1. 6.1 The Greek Letters Lecture 6

2. 6.2 Example A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S 0 = 49, X = 50, r = 5%,  = 20%, T = 20 weeks, The Black-Scholes value of the option is $240,000 How does the bank hedge its risk?

3. 6.3 Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk

4. 6.4 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

5. 6.5 Delta Delta (  ) is the rate of change of the option price with respect to the underlying Option price A B Slope =  Stock price

6. 6.6 Delta Hedging This involves maintaining a delta neutral portfolio The delta of a European call on a stock paying dividends at rate q is N (d 1 )e – qT –long in a call is delta-hedged by a short position in the stock The delta of a European put is e – qT [N (d 1 ) – 1] - long in a put is delta-hedged by a long position in the stock

7. 6.7 X 1

8. 6.8 Delta Hedging continued The hedge position must be frequently rebalanced Delta hedging a written option involves a “buy high, sell low” trading rule see delta_hedging.xls

9. 6.9 Delta Hedging continued delta_hedging is then equivalent to create a long position in the option synthetically this means that you can create a new way of managing money that replicates the non-linear pay-off of an option

10. 6.10 Theta Theta (  ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time (almost) always negative

11. 6.11 x theta for a european call

12. 6.12 Gamma Gamma (  ) is the rate of change of delta (  ) with respect to the price of the underlying asset See Figure for the variation of  with respect to the stock price for a call or put option

13. 6.13 x gamma for a european call/put

14. 6.14 Gamma Addresses Delta Hedging Errors Caused By Curvature S C Stock price S’S’ Call price C’ C’’

15. 6.15 Relationship Among Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q. From Black-Scholes equation:

16. 6.16 Interpretation of Gamma For a delta neutral portfolio,     t + ½  S 2  SS Negative Gamma viceversa  SS Positive Gamma long call+short stock long put + long stock

17. 6.17 Vega Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility See Figure for the variation of with respect to the stock price for a call or put option

18. 6.18 x vega for a european call/put

19. 6.19 Managing Delta, Gamma, & Vega  can be changed by taking a position in the underlying To adjust  & it is necessary to take a position in an option or other derivative

20. 6.20 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