Chapter 7
Option Greeks
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© 2004 South-Western Publishing
The Black-Scholes option pricing
model
C  SN ( d1 )  Ke  RT N ( d 2 )
where
2
 S  
ln 
R


2
K 
d1 
 T
and
d 2  d1  
2
T


T

Functions and their derivatives

Call price as a function of the following
variables:
–
–
–
–
–
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Stock price
Time
Volatility
Interest rate
Striking price
Stock price and Delta

Measure of Option Sensitivity to small
changes in the stock
For a call option:

For a put option:

4
C
c 
S
P
p 
S
Measure of Option Sensitivity

Delta indicates the number of shares of
stock required to mimic the returns of the
option
–
E.g., a call delta of 0.80 means it will act like 0.80
shares of stock

5
If the stock price rises by $1.00, the call option will
advance by about 80 cents
Call and Put deltas
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
For a European option, the absolute values
of the put and call deltas will sum to one

Delta is exactly equal to N(d1)
S, K and delta



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The delta of an at-the-money option
declines linearly over time and approaches
0.50 at expiration
The delta of an out-of-the-money option
approaches zero as time passes
The delta of an in-the-money option
approaches 1.0 as time passes
Hedge Ratio

Delta is the hedge ratio
–
8
Assume a short option position has a delta of
0.25. If someone owns 100 shares of the stock,
writing four calls results in a theoretically
perfect hedge
Likelihood of Becoming In-theMoney

Delta is a crude measure of the likelihood
that a particular option will be in the money
at option expiration
–
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E.g., a delta of 0.45 indicates approximately a
45 percent chance that the stock price will be
above the option striking price at expiration
Theta

Theta is a measure of the sensitivity of a
call option to the time remaining until
expiration:
C
c 
t
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P
p 
t
Theta (cont’d)
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
Theta is greater than zero because more
time until expiration means more option
value

Because time until expiration can only get
shorter, option traders usually think of
theta as a negative number
Theta (cont’d)
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
The passage of time hurts the option holder

The passage of time benefits the option
writer
Theta (cont’d)
Calculating Theta
For calls and puts, theta is:
Se
 rt
c  
 rKe N (d 2 )
2 2t
.5 ( d1 ) 2
Se
 rt
p 
 rKe N (d 2 )
2 2t
.5 ( d1 ) 2
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Theta (cont’d)
Calculating Theta (cont’d)
The equations determine theta per year. A theta
of –5.58, for example, means the option will lose
$5.58 in value over the course of a year ($0.02 per
day).
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Gamma



15
Gamma is the second derivative of the
option premium with respect to the stock
price
Gamma is the first derivative of delta with
respect to the stock price
Gamma is also called curvature
Gamma (cont’d)
 2C  c
c  2 
S
S
 2 P  p
p  2 
S
S
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Gamma (cont’d)



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As calls become further in-the-money, they
act increasingly like the stock itself
For out-of-the-money options, option prices
are much less sensitive to changes in the
underlying stock
An option’s delta changes as the stock
price changes
Gamma (cont’d)

Gamma is a measure of how often option
portfolios need to be adjusted as stock
prices change and time passes
–

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Options with gammas near zero have deltas
that are not particularly sensitive to changes
in the stock price
For a given striking price and expiration,
the call gamma equals the put gamma
Gamma (cont’d)
Calculating Gamma
For calls and puts, gamma is:
.5 ( d1 ) 2
e
c  p 
S 2t
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Sign Relationships
Delta
Theta
Gamma
Long call
+
-
+
Long put
-
-
+
Short call
-
+
-
Short put
+
+
-
The sign of gamma is always opposite to the sign of theta
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Other Derivatives





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Vega
Rho
The greeks of vega
Position derivatives
Caveats about position derivatives
Vega

Vega is the first partial derivative of the
OPM with respect to the volatility of the
underlying asset:
C
vega c 

P
vega c 

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Vega (cont’d)

All long options have positive vegas
–
–

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The higher the volatility, the higher the value of the option
E.g., an option with a vega of 0.30 will gain 0.30% in value
for each percentage point increase in the anticipated
volatility of the underlying asset
Vega is also called kappa, omega, tau, zeta, and
sigma prime
Vega (cont’d)
Calculating Vega
0.5 ( d12 )
S te
vega 
2
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Rho

Rho is the first partial derivative of the OPM
with respect to the riskfree interest rate:
 c  Kte N (d 2 )
 rt
 p   Kte N (d 2 )
 rt
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Rho (cont’d)

Rho is the least important of the derivatives
–
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Unless an option has an exceptionally long life,
changes in interest rates affect the premium
only modestly
The Greeks of Vega

