1The Option "Greeks"

2

Outline

Introduction

The Option Delta

The Option Gamma

The Option Theta

The Option Vega

The Option Rho

Position Greeks

Summary

3Introduction

4

Objectives

This chapter introduces the sensitivity measures collectively called the option

greeks.

The objective is to understand:

The meaning of each greek (what is it measuring)?

The properties of the greek. For example:

When is it 'large" in value? What is the intuition for this behavior?.

The limitations of the greek.

A final section discusses aggregating the risks at the portfolio level.

5Introduction

Options are instruments whose values are affected by many factors. Option

pricing models value options taking as given information about these factors

at a point in time.

As time passes, changes in the values of these factors (price of the

underlying, time-to-maturity, volatility, . . . ) will cause changes in option

values.

Sensitivity Analysis aims to quantify the impact of a change in each

factor on the option price.

Corresponding to each factor is a sensitivity measure (called the option

greek) that gives the quantitative (i.e., dollar) impact of a change in that

factor.

6

The Factors

Sensitivity analysis focusses on the impact of four factors:

The price of the underlying asset S.

Time left to maturity T t.

Volatility of the underlying .

The rate of interest r.

7The Greeks

Corresponding to these factors are five sensitivity measures or greeks:

1. Delta, denoted :

Measures impact of a "small" change in asset price.

2. Gamma, denoted :

Measures option curvature, and can be used to estimate impact of a "large"

change in asset price.

3. Theta, denoted : Measures impact of the passage of time.

4. Vega, denoted : Measures impact of a change in volatility.

5. Rho, denoted : Measures impact of change in the interest rate.

For simplicity, we refer to the underlying asset as a "stock," though it could

be an exchange rate or index level or other variable as well.

8

The role of the option will be described shortly.

A Broad Overview

9Simple Examples

10

Comments

Two important caveats:

The impact measured by the greeks is only approximate. The greeks

are meant to give a snapshot view of sensitivities to different factors.

Even the approximations are good only for "small" changes in the price

of the concerned factor. For large changes, the approximations do not

always do well.

We elaborate on both these points in the sequel when we study the greeks

in more detail.

Finally, note that all the greeks described above measure the dollar impact

of a change in the given factor, not a proportional change.

11

An Overview of the Gamma

We need a sensitivity measure to measure the option reaction to large "jump"

changes in the underlying price.

The option gamma provides this for us. If the price changes by a "large"

amount dS, then the option value changes by approximately

The term is a "curvature correction" which always improves the

estimated impact of the change dS, but its effect is particularly important for

large dS.

The gamma also has other uses discussed below.

12

The Material to Follow

We examine each greek in turn, proceeding in a series of steps.

Mathematical definition of the sensitivity measure.

Description of its properties, focussing especially on whether it is +ve or

ve.

That is, does an increase in the factor increase or decrease the

option value?

For example: Does more time-to-maturity increase call values? What

about an increase in interest rates?

Plot of the sensitivity measure to illustrate its behavior.

Examples illustrating the use/interpretation of the sensitivity measure.

13

The Black-Scholes Framework: Notation

t : current time

S : current price of underlying

: volatility of underlying

T : maturity date of option (so time-left-to-maturity = T t ).

K : strike price of option

r : riskless interest rates (continuously compounded)

C, P : prices of call and put respectively.

14

Black-Scholes Call and Put Prices

C = S N (d1) PV (K ) N (d2)

P = PV (K ) N (d2) S N (d1)

where N () is the cumulative standard normal distribution, and

15

Black-Scholes Example: Parameter Values

Throughout we use a Black-Scholes example to illustrate computing and

working with the greeks.

Contract parameters:

Strike price K = 100.

Time-to-maturity T t = 0.50 years.

Market parameters:

Current price of underlying S.

Volatility of underlying: = 20% = 0.20

Riskless interest rates r = 5% = 0.05.

Figure on next page plots C and P for values of S ranging from 70 to 130.

Note the curvature in option values as S increases.

16

Black-Scholes Option Prices

17

The Option Delta

18

The Option Delta

Delta: single most important sensitivity measure for an option.

Measures sensitivity of option values to changes in the underlying price.

As a sensitivity measure

Delta is typically very accurate for "small" changes in S , but

Needs to be supplemented with the gamma for gauging the impact of

"large" changes in S.

19

Definition

Intuitively, the delta may be thought of as a ratio:

Delta = Change in option value per $1 change in S.

More formally, the delta is defined as the slope of the option pricing function:

20

The Sign of the Delta

The delta of a call is always positive: C 0.

