Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.1
The Greek Letters
Chapter 14
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.2
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,5th edition ? 2002 by John C,Hull
14.3Naked & 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,5th edition ? 2002 by John C,Hull
14.4
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,5th edition ? 2002 by John C,Hull
14.5Delta (See Figure 14.2,page 302)
? 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,5th edition ? 2002 by John C,Hull
14.6Delta 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,5th edition ? 2002 by John C,Hull
14.7
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 14.2 (page 307) and 14.3
(page 308) for examples of delta
hedging
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.8
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,5th edition ? 2002 by John C,Hull
14.9
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
? See Figure 14.5 for the variation of Q with
respect to the stock price for a European call
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.10
Gamma
? Gamma (G) is the rate of change of
delta (D) with respect to the price of the
underlying asset
? See Figure 14.9 for the variation of G
with respect to the stock price for a call
or put option
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.11Gamma Addresses Delta Hedging
Errors Caused By Curvature
(Figure 14.7,page 312)
S
C Stock price
S
’
Call
price
C’
C’’
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.12Interpretation of Gamma
? For a delta neutral portfolio,
dP? Q dt + ?GdS 2
dP
dS
Negative Gamma
dP
dS
Positive Gamma
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.13
Relationship Among Delta,
Gamma,and Theta
For a portfolio of derivatives on a stock
paying a continuous dividend yield at
rate q
Q D G P? ? ? ?( )r q S S r
1
2
2 2s
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.14
Vega
? Vega (n) is the rate of change of the
value of a derivatives portfolio with
respect to volatility
? See Figure 14.11 for the variation of n
with respect to the stock price for a call
or put option
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.15
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,5th edition ? 2002 by John C,Hull
14.16
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,5th edition ? 2002 by John C,Hull
14.17
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,5th edition ? 2002 by John C,Hull
14.18
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 Other Derivatives,5th edition ? 2002 by John C,Hull
14.19
Hedging vs Creation of an
Option Synthetically
? When we are hedging we take
positions that offset D,G,n,
etc.
? When we create an option
synthetically we take positions
that match D,G,& n
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.20Portfolio Insurance
? In October of 1987 many portfolio
managers attempted to create a put
option on a portfolio synthetically
? This involves initially selling enough of
the portfolio (or of index futures) to
match the D of the put option
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.21
Portfolio Insurance
continued
? As the value of the portfolio increases,
the D of the put becomes less negative
and some of the original portfolio is
repurchased
? As the value of the portfolio decreases,
the D of the put becomes more negative
and more of the portfolio must be sold
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
14.22
Portfolio Insurance
continued
The strategy did not work well on
October 19,1987...