Two derivatives measure how vega
changes:
–
–
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Vomma measures how sensitive vega is to
changes in implied volatility
Vanna measures how sensitive vega is to
changes in the price of the underlying asset
Position Derivatives

The position delta is the sum of the deltas
for a particular security
–
–
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Position gamma
Position theta
Caveats About Position
Derivatives

Position derivatives change continuously
–
–
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E.g., a bullish portfolio can suddenly become
bearish if stock prices change sufficiently
The need to monitor position derivatives is
especially important when many different option
positions are in the same portfolio
Delta Neutrality



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Introduction
Calculating delta hedge ratios
Why delta neutrality matters
Introduction

Delta neutrality means the combined deltas
of the options involved in a strategy net out
to zero
–
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Important to institutional traders who establish
large positions using straddles, strangles, and
ratio spreads
Calculating Delta Hedge Ratios
(cont’d)
A Strangle Example
A stock currently trades at $44. The annual volatility of the
stock is estimated to be 15%. T-bills yield 6%.
An options trader decides to write six-month strangles using
$40 puts and $50 calls. The two options will have different
deltas, so the trader will not write an equal number of puts
and calls.
How many puts and calls should the trader use?
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Calculating Delta Hedge Ratios
(cont’d)
A Strangle Example (cont’d)
Delta for a call is N(d1):
.152 
 44  
.5
ln     .06 
2 
 50  
d1 
 .87
.15 .5
N (.87)  .19
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Calculating Delta Hedge Ratios
(cont’d)
A Strangle Example (cont’d)
For a put, delta is N(d1) – 1.
.152 
 44  
.5
ln     .06 
2 
 40  
d1 
 1.23
.15 .5
N (1.23)  1  .11
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Calculating Delta Hedge Ratios
(cont’d)
A Strangle Example (cont’d)
The ratio of the two deltas is -.11/.19 = -.58.
This means that delta neutrality is achieved
by writing .58 calls for each put.
One approximate delta neutral combination
is to write 26 puts and 15 calls.
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Why Delta Neutrality Matters

Strategies calling for delta neutrality are
strategies in which you are neutral about
the future prospects for the market
–
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You do not want to have either a bullish or
a bearish position
Why Delta Neutrality Matters
(cont’d)

The sophisticated option trader will revise
option positions continually if it is
necessary to maintain a delta neutral
position
–
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A gamma near zero means that the option
position is robust to changes in market factors
Two Markets: Directional and
Speed



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Directional market
Speed market
Combining directional and speed markets
Directional Market

Whether we are bullish or bearish indicates
a directional market

Delta measures exposure in a directional
market
–
–
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Bullish investors want a positive position delta
Bearish speculators want a negative position
delta
Speed Market

The speed market refers to how quickly we
expect the anticipated market move to
occur
–
40
Not a concern to the stock investor but to the
option speculator
Speed Market (cont’d)
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
In fast markets you want positive gammas

In slow markets you want negative gammas
Combining Directional and
Speed Markets
Directional Market
Down
Speed
Market
Up
Slow
Write calls
Write
straddles
Write puts
Neutral
Write calls;
buy puts
Spreads
Buy calls;
write puts
Buy puts
Buy
straddles
Buy calls
Fast
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Neutral
Dynamic Hedging



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Introduction
Minimizing the cost of data adjustments
Position risk
Introduction

A position delta will change as
–
–
–
–

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Interest rates change
Stock prices change
Volatility expectations change
Portfolio components change
Portfolios need periodic tune-ups
Minimizing the Cost of Data
Adjustments

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It is common practice to adjust a portfolio’s
delta by using both puts and calls to
minimize the cash requirements associated
with the adjustment
Position Risk
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
Position risk is an important, but often
overlooked, aspect of the riskiness of
portfolio management with options

Option derivatives are not particularly
useful for major movements in the price
of the underlying asset
Position Risk (cont’d)
Position Risk Example
Assume an options speculator holds an aggregate portfolio
with a position delta of –155. The portfolio is slightly bearish.
Depending on the exact portfolio composition, position risk
in this case means that the speculator does not want the
market to move drastically in either direction, since delta is
only a first derivative.
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Position Risk (cont’d)
Position Risk Example (cont’d)
Profit
Stock Price
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Position Risk (cont’d)
Position Risk Example (cont’d)
Because of the negative position delta, the curve moves into
profitable territory if the stock price declines. If the stock
price declines too far, however, the curve will turn down,
indicating that large losses are possible.
On the upside, losses occur if the stock price advances a
modest amount, but if it really turns up then the position delta
turns positive and profits accrue to the position.
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