Greek The Letters
Chapter 15
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.1
Example
A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S = 49, K = 50, r = 5%, = 20%, 0 T = 20 weeks, = 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
15.2
Naked & Covered Positions
Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.3
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 This deceptively simple hedging strategy does not work well
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.4
Delta (See Figure 15.2, page 345)
Delta () is the rate of change of the option price with respect to the underlying
Option price Slope = A Stock price
15.5
B
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
15.6
Delta Hedging continued
The
hedge position must be frequently rebalanced Delta hedging a written option involves a "buy high, sell low" trading rule See Tables 15.2 (page 350) and 15.3 (page 351) for examples of delta hedging
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.7
Using Futures for Delta Hedging
The
delta of a futures contract 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
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.8
Theta
Theta () 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
15.9
Gamma
Gamma
() is the rate of change of delta () with respect to the price of the underlying asset Gamma is greatest for options that are close to the money (see Figure 15.9, page 358)
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.10
Gamma Addresses Delta Hedging Errors Caused By Curvature
(Figure 15.7, page 355)
Call price
C'' C' C
Stock price S S'
15.11
Options, and Futures, Other Derivatives, 6th Edition, Copyright John C. Hull 2005
Interpretation of Gamma
For a delta neutral portfolio, t + S 2
S S
Positive Gamma
Negative Gamma
15.12
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
1 2 2 + (r - q ) S + S = r 2
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.13
Vega
( ) 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 (See Figure 15.11, page 361)
Vega
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.14
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
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.15
Rho
Rho
is the rate of change of the value of a derivative with respect to the interest rate currency options there are 2 rhos
For
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
15.16
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
15.17
Scenario Analysis
A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities
Options, Futures, and Othe...