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Pricing Discrete Lookback Options Under A Jump Diffusion Model
Department： NTU FinanceSupervisor：傅承德 教授Student：顏汝芳
I. BackgroundI. Background
II. The Model II. The Model
III. Numerical Results III. Numerical Results
Agenda
IV. Conclusion IV. Conclusion
I. BackgroundI. Background - Introduction - Motivation - Pricing Issues
- Literature Review
Introduction
• Popular Products: (options)– Maturity style
• European• American
– Path-dependent Payoff• Lookback option• Barrier option• Asian option …etc.
Introduction
• Lookback Option:
a contract whose payoffs depend on the maximum or the minimum of the underlying assets price during the lifetime of the options.
Introduction
• Two types:– Floating strike price.
– Fixed strike price.
• Two cases– continuous monitoring (analytical use).
– discrete monitoring (practical use).
Introduction
• Time-T payoffs can be expressed as
– For European floating strike lookback calls and puts respectively:
and
– For European fixed strike lookback calls an
d puts respectively:
and
0minT t
t T
S S
0
max t Tt T
S S
0
( )max tt T
S K
0
( )min tt T
K S
Introduction
• Under the Black-Scholes modelThe value process of a Lookback put option (LBP) is
given by (continuous case)
where
220 0
0 1 1
2( , , ) [ ( ) ( ) ( )].
2
rrt rts s r
A s s t e e N d N d Tr s
0 1 0 1 0( ) ( ) ( , , )rtLBP s e N d s N d T A s s t
P( s, s+, t )
Motivation
• Empirical Phenomena
– Asymmetric leptokurtic
• left-skewed; high peak, heavy tails
– Volatility smile
• Models which capture these features:
– SV (stochastic volatility) model
– CEV (constant elasticity volatility) model
– Jump diffusion models
Introduction
• For continuous-version lookback options
– Under the Black-Scholes model
• Goldman,Sosin and Gatto (1979)
• Xu and Kwok (2005)
• Buchen and Konstandatos (2005)
– Under the jump diffusion model
• Kou and Wang (2003, 2004)
– In general exponential Levy models
• Nguyen-Ngoc (2003)
Motivation
• In practice, many contracts with lookback features are settled by reference to a discrete sampling of the price process at regular time intervals (daily at 10:00 am).
• These options are usually referred to as discrete lookback options. In these circumstances the continuous-sampling formulae are inaccurate.
• The values of lookback options are quite sensitive to whether the extrema are monitored discretely or continuously.
Motivation
• For discrete lookback option
– Essentially, there are no closed solutions.
– Direct Monte Carlo simulation or standard binomial trees may be difficult.
– Numerically, the difference between discretely and continuously monitored lookback options can be surprisingly large, even for high monitoring frequency, see Levy and Mantion (1998).
Pricing Issues
Can we price discrete lookback options under a jump diffusion model by using the continuous one ?
Literature Review
• For discrete-version lookback options
Broadie, Glasserman and Kou provided in 1999
a technique for approximately pricing discrete
lookback options under Black-Scholes model.
They use Siegmund’s corrected diffusion
approximation, refer to Siegmund (1985).
Literature Review
Theorem 3. The price of a discrete lookback at the kth fixing date and the price of a continuous lookback at time t=kΔt satisfy
Where, in and , the top for puts and the bottom for calls; the constant , ζ the Rimann zeta function.
Otherwise and .
( Cited from Broadie, Glasserman and Kou, (1999), “Connecting discrete and
continuous path-dependent options”.)
1( ) [ ( ) ( 1) ] ( )t t t
m tV S e V S e e S om
( (1/ 2) / 2 ) 0.5862
0max u t uS S 0min u t uS S
Literature Review
Table 4. Performance of the approximation of Theorem 3 for pricing a discrete lookback put option with a predetermined maximum. The parameters are: S=100, r=0.1, σ=0.3, T=0.5, with the number of monitoring points m and the predetermined maximum S+ varying as indicated. The option in the left panel has a continuously monitored option price of 16.84677, the right panel is 21.06454.
S+=110 S+=120 m True Approx. Error True Approx. Error 5 13.29955 12.79091 -0.50864 18.83723 18.44999 -0.38724 10 14.12285 13.85570 -0.26715 19.32291 19.11622 -0.20669 20 14.80601 14.66876 -0.13725 19.74330 19.63509 -0.10821 40 15.34459 15.27470 -0.06990 20.08297 20.02718 -0.05579 80 15.75452 15.71899 -0.03553 20.34598 20.31747 -0.02851 160 16.05908 16.04117 -0.01791 20.54389 20.52942 -0.01447
II. The ModelII. The Model - Continuity Correction
- Continuous-Monitoring Case
- Discrete-Monitoring Case
- Some known results
Continuity Correction
Theorem 2.1 For 0 < δ < 1 the discrete-version at kth an
d continuous-version at time t = kΔt floating strike LBP o
ption satisfy
and for δ >1 floating strike LBC have the approximation
The constant .
