The Greek Letters
Chapter 15
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
15.1
Example
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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
15.2
Naked & Covered Positions
Naked position
Take no action
(Good strategy if ST < 50)
(Very bad if ST = 60, Loss of 700.000)
Covered position
Buy 100,000 shares today
(Good strategy if ST > 50)
(Very bad if ST 40, Loss of 600.000)
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 (D) is the rate of change of the
option price with respect to the underlying
Option
price
Slope = D
B
A
Stock price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
15.5
Delta Hedging
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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
15.6
Delta Hedging
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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
15.7
Delta Hedging
continued
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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
15.8
Theta
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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
15.9
Gamma
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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
15.10
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
15.11
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
15.12
Relationship Between Delta,
Gamma, and Theta
2P

P rS P  1s 2S 2
rP
2
t
S 2
S
2P

Q P ,..D P ,..G
t
S
S 2
QrSD 1s 2S 2GrP
2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
15.13
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
Q  (r  q ) SD  s S G  rP
2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
15.14
Vega
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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
15.15
Managing Delta, Gamma, & Vega

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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
15.16
Rho
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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
15.17
Hedging in Practice
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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
15.18