Global Research & Analytics Dpt. · 2019. 9. 19. · 1 The pricing formulas and codes are from his book: “The complete guide to option pricing formulas”, edited by McGraw-Hill

International Business Solutions Advisors

Global Research & Analytics Dpt.

Valuation & Pricing Solutions

By David REGO -Paris Office- Supported by Benoit GENEST -London Office- and Ziad Fares -Paris Office-

Free Pricer Content Detail of Generic Closed Formulas Solutions

April, 2013


Chappuis Halder & Cie Global Research & Analytics Dpt.

GENESIS

PHILOSOPHY

The department of “Global Research &

Analytic” (GRA) is a team of passionate

people. One unifying criteria in the GRA

remains the dominant quantitative topics,

including the risk modeling part.

As such, each member works regularly on

topics likely to be of interest to the

financial community. The results of this

work are always freely downloadable and

fully shared with anyone interested.

Because we consider “risk modeling” as a

hobby, we try to share ideas or researches

that we found useful within our day to day

practice.

INTRODUCTION

The following document is in response to

repeated requests from various players in

the market and asking for quick access to

a conventional financial pricing library.

Formerly available on the internet, it is

now more difficult to find on the web.

Our approach is to bring up to date all the

work done by Espen Gaarder HAUG1 and

to complete it with a summary document

to assist the reader. This document is

based on his great work. Moreover, we

would like to thank him for his significant

contribution in options pricing field and to

share it with the financial community.

In an initiative to promote knowledge and

expertise sharing, Chappuis Halder & Cie

decided to put this Options Pricer on free

access. It contains a charts generator and

the detail sheets of each type of options.

1 The pricing formulas and codes are from his book: “The

complete guide to option pricing formulas”, edited by McGraw-Hill (second edition).

WARNING OF NO PROPERTY

This document and all its contents,

including texts, formulas, charts and any

other material, are not the property of

CH&Cie.

WARNING OF NO

RESPONSIBILITY

The information, formulas and codes

contained in this document are merely

informative.

There is no guarantee of any kind, express

or implied, about the completeness or

accuracy of the information provided via

this document. Any reliance you place on

the descriptions, mathematical formulas

or related graphs is therefore strictly at

your own risk.

TABLE OF CONTENTS

1.A. The Generalized Black & Scholes Formula ....................................................................................... 1

1.B. The generalized Black and scholes options sensitivities .................................................................. 2

2. European option on a stock with cash dividends ................................................................................ 8

3. The Black-Scholes model adjusted for trading day volatility (French) ................................................ 9

4. The merton’s Jump Diffusion Model option pricing.......................................................................... 10

5. American Calls on stocks with known dividends ............................................................................... 11

6.A. American approximations: The Barone-Adesi and Whaley approximation .................................. 12

6.B. American approximations: The Bjerksund and Stensland approximation ..................................... 14

7. The Miltersen and Schwartz commodity option model .................................................................... 16

8. Executive stock options ..................................................................................................................... 18

9. Forward start options ........................................................................................................................ 19

10. Time switch options ........................................................................................................................ 20

11.A. Simple chooser options ................................................................................................................ 21

11.B. Complex chooser optionS ............................................................................................................ 22

12. Options on options .......................................................................................................................... 24

13. Writer extendible options ............................................................................................................... 26

14. Two assets correlation options ....................................................................................................... 28

15. Option to exchange one asset for another ..................................................................................... 29

16. Exchange options on exchange options .......................................................................................... 31

17. Options on the maximum or the minimum of two risky assets ...................................................... 34

18. Spread option approximation ......................................................................................................... 36

19. Floating strike lookback options ...................................................................................................... 38

20. Fixed strike lookback options .......................................................................................................... 40

21. Partial-Time Floating-Strike Lookback Options ............................................................................... 42

22. Partial-Time Fixed-Strike Lookback Options .................................................................................... 44

23. Extreme-spread options .................................................................................................................. 46

24. Standard barrier options ................................................................................................................. 48

25. Double barrier options .................................................................................................................... 52

26. Partial-time single asset barrier options ......................................................................................... 55

27. Two asset barrier options ................................................................................................................ 60

28. Partial time two asset barrier options ............................................................................................. 63

29. Look-barrier options ........................................................................................................................ 66

30. Soft-barrier options ......................................................................................................................... 68



31. Gap options ..................................................................................................................................... 70

32. Cash-or-nothing options .................................................................................................................. 71

33. Two asset cash-or-nothing options ................................................................................................. 72

34. Asset-or-nothing options ................................................................................................................. 74

35. Supershare options ......................................................................................................................... 75

36. Binary barrier options...................................................................................................................... 76

37. Asian Options 1: Geometric average rate options .......................................................................... 86

38. Asian Options 2: The Turnbull and Wakeman arithmetic average approximation ......................... 87

39. Asian Options 3: Levy's arithmetic average approximation ............................................................ 88

40. Foreign equity options struck in domestic currency (Value in domestic currency) ........................ 90

41. Fixed exchange rate foreign equity options - Quantos (Value in domestic currency) .................... 92

42. Equity linked foreign exchange options (Value in domestic currency) ........................................... 94

43. Takeover foreign exchange options ................................................................................................ 96

44. European swaptions in the Black-76 model .................................................................................... 97

45. The Vasicek model for european options on zero coupon bonds .................................................. 98


1


1.A. THE GENERALIZED BLACK & SCHOLES FORMULA

DESCRIPTION

This function allows to price plain vanilla European call and put options,

using the Generalized Black and Scholes formula.

MATHEMATICAL FORMULA

The Generalized Black & Scholes formulas for a call and put are

( )

1 2. ( ) . ( )b r T rTCall S e CND d X e CND d

( )

2 1. ( ) . ( )rT b r TPut X e CND d S e CND d

Where d1 and d2 are defined by the following formulas

2

1

ln2

Sb T

Xd

T

2 1d d T

And

S = Forward Asset price

X = Strike price

r = Risk-free rate

T = Time to maturity (Years)

b = Cost of carry

= Volatility

CND(x)= The Cumulative Normal Distribution Function

PAYOFFS

The payoffs of this model can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)

NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas

INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)


2


1.B. THE GENERALIZED BLACK AND SCHOLES OPTIONS SENSITIVITIES

DELTA

DESCRIPTION

The parameter Delta, noted , is the sensitivity of the plain vanilla option’s

price to the underlying asset price.


( ).

1. ( )b r T

Call e CND d

( ).

1.( ( ) 1)b r T

Put e CND d

With:

2

1

log .2

.

Sb T

Xd

T

and 2 1d d T

DELTA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE

Buying a call position is in the left side while buying a put position is in the right side.

DELTA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE

TIME TO MATURITY



3


GAMMA

DESCRIPTION

The parameter Gamma, noted , is the sensitivity of the plain vanilla option’s

delta to the underlying asset price. It measures the acceleration and

curvature of the option’s price evolution.


( ).

1. ( )

.

b r T

option

e CND d

S T

With:

2

1

log .2

.

Sb T

Xd

T

and 2 1d d T

GAMMA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE

The gamma is the same for a call or a put.

GAMMA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND

THE TIME TO MATURITY

The gamma is the same for a call or a put.

0,02

0,18

0,34

0,50,660,820,98

00000000

0

0

0

5060

7080 90 100 110120130140150

Time to Maturity

Spot


4


VEGA

DESCRIPTION

The parameter Vega, noted , is the sensitivity of the plain vanilla option’s

price to the underlying asset volatility.


( ).

1. . ( ).b r T

optionvega S e CND d T

With:

2

1

log .2

.

Sb T

Xd

T

and 2 1d d T

VEGA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE

The vega is the same for a call or a put.

VEGA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE

TIME TO MATURITY

The vega is the same for a call or a put.

0

1

2

3

4

5

6

7

8

9

10

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

05

1015

20

25

30

35

40

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


5


THETA

DESCRIPTION

The parameter Theta, noted , is the sensitivity of the plain vanilla option’s

price to the time to maturity.


( ).

1

( ). .

1 2

. ( ).( ).

2

. . ( ) . . . ( )

b r T

Call

b r T r T

S e CND db r

T

S e CND d r X e CND d

( ).( ).1

1

.

2

. ( ).( ). . . ( )

2

. . . ( )

b r Tb r T

Put

r T

S e CND db r S e CND d

T

r X e CND d

With:

2

1

log .2

.

Sb T

Xd

T

and 2 1d d T

THETA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE


THETA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE

TIME TO MATURITY


-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

50 60 70 80 90 100 110 120 130 140 150

Spot

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

50 60 70 80 90 100 110 120 130 140 150

Spot

0,02

0,18

0,34

0,50,660,820,98

-50-45-40-35-30-25-20-15-10-505

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

-50-45-40-35-30-25-20

-15-10-50

5

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


6


RHO

DESCRIPTION

The parameter Rho, noted , is the sensitivity of the plain vanilla option’s

price to the interest rate.


2

b 0: . . . ( )

.Call (S,X,T,r,b, )

rT

call

call Generalized BS

if T X e CND d

else T

2

b 0: . . . ( )

.Put (S,X,T,r,b, )

rT

put

put Generalized BS

if T X e CND d

else T

With:

2

1

log .2

.

Sb T

Xd

T

and 2 1d d T

RHO VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE


RHO VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE

TIME TO MATURITY


0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

2,2

50 60 70 80 90 100 110 120 130 140 150Spot

-2

-1,8

-1,6

-1,4

-1,2

-1

-0,8

-0,6

-0,4

-0,2

0

0,2

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

01020304050607080

90

100

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

-100-90-80-70

-60-50

-40-30

-20

-10

0

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


7


COST OF CARRY

DESCRIPTION

The parameter Rho, noted b , is the sensitivity of the plain vanilla option’s

price to the cost of carry.


( ).

1. . . ( )b r T

Callb T S e CND d

( ).

1. . . ( )b r T

Putb T S e CND d

With:

2

1

log .2

.

Sb T

Xd

T

and 2 1d d T

CARRY SENSITIVITY VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT

PRICE


CARRY SENSITIVITY VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT

PRICE AND THE TIME TO MATURITY


00,20,40,60,811,21,41,61,822,22,42,62,83

50 60 70 80 90 100 110 120 130 140 150Spot

-3-2,8-2,6-2,4-2,2

-2-1,8-1,6-1,4-1,2

-1-0,8-0,6-0,4-0,2

00,2

50 60 70 80 90 100 110 120 130 140 150

Spot

0,02

0,18

0,34

0,50,660,820,98

0102030405060708090100110120130140

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

-60

-50

-40

-30

-20

-10

0

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


8


2. EUROPEAN OPTION ON A STOCK WITH CASH DIVIDENDS

DESCRIPTION

This function allows to price plain vanilla European call and put options with

cash dividend, using the original Black Scholes formula. Although simple, this

approach can lead to significant mispricing and arbitrage opportunities. In

particular, it will underprice options where the dividend is close to the

option's expiration date.


1 2. ( ) . ( )rTCall S CND d X e CND d

2 1. ( ) . ( )rTPut X e CND d S CND d

2

1 2 1

ln2

Where ;

Sr T

Xd d d T

T

31 2

1 2 3 . . .rtrt rt

DividendsWith S stock price NPV s D e D e D e

Where

s is the Stock price

1 2, D D and 3D are dividends for 1 2t , t and 3t .

X = Strike price

r = Risk-free rate

T = Time to maturity (Years)

= Volatility

CND(x)= The Cumulative Normal Distribution Function (CND)

PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


9


3. THE BLACK-SCHOLES MODEL ADJUSTED FOR TRADING DAY VOLATILITY (FRENCH)

DESCRIPTION

This function allows to price plain vanilla European call and put options,

using the adjusted Generalized Black and Scholes formula. This adjustment

was done by French in 1984 to take into consideration that the volatility is

usually higher on trading days than on non-trading days. If trading days to

maturity are equals to calendar days to maturity, the output theoretical price

would be the same as the one generated by the Generalized Black Scholes

formula.


