The Greeks

THE GREEKS: A MEASURE
OF RISK FOR OPTIONS

ALAN ANDERSON, Ph.D.
ECI RISK TRAINING
www.ecirisktraining.com

(c) ECI Risk Training 2009
1

THE GREEKS

The Greeks are risk measures that describe the
sensitivity of option prices to changes in:

  the underlying asset price
  the volatility of the underlying asset
  the risk-free rate of interest
  the time to maturity of the option
2

The Greeks are:

• delta
• gamma
• theta
• vega
• rho
3

DELTA

The delta of an option is the sensitivity of
the option’s price with respect to a change in
the price of the underlying asset

4

For a call option, delta is defined as:

∂C
ΔC =
∂S

5

This represents the change in C
with respect to a change in S

The delta of a call option can
assume a value between 0 and 1

6

A call’s delta equals the slope of its price curve:

7

8

Delta is close to zero when the call is deep out
of the money, rises to 0.5 when the call is at
the money, then moves close to one as the call
moves deep into the money

9

For a put option, delta is defined as:

∂P
ΔP =
∂S

10

The delta of a put option can assume a value
between -1 and 0.

A put’s delta equals the slope of its price curve;
the following diagram shows a European put:

11

12

Delta is close to -1 when the put is deep in the
money, moves to -0.5 when the put is at the
money, then moves close to zero as the put
moves deep out of the money

13

The price curve of an American put
is shown in the following diagram:

14

15

PORTFOLIO DELTA

Since delta is a linear measure, the delta of a
portfolio of assets is a weighted average of
the deltas of the assets in the portfolio

16

This is computed as follows:

n
Δ π = ∑ wi Δ i
i =1

17

where:

π = portfolio delta
wi = weight of asset i

i = delta of asset i

18

DELTA NEUTRAL

A portfolio with a delta of zero is perfectly
hedged; its value is unaffected by changes in
market prices

This portfolio is said to be delta neutral

19

GAMMA

The gamma of an option is the
sensitivity of the option’s price
with respect to a change in the
delta of the option

20

CALL GAMMA

For a call option, gamma is defined as:

∂C
ΓC =
∂ Δ( )
=
∂( )
∂S = ∂ C
2

∂S ∂S ∂S 2

21

PUT GAMMA

For a put option, gamma is defined as:

∂P
ΓP =
( )
∂ Δ
=
∂( )
∂S = ∂ P
2

∂S ∂S ∂S 2

22

NOTE

Gamma is identical for a call and a put option
with the same strike, maturity and underlying
asset.

Gamma’s value is a function of the moneyness
of the option:

23

Gamma reaches its maximum value when an
option is close to being at the money, and
declines as the option moves further into or
out of the money.

These features of gamma can be seen by
noting that gamma is the slope of the delta
function for both the call and the put option.

24

Since the delta of the call and put differ by a
constant, the slopes of their delta functions are
equal.

In both cases, the slope of the curve reaches
its maximum value near the strike price of the
option.

25

Since the call and put delta function have
positive slopes throughout; therefore, gamma
is always positive.

26

Gamma

0.03

0.025

0.02
Gamma

0.015

0.01

0.005

0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Stock Price ($)

27

NOTE

The gamma of the underlying asset is zero.
Since a forward contract is a linear instrument,
its delta is a constant; therefore, its gamma is
also zero.

28

THETA

The theta of an option is the sensitivity of the
option’s price with respect to a change in the time
to maturity.

Theta is also known as the option’s time decay.

29

NOTE

Theta is usually negative; it can be positive
for an in-the-money European put on a non-
dividend paying stock due to the possibility
that it is currently selling for less than its
intrinsic value.

30

Theta’s value declines continuously
with the option’s time to maturity.

31

Call Theta

0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

-0.5

-1

-1.5
Theta

-2

-2.5

-3

-3.5
Stock Price ($)

32

Put Theta

0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

-0.5

-1

-1.5
Theta

-2

-2.5

-3

-3.5
Stock Price ($)

33

VEGA

The vega (sometimes known as lambda or
kappa) of an option is the sensitivity of the
option’s price with respect to a change in the
volatility of the underlying asset.

34

NOTE

Vega is identical for a call and a put
option with the same strike, maturity and
underlying asset.

Vega is always positive and is a function
of the option’s moneyness.

35

Vega reaches its maximum value when an
option is close to being at the money, and
declines as the option moves further into or
out of the money

36

Vega

20

18

16

14

12
Vega

10

8

6

4

2

0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Stock Price ($)

37

RHO

The rho of an option is the sensitivity of
the option’s price with respect to a change
in the risk-free rate of interest.

For a call option, rho is positive; for a put
option, rho is negative.

38

Call Rho

50

45

40

35

30
Rho

25

20

15

10

5

0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Stock Price ($)

39

Put Rho

0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

-5

-10

-15

-20
Rho

-25

-30

-35

-40

-45

-50
Stock Price ($)

40

The Greeks

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