GRA Pricer - 2013 - Valuation & Pricing Solutions

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

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
Chappuis Halder & Cie
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 rT
Call S e CND d X e CND d 
 
( )
2 1. ( ) . ( )rT b r T
Put X e CND d S e CND d 
   
Where d1 and d2 are defined by the following formulas
2
1
ln
2
S
b T
X
d
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:

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
.
S
b T
X
d
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
right side.

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
.
S
b T
X
d
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
0,5
0,66
0,82
0,98
0
0
0
0
0
0
0
0
0
0
0
50
60
70
80 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
.
S
b T
X
d
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 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
5
10
15
20
25
30
35
40
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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 d
b r
T
S e CND d r X e CND d



 

  

( ).
( ).1
1
.
2
. ( ).
( ). . . ( )
2
. . . ( )
b r T
b r T
Put
r T
S e CND d
b r S e CND d
T
r X e CND d






   
 
With:
2
1
log .
2
.
S
b T
X
d
T


  
    
    and 2 1d d T 
THETA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE
right side.
THETA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE
TIME TO MATURITY
right side.
-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
0,5
0,66
0,82
0,98
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
50
60
70
80 90 100110120130140150
Timeto
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
50 60 70 80 90 100 110 120
130
140
150
Timeto
Maturity
Spot

6
RHO
DESCRIPTION
The parameter Rho, noted  , is the sensitivity of the plain vanilla option’s
price to the interest rate.
2b 0: . . . ( )
.Call (S,X,T,r,b, )
rT
call
call Generalized BS
if T X e CND d
else T

 

 
 
2b 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
.
S
b T
X
d
T


  
    
    and 2 1d d T 
RHO VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE
right side.
RHO VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE
TIME TO MATURITY
right side.
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 90 100 110 120 130 140 150
Spot
-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 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
80
90
100
50
60
70
80 90 100110120130140150
Timeto
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
50 60 70 80 90 100 110 120
130
140
150
Timeto
Maturity
Spot

7
COST OF CARRY
DESCRIPTION
The parameter Rho, notedb , 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
.
S
b T
X
d
T


  
    
    and 2 1d d T 
CARRY SENSITIVITY VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT
PRICE
right side.
CARRY SENSITIVITY VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT
PRICE AND THE TIME TO MATURITY
right side.
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
2,2
2,4
2,6
2,8
3
50 60 70 80 90 100 110 120 130 140 150
Spot
-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
0
0,2
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
50
60
70
80 90 100110120130140150
Timeto
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
-60
-50
-40
-30
-20
-10
0
50 60 70 80 90 100 110 120
130
140
150
Timeto
Maturity
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. ( ) . ( )rT
Call S CND d X e CND d
 
2 1. ( ) . ( )rT
Put X e CND d S CND d
   
2
1 2 1
ln
2
Where ;
S
r T
X
d 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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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
S
bT t
X
d
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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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 T
Call Call S X T r
i





 
0
( )
( ; ; ; ; )
!
T i
i i
i
e T
Put Put S X T r
i





 
With :
2 2
i
i
z
T
 
 
   
 
;
2




and
2 2
z   
NB: iCall and iPut are calculated with the Generalized Black Scholes
Function.
With :
 S = Stock Price
 T = Calendar Time (time to Expiration on years)
PAYOFFS
INSTRUMENT PRICE
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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
( ). ( ) ( ). , ;
. . , ; ( ). ( )
ln
2
With ;
ln
2
rt rt
rT rt
rt
rt
c
t
Call S De CND b S De M a b
T
t
X e M a b X D e CND b
T
S De
r T
X
a a a T
T
S De
r T
S
b
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 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
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
S
Call S X T A S S
Call S X T S
S X
  
      
 
1
**
1 **
( , , ) when
( , , )
else
Q
GBS
S
Put 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
S
A e CND d S
Q
S
A e CND d S
Q


     
    
2 2
1 2
( 1) ( 1) 4 ( 1) ( 1) 4
;
2 2
M M
N 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
 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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Timeto
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Timeto
Maturity
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 1
Where ( ) and
2 2
b b
I X I


