Ewa Dziawgo

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
DYNAMIC ECONOMETRIC MODELS 
Vol. 7 – Nicolaus Copernicus University – Toruń – 2006  
 
 
 
 
 
Ewa Dziawgo 
Nicolaus Copernicus University in Toruń 
 
 
 
Sensitivity Model Analysis of the Floating–strike  
Lookback Call Option Pricing 
 
 
1. Introduction 
 
  Lookback options are the extremum–dependent options in the class of ex-
otic options. The value of the extremum–dependent options is influenced by the 
extreme value reached by the underlying instrument in the exercise period of 
the option.  
The buyer of the floating–strike lookback call option is entitled to the purchase 
of the underlying instrument at the lowest price reached by the underlying in-
strument in the option exercise time.  
Conversely, the buyer of the floating–strike lookback put option is entitled to 
sell the underlying instrument at the highest price reached by the underlying in-
strument in the option exercise time.  
  The article presents the issues connected with the floating–strike lookback 
options: the types of the options were characterized, the pricing model was de-
scribed and the influence of selected factors on the pricing of the analysed look-
back options was shown. The empirical research covered in the article deals 
with the impact of volatility, minimum of the underlying asset price, and time of 
expiration on the pricing of the lookback call options.  
The study of the prices was carried out on the examples of the lookback and 
standard options pricing on EUR as well as the influence of the selected factors 
on the price of the analysed options was examined. On the basis of the per-
formed simulations the comparative analysis of the prices for the standard and 
lookback options was demonstrated. 
 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Ewa Dziawgo  126
2. Pricing Model of the Floating–strike Lookback Call Option 
 
 The  floating–strike lookback call options can be of the in–the–money type, 
where the current price of the underlying instrument is higher than the strike 
price, or of the at–the–money type, where the current price of the underlying in-
strument amounts to the strike price. The floating–strike lookback options can 
never be out–of–the–money.
1 If the current price of the underlying instrument 
was lower than the strike price, it would become the strike price and the call op-
tion would become at–the–money. 
At the expiration date the payoff function of the floating–strike lookback call 
ption is the following:  o  
S
T T
S
T T m S m S − = − ] , 0 max[  (1) 
 
where: 
ST – price of the underlying instrument at the expiration date, 
mT
S – the lowest price of the underlying instrument in the option exercise time. 
In time tT ∈[; ] 0 the price of the option amounts to:
2
 
( )
( ) ( )
() () []
() [] () [] = − Ι − − =
= + − − =
= − =
= ⎟
⎠
⎞ ⎜
⎝
⎛ ⎟
⎠
⎞ ⎜
⎝
⎛ − ≡
≡ − = − =
− −
≥
− − −
− − −
− − −
− − − −
− − − − − −
−
−
−
−
S
t
t T r
z M
S
t
M
t
Q t T r
t
S
t
S
t
M
t
S
t
Q t T r
t
M
t
S
t
Q t T r
t
M
t
S
t
Q t T r
t T
rT Q rt
t
S
T
Q t T r
t T
Q t T r
t
S
T T
Q t T r W
t
m e m e S E e S
m m e S m E e S
e S m E e S
e S m E e F S e E e
F m E e F S E e F m S E e C
X
t T
X
t T
X
t T
X
t T
X
T t
) ( ) (
) (
) (
) (
) ( ) ( ) (
, min
, min
, min ) (
) ( ) (
,
 
[] []
[]
() [] () []
() () = ⎥
⎦
⎤
⎢
⎣
⎡
Ι − −
− Ι − − =
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
− + ≥ − −
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
≥ − − =
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
≥ − − =
− + ≥
− − − − − −
≥
− − − −
∞
−
− − −
∞
−
− − −
∞
−
− − −
−
−
−
−
−
−
−
∫
∫ ∫
)] ( 2 [
2 )) ( 2 ( 2
2
) (
) (
2 ) (
) ( ) (
2 2
2
2
) ( 2
t T z X
z r t T X r Q t t T r
z X
z X Q
t
t T r
z
t T
y r
t
t T r
z
t T
y
t
t T r
z
X
t T
y
t
t T r
t T
t T
t T
t T
e e E
r
S
e
e e E S e
dy t T y X Q e S e
dy y X Q e S e dy y M Q e S e
λ
σ λ σ
σ
σ
λ
 
[] ( ) [ ]− ≥ − Ι − = − ≥
− − − − −
−
− ] [
) ( 5 , 0
~
) ( ) (
2
z X Q m e E e S e t T
S
t z X
t T B Q t T r
t
t T r
t T
t T σ σ  
                                                      
1 The call option is out–of–the–money when the current price of the underlying in-
strument is lower than the strike price. 
2 See: Goldman, Sosin, Gatto (1979), Musiela, Rutkowski (1998). © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Sensitivity model analysis of the floating–strike lookback call option pricing  127
( ) + ⎥
⎦
⎤
⎢
⎣
⎡
Ι − − + ≥
− − − − −
−
−
−
)] ( 2 [
)) ( 2 ( 2
2
) (
2
2
t T z X
t T X r Q t t T r
t T
t T e E
r
S
e λ
λ σ σ
 