( )

1 2

( )

2 1

. . ( ) . ( )

. ( ) . . ( )

b r T rT

rT b r T

Call S e CND d X e CND d

Put X e CND d S e CND d

Where :

2

1

ln .2

SbT t

Xd

t

and 2 1d d t With:

S = Stock Price

X = Strike Price

r = Risk-Free Rate

t = Trading time= Trading days until maturity / Trading days per year

T = Calendar Time = Calendar days until maturity / Calendar days per

year


= Standard Deviation

PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


10


4. THE MERTON’S JUMP DIFFUSION MODEL OPTION PRICING

DESCRIPTION

This Model allows to price plain vanilla European call and put options, using

the Merton’s Jump Diffusion formula. This alternative model supposes a non-

correlated Brownian motion and jumps.


0

( )( ; ; ; ; )

!

T i

i i

i

e TCall Call S X T r

i

0

( )( ; ; ; ; )

!

T i

i i

i

e TPut Put S X T r

i

With : 2 2

i

iz

T

;

2

and 2 2z

NB: iCall and iPut are calculated with the Generalized Black Scholes

Function.

With :

S = Stock Price

X = Strike Price

r = Risk-Free Rate

T = Calendar Time (time to Expiration on years)



PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


11


5. AMERICAN CALLS ON STOCKS WITH KNOWN DIVIDENDS

DESCRIPTION

This Model allows to price American Calls on stocks with known dividends,

using the Roll-Geske-Whaley approximation formula. We consider here that

the stock is paying a single discrete dividend yield. The method can be

extended to a multiple dividends.


1 1 1

2 2 2

2

1 2 1

2

1

( ). ( ) ( ). , ;

. . , ; ( ). ( )

ln2

With ;

ln2

rt rt

rT rt

rt

rt

c

tCall S De CND b S De M a b

T

tX e M a b X D e CND b

T

S Der T

Xa a a T

T

S Der T

Sb

T

2 1; b b T

With:

S = Stock Price; X = Strike Price; = Standard Deviation; r = Risk-Free Rate;

D = Cash Div.; T = Time to option expiration; t = time to dividend payout

CND(x)= The Cumulative Normal Distribution Function; M(a,b ; ρ) = The

Cumulative Bivariate Normal Distribution Function with upper integral limits

a and b and correlation coefficient ρ.

cS is the critical ex-dividend stock price that solves:

2 1, ,c cCall S X T t S D X

Where 2 1, ,cCall S X T t = the price of European call with stock

price of I and time to maturity 2 1T t

PAYOFFS

The payoff of this model can be represented as follows (for buying a call)

NB: "Payoff" Chart represents prices seven days before expiry, not payoffs formulas

INSTRUMENT PRICE

The price of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for buying a call)

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot


12


6.A. AMERICAN APPROXIMATIONS: THE BARONE-ADESI AND WHALEY APPROXIMATION

DESCRIPTION

This quadratic approximation method by Barone-Adesi and Whaley (1987)

allows to price American call and put options on an underlying asset with

cost-of-carry rate b. When b > r, the American call value is equal to the

European call value and can then be found by using the generalized Black-

Scholes-Merton (BSM) formula. This model is fast and accurate for most

practical input values.


2

*

2 *( , , ) when

( , , )

else

Q

GBS

SCall S X T A S S

Call S X T S

S X

1

**

1 **( , , ) when

( , , )

else

Q

GBS

SPut S X T A S S

Put S X T S

X S

Where:

GBSCall and GBSPut are respectively the values of Europeans Call

and put options computed by General Black Scholes formula.

**( ) **

1 1

1

*( ) *

2 1

2

1 ( )

1 ( )

b r T

b r T

SA e CND d S

Q

SA e CND d S

Q

2 2

1 2

( 1) ( 1) 4 ( 1) ( 1) 4

; 2 2

M MN N N N

K KQ Q

2 2

2 2 ; N= ; 1 rTr b

M K e

With:

S = Stock Price

b = cost of carry rate

X = Strike Price

r = Risk-Free Rate

T = Time to option expiration

CND(x)= The Cumulative Normal Distribution Function


**S = the critical commodity price for put options

*S = the critical commodity price for call options

*S and

**S are determined by using the Newton-Raphson algorithm.


13


PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


14


6.B. AMERICAN APPROXIMATIONS: THE BJERKSUND AND STENSLAND APPROXIMATION

DESCRIPTION

The Bjerksund and Stensland (1993) approximation can be used to price

American options on stocks, futures, and currencies. The method is analytical

and extremely computer-efficient. Bjerksund and Stensland's approximation

is based on an exercise strategy corresponding to a flat boundary / (trigger

price). It is demonstrated that the Bjerksund and Stensland approximation is

somewhat more accurate for long-term options than the Barone-Adesi and

Whaley approximation.


2 2

Call(X,S,T,r,b, ) = S (S, T, ,I, I) + (S, T, 1, I, I) - (S, T, 1, X, I)

- X (S, T, 0, I, I) + X (S,T, 0, X, I)

1 1Where ( ) and

2 2

b bI X I

2

22

r

The function (S, T, ,H, I) is given by

2ln( / )(S, T, ,H, I)=e ( )

kI I S

S CND d N dS T

2

2

1( 1)

2

1ln( / ) ( )

2

r b T

S H b T

dT

2

2(2 1)

bk

And the trigger price I is defined as

( ) 00 0

0

0

( )(1 ) and ( ) ( 2 )

and max ,1

h T BI B B B e h T bT T

B B

rB X B X X

r b

If S I , it is optimal to exercise the option immediately, and the value

must be equal to the intrinsic value of S-X. on the other hand, if b r , it will

never be optimal to exercice the American call option before expiration, and

the value can be found using the generalized black-scholes formula. The

value of the American put is given by the Bjerksund and Stensland put-call

transformation:

Put (S,X,T,r,b, ) ( , ,T, r-b, b, )Call X S

Where Call(.) is the value of an American call with risk-free rate r-b and drift

–b. With the use of this transformation, it is not necessary to develop a

separate formula for an American put option.


15


PAYOFFS


NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas

INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


16


7. THE MILTERSEN AND SCHWARTZ COMMODITY OPTION MODEL

DESCRIPTION

Miltersen and Schwartz (1998) developed an advanced model for pricing

options on commodity futures. The model is a three-factor model with

stochastic futures price, a term structure of convenience yields and interest

rates. The model assumes commodity prices are log-normally distributed

and that continuously compounded forward interest rates and future

convenience yields are normally distributed (aka Gaussian).

Investigations using this option pricing model show that the time lag

between the expiration on the option and the underlying futures will have a

significant effect on the option value. Even with three stochastic variables,

Miltersen and Schwartz manage to derive a closed-form solution similar to a

BSM-type formula. The model can be used to price European options on

commodity futures.


1 2( ) ( )xz

t TCall P F e CND d XCND d

Where t is the time to maturity of the option, TF is a futures price with time

to expiration T, and tP is a zero coupon bond that expires on the option’s

maturity.

2

1 2 1

ln( / ) / 2, T xz z

z

z

F Xd d d

And the variances and covariance can be calculated as

2

22

0 0

0

0

( ) ( , ) ( , ) ( )

( , ) . ( ) ( , ) ( , )

( ). ( ) .

T

t t

t T t

z s f e F

u

t t T

xz f s f e

u u

t

P F

u u s u s ds du u du

u s ds u u s u s ds du

u u du

Where

( ) ( , )

( ) ( ) ( , ) ( , )

t

t

T

P f

t

T

F s f e

t

t t s ds

t t t s t s ds

This is an extremely flexible model where the variances and covariances

admits several specifications.


17


PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


18


8. EXECUTIVE STOCK OPTIONS

DESCRIPTION

Executive stock options are priced by the Jennergren and Naslund (1993)

formula which takes into account that an employee or executive often loses

his options if he has to leave the company before the option's expiration.


( )

1 2( ) ( )T b r T rTCall e Se CND d Xe CND d

( )

2 1( ) ( )T rT b r TPut e Xe CND d Se CND d

Where:

2

1 2 1

ln( / ) ( / 2) d

S X b Td d T

T

is the jump rate per year. The value of the executive option equals the

ordinary Black-Scholes option price multiplied by the probability Te

that

the executives will stay with the firm until the option expires.

PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


19


9. FORWARD START OPTIONS

DESCRIPTION

Forward start options with time to maturity T starts at-the-money or

proportionally in- or out-of-the-money after a known time t in the future.

The strike is set equal to a positive constant times the asset price S after

the known time t. If is less than unity, the call (put) will start 1 -

percent in-the-money (out-of-the money); if is unity, the option will start

at-the-money; and if is larger than unity, the call (put) will start - 1

percentage out-of-the money (in-the-money). A forward start option can be

priced using the Rubinstein (1990) formula.


( ) ( )( ) ( )

1 2( ) ( )b r t b r T t r T tCall Se e CND d e CND d

( ) ( ) ( )( )

2 1( ) ( )b r t r T t b r T tPut Se e CND d e CND d

Where:

2

1 2 1

ln(1/ ) ( / 2)( ) ; d

b T td d T t

T t

With: t= t1= Starting time of the option

PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


20


10. TIME SWITCH OPTIONS

DESCRIPTION

A discrete time-switch call option, introduced by Pechtl (1995), pays an

amount A t at maturity T for each time interval t the corresponding

asset price i tS has exceeded the strike price X. The discrete time-switch

put option gives a similar payoff A t at maturity T for each time interval t

the asset price i tS has been below the strike price X.


2

1

ln( / ) ( / 2)nrT

i

S X b i tCall Ae N t

i t

2

1

ln( / ) ( / 2)nrT

i

S X b i tPut Ae N t

i t

With:

A: accumulated amount

/n T t

If some of the option's total lifetime has already passed, it is necessary to

add a fixed amount At Ae -rT m to the option pricing formula, where m is the

number of time units where the option already has fulfilled its condition.

PAYOFFS



INSTRUMENT PRICE



21


11.A. SIMPLE CHOOSER OPTIONS

DESCRIPTION

A simple chooser option gives the right to choose whether the option is to be

a standard call or put after a time t1, with strike X and time to maturity T2.

The payoff from a simple chooser option at time t1 (t1 < T2) is

1 2 2 2( , , , ) max ( , , ), ( , ,GBS GBSw S X t T Call S X T Put S X T

Where 2( , , )GBSCall S X T and 2( , , )GBSPut S X T

are the general Black-

Scholes call and put formulas.


A simple chooser option can be priced using the formula originally published

by Rubinstein (1991c):

2 2

2 2

( )

2

( )

1

( ) ( )

( ) ( )

b r T rT

b r T rT

Payoff w Se CND d Xe CND d T

Se CND y Xe CND y t

Where

2 2

2 2 1

2 1

ln( / ) ( / 2) ln( / ) / 2 ; y =

S X b T S X bT td

T t

PAYOFFS

The payoff of this model can be represented as follows

NB : "Payoff" Chart represents prices seven days before expiry, not payoffs formulas

INSTRUMENT PRICE

The price of this model according to the price of the underlying asset and the time to maturity can be represented as follows

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot


22


11.B. COMPLEX CHOOSER OPTIONS

DESCRIPTION

A Complex chooser option gives the right to choose whether the option is to

be a standard call option after a time t, with time to expiration CT and strike

CX, or a put option with time to maturity PT

and strike PX. The difference

with regard to simple chooser options is that the calls and the puts will have

different strikes ( CX and PX

) and maturities ( CT and PT

).

The payoff from a complex chooser option at time t (t < CT, T) is

( , , , , , ) max ( , , ), ( , ,C P C P GBS C C GBS P Pw S X X t T T Call S X T Put S X T

Where ( , , )GBS C CCall S X T

and ( , , )GBS P PPut S X T

are the general Black-

Scholes call and put formulas.


