© 2002 South-Western Publishing 1

Chapter 7

Option Greeks

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Outline

Introduction The principal option pricing derivatives Other derivatives Delta neutrality Two markets: directional and speed Dynamic hedging

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Introduction

There are several partial derivatives of the BSOPM, each with respect to a different variable:

– Delta– Gamma– Theta– Etc.

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The Principal Option Pricing Derivatives

Delta Measure of option sensitivity Hedge ratio Likelihood of becoming in-the-money Theta Gamma Sign relationships

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Delta

Delta is an important by-product of the Black-Scholes model

There are three common uses of delta Delta is the change in option premium

expected from a small change in the stock price

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Measure of Option Sensitivity

For a call option:

For a put option:S

Cc

S

Pp

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Measure of Option Sensitivity (cont’d)

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

If the stock price rises by $1.00, the call option will advance by about 80 cents

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Measure of Option Sensitivity (cont’d)

For a European option, the absolute values of the put and call deltas will sum to one

In the BSOPM, the call delta is exactly equal to N(d1)

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Measure of Option Sensitivity (cont’d)

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

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Hedge Ratio

Delta is the hedge ratio

– 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

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Likelihood of Becoming In-the-Money

Delta is a crude measure of the likelihood that a particular option will be in the money at option expiration

– 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

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Theta

Theta is a measure of the sensitivity of a call option to the time remaining until expiration:

t

Pt

C

p

c

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Theta (cont’d)

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

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Theta (cont’d)

The passage of time hurts the option holder

The passage of time benefits the option writer

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Theta (cont’d)

Calculating Theta

For calls and puts, theta is:

)(22

)(22

2

)(5.

2

)(5.

21

21

dNrKet

eS

dNrKet

eS

rtd

p

rtd

c

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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

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

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Gamma (cont’d)

SS

P

SS

C

pp

cc

2

2

2

2

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Gamma (cont’d)

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

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Gamma (cont’d)

Gamma is a measure of how often option portfolios need to be adjusted as stock prices change and time passes– 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

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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

Vega Rho The greeks of vega Position derivatives Caveats about position derivatives

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Vega

Vega is the first partial derivative of the OPM with respect to the volatility of the underlying asset:

P

C

c

c

vega

vega

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Vega (cont’d)

All long options have positive vegas– 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 or lambda

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Vega (cont’d)

Calculating Vega

2vega

)(5.0 21detS

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Rho

Rho is the first partial derivative of the OPM with respect to the riskfree interest rate:

)(

)(

2p

2c

dNKte

dNKtert

rt

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Rho (cont’d)

Rho is the least important of the derivatives

– Unless an option has an exceptionally long life, changes in interest rates affect the premium only modestly

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The Greeks of Vega

Two derivatives measure how vega changes:

– 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

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Position Derivatives

The position delta is the sum of the deltas for a particular security

– Position gamma– Position theta

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Caveats About Position Derivatives

Position derivatives change continuously

– 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

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Delta Neutrality

Introduction Calculating delta hedge ratios Why delta neutrality matters

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Introduction

Delta neutrality means the combined deltas of the options involved in a strategy net out to zero

– Important to institutional traders who establish large positions using straddles, strangles, and ratio spreads

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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):

19.)87.(

87.5.15.

5.215.

06.5044

ln

1

N

d

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Calculating Delta Hedge Ratios (cont’d)

A Strangle Example (cont’d)

For a put, delta is N(d1) – 1.

11.1)23.1(

23.15.15.

5.215.

06.4044

ln

1

N

d

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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

– You do not want to have either a bullish or a bearish position

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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

– A gamma near zero means that the option position is robust to changes in market factors

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Two Markets: Directional and Speed

Directional market Speed market Combining directional and speed markets

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Directional Market

Whether we are bullish or bearish indicates a directional market

Delta measures exposure in a directional market– Bullish investors want a positive position delta– Bearish speculators want a negative position

delta

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Speed Market

The speed market refers to how quickly we expect the anticipated market move to occur

– Not a concern to the stock investor but to the option speculator

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Speed Market (cont’d)

In fast markets you want positive gammas

In slow markets you want negative gammas

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Combining Directional and Speed Markets

Directional Market

Down Neutral Up

Speed Market

Slow Write calls Write straddles

Write puts

Neutral Write calls; buy puts

Spreads Buy calls; write puts

Fast Buy puts Buy straddles

Buy calls

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Dynamic Hedging

Introduction Minimizing the cost of data adjustments Position risk

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Introduction

A position delta will change as– Interest rates change– Stock prices change– Volatility expectations change– Portfolio components change

Portfolios need periodic tune-ups

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Minimizing the Cost of Data Adjustments

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

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Position Risk

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

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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.