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

2. 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

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

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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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

16. 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

17. 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

18. 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