A Complex chooser option can be priced using the formula originally

published by Rubinstein (1991c):

( )

1 1 1 2 1 1

( )

1 2 2 2 2 2

( , , ) ( , , )

( , , )+ ( , , )

C C

P P

b r T rT

C C

b r T rT

P P

w Se M d y X e M d y T

Se M d y X e M d y T

Where 2

1 2 1

ln( / ) ( / 2) d

S I b td d t

t

2 2

1 2

1 2

ln( / ) ( / 2) ln( / ) ( / 2) y

/ /

C C P P

C P

C P

S X b T S X b Ty

T T

t T t T

S = The spot of the underlying asset

b = The cost of carry

r = The risk free rate

X = The strike price

1t = Time to when the holder must choose call or put

2T= Time to maturity

CT= The time to maturity of the call.

PT= The time to maturity of the put.

M(a,d; ρ) = The cumative bivariate normal distribution function.

N(x) = The normal distribution function

And I is the solution to ( )( )( ) ( ) ( )( )

1 1 2 2

2 2

1 2

( ) ( ) ( ) ( ) 0

ln( / ) ( / 2)( ) ln( / ) ( / 2)( )With and z

pC C Pr T tb r T t r T t b r T t

C C P p

C C P P

C P

Ie N z X e N z T t Ie N z X e N z T t

I X b T t I X b T tz

T t T t


23


PAYOFFS

The payoff of this model can be represented as follows (for buying the option):

NB : "Payoff" Chart represent prices seven days before expiry, not payoffs formulas

INSTRUMENT PRICE

The price of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for buying the option)

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


24


12. OPTIONS ON OPTIONS

DESCRIPTION

This pricer allows to price options on options, namely, call on call, put on call,

call on put and put on put. The pricing of such options is based on works of

Geske(1979), Hodges and selby (1987) and Rubinstein (1999).


CALL ON CALL

2 2 1

1 2 2

( )

1 1 1 2 2 2 2

( , , ) ;0

( , , ) ( , , ) ( )

GBS

b r T rT rt

call

Payoff Max Call S X T X

Call Se M z y X e M z y X e N y

2

11 2 1 1

1

2

1 21 2 1 2

2

1 2

ln( / ) ( / 2) y

ln( / ) ( / 2) z

/

S I b ty y t

t

S X b Tz z T

T

t T

1X : strike price of the underlying option

2X : strike price of the option on the option

2T : time to maturity of the underlying option

1t : time to maturity of the option on option

1 2( , , )GBSCall S X T: the black-scholes generalized formula with

strike 1X and time to maturity 2T

M(a,d; ρ) = The cumative bivariate normal distribution function

PUT ON CALL

2 2 1

2 1 2

( )

1 2 2 1 1 2 2

( , , );0

( , , ) ( , , ) ( )

GBS

rT b r T rt

Call

Payoff Max X Call S X T

Put X e M z y Se M z y X e N y

Where the value I is found by solving the equation

1 2 1 2( , , )GBSCall I X T t X

CALL ON PUT

2 2 1

1 2 2

( )

1 2 2 1 1 2 2

( , , ) ;0

( , , ) ( , , ) ( )

GBS

rT b r T rt

put

Payoff Max Put S X T X

Call X e M z y Se M z y X e N y

PUT ON PUT

2 2 1

2 1 2

( )

1 1 1 2 2 2 2

( , , );0

( , , ) ( , , ) ( )

GBS

b r T rT rt

put

Payoff Max X Put S X T

Put Se M z y X e M z y X e N y

Where the value I is found by solving the equation1 2 1 2( , , )GBSPut I X T t X


25


PAYOFFS

The payoffs of this model can be represented as follows: (for 4 positions: buying a call on call, buying a call on put, buying a put on call, buying a put on put )

Call on Call Call on Put

Put on Call Put on Put NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas

INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity could be represented as follows (for 4 positions: buying a call on call, buying a call on put, buying a put on call, buying a put on put )

Call on Call Call on Put

Put on Call Put on Put

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


26


13. WRITER EXTENDIBLE OPTIONS

DESCRIPTION

In general, extendible options are options where maturity can be extended.

Such options can be found embedded in several financial contracts. For

example, corporate warrants have frequently given the corporate issuer the

right to extend the life of the warrants. Another example is options on real

estate where the holder can extend the expiration by paying an additional

fee. Pricing of such extendible options was introduced by Longstaff (1990). In

particular, Writer extendible options can be exercised at their initial maturity

date 1t but are extended to 2T if the option is out-of-the-money at 1t .


EXTENDIBLE CALL

Payoff

1 1

1 2 1 2

2 2 1

( ) if ( , , , , )

Call (S,X ,T -t ) elseGBS

S X S XCall S X X t T

Value

2

2

( )

1 1 1 2

2 1 2 2 1

( , , ) ( , ; )

( , ; )

b r T

GBS

rT

Call Call S X t Se M z z

X e M z T z t

EXTENDIBLE PUT

Payoff

1 1

1 2 1 2

2 2 1

( ) if ( , , , , )

(S,X ,T -t ) elseGBS

X S S XPut S X X t T

Put

Value

2

2

1 1 2 1 2 2 1

( )

1 2

( , , ) ( , ; )

( , ; )

rT

GBS

b r T

Put Put S X t X e M z T z t

Se M z z

Where 2 2

2 2 1 11 2 1 2

2 1

ln( / ) ( / 2) ln( / ) ( / 2) ; z ; /

S X b T S X b tz t T

T t

All formulas with

1X : strike price of the original maturity

2X : strike price of the extendible maturity

2T : time to maturity of the extendible maturity

1t : time to maturity of the extendible option

1 2( , , )GBSCall S X T: the black-scholes generalized formula with

strike 1X and time to maturity 2T



27


PAYOFFS



INSTRUMENT PRICE


0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


28


14. TWO ASSETS CORRELATION OPTIONS

DESCRIPTION

This call option pays off max(S2 - X2; 0) if S1 > X1 and 0 otherwise. The put

pays off max(X2 - S2) if S1 < X1 and 0 otherwise. These options are priced

using the formulas of Zhang (1995).


2( )

2 2 2 1 2 2 2 1( , ; ) ( , ; )b r T rTCall S e M y T y T X e M y y

2( )

2 2 1 2 2 2 1 2( , ; ) ( , ; )b r TrTPut X e M y y S e M y T y T

Where

2 2

1 1 1 1 2 2 2 21 2

1 2

ln( / ) ( / 2) ln( / ) ( / 2) ;

S X b T S X b Ty y

T T

With

1S = The spot of the asset 1; 2S = The spot of the asset 2

1X = Strike of asset 1;

2X = Strike of asset 2

1b = The cost of carry of asset 1 ;

2b = The cost of carry of asset 2;

1 = The volatility of the asset 1;


r = The risk free rate; = Correlation between assets 1 and 2;

T = Time to expiry of the option


PAYOFFS



INSTRUMENT PRICE



29


15. OPTION TO EXCHANGE ONE ASSET FOR ANOTHER

DESCRIPTION

An exchange-one-asset-for-another option gives the holder the right, as its

name indicates, to exchange one asset 2S for another

1S at expiration. The

payoff from an exchange-one-asset-for-another option is

1 1 2 2( ;0)Max Q S Q S .


EUROPEAN CALL

1 2( ) ( )

1 1 1 2 2 2( ) ( )b r T b r TCall Q S e CND d Q S e CND d

where

2

1 1 2 2 1 21 2 1

ln( / ) ( / 2) ;

Q S Q S b b Td d d T

T

2 2

1 2 1 22

and where

1S = The spot of the underlying asset 1



2b = The cost of carry of asset 2


2 = The volatility of the asset 2



= Correlation between assets 1 and 2

1Q= Quantity of asset 1

2Q= Quantity of asset 2

CND = The cumulative normal distribution function

AMERICAN CALL

Bjerksund and Stensland (1993) showed that an American Exchange one asset

for another option (S2 for S1) can be priced using a formula for pricing a plain

vanilla American option, with the underlying asset S1 with a risk-adjusted drift equal

to b1-b2, the strike price equal to S2 , time to maturity T, risk free rate equal to r-b2,

and volatilityequal to (defined in the same way as it is for the European option).


30


PAYOFFS

The payoffs of this model can be represented as follows (for 2 positions: buying a European call in the left side and buying an American call in the right side)


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows for 2 positions: buying a European call in the left side and buying an American call in the right side)

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

5060

7080 90 100 110120130140150

Time to Maturity

Spot

0,02

0,22

0,42

0,620,82

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120130

140150

Time toMaturity

Spot


31


16. EXCHANGE OPTIONS ON EXCHANGE OPTIONS

DESCRIPTION

An Exchange options on exchange options can be found embedded in

sequential exchange opportunities. An example described by Carr (1988) is a

bond holder converting into a stock and later exchanging the shares received

for stocks of an acquiring firm. Those options can be priced analytically using

a model introduced by Carr (1988).


[1] Option to exchange Q*S2 for the option to exchange S2 for S1

The value of the option to exchange the option to exchange a fixed

quantity Q of asset 2S for the option to exchange asset

2S for 1S is :

1 2 2 2

2 1

( ) ( )

1 1 1 1 2 2 2 2 1 2

( )

2 2

( , ; / ) ( , ; /

( )

b r T b r T

b r t

Call S e M d y t T S e M d y t T

QS e CND d

where

2

1 2 1 2 11 2 1 1

1

2

2 1 2 1 13 4 3 1

1

ln( / ) ( / 2) ;

ln( / ) ( / 2);

S IS b b td d d t

t

IS S b b td d d t

t

2

1 2 1 2 21 2 1 2

2

2

2 1 2 1 23 4 3 2

2

ln( / ) ( / 2) ; y

ln( / ) ( / 2);

S S b b Ty y T

T

S S b b Ty y y T

T

2 2

1 2 1 22

[2] Option to exchange the option to exchange S2 for S1 in return for Q*S2

The value of the option to exchange asset 2S for

1S in return for a

fixed quantity Q of asset 2S is :

2 2 1 2

2 1

( ) ( )

2 3 2 1 2 1 4 1 1 2

( )

2 3

( , ; / ) ( , ; / )

( )

b r T b r T

b r t


QS e CND d

I is the unique critical price ratio1 2 1

2 2 1

( )( )

11 ( )( )

2

b r T t

b r T t

S eI

S e

solving

1 1 2

2

1 2 1

1 2 1 2 1

2 1

( ) ( )

ln( ) ( ) / 2;

I N z N z Q

I T tz z z T t

T t


32


[3] Option to exchange Q*S2 for the option to exchange S1 for S2

The value of the option to exchange a fixed quantity Q of asset 2S for the

option to exchange asset 1S for

2S is:

2 2 1 2

2 1

( ) ( )

2 3 3 1 2 1 4 4 1 2

( )

2 3

( , ; / ) ( , ; / )

( )

b r T b r T

b r t


QS e CND d

[4] Option to exchange the option to exchange S1 for S2 in return for Q*S2

The value of the option to exchange the option to exchange asset 1S for

2S

in return for a fixed quantity Q of asset 2S is :

1 2 2 2

2 1

( ) ( )

1 1 4 1 2 2 2 3 1 2

( )

2 2

( , ; / ) ( , ; / )

( )

b r T b r T

b r t


QS e CND d

where I is now the unique critical price ratio 2 2 1

1 2 1

( )( )

22 ( )( )

1

b r T t

b r T t

S eI

S e

that solves

1 2 2

2

2 2 1

1 2 1 2 1

2 1

( ) ( )

ln( ) ( ) / 2;

N z I N z Q

I T tz z z T t

T t

where



1b = The cost of carry of the asset 1

2b = The cost of carry of the asset 2


1 = Volatility of asset 1

2 = Volatility of asset 2

1t = Time to expiration of the "original" option.

2T = Time to expiration of the underlying option (T2 > t1)

= Correlation between assets 1 and 2.

