1 The Greek Letters Chapter 17. 2 The Greeks are coming! Parameters of SENSITIVITY Delta = Theta = ...

1

The Greek Letters

Chapter 17

2

The Greeks are coming!

The Greeks are coming!

Parameters of SENSITIVITY

Delta =

Theta =

Gamma =

Vega =

Rho =

3

c = SN(d1) – Ke–r(T – t)N(d2)

p = Ke–r)T –t)N(-d2) – SN(-d1)

Notationally:

c = c(S; K; T-t; r; σ)

p = p(S; K; T-t; r; σ)

Once c and p are calculated, WHAT IF?

4

The GREEKS are measures of sensitivity. The question is how sensitive a position’s value is to

changes in any of the variables that contribute to the position’s market

value.These variables are:

S, K, T-t, r and .

Each one of the Greek measures indicates the change in the value of the position as a result of a “small”

change in the corresponding variable.

Formally, the Greeks are partial derivatives.

5

Delta =

In mathematical terms DELTA is the first derivative of the option’s

premium with respect to S. As such, Delta carries the units of the option’s

price; I.e., $ per share.

For a Call: (c)= c/S

For a Put: (p)= p/S

Results: (p) = (c) - 1

For the (S) = S/S = 1

6

THETA

Theta measures are given by:

(c)= c/(T-t) (p)= p/(T-t)

s are positive but the they are reported as negative values. The

negative sign only indicates that as time passes, t increases, time to

expiration, T – t, diminishes and so does the option’s value, ceteris

paribus. This loss of value is labeled the option’s “time decay.”

Also, (S) = 0.

7

GAMMA

Gamma measures the change in delta when the price of the underlying asset

changes.

Gamma is the second derivative of the option’s price with respect to the

underlying price.

(c) = (c)/S = 2c/ S2

(p) = (p)/S = 2p/ S2

Results: (c) = (p)

(S) = 0.

8

VEGA

Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the

underlying asset’s return. (c)= c/

(p)= p/

Thus, Vega is in terms of

$/1% change in .

(S)= 0.

9

RHO Rho measures the sensitivity of the option’s price to “small” changes in

the rate of interest.

(c) = c/r

(p)= p/r

Rho is in terms of $/%change of r.

(S) = 0.

10

Example:

S=100; K = 100; r = 8%; T-t =180 days;

= 30%. Call Put

Premium $10.3044 $6.4360

The Greeks:

Delta = 0.6151 -0.3849

Theta = -0.03359 -0.01252Gamma = 0.0181

0.0181Vega = .268416

.268416Rho = .252515

-.221559

11

Again. the Delta of any position measures the $ change/share in the position’s value that ensues a “small” change in the value of

the underlying.

(c)= 0.6151

(p) = - 0.3849

12

Call Delta (See Figure 15.2, page 345)

• Delta is the rate of change of the call price with respect to the underlying

Call price

A

CSlope =

Stock price

13

THETA

Theta measures the sensitivity of the option’s price to a “small” change in the time

remaining to expiration:

(c) = c/(T-t) (p)= p/(T-t)Theta is given in terms is $/1 year.

(c) = - $12.2607/year if time to expiration increases (decreases) by one year, the call price will increase (decrease) by $12.2607. Or, 12.2607/365 = 3.35 cent per day.

14

GAMMA

Gamma measures the change in delta when the price of the underlying asset

changes.

= 0.0181. (c) = .6151; (p) = -.3849.

If the stock price increases to $101:(c) increases

to .6332 (p) increases to -.3668.

If the stock price decreases to $99:(c) decreases to .5970(p) decreases to -.4030.

15

VEGA

Vega measures the sensitivity of the option’s market price to

“small” changes in the volatility of the underlying asset’s return.

= .268416

(Check on Computer)

16

RHO Rho measures the sensitivity of

the option’s price to “small changes in the rate of interest.

Rho = Call Put.252515

-.221559


(check on computer)

17

DELTA-NEUTRAL POSITIONS

A market maker wrote n(c) calls and wishes to protect the revenue against

possible adverse move of the underlying asset price. To do so,

he/she uses shares of the underlying asset in a quantity that GUARANTEES

that a small price change will not have any impact on the call-shares position.

Definition: A portfolio is Delta-neutral if

(portfolio) = 0

18

DELTA neutral position in the simple case of call-stock portfolios.