[] = ⎥
⎦
⎤
⎢
⎣
⎡
− + ≥ + −
− − −
−
) ( 2
2
2
2 ) (
2
t T z X Q
r
S
e e t T
t z r t T r λ
σ σ  
[] + − + ≥ − ≥ + ≥ − = − −
− −
− ) ( 2
2
] [ ] [
2
) ( t T z X P
r
S
z X Q m e z X P S t T
t
t T
S
t
t T r
t T t λ
σ
 
[] = − + ≥ + −
− − −
−
) ( 2
2
2
2 ) (
2
t T z X Q
r
S
e e t T
t z r t T r λ
σ σ  
− ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
− − −
+
+ ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
− + −
− =
− −
t T
t T r S m
N m e
t T
t T r S m
N S
t
S
t S
t
t T r
t
S
t
t
σ
σ
σ
σ
) )( 5 , 0 ( ) / ln(
) )( 5 , 0 ( ) / ln(
2
) (
2
 (2) 
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
− − +
+
+ ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
− + −
−
− − − −
t T
t T r S m
N
r
S
e e
t T
t T r S m
N
r
S
t
S
t t S m r t T r
t
S
t t
t
S
t
σ
σ σ
σ
σ σ
σ ) )( 5 , 0 ( ) / ln(
2
) )( 5 , 0 ( ) / ln(
2
2 2
) / ln( 2 ) (
2 2
2
 
 
where: 
W
t C – the price of the floating–strike lookback call option, 
σ – price volatility of the underlying instrument, 
T – time to the option expiry date,  
r – risk–free rate of interest, 
, min
,
] , [
,
X
T t M
t u
T t u
S
T t e S S m
−
∈
= ≡ ( ), max ] , [ , t u T t u
X
T t X X M − = ∈
⎟
⎠
⎞ ⎜
⎝
⎛ = =
−
X
T t M
t
S
t
S
T t
S
t
S
T e S m m m m
, , min ) , min( , , 
( ) ( )( ) ( )
) ( 2 5 , 0
~ ~
exp
t u X X
t t u t u e S t u r B B S S
− − = − − + − = σ σ , 
t B X t t λ σ + − =
~
,  ,  r − =
2 5 , 0 σ λ t B
~
 is the standard Brownian motion against 
measure Q. 
Finally, the price amounts to
3:     
  ⎥
⎦
⎤
⎢
⎣
⎡
− − − − − =
− − ) (
2
) ( ) (
2
) ( 3
2
2
) (
1
2
1
1 d N e
r
d N m e d N
r
S d N S C
a S
t
t T r
t t
W
t
σ σ
 (3) 
 
where: 
                                                      
3 See: Hull (1989), 464. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Ewa Dziawgo  128
t T
t T r
m
S
d
S
t
t
−
− + + ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
=
σ
σ ) )( 5 , 0 ( ln
2
1 ,  t T d d − − = σ 1 2 , 
t T
t T r
m
S
d
S
t
t
−
− + − + ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
=
σ
σ ) )( 5 , 0 ( ln
2
3 , 
2
2
1
ln ) 5 , 0 ( 2
σ
σ S
t
t
m
S
r
a
− −
= , 
 
N(d) – cumulative probability function of the standardised normal distribution. 
 
 
3. Empirical Example 
 
  The empirical research is concerned with the pricing simulation of the 
European currency call options. The options are on EUR. One of the options is 
the floating–strike lookback option (look. op.), the other is the standard option 
(stand. op.). These are 4–month options. The considered period is between 
03.01.2005 and 05.05.2005. 
Fig. 1 shows the prices of the discussed options. 
 
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0
3
.
0
1
1
2
.
0
1
2
1
.
0
1
0
1
.
0
2
1
0
.
0
2
2
1
.
0
2
0
2
.
0
3
1
1
.
0
3
2
2
.
0
3
0
1
.
0
4
1
3
.
0
4
2
2
.
0
4
0
4
.
0
5
Date
P
r
i
c
e
 
[
P
L
N
]
lookb. op.
stand. op.
 
 
Fig. 1. The price of the floating–strike lookback call option and of the standard option, 
on EUR. The considered period: 03.01.2005 – 05.05.2005.  
Source: Author’s calculations. 
 
The price analysis shows that the floating–strike lookback call option is more 
expensive than the standard option. A slump in the price of the underlying in-
strument considerably influenced the fall in the price of the standard call option 
as well as the increase in the price of the floating–strike lookback call option. 
The approaching expiration date narrowed the gap between the prices of the 
considered options. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Sensitivity model analysis of the floating–strike lookback call option pricing  129
  Fig. 2 shows the influence of the expiration date on the price of the floating–
strike lookback call option and standard call option.  
0
0,1
0,2
0,3
0,4
0,5
12346 1 2
Time to expiration [month]
P
r
i
c
e
 
[
P
L
N
]
lookb. op.
stand. op.
   