     
 
 
 


  
      
  
2
2
2
r


 

The function (S, T, ,H, I)  is given by
2ln( / )
(S, T, ,H, I)=e ( )
k
I I S
S CND d N d
S T
 
 

   
     
     
2
2
1
( 1)
2
1
ln( / ) ( )
2
r b T
S H b T
d
T
    
 

 
      
 
     
2
2
(2 1)
b
k 

  
And the trigger price I is defined as
( ) 0
0 0
0
0
( )(1 ) and ( ) ( 2 )
and max ,
1
h T B
I B B B e h T bT T
B B
r
B 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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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 X
d d d
 


 
  
And the variances and covariance can be calculated as
2
2
2
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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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 rT
Call e Se CND d Xe CND d  
   
( )
2 1( ) ( )T rT b r T
Put e Xe CND d Se CND d  
     
Where:
2
1 2 1
ln( / ) ( / 2)
d
S X b T
d 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
T
e 
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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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 t
Call Se e CND d e CND d    
   
( ) ( ) ( )( )
2 1( ) ( )b r t r T t b r T t
Put Se e CND d e CND d    
     
Where:
2
1 2 1
ln(1/ ) ( / 2)( )
; d
b T t
d 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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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)n
rT
i
S X b i t
Call Ae N t
i t




   
  
 

2
1
ln( / ) ( / 2)n
rT
i
S X b i t
Put 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 t
d
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 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
0,62
0,82
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
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 t
d 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 T
y
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 P
r 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 t
z
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 time to maturity can be represented as follows (for buying the
option)
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,22
0,42
0,62
0,82
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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
1
1 2 1 1
1
2
1 2
1 2 1 2
2
1 2
ln( / ) ( / 2)
y
ln( / ) ( / 2)
z
/
S I b t
y y t
t
S X b T
z 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 equation 1 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
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
0,62
0,82
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,22
0,42
0,62
0,82
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
Spot
0,02
0,22
0,42
0,62
0,82
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,22
0,42
0,62
0,82
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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 X
Call 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 X
Put 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 1
1 2 1 2
2 1
ln( / ) ( / 2) ln( / ) ( / 2)
; z ; /
S X b T S X b t
z 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
0,62
0,82
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Timeto
Maturity
Spot
0,02
0,22
0,42
0,62
0,82
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Timeto
Maturity
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 rT
Call S e M y T y T X e M y y    
   
2( )
2 2 1 2 2 2 1 2( , ; ) ( , ; )b r TrT
Put X e M y y S e M y T y T   
       
Where
2 2
1 1 1 1 2 2 2 2
1 2
1 2
ln( / ) ( / 2) ln( / ) ( / 2)
;
S X b T S X b T
y 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; 2 = The volatility of the asset 2;
 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 T
Call Q S e CND d Q S e CND d 
 
where
2
1 1 2 2 1 2
1 2 1
ln( / ) ( / 2)
;
Q S Q S b b T
d d d T
T



  
  
2 2
1 2 1 22      
and where
 1S = The spot of the underlying asset 1
 1b = The cost of carry of asset 1; 2b = The cost of carry of asset 2
 1 = The volatility of the asset 1; 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
buying a European call in the left side and buying an American call in the
right side)
INSTRUMENT PRICE
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
0,62
0,82
0
10
20
30
40
50
60
50
60
70
80 90 100110120130140150
Time to
Maturity
Spot
0,02
0,22
0,42
0,62
0,82
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120
130
140
150
Time to
Maturity
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 1
1 2 1 1
1
2
2 1 2 1 1
3 4 3 1
1
ln( / ) ( / 2)
;
ln( / ) ( / 2)
;
S IS b b t
d d d t
t
IS S b b t
d d d t
t






  
  
  
  