That is, call values increase when the price of the underlying

increases.

The delta of a put is always negative: P 0.

That is, put values decrease when the price of the underlying

increases.

Moreover, the delta of an option is always less than one in absolute

value: C 1 and P 1.

Summing up:

0 C 1 1 P 0.

21

The Delta and "Moneyness"

When is the delta of an option "small" (close to zero)? When is it "large"

(close to one in absolute value)?

If an option is deep out-of-the-money, its value is close to zero and not very

responsive to changes in the price of the underlying.

Deep OTM options have deltas close to zero.

If an option is deep in-the-money, its value responds almost one-for-one to

changes in the price of the underlying.

Deep ITM options have deltas close to one (in absolute value).

In general, as an option moves from deep OTM to deep ITM, its delta (in

absolute value) moves from zero towards one.

22

The Delta and Put-Call Parity

Put-call parity:

C P = S PV (K ).

Differentiating both sides with respect to S :

(call) (put) = 1,

or, equivalently,

(call) +|(put)| = 1.

Intuitively, when S changes by a dollar, the left-hand side of put-call parity too must change by a dollar.

23

Delta in the Black-Scholes Framework

The delta of a call in the Black-Scholes framework is given by the quantity

N (d1), and the delta of a put by the term N (d1):

C = N (d1) P = N (d1)

The figure on the next page plots call and put deltas in the Black-Scholes

setting.

Parameter choices same as those used earlier in plotting call and put

prices.

K = 100.

T t = 0.50.

r = 0.05.

= 0.20.

S varies from 72 to 128.

24

Plot of Call and Put Deltas

25

Using the Option Delta

If the stock price changes by a small amount dS , the estimated change in an

option price is the option delta dS :

dC = C dS dP = P dS

For example, suppose a put is trading at $11.45 and has a delta of 0.70.

Suppose the price of the underlying increases by $0.50.

Then: dS = +0.50 and P = 0.70, so the estimated change in the put

value:

(0.70) (+0.50) = 0.35.

Estimated new put value: 11.45 0.35 = 11.10.

26

How Good is the Estimate?

In general, these estimates do very well for "small" dS.

However, they become progressively less accurate as dS increases.

To illustrate these points, we turn to numerical examples.

For this purpose, we use a Black-Scholes model with S = K = 100, r = 0.05,

T = 0.50 and = 0.20.

At these numbers, we have from the Black-Scholes formula:

C = 6.889 C = +0.598

27

Very Good for "Small" Changes dS ...

For a change of dS = +1, the delta estimates that the call price should

change by

dC = C x dS = +0.598.

That is, the new call price at S = 101 should be

6.889 + 0.598 = 7.487.

In fact, if we use the Black-Scholes formula to calculate the new price at

S = 101 with the other parameters unchanged, we get C = 7.500.

Thus, the estimated and actual values differ only by just over a penny.

28

... But Not For Large Changes

However, the delta becomes progressively less accurate for "large" dS.

Consider the same value of S = 100, but a larger change of dS = +5.

Using the delta, the estimated change in the call value is

(0.598)(+5) = +2.990.

Thus, the estimated new price at S = 105 is 6.889 + 2.990 = 9.879.

Actual new price at S = 105 according to the Black-Scholes formula:

C = 10.201.

The delta underestimates the change by 0.32 or over 10%.

29

The Delta Approximation Error

30

Source of the Error

What causes this error?

Curvature in the option price: the option price is not linear in S (see the next

page).

When we use dC = C dS, an implicit assumption of linearity is made.

The smaller the curvature, the more accurate is delta as a measure of option

risk (for example, deep OTM and ITM options).

Natural question: how do we quantify and account for this curvature in

measuring option sensitivity?

The option gamma.

31

The Effect of Curvature

32

The Option Gamma

33

The Option Gamma

Gamma measures the curvature of the option pricing function.

Curvature is the change in the slope of the function: a function whose

slope never changes has no curvature (it is a straight line).

Since the slope is measured by the delta, gamma is the rate of change of

the delta.

Intuitively,

= Change in Option Delta per $1 change in S

More formally,

34

Sign of the Gamma

First, consider the sign of the gamma.

Call and put deltas both increase as S increases.

This means the gammas of both puts and calls are positive.

This just says that option prices have positive curvature, or, in mathematical

terms, option prices are convex in S.

35

Gamma and Moneyness

How does depth-in-the-money of the option matter?

For both calls and puts, the option delta is:

0 and insensitive to changes in S if option is deep OTM.

1 in absolute value and insensitive to changes in S if option is deep

ITM.