Vportfolio = Sn(S) + cn(c;S)

(portfolio) = (S)n(S) + (c)n(c;S)

(portfolio) = 0 n(S) + (c)n(c;S) = 0.

n(S) = - n(c;S)(c).

The call delta is positive. Thus, the negative sign indicates that the calls and the shares of the underlying asset must be held in opposite direction.

19

EXAMPLE: call - stock portfolio

We just sold 10 CBOE calls whose delta is $.54/shares. Each call covers 100 shares.

n(S) = - n(c;S)(c).

(c) = 0.54 and n(c) = -10.

n(c;S) = - 1,000 shares.

n(s) = - [ - 1,000(0.54)] = 540.

The DELTA-neutral position consists of the 10 short calls and 540 long shares.

20

The Hedge Ratio c

Definition: Hedge ratio.

options by the covered shares ofnumber The

portfolio theneutralize torequired shares ofnumber The

Ratio Hedge

In the example:

Hedge ratio = 540/1,000 = .54

Notice that this is nothing other than (c).

21

In the numerical example, Slide 9:

The hedge ratio: (c) = 0.6151

With 100 CBOE short calls:

n(S) = -(c)n(c;S).

n(c;S) = -10,000.

n(S) = -(.6151)[-10,000] = +6,151 shares The value of this portfolio is:

V = -10,000($10.3044) + 6,151($100)

V = $512,056

22

Suppose that the stock price rises by $1.

SNEW = 100 + 1 = $101/share.

V = - 10,000($10.3044 + $.6151)

+6,151($101)

V = - 10,000($10.3044) + 6,151($100) - 10,000($.6151) + $1(6,151)

V = $512,056 - $6,151 + $6,151

V = $512,056.

23

Suppose that the stock price falls by $1.

SNEW = 100 - 1 = $99/share.

V = - 10,000($10.3044 - $.6151)+6,151($99)

V = - 10,000($10.3044) + 6,151($100) - 10,000( - $.6151) - $1(6,151)

V = $512,056 + $6,151 – $6,151

V = $512,056.

24

In summary:

The portfolio consisting of 100 short calls and 6,151 long shares is

delta- neutral.Price/share: +$1 -$1

shares +$6,151 -$6,151

calls +(-$6,151) -(-$6,151)

Portfolio $0 $0

25

DELTA neutral position in the simple case of put-stock portfolios.

Vportfolio = Sn(S) + pn(p;S)

(portfolio) = (S)n(S) + (p)n(p;S)

(portfolio) = 0 n(S) + (p)n(p;S) = 0.

n(S) = - n(p;S)(p)

Since the put delta is negative, then the negative sign indicates that the puts and the underlying asset must be held in the same direction.

26

EXAMPLE: put – stock portfolio.

We just bought 10 CBOE puts whose delta is -$.70/share. Each put covers 100 shares. n(S) = - n(p;S)(p).

(p) = -.70 and n(p) = 10.

n(p;S) = 1,000 shares.

n(S) = - 1,000(-.70) = 700.

The DELTA-neutral position consists of the 10 long puts and 700 long shares.

27

Portfolio: The portfolio consisting of 10 long puts and 700 long shares is


shares +$700 -$700

puts -$700 $700

Portfolio $0 $0

28

In the numerical example, Slide 9:

The hedge ratio: (p) = -0.3849

The Delta neutral position with 100 CBOE long puts requires the holding

of:

n(S) = -(p)n(p;S)

n(S) = -(-.3849)[10,000] = +3,849shares The value of this

portfolio is:

V = 10,000($6.4360) + 3,849($100)

V = $449,260.

29

Suppose that the stock price rises by $1.

SNEW = 100 + 1 = $101/share.

V = 10,000($6.4360 - .3849)

+3,849($101)

V = 10,000( $6.4360) – 3,849($100) - 10,000(.3849) + $1(3,849)

V = $449,260- $3,849 + $3,849

V = $449,260.

30

Suppose that the stock price falls by $1.

SNEW = 100 - 1 = $99/share.

V = 10,000($6.4360 + $.3849)+3,849($99)

V = 10,000( $6.4360) + 3,849($100) 10,000($.3849) - $1(3,849)

V = $449,260+ $3,849 - $3,849

V = $449,260.

31

In summary:

The portfolio consisting of 100 long puts and 6,151 long shares is


shares +$3,849 -$3,849

calls +(-$3,849) -(-$3,849)

Portfolio $0 $0

32

An extension:

calls, puts and the stock position

Vportfolio = Sn(S) + cn(c;S) + pn(p;S)

portfolio = (S)n(S) + (c)n(c;S) + (p)n(p;S).

