Chapter 18The Greek Letters,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,1,Example,A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stock S0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks, m = 13% The Black-Scholes-Merton value of the option is $240,000 How does the bank hedge its risk to lock in a $60,000 profit?,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,2,Naked & Covered Positions,Naked position Take no action Covered position Buy 100,000 shares today What are the risks associated with these strategies?,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,3,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,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,4,Stop-Loss Strategy continued,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,5,Ignoring discounting, the cost of writing and hedging the option appears to be max(S0K, 0). What are we overlooking?,Delta (See Figure 18.2, page 381),Delta (D) is the rate of change of the option price with respect to the underlying,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,6,Hedge,Trader would be hedged with the position: short 1000 options buy 600 shares Gain/loss on the option position is offset by loss/gain on stock position Delta changes as stock price changes and time passes Hedge position must therefore be rebalanced,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,7,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,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,8,The Costs in Delta Hedgingcontinued,Delta hedging a written option involves a “buy high, sell low” trading rule,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,9,First Scenario for the Example: Table 18.2 page 384,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,10,Second Scenario for the Example Table 18.3, page 385,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,11,Theta,Theta (Q) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of a long call or put option declines,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,12,Theta for Call Option: K=50, s = 25%, r = 5% T = 1,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,13,Gamma,Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset Gamma is greatest for options that are close to the money,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,14,Gamma for Call or Put Option: K=50, s = 25%, r = 5% T = 1,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,15,Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 18.7, page 389),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,16,S,C,Stock price,S,Call price,C,C,Interpretation of Gamma,For a delta neutral portfolio, DP Q Dt + GDS 2,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,17,Relationship Between Delta, Gamma, and Theta (page 393),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,18,For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q it follows from the Black-Scholes-Merton differential equation that,Vega,Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,19,Vega for Call or Put Option: K=50, s = 25%, r = 5% T = 1,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,20,Taylor Series Expansion (Appendix to Chapter 18),The value of a portfolio of derivatives dependent on an asset is a function of of the asset price S, its volatility s, and time t,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,21,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,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,22,Example,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C.