The Greek Letters Chapter 15 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Example A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks, m = 13% The Black-Scholes 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, 6th Edition, Copyright © John C. Hull 2005

Naked & Covered Positions Naked position Take no action (Good strategy if ST < 50) (Very bad if ST = 60, Loss of 700.000) Covered position Buy 100,000 shares today (Good strategy if ST > 50) (Very bad if ST 40, Loss of 600.000) Both strategies leave the bank exposed to significant risk Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Delta (See Figure 15.2, page 345) Delta (D) is the rate of change of the option price with respect to the underlying Option price A B Slope = D Stock price Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Delta Hedging Suppose the stock price is 100 and the option price is 10. Δ=0.6 An investor has sold 20 call option contracts (option to buy 2000 shares). The investor’s position could be hedged by buying 0.6*2000=1200 shares. The gain (loss) on the option position would tend to be offset by the loss (gain) on the stock position The delta of the options position is 0.6(-2000)=-1200 The delta of the stock is 1, so the delta of 1200 stocks is 1200. Delta of overall position is 0 (Delta neutral). Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

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 The delta of a European put is e– qT [N (d 1) – 1] Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Delta Hedging continued The hedge position must be frequently rebalanced Delta hedging a written option involves a “buy high, sell low” trading rule Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

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 the option declines Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Gamma Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset If gamma is small, delta changes slowly and adjustments to keep a portfolio delta neutral need only be made relatively infrequently. If gamma is large, delta is highly sensitive to the price of the underlying asset. It is then quite risky to leave a delta-neutral portfolio unchanged for any length of time. Gamma is greatest for options that are close to the money Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 15 Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 15.7, page 355) Call price C'' C' C Stock price S S' Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Interpretation of Gamma For a delta neutral portfolio, DP » Q Dt + ½GDS 2 DP DP DS DS Positive Gamma Negative Gamma Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Relationship Between Delta, Gamma, and Theta Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Relationship Between Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Vega Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility Vega tends to be greatest for options that are close to the money Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Managing Delta, Gamma, & Vega D can be changed by taking a position in the underlying To adjust G & n it is necessary to take a position in an option or other derivative Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Rho Rho is the rate of change of the value of a derivative with respect to the interest rate For currency options there are 2 rhos Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005