2
1 2 1 2 2
1 2 1 2
2
2
2 1 2 1 2
3 4 3 2
2
ln( / ) ( / 2)
; y
ln( / ) ( / 2)
;
S S b b T
y y T
T
S S b b T
y 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 ratio
1 2 1
2 2 1
( )( )
1
1 ( )( )
2
b r T t
b r T t
S e
I
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 t
z 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
( )( )
2
2 ( )( )
1
b r T t
b r T t
S e
I
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 t
z 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
[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 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 1
1
1
2
2 2 2
2
2
ln( / ) ( / 2) ln( / ) ( / 2)
Where ;
ln( / ) ( / 2)
S S b b T S X b T
d y
T T
S X b T
y
T
 
 


    
 
 

2 2 1 2 2 1
1 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, ) ( , , , )rT
Put 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, ) ( , , , )rT
Put 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
[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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot 0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
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
1
1 2 2
2
: ( ,0) max 1,0 ( )
S
Payoff Max S S X S X
S X
 
      
 
 2( )
2 2 1 2( ) ( ) ( )b r T rT
Call Q S e Xe SN d N d 
  
PUT SPREAD
1
1 2 2
2
: ( ,0) max 1 ,0 ( )
S
Payoff Max X S S S X
S X
 
      
 
 2( )
2 2 2 1( ) ( ) ( )b r T rT
Put Q S e Xe N d SN d 
    
Where
2
1 2 1
ln( ) ( / 2)
;
S T
d d d T
T




  
1
2
( )
1 1
( )
2 2
b r T
b r T rT
Q S e
S
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 e
F
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
 1b = The cost of carry of asset 1; 2b = The cost of carry of asset 2
 1 = The volatility of the asset 1; 2 = The volatility of the asset 2
  = Correlation between assets 1 and 2
 CND = Cumulative Normal Distribution
1S
2S
1Q
2Q

37
PAYOFFS
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)
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
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
2
2
1 1
min
if b 0 then
( ) ( )
2
( )
2
b r T rT
b
bTrT
Call Se N a S e N a
S b
Se N a T e N a
b S


 



 
 
                
 1 min 2 1 1 1
And if b=0 we have
( ) ( ) ( ) ( ( ) 1)rT rT rT
Call Se N a S e N a Se T n a a N a  
    
Where
2
min
1 2 1
ln( / ) ( / 2)
a
S S b T
a 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 b
Se N b T e N b
b S


 



   
 
               
 ( )
max 2 1 1 1 1
And if b=0 we have
( ) ( ) ( ) ( ) )rT b r T rT
Put S e N b Se N b Se T n b N b b  
     
Where
2
max
1 2 1
ln( / ) ( / 2)S S b T
b b b T
T



 
  
PAYOFFS

39
INSTRUMENT PRICE
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)

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, maxS and 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
2
2
1 1
( ) ( )
2
( )
2
b r T rT
b
bTrT
Call Se N d Xe N d
S b
Se N d T e N d
b X


 


 
 
               
 
Where
2
1 2 1
ln( / ) ( / 2)
;
S X b T
d 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 b
Se N e T e N e
b S


  


    
 
               
2
max
1 2 1
ln( / ) ( / 2)
Where and e
S S b T
e e T
T



 
  
FIXED STRIKE LOOKBACK PUT
min: ( ;0)Payoff Max X S
2
( )
2 1
2
2
1 1
( ) ( )
2
( )
2
rT b r T
b
bTrT
Put Xe N d Se N d
S b
Se N d T e N d
b X


 


   
 
               
 
2
( )
min min 1 min 2
2
2
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 b
Se N f T e N f
b S


  


      
 
                
2
min
1 2 1
ln( / ) ( / 2)
Where and
S S b T
f 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
2
2
2
( )
1 1 min 2 1
2
1 2
2 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 TS
M f d g t T
SSe
b
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 f
b




  
   
 
    
 
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 2
1 2 1 2
2
ln( / ) ( / 2)S M b T
d d d T
T



 
  
2
2 1
1 2 1 2 1
2 1
( / 2)( )
e
b T t
e e T t
T t



 
   

2
0 1
1 2 1 1
1
ln( / ) ( / 2)S M b t
f f f t
t



 
  