Q = Quantity of asset delivered if option is exercised

CND = The cumulative normal distribution function


33


PAYOFFS

The payoffs of this model can be represented as follows:

[1] Q2S2 for Option (S2 for S1) [2] Q2S2 for Option (S1 for S2)

[3] Option (S2 for S1) for Q2S2 [4] Option (S1 for S2) for Q2S2


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows:

[1] Q2S2 for Option (S2 for S1) [2] Q2S2 for Option (S1 for S2)

[3] Option (S2 for S1) for Q2S2 [4] Option (S1 for S2) for Q2S2


34


17. OPTIONS ON THE MAXIMUM OR THE MINIMUM OF TWO RISKY ASSETS

DESCRIPTION

These options on the minimum or maximum of two risky assets are priced by

using the formula of Stulz (1982) witch have later been extended and

discussed by Johnson (1987), Rubinstein (1991) and others.


[1] CALL ON THE MAXIMUM OF TWO ASSETS

1 2: min( , ) ,0Payoff Max S S X

1

2

( )

min 1 2 1 1 1

( )

2 2 2 1 1 2 2

( , , , ) , ;

( , ; ) ( , ; )

b r T

b r T rT

Call S S X T S e M y d

S e M y d T Xe M y T y T

2 2

1 2 1 2 1 1 11

1

2

2 2 22

2

ln( / ) ( / 2) ln( / ) ( / 2)Where ;

ln( / ) ( / 2)

S S b b T S X b Td y

T T

S X b Ty

T

2 2 1 2 2 11 2 1 2 1 22 ; ;

[2] CALL ON THE MAXIMUM OF TWO ASSETS

1 2: max( , ) ,0Payoff Max S S X

1 2( ) ( )

min 1 2 1 1 1 2 2 2

1 1 2 2

( , , , ) , ; ( , ; )

1 ( , ; )

b r T b r T

rT

Call S S X T S e M y d S e M y d T

Xe M y T y T

[3] PUT ON THE MINIMUM OF TWO ASSETS

1 2: min( , ),0Payoff Max X S S

min 1 2 min 1 2 min 1 2( , , , ) ( , ,0, ) ( , , , )rTPut S S X T Xe Call S S T Call S S X T

1 1 2( ) ( ) ( )

min 1 2 1 1 2Where ( , ,0, ) ( ) ( )b r T b r T b r

Call S S T S e S e CND d S e CND d T

[4] PUT ON THE MAXIMUM OF TWO ASSETS

1 2: max( , ),0Payoff Max X S S

max 1 2 max 1 2 max 1 2( , , , ) ( , ,0, ) ( , , , )rTPut S S X T Xe Call S S T Call S S X T

2 1 2( ) ( ) ( )

max 1 2 2 1 2Where ( , ,0, ) ( ) ( )b r T b r T b r

Call S S T S e S e CND d S e CND d T


35


PAYOFFS

The payoffs of this model can be represented as follows:

[1] Call on Minimum [2] Call on Maximum

[3] Put on Minimum [4] Put on Maximum NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formula

INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows:

[1] Call on Minimum [2] Call on Maximum

[3] Put on Minimum [4] Put on Maximum

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot 0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080

90 100 110 120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot


36


18. SPREAD OPTION APPROXIMATION

DESCRIPTION

A European spread option is constructed by buying and selling equal number

of options of the same class on the same underlying asset but with different

strike prices or expiration dates. They can be valued using the standard Black

Scholes (1973) model by performing the following transformation, as

originally shown by Kirk(1995).


CALL SPREAD

11 2 2

2

: ( ,0) max 1,0 ( )S

Payoff Max S S X S XS X

2( )

2 2 1 2( ) ( ) ( )b r T rTCall Q S e Xe SN d N d

PUT SPREAD

11 2 2

2

: ( ,0) max 1 ,0 ( )S

Payoff Max X S S S XS X

2( )

2 2 2 1( ) ( ) ( )b r T rTPut Q S e Xe N d SN d

Where 2

1 2 1

ln( ) ( / 2) ;

S Td d d T

T

1

2

( )

1 1

( )

2 2

b r T

b r T rT

Q S eS

Q S e Xe

And the volatility can be approximated by

2 2

1 2 1 2( ) 2F F

Where 2

2

( )

2 2

( )

2 2

b r T

b r T rT

Q S eF

Q S e Xe

where

= The spot of the underlying asset 1

= The spot of the underlying asset 2

= Quantity of asset 1

= Quantity of asset 2


2b = The cost of carry of asset 2


2 = The volatility of the asset 2

= Correlation between assets 1 and 2



CND = Cumulative Normal Distribution

1S

2S

1Q

2Q

http://en.wikipedia.org/wiki/Strike_price


37


PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080

90 100 110 120130140150

Time to Maturity

Spot0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot


38


19. FLOATING STRIKE LOOKBACK OPTIONS

DESCRIPTION

A floating strike lookback call gives the holder the right to buy the underlying

asset at the lowest price observed, minS , during the option’s lifetime.

Similarly, a floating-strike put gives the holder the right to sell the underlying

asset at the higher price observed, maxS , during the option’s lifetime.


Floating Strike Lookback Call

min: ( ;0)Payoff Max S S

2

( )

1 min 2

22

1 1

min

if b 0 then

( ) ( )

2( )

2

b r T rT

b

bTrT

Call Se N a S e N a

S bSe N a T e N a

b S

1 min 2 1 1 1

And if b=0 we have

( ) ( ) ( ) ( ( ) 1)rT rT rTCall Se N a S e N a Se T n a a N a

Where

2

min1 2 1

ln( / ) ( / 2) a

S S b Ta a T

T

Floating Strike Lookback Put

max: ( ;0)Payoff Max S S

2

( )

max 2 1

2

2

1 1

max

if b 0 then

( ) ( )

2( )

2

rT b r T

b

bTrT

Put S e N b Se N b

S bSe N b T e N b

b S

( )

max 2 1 1 1 1

And if b=0 we have

( ) ( ) ( ) ( ) )rT b r T rTPut S e N b Se N b Se T n b N b b

Where

2

max1 2 1

ln( / ) ( / 2)

S S b Tb b b T

T

PAYOFFS



39



INSTRUMENT PRICE



40


20. FIXED STRIKE LOOKBACK OPTIONS

DESCRIPTION

In a fixed-strike lookback call, the strike is fixed in advance. At expiration, the option pays out the maximum of the difference between the highest

observed price during the option's lifetime, maxSand the strike X, and 0.

Similarly, a put at expiration pays out the maximum of the difference

between the fixed-strike X and the minimum observed price minS , and 0.

Fixed-strike lookback options can be priced using the Conze and Viswanathan (1991) formula.


FIXED-STRIKE LOOKBACK CALL

max: ( ;0)Payoff Max S X

2

( )

1 2

22

1 1

( ) ( )

2( )

2

b r T rT

b

bTrT

Call Se N d Xe N d

S bSe N d T e N d

b X

Where2

1 2 1

ln( / ) ( / 2) ;

S X b Td d d T

T

2

( )

max max 1 max 2

2

2

1 1

max

When X S : ( ) ( ) ( )

2( )

2

rT b r T rT

b

bTrT

Call e S X Se N e S e N e

S bSe N e T e N e

b S

2

max1 2 1

ln( / ) ( / 2)Where and e

S S b Te e T

T

FIXED STRIKE LOOKBACK PUT

min: ( ;0)Payoff Max X S

2

( )

2 1

22

1 1

( ) ( )

2( )

2

rT b r T

b

bTrT

Put Xe N d Se N d

S bSe N d T e N d

b X

2

( )

min min 1 min 2

22

1 1

min

When X S : ( ) ( ) ( )

2( )

2

rT b r T rT

b

bTrT

Put e X S Se N f S e N f

S bSe N f T e N f

b S

2

min1 2 1

ln( / ) ( / 2)Where and

S S b Tf f f T

T


41


PAYOFFS



INSTRUMENT PRICE



42


21. PARTIAL-TIME FLOATING-STRIKE LOOKBACK OPTIONS

DESCRIPTION

In the partial-time floating-strike lookback options, the lookback period is at the beginning of the option's lifetime. Time to expiration is T2, and time to the end of the lookback period is t1 (t1 < T2). Except for the partial lookback period, the partial-time floating-strike lookback option is similar to a standard floating-strike lookback option. However, a partial lookback option must naturally be cheaper than a similar standard floating-strike lookback option. Heynen and Kat (1994) have developed formulas for pricing these options.


PARTIAL TIME FLOATING-STRIKE LOOKBACK CALL

2 2

2

2

22

2

( )

1 1 min 2 1

2

1 22 1 1 1 1 2

min

2

1 1 1 2 1 2

( )

1 1 1 2 1 2

( ) ( )

2 2; /

2

, ; 1 /

, ; 1 /

b r T rT

b

rT

b

bT

b r T

Call Se N d g S e N d g

b t b TSM f d g t T

SSeb

e M d g e g t T

Se M d g e g t T

2

2 1 2

min 2 2 1 1 2

2( ) ( )

2 2 1

( , ; / )

1 ( ) ( )2

rT

b T t b r T

S e M f d g t T

e Se N e g N fb

The factor enables the creation of so called “fractional” lookback options

where the strike is fixed at some percentage above or below the actual

extreme, 1 for calls and 0 1 for puts.

Where

2

0 21 2 1 2

2

ln( / ) ( / 2)

S M b Td d d T

T

2

2 11 2 1 2 1

2 1

( / 2)( ) e

b T te e T t

T t

2

0 11 2 1 1

1

ln( / ) ( / 2)

S M b tf f f t

t

1 2

2 2 1

ln( ) ln( ) gg

T T t

Where

min

0

max

if call

if put

SM

S


43


PARTIAL TIME FLOATING-STRIKE LOOKBACK PUT

2 2

2

2

22

2

2

( )

max 2 1 1 1

2

1 22 1 1 1 1 2

max

2

1 1 1 2 1 2

( )

1 1 1 2 1 2

max 2

( ) ( )

2 2; /

2

, ; 1 /

, ; 1 /

( ,

rT b r T

b

rT

b

bT

b r T

rT

Put S e N d g Se N d g

b t b TSM f d g t T

SSeb

e M d g e g t T

Se M d g e g t T

S e M f

2 1 2

2 1 1 2

2( ) ( )

2 2 1

; / )

1 ( ) ( )2

b T t b r T

d g t T

e Se N e g N fb

PAYOFFS



INSTRUMENT PRICE



44


22. PARTIAL-TIME FIXED-STRIKE LOOKBACK OPTIONS

DESCRIPTION

For Partial-Time Fixed-Strike Lookback option, the lookback period starts at a

predetermined date 1t after the option contract is initiated. The partial time

fixed-strike lookback call payoff is given by the maximum of the highest

observed price of the underlying asset in the lookback period, in excess of

the strike price X, and 0. The put pays off the maximum of the fixed-strike

price X minus the minimum observed asset price in the lookback period

2 1( )T t minS , and 0. This option is naturally cheaper than a similar standard

fixed-strike lookback option. Partial-time fixed strike lookback options can be

priced analytically using a model introduced by Heynen and Kat (1994).


PARTIAL TIME FIXED-STRIKE LOOKBACK CALL

2 2

2

2

2

2 2

2 1

( )

1 2

2

2 12

1 1 1 2

1 1 1 2

( )

1 1 1 2 2 2 1 2

2( ) ( )

( ) ( )

2 2; /

2, ; 1 /

, ; 1 / ( , ; / )

12

b r T rT

b

rT

bT

b r T rT

b T t b r T

Call Se N d Xe N d

b T b tSM d f t T

XSeb

e M e d t T

Se M e d t T Xe M f d t T

e Seb

2

1 2( ) ( )N f N e

PARTIAL TIME FIXED-STRIKE LOOKBACK PUT

2 2

2

2

2

2

2

2 1

( )

2 1

2

2 12

1 1 1 2

1 1 1 2

( )

1 1 1 2

2 2 1 2

2( ) (

( ) ( )

2 2, ; /

2, ; 1 /

, ; 1 /

( , ; / )

12

rT b r T

b

rT

bT

b r T

rT

b T t b

Put Xe N d Se N d

b T b tSM d f t T

XSeb

e M e d t T

Se M e d t T

Xe M f d t T

e Seb

2)

2 1( ) ( )r T

N e N f

Where 1 1 1, and d e f are defined under the floating-strike Lookback options.