Fig. 2. The influence of the time to the option expiry date on the pricing of the floating– 
strike lookback call option and standard call option 
Source: Author’s calculations. 
 
Price analysis proves, that for every expiration date, the prices of the floating–
strike lookback call option are higher than for the standard call option. In the 
case the options have a longer–term expiry date, the differences between the 
prices of the floating–strike lookback call option and standard call option are 
greater. The approaching expiry date narrows down the differences in the price. 
  Fig. 3 illustrates the influence of the minimum price of the underlying in-
trument on the prices of the floating–strike lookback call option.  s  
0
0,1
0,2
0,3
0,4
3.8335 3.8835 3.9235 3.9735
Minimum price of the underlying instrument
P
r
i
c
e
 
[
P
L
N
]
 
 
Fig. 3. The impact of the minimum price of the underlying instrument on the price of 
the floating–strike lookback call option 
Source: Author’s calculations. 
 
The increase in the minimum price of the underlying instrument results in the 
drop in the price of the floating–strike lookback call option. 
  Fig. 4 demonstrates the impact of the volatility on the prices of the lookback 
and standard options. 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Ewa Dziawgo  130
0
0,2
0,4
0,6
0,8
0.0252 0.0752 0.1252 0.1752 0.2252
Volatility
P
r
i
c
e
 
[
P
L
N
]
lookb. op.
stand. op.
 
 
Fig. 4. The influence of the volatility on the prices of the lookback and standard call op-
tions 
Source: Author’s calculations. 
 
Price analysis confirms that the increase in the volatility results in the rise of the 
lookback and standard option. However, in the case of the volatility growth, the 
prices of the lookback options tend to rise more dramatically. In the case of the 
volatility fall, the differences in the prices of the described options tend to get 
smaller. 
  Fig. 5 and 6 show the impact of time to expiry and volatility on the price of 
he lookback call option (see Fig. 1) and standard call option (see Fig. 6).  t  
1 2 3 4 6 12
0.0252
0.0752
0.1252
0.1752
0.2252
0
0,2
0,4
0,6
0,8
1
P
r
i
c
e
 
[
P
L
N
]
Time to expiration [month]
Volatility
   
Fig. 5. The effect of volatility and time to expiry on the prices of the lookback call op-
tion 
Source: Author’s calculations. 
 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Sensitivity model analysis of the floating–strike lookback call option pricing  131
1 2 3 4 6 12
0.0252
0.0752
0.1252
0.1752
0.2252
0
0,1
0,2
0,3
0,4
0,5
0,6
P
r
i
c
e
 
[
P
L
N
]
Time to expiration [month]
Volatility
   
Fig. 6. The effect of volatility and time to expiry on the prices of the standard call op-
tion 
Source: Author’s calculations. 
 
The examined example the lookback options are more expensive than the stan-
dard options for each volatility and time to expiry. The gap in the price de-
creases in case of the fall in volatility and time to expiry. Longer time to expiry 
and the growth in volatility results in the considerable price increase of the 
lookback call option. 
 
 
4. Summary  
 
 The  floating–strike lookback call option entitles the owner to purchase the 
underlying instrument at the lowest price reached by the underlying instrument 
in the option exercise time. The lookback option is more expensive than the 
standard option.  
The value of the payoff function at expiry of the floating–strike lookback call 
option depends on two factors that influence the volatility of the underlying in-
strument: 
– current price of the underlying instrument at expiry, 
– lowest price of the underlying instrument in the option exercise time. 
The prices of the lookback options react dramatically to the fluctuation of the 
volatility. The prices of the lookback options of the longer time to expiry are 
particularly sensitive to the fluctuations of volatility.  
Consequently, the floating–strike lookback option is an exceptionally attractive 
financial instrument for those investors who use the option in speculative trans-
actions on the price volatility market of the underlying instrument. 
 
  © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Ewa Dziawgo  132
References  
 
Goldman, B. M., Sosin, H. B., Gatto, M. A. (1979), Path Dependent Options: ”Buy at 
the Low, Sell at the High”, The Journal of Finance, vol. XXXIV, 1111–1127. 
Hull, J. C. (1989), Options, Futures and Other Derivatives, Prentice Hall International, 
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Jajuga, K., Gudaszewski, W., Mróz, W. (2004), Opcje egzotyczne – wprowadzenie 
(Exotic options – introduction), Rynek Terminowy (Futures Market), 1, 6–11. 
  Musiela, M., Rutkowski, M. (1998), Martingale Methods in Financial Modelling, 
Springer–Verlag, Berlin. 
Napiórkowski, A. (2002), Charakterystyka, wycena i zastosowanie wybranych opcji eg-
zotycznych (Characteristics, pricing and application of the selected exotic op-
tions), NBP Departament Analiz i Badań, Warszawa. 
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An introduction to the Mathematics of Financial Derivatives,

Charakterystyka, wycena i zastosowanie wybranych opcji egzotycznych (Characteristics, pricing and application of the selected exotic options), NBP Departament Analiz i Badań,

Sensitivity Model Analysis of the Floating-strike Lookback Call Option Pricing

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