Continuity Correction( ) ( ) ( 1) (1/ )t tC C C tm tV S e V S e e S o m
( ) ( ) ( 1) (1/ )tC C Ct tm tV S e V S e e S o m
2 2(log )C
Continuous-Monitoring Case
• Incomplete Market – Change the measure from original probability to a
risk-neutral probability measure, see, for example, Shreve (2004);
– Choose the only market pricing measure among risk-neutral probabilities, we refer to Brockhaus et al. (2000) which is focusing on risk minimizing strategy and its associated minimal martingale measure under the jump-diffusion processes.
Continuous-Monitoring Case
• To construct a risk-neutral measure Let θ be a constant and λ be positive number. Define
*
Then the new measure is defined as follows,
( ) ( ) , tAP A Z t dP A F
21
( )2
1 2
1( ) exp[ ( ) ]
2
( ) exp[( ) ]( )
( ) ( ) ( ).
N t
Z t W t t
Z t t
Z t Z t Z t
Continuous-Monitoring Case
• Under the probability measure P*,
– the process is a Brownian motion,
– is a Poisson process with intensity λ , and
– and are independent.
( ) ( )W t W t t
( )N t
( )N t( )W t
Continuous-Monitoring Case
• Under the original measure P,
where is the compensated Poisson process and is a martingale.
• P* is risk-neutral if and only if
( ) ( ) ( ) ( ) ( 1) ( ) ( )
[ ( 1) ] ( ) ( ) ( ) ( 1) ( ) ( )
dS t rS t dt S t dW t S t dM t
r S t dt S t dW t S t dN t
( ) ( ) ( ) ( ) ( 1) ( ) ( )
[ ( 1) ] ( ) ( ) ( ) ( 1) ( ) ( ),
dS t S t dt S t dW t S t d M t
S t dt S t dW t S t dN t
( ) ( )M t N t t
• By contrast, we can get the relation
• Since there are one equation and 2 unknowns, θ and λ, there are multiple risk-neutral measures.
• Extra stocks would help determine a unique risk-neutral measure.
( 1) ( 1)r
Continuous-Monitoring Case
• On ‘the’ probability space (Ω,F,P*)
where and δ > 0, δ ≠ 1.
• .
0
( ) ( )0
0
( ) exp{ ( ) (log ) ( )}
exp{ ( )}.
W t t N t
S t s W t t N t
s e
s X t
212r
[0, ]
Let ( , ) inf{ 0 : }
and ( ) max ( ) for some fixed time t.u t
th S t S h
S t S u
Continuous-Monitoring Case
• The price of a continuous floating strike lookback put (LBP) o
ption at arbitrary time 0<t<T is given by ( t=kΔt )
where
• [0, ] [ , ]Rewrite max as max{ ,max },
and then t T t u t T uS S S
*[0, ] ( ) e {max | } r
t T t T tV t E S S F
- .T t
*[ , ]( ) e { (max ) | }.r
u t T u T tV t E S S S S F
Continuous-Monitoring Case
• Then we can use the fact that
to get the continuous value process as follows,
[ , ] 0{max } { ( , ) }Xu t T uS b b s e T
*[ , ]
*0 0 0
[ , ]0
( ) e e {(max ) | }
e e ( ) ,
where ln( ) and denote max ( ).
r rt u t T u t
r r x M xt a
u t T
V t S S E S S F
S S s e P s e s e dx
Sa M X u
s
Remark1
Continuous-Monitoring Case
Remark 1. Focus on
which can be deemed the discounted value of a
Up-and-In barrier call option with barrier and strike
price called the moving barrier option. This issue is
quite interesting and will be open for later discussion.
*e [(max ) | ] rt u T u tE S S F
S
S
[ . ]
*{ }
The present val
ue of Up-In barrier c
e [( ) 1 | ]
all options is :
.t
t o T
rT Max S H tE S K F
Continuous-Monitoring Case
• The floating strike lookback put
• The fixed strike lookback call • The relation between them at an arbitrary time t satisfies
*
[0, ]
*
[ , ]
*
[ , ]
( ) e [(max ) | ]
e [(max{ , max } ) | ]
e [(max ) | ]
( ) e
c ru t
u T
ru t
u t T
ru t
u t T
p rt
F S E S S F
E S S S F
E S S F
V S S S
( )pV S
( )cF S
Discrete-Monitoring Case
01
0
. . . .
Fix 0, let , for 1,2, , , define
exp ( log ) log
exp{ ( )},
Z ~ N(0,1) ~ ( ).
n
n i ii
i i d i i d
i i
i
TT t n m
m
S s tZ t M
s X n
where and N Poisson t
M
0
th0
is the compensated Poisson process.