45


PAYOFFS



INSTRUMENT PRICE



46


23. EXTREME-SPREAD OPTIONS

DESCRIPTION

These options are closer to lookback options than spread options, due to the way the time to maturity is divided. It is divided into two periods: one period

starting today and ending at time 1t , and another period starting at 1t and

ending at the maturity of the option 2T . Extreme spread options can be priced

analytically using a model introduced by Bermin (1996).


EXTREME-SPREAD OPTIONS

1 2 1

1 1 2

( , ) (0, )

max max

(0, ) ( , )

min min

( ) : ( ;0)

( ) : ( ;0)

t T t

t t T

Payoff Call Max S S

Payoff Put Max S S

2 2 1 2( )( )( )

( ) ( ) ( 1) ( )

( ) ( 1) ( )

DT D r T t DT

extreme

Se KN A e Se

Spread KN B N C k e N D

N E k e N F

2 2 2 1 1 2

2 1 2

1 2 1 1 1 1

2 1 1

Where ; ;

; ;

m T m t m TA B C

T t T

m T m t m tD E F

T t t

2

2

2

2 2

1 2

2And where ; 1 ; =

2( )

ln( / ) ; 0.5 ; 0.5

rTe M k

r D

M S r D r D

.

1 if Call 1 if extreme spread 1 ; and = ;

-1 if Put 1 if reverse extreme spread 1

MaximumValue ifM

MinimumValue if

REVERSE EXTREME-SPREAD OPTIONS

1 2 1

1 1 2

( , ) (0, )

min min

(0, ) ( , )

max max

( ) : ( ;0)

( ) : ( ;0)

t T t

t t T

Payoff Call Max S S

Payoff Put Max S S

2

2

2 1 2( )( )

( ) ( )

( 1) ( ) ( )

( 1) ( )

DT

DT

Reverseextreme

D r T t DT

Se KN A N B

Spread k e N C Se KN G

e Se k N H

2 2 1 1 2 1

2 1 2 1

( ) ( )Where ;

T t T tG H

T t T t


47


PAYOFFS

The payoffs of this model can be represented as follows (for 2 positions: buying a call in the left sides and buying a put in the right sides)

Extreme Spread options

Reverse Extreme Spread options


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for 2 positions: buying a call in the left sides and buying a put in the right sides)

Extreme Spread options

Reverse Extreme Spread options


48


24. STANDARD BARRIER OPTIONS

DESCRIPTION

There are four types of single barrier options. The type flag "cdi" denotes a

down-and-in call, "cui" denotes an up-and-in call, "cdo" denotes a down-and-

out call, and "cuo" denotes an up-and-out call. Similarly, the type flags for

the corresponding puts are pdi, pui, pdo, and puo. A down-and-in option

comes into existence if the asset price, S, falls to the barrier level, H. An up-

and-in option comes into existence if the asset price rises to the barrier level.

A down-and-out option becomes worthless if the asset price falls to the

barrier level. An up-and-out option becomes worthless if the asset price rises

to the barrier level. In general a prespecified cash rebate K is included. It is

paid out at option expiration if the option has not been knocked in during its

lifetime for «in» barriers or if the option is knocked out before expiration for

«out » barriers.

European single barrier options can be priced analytically using a model

introduced by Reiner and Rubinstein (1991).


The different formulas use a common set of factors:

( )

1 1

( )

2 2

( ) 2( 1) 2

1 1

( ) 2( 1) 2

2 2

2

2

( ) ( )

( ) ( )

( / ) ( ) ( / ) ( )

( / ) ( ) ( / ) ( )

( ) ( / ) (

b r T rT

b r T rT

b r T rT

b r T rT

rT

A Se N x Xe N x T

B Se N x Xe N x T

C Se H S N y Xe H S N y T

D Se H S N y Xe H S N y T

E Ke N x T H S N

2 )

( / ) ( ) ( / ) ( 2 )

y T

F K H S N z H S N z T

Where

1 2

2

1 2

22

2 2

ln( / ) ln( / )(1 ) ; (1 )

ln( / ) ln( / )(1 ) ; (1 )

ln( / ) / 2 2 ; ;

S X S Hx T x T

T T

H SX H Sy T y T

T T

H S b rz T

T


49


”IN” BARRIERS

Down-and-in Call S>H

: ( ;0) if S H before T else K at expiration

C ( ) =1, 1

C ( ) =1, 1

di

di

Payoff Max S X

X H C E

X H A B D E

Up-and-in Call S<H


C ( ) =-1, =1

C ( ) =-1, =1

ui

ui

Payoff Max S X

X H A E

X H B C D E

Down-and-in put S>H


P ( ) =1, = -1

P ( ) =1, = -1

di

di

Payoff Max X S

X H B C D E

X H A E

Up-and-in Put S<H


P ( ) =-1, = -1

P ( ) =-1, = -1

ui

ui

Payoff Max X S

X H A B D E

X H C E

“OUT” BARRIERS

Down-and-out Call S>H

: ( ;0) if S> H before T else K at hit

C ( ) =1, =1

C ( ) =1, =1

do

do

Payoff Max S X

X H A C F

X H B D F

Up-and-out Call S<H

: ( ;0) if S< H before T else K at hit

C ( ) =-1, =1

C ( ) =-1, =1

uo

uo

Payoff Max S X

X H F

X H A B C D F

Down-and-out put S>H

: ( ;0) if S> H before T else K at hit

( ) =1, =-1

( ) =1, =-1

do

do

Payoff Max X S

P X H A B C D F

P X H F

Up-and-out Put S<H

: ( ;0) if S< H before T else K at hit

( ) =-1, =-1

( ) =-1, =-1

uo

uo

Payoff Max X S

P X H B D F

P X H A C F


50


PAYOFFS

The payoffs of this model can be represented as follows (Buying positions, Rebate = 3):

Call Up and In Call Up and Out

Call Down and In Call Down and Out


Put Up and In Put Up and Out

Put Down and In Put Down and Out


51


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions, Rebate = 3):






52


25. DOUBLE BARRIER OPTIONS

DESCRIPTION

A double-barrier option is knocked either in or out if the underlying price

touches the lower boundary L or the upper boundary U prior to expiration.

The formulas below pertain only to double knock-out options. The price of a

double knock-in call is equal to the portfolio of a long standard call and a

short double knock-out call, with identical strikes and time to expiration.

Similarly, a double knock-in put is equal to a long standard put and a short

double knock-out put. Doublebarrier options can be priced using the Ikeda

and Kuintomo (1992.)


CALL UP-AND-OUT-DOWN-AND-OUT

1 32

1 2

1( )

1 2 3 4

2

1 2

: ( , , , ) ( ;0) if L<S<U before T else 0.

( ) ( ) ( ) ( )

( ) ( )

n nb r T

n nn

n

n

rT

Payoff Call S U L T Max S k

U L LCall Se N d N d N d N d

L S U S

U LN d T N d T

L SXe

L

3 21

3 4( ) ( )nn

nN d T N d T

U S

Where

2 2 2 2 2 2

1 2

2 2 2 2 2 2 2 2

3 4

ln( / ( )) ( / 2) ln( / ( )) ( / 2) ; d

ln( / ( )) ( / 2) ln( / ( )) ( / 2) ; d

n n n n

n n n n

SU XL b T SU FL b Td

T T

L XSU b T L FSU b Td

T T

1

2 1 1

1 22 2

2 1

3 2

2 2 21 ; 2

2 21 ;

T

b nn

b nF Ue

Where 1 and 2 determine the curvature of L and U.

PUT UP-AND-OUT-DOWN-AND-OUT

1 2

3

1 2

2

1 2

21

3 4

( )

: ( , , , ) ( ;0) if L<S<U before T else 0.

( ) ( )

( ) ( )

n

n

rT

nn

n

n

n

b r T

Payoff Put S U L T Max X S

U LN y T N y T

L SPut Xe

LN y T N y T

U S

U L

L SSe

3

1 2

1

3 4

( ) ( )

( ) ( )nn

n

N y N y

LN y N y

U S


53


Where

2

2 2 2 2 2 22

1 2

2 2 2 2 2 2 2 2

3 4

ln( / ( )) ( / 2) ln( / ( )) ( / 2); ;

ln( / ( )) ( / 2) ln( / ( )) ( / 2);

n n n nT

n n n n

SU EL b T SU XL b Ty y E Le

T T

L ESU b T L XSU b Ty y

T T

CALL UP-AND-IN-DOWN-AND-IN

Up-and-Out-Down-and-OutCall GBSCall Call

PUT UP-AND-IN-DOWN-AND-IN

Up-and-Out-Down-and-Out GBSPut Put Put


54


PAYOFFS

The payoffs of this model can be represented as follows (Buying positions):

Call Out Call In

Put Out Put in


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions):

Call out Call In

Put Out Put in

0

10

20

30

40

50

60

70

80

90

100

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

80

90

100

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

80

90

100

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

80

90

100

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

01020

30

40

50

60

70

80

90

100

5060

7080

90 100 110 120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

010

20

30

40

50

60

70

80

90

100

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot

0,02

0,18

0,34

0,50,660,820,98

01020

30

40

50

60

70

80

90

100

5060

7080

90 100 110 120130140150

Time to Maturity

Spot0,02

0,18

0,34

0,50,660,820,98

010

20

30

40

50

60

70

80

90

100

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot


55


26. PARTIAL-TIME SINGLE ASSET BARRIER OPTIONS

DESCRIPTION

For single asset partial-time barrier options, the monitoring period for a

barrier crossing is confined to only a fraction of the option's lifetime. There

are two types of partial-time barrier options: partial-time-start (type A) and

partial-time-end (type B). Partial-time-start barrier options (type A) have the

monitoring period start at time zero and end at an arbitrary date before

expiration. Partial-time-end barrier options (Type B) have the monitoring

period start at an arbitrary date before expiration and end at expiration.

Partial-time-end barrier options (type B) are then broken down again into

two categories: B1 and B2. Type B1 is defined such that only a barrier hit or

crossed causes the option to be knocked out. There is no difference between

up and down options. Type B2 options are defined such that a down-and-out

call is knocked out as soon as the underlying price is below the barrier.

Similarly, an up-and-out call is knocked out as soon as the underlying price is

above the barrier. Partial-time barrier options can be priced analytically

using a model introduced by Heynen and Kat (1994).


PARTIAL-TIME-START-OUT OPTIONS: UP-AND-OUT & DOWN-AND-OUT CALLS

TYPE A

2

2

2( 1)

( )

1 1 1 3

2

2 2 2 4

( , ; ) ( , ; )

( , ; ) ( , ; )

b r T

A

rT

HCall Se M d e M f e

S

HXe M d e M f e

S

Where

1 for an up-and-out call (C )

1 for a down-and-out call (C )

uoA

doA

2

21 2 1 2

2

2

21 2 1 2

2

2

11 2 1 1 3 1

1 1

2

14 3 1 2

2

ln( / ) ( / 2) ;

ln( / ) 2 ln( / ) ( / 2) ;

ln( / ) ( / 2) 2 ln( / ); ;

/ 2; ;

S X b Td d d T

T

S X H S b Tf f f T

T

S H b t H Se e e t e e

t t

tbe e t

T


56


PARTIAL-TIME-START-IN OPTIONS (TYPE A)

The price of "in" options of type A can be found using "out" options in

combination with plain vanilla call options computed by the Generalized

Black-Scholes formula (GBS).