But S = 1.

Thus, for delta-neutral portfolio:

portfolio = 0 and

n(S) = -(c)n(c;S) -(p)n(p;S).

33

EXAMPLE: We short 20 calls and 20 puts whose deltas are $.7/share and -$.3/share, respectively. Every call and every put covers 100 shares.

How many shares of the underlying stock we must purchase in order to create a delta-neutral position?

n(S) = -(c)n(c;S) + [-(p)]n(p;S).

n(S) = -(.7)(-2,000) – [-.3](-2,000)

n(S) = 800.

34

Example continued

The portfolio consisting of 20 short calls, 20 short puts and 800 long

shares is

delta- neutral.

Price/share: +$1 -$1

shares +$800 -$800

calls -$1,400 +$1,400

Puts +$600 -$600

Portfolio $0 $0

35

EXAMPLES

The put-call parity:

Long 100 shares of the underlying stock,

long one put and short one call on this

stock is always delta-neutral:

(position)

= 100 + (p)n(p;S) + (c)n(c;S)

= 100 + [(c) – 1](100) + (c)(-100)

= 0.

36

EXAMPLES

A long STRADDLE:

Long 15 puts and long 15 calls

(same underlying asset, K and T-t),

with: (c) = .64; (p) = - .36.

(straddle) = 15(100)[.64 + (- .36)]

=$420/share.

Long 420 shares to delta neutralize this straddle.

37

Results:

1.The deltas of a call and a put on the same underlying asset, (with the same time to expiration and the

same exercise price) must satisfy the following equality:

(p) = (c) - 1

2. Using the Black and Scholes formula:

(c) = N(d1) 0 < (c) < 1

(p) = N(d1) – 1 -1 < (p) < 0

42

THETA

Theta measures the sensitivity of the option’s price to a “small” change in

the time remaining to expiration:

(c) = c/(T-t)

(p)= p/(T-t)

Theta is given in terms is $/1 year. Thus, if (c) = - $20/year, it means that if time to expiration increases (decreases) by one year, the call price will increase (decreases) by $20. Or, $20/365 = 5.5 cent per day.

47

GAMMA

Gamma measures the change in delta when the market price of the underlying asset changes.

(c) = (c)/S = 2c/ S2

(p) = (p)/S = 2p/ S2

Results:Results:

(c) = (p)

(S) = 0.

48

GAMMA

In general, the Gamma of any portfolio is the change of the portfolio’s delta

due to a “small” change in the underlying asset price.

As the second derivative of the option’s price with respect to S,

Gamma measures the sensitivity of the option’s price to “large”

underlying asset’s price changes.

May be positive or negative.

49

Interpretation of GammaThe delta neutral position with 100

short calls and 6,151 long shares has Γ= -$181

S

Negative Gamma means that the position loses value when the stock price moves more and more away from it initial value.

75 100 125

$512,056Position value

More negative Γ

50

Interpretation of Gamma

S

Negative Gamma

S

Positive Gamma

51

Result: The Gammas of a put and a call are equal. Using the Black and

Shcoles model:

(c) = n(d1) and (p) = n(d1) – 1.

Clearly, the derivatives of these deltas with respect to S are equal.

EXAMPLE: (c) = .70; (p) = - .30;

= .2345.

Holding a long call and a short put has: = .70 - (- .30) = 1.00.

= .2345 – .2345 = 0.

52

EXAMPLE:

(c) = .70, (p) = - .30 and let gamma be .2345.

Holding the underlying asset long, a long put and a short call yields a portfolio with:

= 1 - .70 + (- .30) = 0 and

= 0 - 0,2345 + 0,2345 = 0,

simultaneously! This portfolio is

delta-gamma-neutral.

56

VEGA Vega measures the sensitivity of the option’s

market price to “small” changes in the volatility of the underlying asset’s return.

(c) = c/(p) = p/

Thus, Vega is in terms of $/1% change in

61

RHO Rho measures the sensitivity of the option’s

price to “small changes in the rate of interest.

(c) = c/r

(p) = p/r


66

SUMMERY OF THE GREEKS

Position Delta Gamma Vega Theta Rho

LONG STOCK 1 0 0 00

SHORT STOCK -1 0 0 00

LONG CALL + + + -+

SHORT CALL - - - +-

LONG PUT - + + --

SHORT PUT + - - + +

67

The sensitivity of portfolios, a summary.

