the greeks

THE GREEKS: A MEASURE OF RISK FOR OPTIONS

ALAN ANDERSON, Ph.D. ECI RISK TRAINING

www.ecirisktraining.com

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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

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The Greeks are:

• delta

• gamma

• theta

• vega

• rho

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

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For a call option, delta is defined as:

ΔC = ∂C

∂S

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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

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

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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

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For a put option, delta is defined as:

ΔP = ∂P

∂S

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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:

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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

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

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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

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This is computed as follows:

Δπ = wii=1

n

∑ Δi

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where:

Δπ = portfolio delta

wi = weight of asset i

Δi = delta of asset i

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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

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

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CALL GAMMA

ΓC =

∂ Δ( )∂S

=∂(∂C

∂S)

∂S= ∂2C∂S 2

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For a call option, gamma is defined as:

PUT GAMMA

For a put option, gamma is defined as:

ΓP =

∂ Δ( )∂S

=∂(∂P∂S

)

∂S= ∂2P∂S 2

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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:

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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.

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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.

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

Gamma

0

0.005

0.01

0.015

0.02

0.025

0.03

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

Stock Price ($)

Gam

ma

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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.

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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.

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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.

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

Call Theta

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Stock Price ($)

Th

eta

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Put Theta

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

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

Stock Price ($)

Th

eta

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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.

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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.

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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

Vega

0

2

4

6

8

10

12

14

16

18

20

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

Stock Price ($)

Veg

a

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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.

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Call Rho

0

5

10

15

20

25

30

35

40

45

50

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

Stock Price ($)

Rh

o

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Put Rho

-50

-45

-40

-35

-30

-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 ($)

Rh

o

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