 17 1

2. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Example (Page 359) A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend- paying stock S 0 = 49, K = 50, r = 5%,  = 20%, T = 20 weeks,  = 13% The Black-Scholes-Merton value of the option is $240,000 How does the bank hedge its risk? 2

3. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk 3

4. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Stop-Loss Strategy This involves: Buying 100,000 shares as soon as price reaches $50 Selling 100,000 shares as soon as price falls below $50 This deceptively simple hedging strategy does not work well 4

5. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Delta (See Figure 17.2, page 363) Delta (  ) is the rate of change of the option price with respect to the underlying Option price A B Slope =  Stock price 5

6. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 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] 6

7. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Delta Hedging continued The hedge position must be frequently rebalanced Delta hedging a written option involves a “buy high, sell low” trading rule 7

8. First Scenario for the Example: Table 17.2 page 366 Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 WeekStock price DeltaShares purchased Cost (‘$000) Cumulative Cost ($000) Interest 049.000.52252,2002,557.8 2.5 148.120.458(6,400)(308.0)2,252.32.2 247.370.400(5,800)(274.7)1,979.81.9....... 1955.871.0001,00055.95,258.25.1 2057.251.000005263.3 8

9. Second Scenario for the Example Table 17.3 page 367 Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 WeekStock price DeltaShares purchased Cost (‘$000) Cumulative Cost ($000) Interest 049.000.52252,2002,557.8 2.5 149.750.568 4,600228.92,789.22.7 252.000.70513,700712.43,504.33.4....... 1946.630.007(17,600)(820.7)290.00.3 2048.120.000(700)(33.7)256.6 9

10. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Theta Theta (  ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time 10

11. Theta for Call Option: S 0 =K=50,  = 25%, r = 5% T = 1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 11

12. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Gamma Gamma (  ) is the rate of change of delta (  ) with respect to the price of the underlying asset 12

13. Gamma for Call or Put Option: S 0 =K=50,  = 25%, r = 5% T = 1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 13

14. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 17.7, page 371) S C Stock price S′S′ Call price C′C′ C′′ 14

15. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Interpretation of Gamma For a delta neutral portfolio,     t + ½  S 2  SS Negative Gamma  SS Positive Gamma 15

16. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Relationship Among Delta, Gamma, and Theta For a portfolio of derivatives on a non- dividend-paying stock paying 16

17. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Vega Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility 17

18. Vega for Call or Put Option: S 0 =K=50,  = 25%, r = 5% T = 1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 18

19. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 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 19

20. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Rho Rho is the rate of change of the value of a derivative with respect to the interest rate 20

21. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 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 21

22. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Scenario Analysis A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities 22

23. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Using Futures for Delta Hedging The delta of a futures contract on an asset paying a yield at rate q is e (r-q)T times the delta of a spot contract The position required in futures for delta hedging is therefore e -(r-q)T times the position required in the corresponding spot contract 23

24. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Hedging vs Creation of an Option Synthetically When we are hedging we take positions that offset , ,, etc. When we create an option synthetically we take positions that match  &  24

25. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Portfolio Insurance In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically This involves initially selling enough of the portfolio (or of index futures) to match the  of the put option 25

26. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Portfolio Insurance continued As the value of the portfolio increases, the  of the put becomes less negative and some of the original portfolio is repurchased As the value of the portfolio decreases, the  of the put becomes more negative and more of the portfolio must be sold 26

27. Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010 Portfolio Insurance continued The strategy did not work well on October 19, 1987... 27