1.A portfolio is a combination of securities and options.

2.All the sensitivity measures are partial derivatives.

3.Theorem(Calculus): The derivative of a linear combination of functions is the combination of the derivatives of these functions. Thus, the sensitivity measure of a portfolio of securities is the portfolio of these securities’ sensitivity measures.

68

Example:The DELTA of a portfolio of 5 long CBOE calls, 5 short puts and 100

shares of the stock long:

(portfolio) = (500c - 500p + 100S)

= (500c - 500p + 100S)/S

= 500c/S - 500p/S + 100

= 500c - 500p + 100

This delta reveals the $/share change in the portfolio value as a function of a “small” change in the underlying

price

69

Example: S = $48.57/barrel.

1 call = 1,000bbls.

Call Delta Gamma

A $0.63/bbl $0.22/bbl

B $0.45/bbl $0.34/bbl

C $0.82/bbl $0.18/bbl

Portfolio:

Long: 3 calls A; 2 calls C; 5,000 barrels. Short: 10 calls B.

70

Example:

= (0.63)3,000+ (0.82)2,000

+ (1)5,000 + (0.45)(-10,000)

= (0.22)3,000+ (0.18)2,000

+ (0)5,000 + (0.34)(-10,000)

= $4,030.

= - $2,380.

71

= 4,030 a “small” change of the oil price, say one cent per barrel, will change the value of the above portfolio by $40.30 in the same direction.

= - 2,380 a “small” change in the oil price, say one cent per barrel, will change the delta by $23.80 in the opposite direction.

Also, Gamma is negative when the price per barrel moves away from $48.57, the portfolio value will decrease.

72

A financial institution holds:

5,000 CBOE calls long; delta .4,

6,000 CBOE puts long; delta -.7,

10,000 CBOE puts short; delta -.5,

Long 100,000 shares

(portfolio) = (.4)500,000 + (-.7)600,000

+(-.5)[-1,000,000]

+ 100,000

= $380,000.

73

GREEKS BASED STRATEGIES

Greeks based strategies are opened and maintained in order to attain a specific level of sensitivity. Mostly, these strategies are set to attain zero sensitivity. What follows, is an example of strategies that are:

1.Delta-neutral

2.Delta-Gamma-neutral

3.Delta-Gamma-Vega-Rho-neutral

74

EXAMPLE:

The underlying asset is the a stock. The options on this stock are European.

S = $300; K = $300; T = 1yr; = 18%; r = 8%; q = 3%.

c = $28.25.

= .6245

= .0067

= .0109

= .0159

75

DELTA-NEUTRAL

Short 100 calls. n0 = - 10,000; Long nS = 6,245 shares

Case A1: S increases from $300 to $301.

Portfolio Initial Value New value Change

-100Calls - $282,500 - $288,800- $6,300

6,245S $1,873,500 $1,879,745 $6,245

Error: - $55

Case A2: S decreases from $300 to $299.

Portfolio Initial value New value Change

-100Calls - $282,500 - $276,200+ $6,300

6,245S $1,873,500 $1,867,255 - $6,245

Error: + $55

76

Case B1: S increases from $300 to $310.

Portfolio Initial Value New value Change

-100Calls - $282,500 - $348,100- $65,600

6,245S $1,873,500 $1,935,950 $62,450

Error: -$3,150

The point here is that Delta has changed significantly and .6245 does not apply any more.

S = $300 $301 $310

= .6245 .6311 .6879.

We conclude that the delta-neutral portfolio must be adjusted for “large” changes of the underlying asset price.

77

Call #0 Call #1

S = $300 S = $300

K = $300 K = $305

T = 1yr T = 90 days

= 18% r = 8% q = 3%

c = $28.25 c = $10.02

= .6245 = .4952

= .0067 = .0148

= .0109 = .0059

= .0159 = .0034

78

A DELTA-GAMMA-NEUTRAL PORTFOILO

(portfolio) = 0: nS + n0(.6245) + n1(.4952) = 0

Γ(portfolio)= 0: n0(.0067) + n1(.0148) = 0

Solution:

n0 = -10,000

n1 = - (-10,000)(.0067)/.0148 = 4,527

nS = - (-10,000)(.6245) – (4,527)(.4952) = 4,003

Short the initial call : n0 = -10,000

Long 45.27 of call #1 n1 = 4,527

Long 4,003 shares nS = 4,003

79

THE DELTA-GAMMA-NEUTRAL PORTFOLIO

Case A1: S increases from $300 to $301.