Up-and-in Call uiA GBS uoAC Call C

Down-and-in Call diA GBS doAC Call C

PARTIAL-TIME-END-OUT OPTIONS (TYPE B)

Out Call Type B1: No difference between up-and-out and down-and-out options

When x > H, the knock-out call value is given by:

2

1

2

2( 1)

( )

1 1 1 3

2

2 2 2 4

( , ; ) ( , ; )

( , ; ) ( , ; )

b r T

oB

rT

HC Se M d e M f e

S

HXe M d e M f e

S

When X < H, the knock-out call value is given by:

2

1

2

2

2

2( 1)

( )

1 1 3 3

2

2 2 4 4

2( 1)

( )

1 1 1 3

2

2 2

( , ; ) ( , ; )

( , ; ) ( , ; )

( , ; ) ( , ; )

( , ; ) (

b r T

oB

rT

b r T

rT

HC Se M g e M g e

S

HXe M g e M g e

S

HSe M d e M f e

S

HXe M d e M

S

2

2

2 4

2( 1)

( )

1 1 3 3

2

2 2 4 4

, ; )

( , ; ) ( , ; )

( , ; ) ( , ; )

b r T

rT

f e

HSe M g e M g e

S

HXe M g e M g e

S

Where

2

21 2 1 2

2

3 1 4 3 2

2

ln( / ) ( / 2);

2ln( / );

S H b Tg g g T

T

H Sg g g g T

T


57


Down-and-Out Call type B2 (case of X < H)

2

2

2

2( 1)

( )

1 1 3 3

2

2 2 4 4

( , ; ) ( , ; )

( , ; ) ( , ; )

b r T

doB

rT

HC Se M g e M g e

S

HXe M g e M g e

S

Up-and-Out Call type B2 (case of X < H)

2

2

2

2

2

2( 1)

( )

1 1 3 3

2

2 2 4 4

2( 1)

( )

1 1 3 1

2

2 2

( , ; ) ( , ; )

( , ; ) ( , ; )

( , ; ) ( , ; )

( , ; )

b r T

uoB

rT

b r T

rT

HC Se M g e M g e

S

HXe M g e M g e

S

HSe M d e M e f

S

HXe M d e M

S

4 2( , ; )e f


58


PAYOFFS


Call Up and Out (A) Call Down and Out (A)

Put Up and Out (A) Put Down and Out (A)

Call Out (B1) Put Out (B1)

Call Up and Out (B2) Call Down and Out (B2)

Put Up and Out (B2) Put Down and Out (B2)


0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

50 60 70 80 90 100 110 120 130 140 150Spot


59


INSTRUMENT PRICE


Call Up and Out (A) Call Down and Out (A)

Put Up and Out (A) Put Down and Out (A)

Call Out (B1) Put Out (B1)

Call Up and Out (B2) Call Down and Out (B2)

Put Up and Out (B2) Put Down and Out (B2)

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080

90 100 110 120 130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

5060

7080

90 100 110 120 130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

70

5060

7080

90 100 110 120 130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

70

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

70

5060

7080

90 100 110 120130140150

Time to Maturity

Spot0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

70

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

70

5060

7080

90 100 110 120 130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0

10

20

30

40

50

60

70

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot


60


27. TWO ASSET BARRIER OPTIONS

DESCRIPTION

In a two asset barrier option, the underlying asset S1 determines how much

the option is in or out-of-the-money. The other asset S2 is the trigger asset

which is linked to barrier hits. Two-asset barrier options can be priced

analytically using a model introduced by Heynen and Kat (1994).


1 2

1 1

( )

1 2 1 2 23 32

2

2 22 2 4 42

2

( , ; )

2( ) ln( / )exp ( , ; )

2 ln( / )( , ; ) exp ( , ; )

b r T

rT

M d e

w S e H SM d e

H SXe M d e M d e

2

1 1 11 2 1 1

1

2 23 1 4 2

2 2

ln( / ) ( );

2 ln( / ) 2 ln( / );

S X Td d d T

T

H S H Sd d d d

T T

2 2 1 2 21 2 1 1 3 1

2 2

2 224 1 1 1 1 2 2 2

2

ln( / ) ( ) 2ln( / ); ;

2 ln( / ); / 2; / 2

H S T H Se e e T e e

T T

H Se e b b

T

TWO-ASSET "OUT" BARRIERS

1 2

1 2

Down-and-out call (C ) 1; -1

Payoff:Max(S ;0) if S before T else 0 at hit

Up-and-out call (C ) 1; 1

Payoff:Max(S ;0) if S before T else 0 at hi

do

uo

X H

X H

1 2

1 2

t

Down-and-out put (P ) 1; 1

Payoff: ( ;0) if S before T else 0 at hit

Up-and-out put (P ) 1; 1

Payoff: ( ;0) if S before T else 0 at h

do

uo

Max X S H

Max X S H

it

TWO-ASSET "IN" BARRIERS

1 2

1

Down-and-in call (C )

Payoff:Max(S ;0) if S before T else 0 at expiration

Up-and-in call (C )

Payoff:Max(S ;0)

di di GBS do

ui ui GBS uo

C Call C

X H

C Call C

X

2

1 2

if S before T else 0 at expiration

Down-and-in put (P ) P

Payoff: ( ;0) if S before T else 0 at expiration

Up-and-in put (P ) P

di di GBS do

ui di GBS

H

Put P

Max X S H

Put P

1 2Payoff: ( ;0) if S before T else 0 at expiration

uo

Max X S H


61


PAYOFFS

The payoffs of this model can be represented as follows (Buying positions, Payoff1= 20):







62


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions, Payoff1= 20):






63


28. PARTIAL TIME TWO ASSET BARRIER OPTIONS

DESCRIPTION

Partial-time two-asset barrier options are similar to standard two-asset

barrier options, except that the barrier hits are monitored only for a fraction

of the option's lifetime. The option is knocked in or knocked out if Asset 2

hits the barrier during the monitoring period. The payoff depends on Asset 1

and the strike price. Partial-time two-asset barrier options can be priced

analytically using a model introduced by Bermin (1996).


1 2

1 1 1 2

( )

12 1 2 2

3 3 1 22

2

2 2 1 2

2 24 4 1 22

2

( , ; / )

2( ) ln( / )exp ( , ; / )

( , ; / )

2 ln( / )exp ( , ; / )

b r T

rT

M d e t T

w S e H SM d e t T

M d e t T

Xe H SM d e t T

2

1 1 1 21 2 1 1 2

1 2

2 23 1 4 2

2 2 2 2

ln( / ) ( );

2 ln( / ) 2 ln( / );

S X Td d d T

T

H S H Sd d d d

T T

2 2 1 2 1 21 2 1 1 1 3 1

2 1 2 1

2 224 2 1 1 1 2 2 2

2 1

ln( / ) ( ) 2ln( / ); ;

2 ln( / ); / 2; / 2

H S t H Se e e t e e

t t

H Se e b b

t

TWO-ASSET "OUT" BARRIERS

cf specification Pricer n°27: TWO ASSET BARRIER OPTIONS

TWO-ASSET "IN" BARRIERS

cf specification Pricer n°27: TWO ASSET BARRIER OPTIONS


64


PAYOFFS

The payoffs of this model can be represented as follows (Buying positions, Payoff1= 20):







65


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions, Payoff1= 20):






66


29. LOOK-BARRIER OPTIONS

DESCRIPTION

A look-barrier option is the combination of a forward starting fixed strike

Lookback option and a partial time barrier option. The option’s barrier

monitoring period starts at time zero and ends at an arbitrary date before

expiration. If the barrier is not triggered during this period, the fixed strike

Lookback option will be kick off at the end of the barrier tenor. Lookback

barrier options can be priced analytically using a model introduced by

Bermin (1996).


A

2

22

2

2

2 1 2 2

21 2( )

2 / 2 1 2 2

1 2

1 1 1 2

1 2

2 / 1 1 1 2

1 2

, ;

12 2 2

, ;

, ;

2 2, ;

n

b r T

h

rT

h

m t k TM

t TA Se

b m h t h k Te M

t T

m t k TM

t Te X

m h t h k Te M

t T

2

2

2

2

2 1

2

1 1 1 2

21 2

2

1 1 1 2

1 2

2 2( )2 2 1

2 1( )

, ;

22 2

, ;

( )1 1

2 2

b

rT

b

b T t

b r T

m t k TSS M

X t Te

bm h t h k TH

H MX t T

T tN e

b bT tSe

2

1 2

1 2 1

2 1

( )

rTg e Xg

T tN

T t

Where ( ) ( ) and M ( , ; ) , ;N x N x a b M a b

1 if up-and-out call min( , ) when 1 ;

1 if down-and-out put max( , ) when 1

h km

h k

2 2 11 2

2

ln( / ); ln( / ); / 2; / 2;t

h H S k X S b bT

2 22 22 / 2 /2 1 2 1 2 1 2 1

1

1 1 1 1

2h hh t h t m t m h tg N e N N e N

t t t t

2 22 21 / 1 /1 1 1 1 1 1 1 1

2

1 1 1 1

2h hh t h t m t m h tg N e N N e N

t t t t


67


PAYOFFS


Call Up and Out Call Up and In

Put Down and Out Put Down and In


INSTRUMENT PRICE


Call Up and Out Call Up and In

Put Down and Out Put Down and In


68


30. SOFT-BARRIER OPTIONS

DESCRIPTION

A soft-barrier option is similar to a standard barrier option, except that the

barrier is no longer a single level. Rather, it is a soft range between a lower

level and an upper level. Soft-barrier options are knocked in or knocked out

proportionally. Introduced by Hart and Ross (1994), the valuation formula

can be used to price soft-down-and-in call and soft-up-and-in put options.

Soft-barrier options can be priced analytically using a model introduced by

Hart and Ross (1994).


SOFT "IN" BARRIERS

1A

U L

0.5

2

1 1 20.5( ) 2

0.52

1 1 2

0,52

3 2 40.52( 1)

0,52

3 2 4

( )( )

2( 0.5)( ) ( )

( ) ( )( )

2( 0.5)( ) ( )

b r T

rT

UN d N d

SXSXA Se S

LN e N e

SX

UN d N d

SXSXXe S

LN e N e

SX

Where

1 if down-and-in call

1 if up-and-in put

2 2

2 2

1 2 1 3

2

4 3 1 2 1

2

3 4 3

0.5 ( 0.5)( 0.5) 0.5 ( 0.5

1 2

ln( / ( )) ln( / ( )); ( 0.5) ; ( 1)

ln( / ( ))( 0.5) ; ; ( 0.5)

ln( / ( ))( 1) ; ( 0.5)

;T T

U SX U SXd T d d T d T

T T

L SXd d T e T e e T

T

L SXe T e e T

T

e e

2

)( 1.5)

2

/ 2;

b

SOFT « OUT » BARRIERS

Soft down-and-out call = standard call - soft down-and-in call.

Soft up-and-out put = standard put - soft up-and-in put

Standard Calls and puts are calculated with the generalized Black scholes

formula.


69


PAYOFFS





INSTRUMENT PRICE





70


31. GAP OPTIONS

DESCRIPTION

Gap options are similar to plain options, except for the payoff. The payoff is a

function of the exercise price. The payoff on a gap option depends on all of

the factors of a plain option, but it is also affected by the gap amount, which

can be either positive or negative. A gap call option is equivalent to being

long an asset-or-nothing call and short a cash-or-nothing call. A gap put

option is equivalent to being long a cash-or-nothing put and short an asset-

or-nothing put. Gap options can be priced analytically using a model



2

1

1

( )

1 2 2

0 if S X( ) :

if S > X

( ) ( )b r T rT

Payoff CallS X

Call Se N d X e N d

2

1

1

( )

2 2 1

0 if S X( ) :

if S < X

( ) ( )rT b r T

Payoff PutX S

Put X e N d Se N d

Where

2

1

1 2 1

ln( / ) ( / 2) ;

S X b Td d d T

T

Notice that the payoff from this option can be negative, depending on the

settings of Xi and X2

PAYOFFS



INSTRUMENT PRICE



71


32. CASH-OR-NOTHING OPTIONS

DESCRIPTION

In a cash-or-nothing option, a predetermined amount is paid if the asset is, at

expiration, above for a call or below for a put some strike level, independent

of the path taken. These options require no payment of an exercise price.

Instead, the exercise price determines whether or not the option returns a

payoff. The value of a cash-or-nothing call (put) option is the present value of

the fixed cash payoff multiplied by the probability that the terminal price will

be greater than (less than) the exercise price. Cash-or-nothing options can be

priced analytically using a model introduced by Reiner and Rubinstein (1991).