1 2
2 2 1
ln( ) ln( )
gg
T T t
 
 
 

Where
min
0
max
if call
if put
S
M
S

 


43
PARTIAL TIME FLOATING-STRIKE LOOKBACK PUT
 
 
2 2
2
2
2
2
2
2
( )
max 2 1 1 1
2
1 2
2 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 TS
M f d g t T
SSe
b
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 f
b

  
 
 
    
 
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 1
2
1 1 1 2
1 1 1 2
( )
1 1 1 2 2 2 1 2
2
( ) ( )
( ) ( )
2 2
; /
2
, ; 1 /
, ; 1 / ( , ; / )
1
2
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 tS
M d f t T
XSe
b
e M e d t T
Se M e d t T Xe M f d t T
e Se
b

  

 


 
  
 
                 
  
 
      
 
  
 
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 1
2
1 1 1 2
1 1 1 2
( )
1 1 1 2
2 2 1 2
2
( ) (
( ) ( )
2 2
, ; /
2
, ; 1 /
, ; 1 /
( , ; / )
1
2
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 tS
M d f t T
XSe
b
e M e d t T
Se M e d t T
Xe M f d t T
e Se
b

  

 




  
   
                
    
 
   
  
 
  
 
2)
2 1( ) ( )r T
N e N f
Where 1 1 1, andd 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 T
A B C
T t T
m T m t m t
D E F
T t t
  
  
  
  
    
  
    
  
2
2
2
2 2
1 2
2
And where ; 1 ; =
2( )
ln( / ) ; 0.5 ; 0.5
rT
e 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 if
M
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 t
G H
T t T t
 
 
  
 
 

47
PAYOFFS
buying a call in the left sides and buying a put in the right sides)
Extreme Spread options
Reverse Extreme Spread options
INSTRUMENT PRICE
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
2
2
2 2
ln( / ) ln( / )
(1 ) ; (1 )
ln( / ) ln( / )
(1 ) ; (1 )
ln( / ) / 2 2
; ;
S X S H
x T x T
T T
H SX H S
y T y T
T T
H S b r
z 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 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 n
b r T
n n
n
n
n
rT
Payoff Call S U L T Max S k
U L L
Call Se N d N d N d N d
L S U S
U L
N d T N d T
L S
Xe
L
 
 
 





 
      
        
      
               



3 21
3 4( ) ( )
nn
n
N 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 T
d
T T
L XSU b T L FSU b T
d
T T
 
 
 
 
 
   
 
   
 
   
  1
2 1 1
1 22 2
2 1
3 2
2 2 2
1 ; 2
2 2
1 ; T
b n
n
b n
F 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 L
N y T N y T
L S
Put Xe
L
N y T N y T
U S
U L
L S
Se
 

 
 
 





 
                 
  
  
       
  
   
   
  


 
 
3
1 2
1
3 4
( ) ( )
( ) ( )
nn
n
N y N y
L
N y N y
U S



 
  
 
 
  
   
  


53
Where
2
2 2 2 2 2 2
2
1 2
2 2 2 2 2 2 2 2
3 4
ln( / ( )) ( / 2) ln( / ( )) ( / 2)
; ;
ln( / ( )) ( / 2) ln( / ( )) ( / 2)
;
n n n n
T
n n n n
SU EL b T SU XL b T
y y E Le
T T
L ESU b T L XSU b T
y y
T T
 
 
 
 
 
   
  
   
 