0) -10,000 - $282,500 - $288,800 -$6,300

1) 4,527 $45,360 $47,657 $2,297

S) 4,003 $1,200,900 $1,204,903 $4,003

Error: 0

Case B1: S increases from $300 to $310.


0) -10,000- $282,500 - $348,100 - $65,600

1) 4,527 $45,360 $70,930 $25,570

S) 4,003 $1,200,900 $1,240,930 $40,030

Error: 0

80

- 144 - 82 0 0 Risk

0 0 0 4,003 4,003S

15 27 67 2,242 4,527

- 159 - 109 - 67- 6,245-10,000

RhoVegaGammaDeltaPortfolio

The above numbers reveal that the Delta-Gamma-neutral portfolio is exposed to risk

associated with

the volatility and the risk-free rate

If we examine the exposure level to all parameters, however, we observe that:

81

Case C1:

S increases from $300 to $310

and simultaneously,

r increases from 8% to 9%.


-10,000 - $282,500 - $330,500 - $48,000

4,527 $45,360 $73,166 $27,806

4,003S $1,200,900 $1,240,930 $40,030

Error: - $10,756

82

Delta-Gamma-Vega-Rho-neutral portfolio

CALL 0 1 2 3

K 300 305 295 300

T(days) 365 90 90 180

Volatility 18% 18% 18% 18%

r 8% 8% 8% 8%

Dividends 3% 3% 3% 3%

c $28.25 $10.02 $15.29 $18.59

83

Delta-Gamma-Vega-Rho-neutral portfolio

CALL

0 .6245 .0067 .0109 .0159

1 .4954 .0148 .0059 .0034

2 .6398 .0138 .0055 .0044

3 .5931 .0100 .0080 .0079

S 1.0 0.0 0.0 0.0

84

The DELTA-GAMMA-VEGA-RHO-NEUTRAL-PORTFOLIO

In order to neutralize the portfolio to all risk exposures, following the sale of the initial call, we now determine the portfolio’s holdings such that all

the portfolio’s sensitivity parameters

are zero simultaneously.

= 0 and = 0 and = 0 and = 0simultaneously!

85

= 0nS+n0(.6245)+n1(.4954)+n2(.6398)+n3(.5931)

=0

= 0

n0(.0067)+n1(.0148)+n2(.0138)+n3(.0100)=0

= 0n0(.0109)+n1(.0059)+n2(.0055)+n3(.0080)

=0

= 0n0(.0159)+n1(.0034)+n2(.0044)+n3(.0079)

=0

86

The solution is:

Exact nShort 100 CBOE calls #0; -10,000

Short 339 calls #1; -33,927

Long 265 calls #2; 26,534

Long 204 calls #3; 20,420

Short 6,234 shares. -6,234

87

Case D: S increases from $300 to $310 r increases from 8% to 9% increases from 18% to 24%

Portfolio Initial Value New value

0) - 10,000 - $282,468 - $428,071

S)- 6,234 - $1,870,200 - $1,932,540

1) - 33,927 - $340,023 - $664,552

2)26,534 $405,668 $694,062

3)20,420 $379,677 $622,240

TOTAL - $1,707,356 $1,708,861

Error:$1,505 or .088%.

88

DYNAMIC DELTA - HEDGING

The market stock price keeps changing all

the time. Thus, a static DELTA- neutral

hedge is not sufficient.

A continuous delta adjustment is not

practical.

An adjusted Delta-neutral Position:

1.Every day, week, etc.

2.Following a given % price change.

89

DYNAMIC DELTA HEDGING

Market makers provide traders with the options they wish to trade. For example, if a trader wishes to long (short) a call, a market maker will

write (long) the call. The difference between the buy and sell prices is

the market maker’s

bid-ask spread.bid-ask spread.

The main problem for a market maker who shorts calls is that the premium received, not only may be lost, but

the loss is potentially unlimited.

90


Recall: The profit profile of an uncovered call is:

At expirationP/L

ST

c

K

91


Recall: The profit profile of a covered call is:

Strategy IFC At expirationST < K ST > K

Short call

c 0 -(ST – K)

Long stock

-St ST ST

Total - St + c ST K

P/L ST - St + c

K - St + c

92


Recall: The profit profile of a covered call is:

At expirationP/L

ST

K –St+ c

K

-St + c

93


The Dynamic Delta hedge is based on the

impact of the time decay on the call Delta.