Responds rapidly to changes in S if option is at- or near-the-money.

Thus, the gamma is:

least when the option is deep OTM or deep ITM.

is highest when the option is at- or near-the-money.

36

Gamma and Put-Call Parity

From put-call parity

C P = S PV (K ).

we have already seen that

(call (put) = 1.

It follows that

(call) (put) = 0,

or

(call) = (put).

37

Gamma and Option Prices

38

Gamma and Changes in Delta

39

The Option Gamma

40

Using the Option Gamma

The gamma has many uses:

In estimating changes in option value resulting from large changes in S.

As reflecting a view on volatility.

In estimating changes in delta that result from changes in S.

As an indicator of the frequency with which a delta hedge must be

rebalanced.

We shall examine each of these uses in turn.

41

Gamma as a Correction Factor

Consider a change a in the stock price.

To estimate changes in option value using the delta alone, we use

dC = a

A more accurate estimation of the change is obtained by

Improvement occurs for all a but is especially significant for "large" a.

42

Gamma as a Correction Factor: Example

To illustrate the importance of the correction, consider a Black-Scholes

example.

Let the parameters be as earlier: S = K = 100, r = 0.05, T = 0.5, and

= 0.20.

Applying the formulae, we obtain:

C = 6.889 = 0.598 = 0.0274.

Consider a change of a = +5 in the stock price S.

43

The Curvature Correction Improves Matters

As we saw above, using the delta alone leads to a large error in estimating

the price change:

The estimated new price is 9.877.

The actual new price from the Black-Scholes formula is 10.201.

Using the curvature correction formula, the new price estimation is:

The error is now substantially smaller.

44

Delta-Hedged Portfolios & Price Changes

As a further illustration, consider the performance of a delta-hedged portfolio.

Consider a call with current price C whose delta and gamma are and .

Suppose you are short the call and have delta hedged yourself by holding

units of the stock.

Suppose the stock price registers an unanticipated move of a.

What is the impact on your portfolio of this change in S ?

45

Curvature and Delta-Hedged Portfolios

Change in value of stock held: a.

Change in option value: approximately

So change in portfolio value is approximately

This is negative regardless of a.

Thus, a delta-hedged position in which you are short the option will lose money from an unanticipated change in prices regardless of the direction in which the price moves.

46

Curvature & Delta-Hedged Portfolios: Example

A numerical example will illustrate this point.

Consider the same parameters as in the example above: S = K = 100,

= 0.20, T = 0.5, and r = 0.05.

Then, we have C = 6.889 and = 0.598.

Suppose you hold a portfolio that is short one call and long 0.598 units

of the stock.

What is the impact on the portfolio of a $2 change in the price?

47

The Portfolio Loses a2

Case 1: Price jumps up to S = 102.

Change in option value: +1.250

Change in value of 0.598 shares: +1.195

Change in portfolio value: 1.195 1.250 = 0.055.

Case 2: Price jumps down to S = 98.

Change in option value: 1.140

Change in value of 0.598 shares: 1.195

Change in portfolio value: 1.195 + 1.140 = 0.055.

Loss in either case: approximately a2 where a is the change in S.

48

Delta-Hedged Portfolio Behavior: Short Option

49

Gamma as a View on Volatility

As a measure of curvature, gamma reflects a view on volatility.

As motivation for this point, consider the holder of a call option.

The curvature in the call implies that the holder of a call benefits more

from a price increase than he loses from a corresponding price decrease.

For example, in a Black-Scholes model with K = 100, = 0.2, r = 0.05, and

T t = 0.5 years, we can check the following:

At S = 100, we have C = 6.889.

If S increases to 104, the option price increases by 2.600.

However, if S falls to 96, the option price decreases by only 2.166.

50

Curvature and Asymmetric Responses

51

Gamma and the Asymmetry

Thus, curvature creates asymmetric exposure to price changes.

This is also true for puts: put holders benefit more from price decreases than

they lose from price increases.

The extent of the asymmetry depends on the gamma:

Large considerable curvature substantial asymmetry.

Small option price is nearly linear little asymmetry.

52

Long Gamma Long Volatility

Asymmetric exposure is desirable if you expect an increase in volatility: it

will enable you to benefit more on the upside than you lose on the

downside.

Thus, a positive gamma position can be regarded as a bullish view on

volatility.

Analogously, a negative gamma positionwhich is the gamma of the

short position in the optioncan be regarded as a bearish view on

volatility.

53

Gamma and Changes in Delta

To estimate changes in delta caused by changes in S, we can use

dC = C dS.

For example, at S = 100, we have C = 0.598 and C = 0.0274.