CALL UP-AND-IN-DOWN-AND-IN
Up-and-Out-Down-and-OutCallGBSCall Call 
PUT UP-AND-IN-DOWN-AND-IN
Up-and-Out-Down-and-OutGBSPut 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 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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
80
90
100
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
80
90
100
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
80
90
100
50 60 70 80 90 100 110 120 130 140 150
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
80
90
100
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
80
90
100
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
80
90
100
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
80
90
100
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
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
H
Call Se M d e M f e
S
H
Xe 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
2
1 2 1 2
2
2
2
1 2 1 2
2
2
1
1 2 1 1 3 1
1 1
2
1
4 3 1 2
2
ln( / ) ( / 2)
;
ln( / ) 2ln( / ) ( / 2)
;
ln( / ) ( / 2) 2ln( / )
; ;
/ 2
; ;
S X b T
d d d T
T
S X H S b T
f f f T
T
S H b t H S
e e e t e e
t t
tb
e 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
H
C Se M d e M f e
S
H
Xe 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
H
C Se M g e M g e
S
H
Xe M g e M g e
S
H
Se M d e M f e
S
H
Xe 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
H
Se M g e M g e
S
H
Xe M g e M g e
S



 
 



 
 
  
  
     
   
  
     
   
Where
2
2
1 2 1 2
2
3 1 4 3 2
2
ln( / ) ( / 2)
;
2ln( / )
;
S H b T
g g g T
T
H S
g 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
H
C Se M g e M g e
S
H
Xe 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
H
C Se M g e M g e
S
H
Xe M g e M g e
S
H
Se M d e M e f
S
H
Xe M d e M
S




 
 
 







  
       
   
  
       
   
  
       
   
 
    
 
4 2( , ; )e f 
 
  
  

58
PAYOFFS
The payoffs of this model can be represented as follows (Buying positions):
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 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
50 60 70 80 90 100 110 120 130 140 150
Spot
0
10
20
30
40
50
60
70
50 60 70 80 90 100 110 120 130 140 150
Spot

59
INSTRUMENT PRICE
the time to maturity can be represented as follows (Buying positions):
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
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
50
60
70
80
90 100 110 120130140150
Time to
Maturity
Spot
0,02
0,18
0,34
0,5
0,66
0,82
0,98
0
10
20
30
40
50
60
70
50 60 70 80 90
100 110
120
130
140
150
Time to
Maturity
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 2
3 32
2
2 2
2 2 4 42
2
( , ; )
2( )ln( / )
exp ( , ; )
2 ln( / )
( , ; ) exp ( , ; )
b r T
rT
M d e
w S e H S
M d e
H S
Xe M d e M d e
  
   
  


      



 
 
   
   
  
   
     
   
2
1 1 1
1 2 1 1
1
2 2
3 1 4 2
2 2
ln( / ) ( )
;
2 ln( / ) 2 ln( / )
;
S X T
d d d T
T
H S H S
d d d d
T T
 


 
 
 
  
   
2 2 1 2 2
1 2 1 1 3 1
2 2
2 22
4 1 1 1 1 2 2 2
2
ln( / ) ( ) 2ln( / )
; ;
2ln( / )
; / 2; / 2
H S T H S
e e e T e e
T T
H S
e 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
Payoff1= 20):

62
INSTRUMENT PRICE
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
( )
1 2 1 2 2
3 3 1 22
2
2 2 1 2
2 2
4 4 1 22
2
( , ; / )
2( )ln( / )
exp ( , ; / )
( , ; / )
2 ln( / )
exp ( , ; / )
b r T
rT
M d e t T
w S e H S
M d e t T
M d e t T
Xe H S
M d e t T
  
   
  

  
 
  



 
 
   
   
  
 
 
   
   
  
2
1 1 1 2
1 2 1 1 2
1 2
2 2
3 1 4 2
2 2 2 2
ln( / ) ( )
;
2 ln( / ) 2 ln( / )
;
S X T
d d d T
T
H S H S
d d d d
T T
 


 
 
 
  
   
2 2 1 2 1 2
1 2 1 1 1 3 1
2 1 2 1
2 22
4 2 1 1 1 2 2 2
2 1
ln( / ) ( ) 2ln( / )
; ;
2ln( / )
; / 2; / 2
H S t H S
e e e t e e
t t
H S
e 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
Payoff1= 20):

65
INSTRUMENT PRICE
the time to maturity can be represented as follows (Buying positions,
Payoff1= 20):

GRA Pricer - 2013 - Valuation & Pricing Solutions

GRA Pricer - 2013 - Valuation & Pricing Solutions

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