Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.1
The Greek Letters
Chapter 15
Pages 328 - 343
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.2
Delta (See Figure 15.2, page 329)
Delta () is the rate of change of the option price with respect to the underlying
Option
price
A
BSlope =
Stock price
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.3
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]
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.4
Delta Hedgingcontinued
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 332) and 15.3 (page 333) for examples of delta hedging
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.5
Theta
Theta () of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time
See Figure 15.5 for the variation of with respect to the stock price for a European call
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.6
Gamma
Gamma () is the rate of change of delta () with respect to the price of the underlying asset
See Figure 15.9 for the variation of with respect to the stock price for a call or put option
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.7
Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 15.7, page 337)
S
CStock price
S′
Callprice
C′C′′
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.8
Interpretation of Gamma For a delta neutral portfolio,
t + ½S 2
S
Negative Gamma
S
Positive Gamma
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.9
Relationship Among Delta, Gamma, and Theta
For a portfolio of derivatives on a non-dividend-paying stock paying
rSrS 20
20 2
1
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.10
Vega
Vega () is the rate of change of the value of a derivatives portfolio with respect to volatility
See Figure 15.11 for the variation of with respect to the stock price for a call or put option
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.11
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
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.12
Rho
Rho is the rate of change of the value of a derivative with respect to the interest rate
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.13
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
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 15.14
Suggested Study Problems / Guide
Quiz Questions 15.2,15.3,15.4,15.5,15.7 Greek Definitions on page 540 Ability to use tables on pages 544 & 545 Know: Delta / Theta / Gamma / Vega
Delta – Bottom pg 330 = N (d1)Theta – Top pg 336Gamma – Top pg 340Vega – Bottom Page 342