Recall that:

Δ(c))N(d

andtTσ

t]][T.5σ[r]KS

ln[d

1

2t

1

94


Observe what happens to d1

when T-t 0.

1. For: St > K d1 and

N(d1) = (c) 1

2. For: St < K d1 - and

N(d1) = (c) 0

95


The Dynamic hedge:

1.Write a call and simultaneously, hedge the call by a long Delta shares of the underlying asset. As time goes by, adjust the number of shares periodically.

Result: As the expiration date nears, delta:

goes to 0 in which case you wind up without any shares.

goes to 1 in which case you call is fully covered.

96


St : St < K St > K

(c): 0 1

Call: uncovered fully covered

n(S): 0 n(c;S) 1

97

Table 15.2 Simulation of Dynamic delta - hedging.(p.364) Cost of

Stock Shares shares Cummulative Interest Week przce Delta purchased purchased cost

cost ($000) ($000)

($000) 0 49.00 0.522 52,200 2,557.8 2,557.8

2.5 1 48.12 0.458 (6,400) (308.0) 2,252.3

2.2 2 47.37 0.400 (5,800) (274.7) 1,979.8

1.9 3 50.25 0.596 19,600 984.9 2,966.6

2.9 4 51.75 0.693 9,700 502.0 3,471.5

3.3 5 53.12 0.774 8,100 430.3 3,905.1

3.8 6 53.00 0.771 (300) (15.9) 3,893.0

3.7 7 51.87 0.706 (6,500) (337.2) 3,559.5

3.4 8 51.38 0.674 (3,200) (164.4) 3,398.5

3.3 9 53.00 0.787 11,300 598.9 4,000.7

3.810 49.88 0.550 (23,700) (1,182.2) 2,822.3

2.7 11 48.50 0.413 (13,700) (664.4) 2,160.6

2.1 12 49.88 0.542 12,900 643.5 2,806.2

2.7 13 50.37 0.591 4,900 246.8 3,055.7

2.9 14 52.13 0.768 17,700 922.7 3,981.3

3.8 15 51.88 0.759 (900) (46.7) 3,938.4

3.8 16 52.87 0.865 10,600 560.4 4,502.6

4.3 17 54.87 0.978 11,300 620.0 5,126.9

4.9 18 54.62 0.990 1,200 65.5 5,197.3

5.0 19 55.87 1.000 1,000 55.9 5,258.2

5.1 20 57.25 1.000 0 0.0 5,263.3

98

Table 15.3 Simulation of dynamic Delta - hedging. (p. 365) Cost of

Stock Shares shares Cumulative I nterestWeek price Delta purchased purchased cost cost

($000) ($000) ($000) 0 49,00 0.522 52,200 2,557.8 557.8

2.5 1 49.75 0.568 4,600 228.9 2,789.2 2.7 2 52.00 0.705

13,700 712.4 3,504.3 3.4 3 50.000.579 (12,600) (630.0) 2,877.7 2.8

4 48.38 0.459 (12,000) (580.6) 2,299.9 2.2 5 48.25 0.443 (1,600) (77.2)

2,224.9 2.1 6 48.75 0.475 3,200 156.0 2,383.0 2.3 7 49.63 0.540 6,500

322.6 2,707.9 2.6 8 48.25 0.420 (12,000) (579.0) 2,131.5 2.1 9 48.25 0.410 (1,000) (48.2) 2,085.4 2.0 10 51.12 0.658 24,800 1,267.8 3,355.2 3.2 11 51.50 0.692 3,400 175.1 3,533.5 3.4 12 49.88 0.542(15,000) (748.2) 2,788.7 2.7 13 49.88 0.538 (400) (20.0) 2,771.4 2.7 14 48.75 0.400 (13,800) (672.7)

2,101.4 2.0 15 47.50 0.236 (16,400) (779.0) 1,324.4 1.3 16 48.00 0.261 2,500

120.0 1,445.7 1.4 17 46.25 0.062 (19,900) (920.4) 526.7 0.5 18 48.13 0.183 12,100 582.4 1,109.6 1.1 19 46.63 0.007 (17,600) (820.7) 290.0

0.3 20 48.12 0.000 (700) (33.7) 256.6

1 The Greek Letters Chapter 17. 2 The Greeks are coming! Parameters of SENSITIVITY Delta = Theta = ...

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