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