Let '( , ) : inf 1: inf{ 1: ln }
and max at the k fixing date (known).
i
n n
n k n
N t
hh S n S h n X
s
S S
Discrete-Monitoring Case
• The price of a discrete floating strike LBP option at the kth fixing date is given by
• .
*[ , ]
[ , ] [ , ]
*[ , ]
( ) e {max | }
where - .
Rewrite max as max{ ,max }
then
( ) e { (max ) | }.
rm n o m n m t
n o m n n k m
rm n k m n m t
V t E S S F
T t
S S
V t E S S S S F
Discrete-Monitoring Case
• Similarly, we can use the fact that
to get the discrete value process as follows,
*m 0 0 0
*0
0
V e e ( ))
e e ( '( , ) ) ,
where ln( ).
mMr r x xt a
r r xt a
S S S e P s e s e dx
S S S e P x X t T dx
Sa
s
[ , ] 0{max } { '( , ) }Xn k m nS b b s e t T
Comparison
• Discrete-monitoring case
• Continuous-monitoring case
*0 ( ) e e ( ( , ) ) r r x
t aV t S S s e P x X T dx
*0e e ( '( , ) )r r x
m t aV S S S e P x X t T dx
What’s the connection between them ?
Some known results
• From Fuh and Luo (2007) we have the relations between the distributions of and as follows.
Proposition 1.2. For a fixed constant b > 0, we have
where “ ” means converging in distribution, moreover
.
'( , )x X t ( , )x X
'( , ) ( , ) 0d
b X t b C t X
d
* * 1( '( , ) ) ( ( , ) ) ( )P b X t T P b C t X T o
m
Continuity Correction
• We need to extend the results
fixed constant b r.v.
• That is, we have to discuss the uniform convergenc
e of the distribution of stopping time when the consta
nt b is a variable number.
0
[ ln( ), )S
y as
Continuity Correction
Lemma 1.3 Suppose that y is a flexible number,
and . Then we have that as m ∞
holds for all .
* * 1( '( , ) ) ( ( , ) ) ( )P y X t T P y C t X T o
m
[ , )y a
0
ln( )S
as
Tt
m
Continuity Correction
Theorem 2.1 For 0 < δ < 1 the discrete-version at kth and continuous-version at time t = kΔt floating strike LBP option satisfy
and for δ >1 the floating strike LBC option satisfy
The constant .
Continuity Correction( ) ( ) ( 1) (1/ )t tC C C t
m tV S e V S e e S o m
( ) ( ) ( 1) (1/ )tC C Ct tm tV S e V S e e S o m
2 2(log )C
Continuity Correction
Theorem 2.2 For 0 < δ < 1 the discrete-version at kth and continuous-version at time t = kΔt fixed strike LBC option satisfy
and for δ >1 the fixed strike LBP option satisfy
The constant .
Continuity Correction( ) ( ) (1/ )t
mC CtF S e F S e o m
( ) ( ) (1/ )mC t tCF S e F S e o m
2 2(log )C
Continuity Correction
• Overshoot– Due to the jump part– Due to discretization effect
• Thus our formula coincides with Broadie et al. (1999) when δ =1.
• For 0 < δ < 1 Spectrally negative jump processes; For δ > 1 Spectrally positive jump processes.
S+
overshoot
' t
St
III. III. Numerical Results - Continuous LBP options
- Results
Continuous LBP options
• Let be the cumulant generating
functions of X(t). And then it is given by
Denote g(． ) as the inverse function of G( ． ).
* ( )( ) log [ ]X tt E e
2 2 log1( ) { ( 1)}
2 ( )
t e t
G t
Continuous LBP options
• Laplace transform
Proposition 1.4 For α such that α +r >0 the Laplace transform w.r.t. T of the LBP option is given by
0
[1 ( )]0
( ) ( )
( )[ ( ) 1]
T
rt a g rrtt
Lp e V T dT
S s eS e
r r g r
Continuous LBP options
• Inverse Laplace transform– Gaver-Stehfest algorithm for numerical
Results
• The LBP parameters we used here are:m = 250, s = 90, s+ = 90, st = 80, r = 0.1, σ = 0.3,
δ = 0.9, λ = 1, T = 1(year), t = 0.8.
Discrete : use Monte-Carlo simulation method with 105 replications and we get the value is 8.4029.
Continuous : use Mathematica4.0 and we get 9.9901.
Corrected continuity : use Mathematica4.0 and the approximation discrete value (theorem 1.1) is 8.46003
Absolute error : 0.0571. Relative err : 0.67 %
Results
Results
IV. IV. Conclusion
Further Works
• How about uniform convergence of the distribution of stopping times ? (Lemma 3.3)
• What if the condition becomes δ >1 for LBP while 0 <δ <1 for LBC ?
• Holds for other Jump-diffusion models ? e.g. Double exponential jump-diffusion model
Thanks sincerely
for listening and advising