0 if S K( )

if S > K

( )rT

Payoff CallK

Call Ke N d

0 if S K( )

if S < K

( )rT

Payoff PutK

Put Ke N d

Where 2ln( / ) ( / 2)S X b T

dT

PAYOFFS



INSTRUMENT PRICE



72


33. TWO ASSET CASH-OR-NOTHING OPTIONS

DESCRIPTION

Two-asset cash-or-nothing options can be useful building blocks for

constructing more complex exotic options. There are four types of two-asset

cash-or-nothing options:

1. A two-asset cash-or-nothing call pays out a fixed cash amount if

Asset 1 is above Strike 1 and Asset 2 is above Strike 2 at expiration.

2. A two-asset cash-or-nothing put pays out a fixed cash amount if

Asset 1 is below Strike 1 and Asset 2 is below Strike 2 at expiration.

3. A two-asset cash-or-nothing up-down pays out a fixed cash amount

if Asset 1 is above Strike 1 and Asset 2 is below Strike 2 at

expiration.

4. A two-asset cash-or-nothing down-up pays out a fixed cash amount

if Asset 1 is below Strike 1 and Asset 2 is above Strike 2 at

expiration.

Two-asset cash-or-nothing options can be priced analytically using a model

introduced by Heynen and Kat (1996).


Payoffs

1 1 2 2

1 1 2 2

1 1 2 2

1 1 2 2

if S and S1 :

0 else

if S and S2 :

0 else

if S and S3 :

0 else

if S and S4 :

0 else

K X XPayoff

K X XPayoff

K X XPayoff

K X XPayoff

Values

1,1 2,2

1,1 2,2

1,1 2,2

1,1 2,2

1 ( , ; )

2 ( , ; )

3 ( , ; )

4 ( , ; )

rT

rT

rT

rT

Value Ke M d d

Value Ke M d d

Value Ke M d d

Value Ke M d d

Where

2

,

ln( / ) ( / 2)i j i i

i j

i

S X b Td

T


73


PAYOFFS


(1)Call (asset 1) (2) Put (Asset 2)

(3)Up Down(Asset1) (4) Down Up(Asset2)


INSTRUMENT PRICE


(1)Call (asset 1) (2) Put (Asset 2)

(3)Up Down(Asset1) (4) Down Up(Asset2)


74


34. ASSET-OR-NOTHING OPTIONS

DESCRIPTION

In an asset-or-nothing option, the asset value is paid if the asset is, at

expiration, above for a call or below for a put some strike level, independent

of the path taken. The exercise price is never paid. Instead, the value of the

asset relative to the exercise price determines whether or not the option

returns a payoff. The value of an asset-or-nothing call (put) option is the

present value of the asset multiplied by the probability that the terminal

price will be greater than (less than) the exercise price. Asset-or-nothing

options can be priced analytically using a model introduced by Cox and

Rubinstein (1985).


( )

0 if S X( )

S if S > K

( )b r T

Payoff Call

Call Se N d

( )

0 if S X( )

S if S < X

( )b r T

Payoff Put

Put Se N d

Where 2ln( / ) ( / 2)S X b T

dT

PAYOFFS



INSTRUMENT PRICE


0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

50 60 70 80 90 100 110 120 130 140 150Spot

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0102030405060708090100110120130140150

5060

7080

90 100 110 120130140150

Time to Maturity

Spot

0,02

0,18

0,34

0,50,660,820,98

0102030405060708090

100110120130140150

50 60 70 80 90100 110

120130

140150

Time toMaturity

Spot


75


35. SUPERSHARE OPTIONS

DESCRIPTION

A supershare is a financial instrument that represents a contingent claim on

a fraction of the underlying portfolio. The contingency is that the value of the

portfolio must lie between a lower LX and an upper bound

HX on its

expiration date. If the value lies within these boundaries, the supershare is

worth a proportion of the assets underlying the portfolio, else the

supershare expires worthless. A supershare has a payoff that is basically like

a spread of two asset-or-nothing calls, in which the owner of a supershare

purchases an asset-or-nothing call with an strike price of Lower Strike and

sells an asset-or-nothing call with an strike price of Upper Strike. Supershare

options can be priced analytically using a model introduced by Hakansson

(1976).


/ if X

0 else

L L HS X S XPayoff

( )

1 2( / )b r T

Lw Se X N d N d

Where 2 2

1 2

ln( / ) ( / 2) ln( / ) ( / 2); L HS X b T S X b T

d dT T

PAYOFFS

The payoff of this model can be represented as follows (buying position):

NB: "Payoff" Chart represents prices seven days before expiry, not payoff formula

INSTRUMENT PRICE

The price of this model according to the price of the underlying asset and the time to maturity can be represented as follows (buying position):

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

50 60 70 80 90 100 110 120 130 140 150Spot

0,02

0,18

0,34

0,50,660,820,98

0001

1

1

1

1

2

2

2

5060

7080

90 100 110 120130140150

Time to Maturity

Spot


76


36. BINARY BARRIER OPTIONS

DESCRIPTION

Binary-barrier options combine characteristics of both binary and barrier

options. They are path dependent options with a discontinuous payoff.

Similar to barrier options, the payoff depends on whether or not the asset

price crosses a predetermined barrier. There are 28 different types of binary

barrier options, which can be divided into two main categories: Cash-or-

nothing and Asset-or-nothing barrier options.

Cash-or-nothing barrier options pay out a predetermined cash amount or

nothing, depending on whether the asset price has hit the barrier.

Asset-or-nothing barrier options pay out the value of the asset or nothing,

depending on whether the asset price has crossed the barrier.

The barrier monitoring frequency can be adjusted to account for discrete

monitoring using an approximation developed by Broadie, Glasserman, and

Kou (1995). Binary-barrier options can be priced analytically using a model



We begin by introducing 9 factors:

( )

1 1

1 1

( )

2 2

2 2

2( 1)( )

3 1

2

3 1

2( 1)( )

4 2

2

4 2

5

/

/

/

/

/ / 2

b r T

rT

b r T

rT

b r T

rT

b r T

rT

A Se N x

B Ke N x T

A Se N x

B Ke N x T

A Se N H S N y

B Ke H S N y T

A Se N H S N y

B Ke H S N y T

A K H S N z H S N z T

Where K is a prespecified cash amount. The binary variables and

each take the value 1 or —1. Moreover:

1 2

2

1 2

22

2 2

ln( / ) ln( / )1 ; 1

ln( / ) ln( / )1 ; 1

ln( / ) / 2 2; ;

S X S Hx T x T

T T

H SX H Sy T y T

T T

H S b rz T

T

By using 1A to

5A and1B to

4B in different combinations, we can price

the 28 binary barrier options described below :


77


Binary Barrier Options case X>H case X<H

[1] Down-and-in cash-(at-hit)-or-nothing (S>H) 1 A5 A5

[2] Up-and-in cash-(at-hit)-or-nothing (S<H) -1 A5 A5

[3] Down-and-in asset-(at-hit)-or-nothing (K=H) (S>H) 1 A5 A5

[4] Up-and-in asset-(at-hit)-or-nothing (K=H)(S<H) -1 A5 A5

[5] Down-and-in cash-(at-expiry)-or-nothing (S>H) 1 -1 B2+B4 B2+B4

[6] Up-and-in cash-(at-expiry)-or-nothing (S<H) -1 1 B2+B4 B2+B4

[7] Down-and-in asset-(at-expiry)-or-nothing (S>H) 1 -1 A2+A4 A2+A4

[8] Up-and-in asset-(at-expiry)-or-nothing (S<H) -1 1 A2+A4 A2+A4

[9] Down-and-out cash-(at-expiry)-or-nothing (S>H) 1 1 B2-B4 B2-B4

[10] Up-and-out cash-(at-expiry)-or-nothing (S<H) -1 -1 B2-B4 B2-B4

[11] Down-and-out asset-(at-expiry)-or-nothing (S>H) 1 1 A2-A4 A2-A4

[12] Up-and-out asset-(at-expiry)-or-nothing (S<H) -1 -1 A2-A4 A2-A4

[13] Down-and-in cash-(at-expiry)-or-nothing call (S>H) 1 1 B3

B1-B2+B4

[14] Up-and-in cash-(at-expiry)-or-nothing call (S<H) -1 1 B1

B2-B3+B4

[15] Down-and-in asset-(at-expiry)-or-nothing call (S>H) 1 1 A3

A1-A2+A4

Binary Barrier Options case X>H case X<H [16] Up-and-in asset-(at-expiry)-or-

nothing call (S<H) -1 1 A1 A2-A3+A4

[17] Down-and-in cash-(at-expiry)-or-nothing put (S>H) 1 -1 B2-B3+B4 B1

[18] Up-and-in cash-(at-expiry)-or-nothing put (S<H) -1 -1 B1-B2+B4 B3

[19] Down-and-in asset-(at-expiry)-or-nothing put (S>H) 1 -1 A2-A3+A4 A1

[20] Up-and-in asset-(at-expiry)-or-nothing put (S<H) -1 -1 A1-A2+A3 A3

[21] Down-and-out cash-(at-expiry)-or-nothing call (S>H) 1 1 B1-B3 B2-B4

[22] Up-and-out cash-(at-expiry)-or-nothing call (S<H) -1 1 0

B1-B2+B3-B4

[23] Down-and-out asset-(at-expiry)-or-nothing call (S>H) 1 1 A1-A3 A2-A4

[24] Up-and-out asset-(at-expiry)-or-nothing call (S<H) -1 1 0

A1-A2+A3-A4

[25] Down-and-out cash-(at-expiry)-or-nothing put (S>H) 1 -1

B1-B2+B3-B4 0

[26] Up-and-out cash-(at-expiry)-or-nothing put (S<H) -1 -1 B2-B4 B1-B3

[27] Down-and-out asset-(at-expiry)-or-nothing put (S>H) 1 -1

A1-A2+A3-A4 0

[28] Up-and-out asset-(at-expiry)-or-nothing put (S<H) -1 -1 A2-A4 A1-A3


78


PAYOFFS


Down and In Cash (hit) or Nothing Up and In Cash (hit) or Nothing

Down and In Asset (hit) or Nothing Up and In Asset (hit) or Nothing


Down and In Cash (expiry) or Nothing Up and In Cash (expiry) or Nothing

Down and In Asset (expiry) or Nothing Up and In Asset (expiry) or Nothing


79


PAYOFFS


Down and Out Cash (expiry) or Nothing Up and Out Cash (expiry) or Nothing

Down and In Cash or Nothing Call Up and In Cash or Nothing Call


Down and Out Asset (expiry) or Nothing Up and Out Asset (expiry) or Nothing

Down and In Asset or Nothing Call Up and In Asset or Nothing Call


80


PAYOFFS


Down and In Cash or Nothing Put Up and In Cash or Nothing Put

Down and Out Cash or Nothing Call Up and Out Cash or Nothing Call


Down and In Asset or Nothing Put Up and In Asset or Nothing Put

Down and Out Asset or Nothing Call Up and Out Asset or Nothing Call


81


PAYOFFS


Down and Out Cash or Nothing Put Up and Out Cash or Nothing Put


Down and Out Asset or Nothing Put Up and Out Asset or Nothing Put


82


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Down and In Cash (hit) or Nothing Up and In Cash (hit) or Nothing

Down and In Asset (hit) or Nothing Up and In Asset (hit) or Nothing

Down and In Cash (expiry) or Nothing Up and In Cash (expiry) or Nothing

Down and In Asset (expiry) or Nothing Up and In Asset (expiry) or Nothing


83


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Down and Out Cash (expiry) or Nothing Up and Out Cash (expiry) or Nothing

Down and In Cash or Nothing Call Up and In Cash or Nothing Call

Down and Out Asset (expiry) or Nothing Up and Out Asset (expiry) or Nothing

Down and In Asset or Nothing Call Up and In Asset or Nothing Call


84


INSTRUMENT PRICE


Down and In Cash or Nothing Put Up and In Cash or Nothing Put

Down and Out Cash or Nothing Call Up and Out Cash or Nothing Call

Down and In Asset or Nothing Put Up and In Asset or Nothing Put

Down and Out Asset or Nothing Call Up and Out Asset or Nothing Call


85


INSTRUMENT PRICE

The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Down and Out Cash or Nothing Put Up and Out Cash or Nothing Put

Down and Out Asset or Nothing Put Up and Out Asset or Nothing Put


86


37. ASIAN OPTIONS 1: GEOMETRIC AVERAGE RATE OPTIONS

DESCRIPTION

Asian options are path-dependent options, with payoffs that depend on the average price of the underlying asset or the average exercise price. There are two categories or types of Asian options: average rate options (also known as average price options) and average strike options. The payoffs depend on the average price of the underlying asset over a predetermined time period. An average option is less volatile than the underlying asset, therefore making Asian options less expensive than standard European options. Asian options are commonly used in currency and commodity markets. Asian options are of interest in markets with thinly traded assets. Due to the little effect it will have on the option’s value, options based on an average, such as Asian options, have a reduced incentive to manipulate the underlying price at expiration.