Taking dS = 4, the estimated values of delta at S = 96 and S = 104 are

(96) = 0.489 (104) = 0.707.

These are quite close to the actual values of (96) = 0.484 and

(104) = 0.700.

54

Gamma and Hedge Rebalancing Frequency

Final use of option gamma: indicator of the frequency with which a delta

hedge needs to be rebalanced.

"Small" delta does not change much for changes in S.

Thus, a delta hedged position will remain approximately delta hedged even as

S changes.

Large even small change in S can create a substantial change in the

delta.

Thus, a delta-hedged position may become risky following changes in S and

the hedge will have to be rebalanced more frequently.

55

The Option Theta

56

The Option Theta

Options are finitely-lived instruments.

Thus, the time-left-to-maturity plays a major role in determining option

values.

The option theta measures the impact of the passage of time on option

values.

57

Definitions

Let t denote the current time and T the maturity date of the option.

Then, the theta is defined by

Intuitively, these expressions measure the change in the option values for a

small move forward in current time (i.e., for a small reduction in the time-

to-maturity).

Theta is often referred to as the time-decay in an option.

58

Properties

For American options, a longer time-to-maturity is unambiguously good

(why?).

Thus, theta is negative for American options, i.e., they are subject to time-

decay.

Also true for European call options on NDP stocks: smaller time-to-maturity

reduces time value and insurance value of the option.

For European put options, however, there is ambiguity: a lower time-to-

maturity reduces insurance value but increases time value. Thus, theta may

be positive in some cases.

59

Call Values and Time

60

Put Values and Time

61

Theta and "Moneyness"

Theta is

Least important (smallest in absolute value) for deep OTM options.

Somewhat more important for deep ITM options.

Most important for ATM/NTM options.

This is reflected in the graphs on the next page.

Intuition?

Time matters through the insurance value and time value of an

option.

How do these depend on moneyness?

62

Call and Put Thetas

63

Theta in the Black-Scholes Framework

In the BlackScholes model, we have

The two terms in each expression have intuitive interpretations:

First term on the RHS measures the insurance value effects of as mall

increase in t . This is ve for both calls and puts.

Second term measures the time value effect. This is ve for a call, but

+ve for a put.

64

Call Theta Decomposition

65

Put Theta Decomposition

66

Using the Option Theta

The option theta estimates that for a given change dt, the change in option

values is given by

dC = c dt dP = p dt For example, if dt = 0.004 (approximately one trading day), the option

value will fall by (0.004) . The call in the Black-Scholes example used above has a theta of 8.116.

Thus, this call will lose value at a rate of 0.032 per day.

Note the severity of the time-decay: the call loses roughly 47bps of its

value in one day, even though it has 6 months to maturity!

67

The GammaTheta Trade-Off

The gamma of an option is always positive, but the theta is (typically)

negative.

Long option position: Profits from convexity, but typically incurs time-

decay.

Short option position: Appreciates over time, but has negative convexity.

This is the gamma-theta trade-off: a portfolio which profits from volatility

cannot avoid time-decay.

Put differently, in a long option position, you pay in time and are

compensated in volatility; a short option position works the other way.

68

The GammaTheta Trade-Off and Moneyness

The gamma-theta trade-off is most important for options that are

ATM/NTM, since this is where gamma and theta are both highest in

absolute terms.

Moreover, the trade-off for ATM options becomes particularly acute as

maturity approaches, since theta and gamma both rise.

For example, in a Black-Scholes framework with S = K = 100, r = 0.05,

and = 0.20, we have the following values:

T = 0.50: c = 0.0274 and c = 8.116. T = 0.25: c = 0.0393 and c = 10.474. T = 0.10: c = 0.0627 and c = 15.120. T = 0.05: c = 0.0889 and c = 20.347. T = 0.01: c = 0.1994 and c = 42.399.

69

Gamma as Maturity Shortens

70

Theta as Maturity Shortens

71

The Option Vega

72

The Option Vega

Volatility is a primary determinant of option value. The option vega measures

the quantitative impact of a change in volatility.

Vega is not a letter in the Greek alphabet, so there is no unique way to notate

it. We shall denote the vega of an option by. The vega is defined by:

Intuitively, the vega measures the impact on call and put values of a small

increase d in current volatility.

73

Properties of the Vega

The vega is always positive for both calls and puts: both calls and puts

increase in value as volatility increases.

For European options, the vega of a call always equals the vega of an

otherwise identical put.

This follows from put-call parity: C P = S PV (K ).

Differentiating both sides with respect to , we get

which says precisely that C = P.