( )

1 2

( ) ( ;0)

( ) ( )rT

Adjusted

Average

b r T

Payoff call Max S X

Call Se N d Xe N d

( )

2 1

( ) ( ;0)

( ) ( )rT

Adjusted

Average

b r T

Payoff Put Max X S

Put Xe N d Se N d

Where

2

1 2 1

2

ln( / ) ( / 2);

1 ; b

2 63

Adjusted Adjusted

Adjusted

Adjusted

Adjusted Adjusted

S X b Td d d T

T

b

PAYOFFS



INSTRUMENT PRICE



87


38. ASIAN OPTIONS 2: THE TURNBULL AND WAKEMAN ARITHMETIC AVERAGE

APPROXIMATION

DESCRIPTION

It is not possible (or very hard) to find a closed-form solution for the value of options on an arithmetic average. The main reason is that when the asset is assumed to be lognormally distributed, the arithmetic average will not itself have a lognormal distribution. Arithmetic average rate options can be priced by the analytical approximation of Turnbull and Wakeman (1991). This approximation adjusts the mean and variance so that they are consistent with the exact moments of the arithmetic average. The adjusted mean,

Ab , and

variance, 2

A , are then used as input in the generalized BSM formula.


( )

1 2

( )

2 1

( ) ( )

( ) ( )

A

A

b r T rT

b r TrT

Call Se N d Xe N d

Put Xe N d Se N d

Where 2

1 2 1

ln( / ) ( / 2); dA A

A

A

S X b Td d T

T

where T is the time to maturity in years.

2 1ln( ) ln( )2 ; bA A A

M Mb

T T

With22

1 1 1(2 ) ( )(2 )

1 2 2 2 2 2 2 2

1 1 1

2 2 1; M

( ) ( )(2 )( ) ( ) 2

bt b t b T tbT b Te e e e eM

b T t b b T t b T t b b

Where 1t is the time to the beginning of the average period.

PAYOFFS



INSTRUMENT PRICE



88


39. ASIAN OPTIONS 3: LEVY'S ARITHMETIC AVERAGE APPROXIMATION

DESCRIPTION

Levy's arithmetic average approximation (1992) is an alternative to the

Turnbull and Wakeman formula described below.


2*

1 2( ) ( )rT

ECall S N d X e N d

Where

2 2

22 2

( )

*

1 2 1

* 2

2 2

(2 )2

2 2

( )

1 ln( )ln( ) ;

2

; ln( ) 2 ln( ) ; D=

2 1 1

2

b r T rT

E

b

A E

b T bT

SS e e

T

Dd X d d V

V

T T MX X S V D rT S

T T

S e eM

bb b

The Asian put value can be found by using the following put-call parity:

2* rT

EPut Call S X e

Where

AS = Arithmetic average of the known asset price fixings.

S = Asset price.

X = Strike price.

r = Risk-free interest rate.

b = Cost-of-carry rate.

T2 = Remaining time to maturity.

T = Original time to maturity.

a = Volatility of natural logarithms of return of the underlying asset.

NB : The formula does not allow for b = 0


89


PAYOFFS



INSTRUMENT PRICE



90


40. FOREIGN EQUITY OPTIONS STRUCK IN DOMESTIC CURRENCY (VALUE IN DOMESTIC

CURRENCY)

DESCRIPTION

As the name indicates, these are options on foreign equity where the strike is

denominated in domestic currency. At expiration, the foreign equity is

translated into the domestic currency. Valuation of these options is achieved

using the formula attributed to Reiner (1992).


The payoff to a European investor for an option linked to the DowJones

index is : *

/ ( / ) ( / ) ( / )

*

/ ( / ) ( / ) ( / )

( ) ( ;0)

( ) ( ;0)

EUR share EUR USD USD share EUR share

EUR share EUR share EUR USD USD share

Payoff Call Max E S X

Payoff Put Max X E S

*

1 2

*

2 1

( ) ( )

( ) ( )

qT rT

rT qT

Call ES e N d Xe N d

Put Xe N d ES e N d

Where

*

*

* 2

1 2 1 *

2 2

* * * *

ln( / ) ( / 2);

2

ES

ES

ES

ES E S ES E S

ES X r q Td d d T

T

S* = Underlying asset price in foreign currency.

X = Delivery price in domestic currency.

r = Domestic interest rate.

q = Instantaneous proportional dividend payout rate of the

underlying asset.

E = Spot exchange rate specified in units of the domestic currency per

unit of the foreign currency.

*

S = Volatility of the underlying asset.

E = Volatility of the domestic exchange rate.

*ES = Correlation between asset and domestic exchange rate.


91


PAYOFFS



INSTRUMENT PRICE



92


41. FIXED EXCHANGE RATE FOREIGN EQUITY OPTIONS - QUANTOS (VALUE IN DOMESTIC

CURRENCY)

DESCRIPTION

A fixed exchange-rate foreign-equity option (Quanto) is denominated in

another currency than that of the underlying equity exposure. The face value

of the currency protection expands or contracts to cover changes in the

foreign currency value of the underlying asset. Quanto options can be priced

analytically using a model published by Dravid, Richardson, and Sun (1993)


*

/ ( / ) ( / ) ( / )

*

/ ( / ) ( / ) ( / )

( ) ( ;0)

( ) ( ;0)

EUR share P EUR USD USD share USD share

EUR share P EUR USD USD share USD share

Payoff Call E Max S X

Payoff Put E Max X S

*

*

( )* *

1 2

( )* *

2 1

( ) ( )

( ) ( )

f S E

f S E

r r q T rT

p

r r q TrT

p

Call E S e N d X e N d

Put E X e N d S e N d

Where

* *

*

* * 2

1 2 1 *

ln( / ) ( / 2);

f ES S

S

S

S X r q Td d d T

T


X* = Delivery price in foreign currency.


rf = Foreign interest rate.


underlying asset.

Ep = Predetermined exchange rate specified in units of domestic

currency per unit of foreign currency.

E* = Spot exchange rate specified in units of foreign currency per

unit of domestic currency.

*


E = = Volatility of the domestic exchange rate.

= Correlation between asset and domestic exchange rate.


93


PAYOFFS



INSTRUMENT PRICE



94


42. EQUITY LINKED FOREIGN EXCHANGE OPTIONS (VALUE IN DOMESTIC CURRENCY)

DESCRIPTION

An equity-linked foreign-exchange option is an option on the foreign

exchange rate and is linked to the forward price of a stock or equity index.

This option can be priced analytically using a model introduced by Reiner

(1992).


*

/ ( / ) ( / ) ( / )

*

/ ( / ) ( / ) ( / )

( ) ( ;0)

( ) ( ;0)

EUR share USD share EUR USD EUR USD

EUR share USD share EUR USD EUR USD

Payoff Call S Max E X

Payoff Put S Max X E

*

*

( )* *

1 2

( )* *

2 1

( ) ( )

( ) ( )

f S E

f S E

r r q TqT

r r q T qT

Call ES e N d XS e N d

Put XS e N d ES e N d

Where

*

2

1 2 1

ln( / ) ( / 2);

f E ES

E

E

E X r r Td d d T

T


X = Currency strike price in domestic currency.


rf = Foreign interest rate.


underlying asset.

E = Spot exchange rate specified in units of the domestic currency

per unit of the foreign currency.

E* = Spot exchange rate specified in units of the foreign

currency per unit of the domestic currency.

*


E =- Volatility of the domestic exchange rate.

= Correlation between asset and the domestic exchange rate.


95


PAYOFFS



INSTRUMENT PRICE



96


43. TAKEOVER FOREIGN EXCHANGE OPTIONS

DESCRIPTION

A takeover foreign exchange call option gives the buyer the right to purchase a specified number of units of foreign currency at a strike price if the corporate takeover is successful. This option can be priced analytically using a model introduced by Schnabel and Wei (1994).


2 1 1 2( , ; ) ( , ; )fr T rT

E ECall N Ee M a T a T Xe M a a

Where

2 2

1 2

ln( / ) ( / 2) ln( / ) ( / 2);

f E V V f E

V E

V N r T E X r r Ta a

T T

Both the strike price X and the currency price E are quoted in units of the

domestic currency per unit of the foreign currency.

PAYOFFS

The payoff of this model can be represented as follows (buying position):


INSTRUMENT PRICE

The price of this model according to the price of the underlying asset and the time to maturity can be represented as follows (buying position):


97


44. EUROPEAN SWAPTIONS IN THE BLACK-76 MODEL

DESCRIPTION

A Swaption (option on Interest Rate Swap IRS) reserves the right for its

holder to purchase a swap at a prescribed time and interest rate in the

future (European Option).

The holder of such a call option has the right, but not the obligation to pay a

fixed interest rate in exchange of a variable interest rate. This option is also

known as “Payer Swaption”. The holder of the equivalent put option has the

right, but not the obligation to receive the fixed interest rate (Receiver

Swaption) and pay the variable interest rate.


1

1

1 2

2 1

11

(1 / )( ) ( ) ( )

11

(1 / )( ) ( ) ( )

t m

rT

t m

rT

F mCall Payer swaption e FN d XN d

F

F mPut Receiver swaption e XN d FN d

F

Where 2

1 2 1

ln( / ) ( / 2);

F X Td d d T

T

t1= Tenor of swap in years; m= compounding per year

F= Underlying Swap Rate; X =Strike Price

T= Time to Maturity; r =Risk-free Rate; = Swap Rate Volatility

PAYOFFS

The payoffs of this model can be represented as follows (for buying position):


INSTRUMENT PRICE

The price of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for buying positions):


98


45. THE VASICEK MODEL FOR EUROPEAN OPTIONS ON ZERO COUPON BONDS

DESCRIPTION

The Vasicek (1977) model is a yield-based one-factor equilibrium model that

assumes that the short rate is normally distributed. The model incorporates

mean reversion and is popular in the academic community—mainly due to

its analytic tractability. The model is not used much by market participants

because it is not ensured to be arbitrage-free relative to the underlying

securities already in the marketplace. This model is given by the following

general formula :

( ) zdr k r dt d

K is the speed of the mean reversion, and is the mean reversion level.


ZERO COUPON BOND VALUE

The price at time t of a discount bond maturing at time T is :

( , ) ( )( , ) ( , ) B t T r tP t T A t T e

Where

( )

2 2 2 2

2

1( , )

( ( , ) )( / 2) ( , )( , ) exp

4

T teB t T

B t T T t B t TA t T

OPTION VALUE

The value of a European option maturing at time T on a zero-coupon bond

that matures at time is:

( , ) ( ) ( , ) ( )

( , ) ( ) ( , ) ( )

p

p

Call P t N h XP t T N h

Put XP t T N h P t N h

Where r(t) is the rate at time t and

2 2 ( )

1 ( , )ln

( , ) 2

(1 )( , )

2

P

P

T t

P

P th

P t T X

eB t


99


PAYOFFS



INSTRUMENT PRICE


Documents

Global Research & Analytics Dpt. · 2019. 9. 19. · 1 The pricing formulas and codes are from his book: “The complete guide to option pricing formulas”, edited by McGraw-Hill