74

Vega and Moneyness

How does the option vega depend on depth-in-the-money?

A change in volatility affects the options insurance value.

For deep ITM options, insurance value is not a big component of overall

value, so vega is likely to be small.

Options that are ATM/NTM react very quickly to changes involatility, so

their vegas are large.

Options that are deep OTM derive most of their value from insurance

value. Thus, they will react to a change in volatility faster than ITM

options, but slower than ATM options.

On the other hand, relatively speaking, volatility is most important for deep

OTM options, and least important for deep ITM options.

75

Calls and Volatility

76

The Vega

77

Using the Option Vega

Given d, vega estimates change in option prices of d . Formula works well for "small" d .

For example, at S = 100 and = 0.2: C = 6.889 & = 25.36. Consider a change d = 0.01.

Then, the estimated change in call value is (25.36)(0.01) = 0.2536.

Thus, the estimated call prices are C ( = 0.19) = 6.635 and C ( = 0.21)

= 7.143, respectively.

The actual prices at these levels are 6.615 and 7.162, respectively.

78

The Option Rho

79

The Option Rho

Options are securities with deferred payoffs, so their values are affected by

the rate of interest.

The option rho measures the sensitivity of option prices to changes in

interest

rates.

The rho is denoted and is defined by

Intuitively, the rho measures the impact on call and put values of a small

increase dr in the risk-free rate.

80

Properties

Interest rates affect option prices through the time value factor.

The time value of a call is positive, but that of a put is negative.

Therefore:

The rho of a call is positive (C if r ). The rho of a put is negative (P if r ).

81

Calls and Interest Rates

82

Puts and Interest Rates

83

Rho and Moneyness

How does the rho depend on depth in-the-money?

Consider calls, for instance.

A change in interest rates affects the calls time value.

In absolute terms, time value is most important for in-the-money calls,

since the probability of ultimate exercise is highest here.

Thus, the call rho increases as depth-in-the-money increases.

What is the analogous property for puts (i.e., how do put rhos depend on

depth in-the-money)?

84

The Call Rho

85

The Put Rho

86

Rhos in the Black-Scholes Framework

In the Black-Scholes model,

c = e r (T t )KN (d2)(T t ). (1)

p = er (T t )KN (d2)(T t ). (2)

87

Using the Option Rho

For a given dr, the rho estimates that option prices will change by c dr. Formula works well for "small" dr.

E.g., at S = 100 and r = 0.05: C = 6.889 and = 26.44. Consider a change of 25 basis points: dr = 0.0025.

Estimated change in call value: (26.44)(0.0025) = 0.0661.

Thus, the estimated call prices: C [r = 0.0475] = 6.823 and

C [r = 0.0525] = 6.955, resply.

Actual prices: C [r = 0.0475] = 6.823 and C [r = 0.0525] = 6.955.

88

Position Greeks

89

Position Greeks

The greeks are easily extended from individual options to portfolios consisting

of options and the underlying.

The position greek is simply equal to the sum of the greeks of each option

weighted by the number of options, plus the sensitivity of the under lying to

that parameter.

For example, the position delta is the sum of the deltas of each option in

the portfolio weighted by the number of options of each type) plus the

delta of the position in the underlying in the portfolio.

In this process, note that:

The delta of the underlying is equal to +1.

The gamma, vega, and rho of the underlying stock are all equal to zero.

90

Position Greeks and Implied Views

The position greeks are interpreted in the usual way.

Position delta: Implied view on direction.

Position gamma: Implied view on jump-risk/volatility.

By adding and subtracting different kinds of options, one can set up a

portfolio with almost any desired kind of exposure.

Note that

By adding positions in the underlying, only the position delta is

affected.

To change any of the other greeks, positions in options are needed.

91

Position Greeks: Example

Consider a calendar spread: short a one-month call, long a two-month call.

Parameter values:

Current stock price: S

Strike price: K = 100

Volatility: = 40%

Interest rate: r = 5%.

The graphs on the following pages plot the spreads value, delta, gamma,

and theta.

92

Calendar Spread: Spread Value

93

Calendar Spread: Spread Delta

94

Calendar Spread: Spread Gamma

95

Calendar Spread: Spread Theta

96

Calendar Spread: Spread Rho

97

Summary

98

Summary

This chapter has examined the five principal option "greeks:"

The option delta.

The option gamma.

The option theta.

The option vega.

The option rho.

Each of these greeks has well-defined and regular properties that we have

identified and discussed.

In the next two chapters, we look at various categories of exotic options and

contrast the behavior of exotic greeks with vanilla greeks.