Post on 15-Jan-2016
transcript
Price an Asian option by PDE approach
5/24 2007
PDE & the pricing of an option
• The advantages of the PDE approach are that it is generally faster than Monte Carlo methods and that it gives the results for all initial prices (and even for all strikes or all maturities T in some cases).
• The drawback is that the numerical methods are usually more complicated to implement for PDE.
-- Francois Dubois
& Tony Lelievre
B.S. PDE
• Suppose an option has payoff function
the value of the option at time t is
• By Black-Scholes assumption,
where dz is a standard Brownian motion.
• By Ito lemma,
tTtTr
t FSgEeStV )(),( )(
)( TSg
SdzSdtdS
dzSVdtVS
VSVdV SSStS
2
22
B.S. PDE
dtVS
VdSVdV
dzSVdtVS
VSVdV
Sdz SdtdS
SStS
SSStS
2
2
22
22
Long: 1 share optionShort: Vs share underlying asset
the value of this portfolioSSVV
02
2
22
22
rVrSVVS
V
dtrSVrVdtrd
dtVS
VdSVdVd
SSSt
S
SStS
is riskless
This is a PDE for vanilla option
Pricing an Asian option by PDE approach
• Suppose an option has payoff function
the value of the option at time t is
where
• By Black-Scholes assumption,
where dz is a standard Brownian motion.• By multi-dimension Ito lemma,
tTTtTr
tt FASgEeAStV ),(),,( )(
),( TT ASg
dtt
AS dA
t
duSA
t
u
0
SdzSdtdS
dzSVdtVt
ASV
SVSVdV SASStS
2
22
Pricing an Asian option by PDE approach
dtVt
ASV
SVdSVdV
dzSVdtVt
ASV
SVSVdV
Sdz SdtdS
ASStS
SASStS
2
2
22
22
Long: 1 share Asian optionShort: Vs share underlying asset
the value of this portfolioSSVV
02
2
22
22
rVVt
ASrSVV
SV
dtrSVrVdtrd
dtVt
ASV
SVdSVdVd
ASSSt
S
ASStS
is riskless
This is a PDE for Asian option
PDE & the pricing of an option
• the PDE approach gives the results for all the time, all initial prices, all running average.
Pricing an Asian option by PDE approach
• Solving this PDE using finite difference take time for V is a function with 3 variables.
3nO
ASgASTV
rVVt
ASrSVV
SV ASSSt
1
),(),,(
02
22
Change of variable Francois Dubois & Tony Lelievre (2005)
• Change of variable
xtSfASt VS
TtAKx ,),,(,
/
),(
),(
),(),(),(
),(),(
),(),(
2
xtSf V
xtfT
t V
xtfS
xxtf
S
xxtf
S
x V
xtxfxtf V
xtfT
AxtSfV
xA
xxxxSS
xS
xtt
• PDE (1) with boundary condition is reduced to:
• Solving this PDE by finite difference method take time
• Note that this PDE has been obtained by Rogers & Shi (1995) by using some different approach.
xxxTf
frxT
fx
f xxxt
)(),(
01
2
22
)(),( KAASg
)( 2nO
)2(
Change of variable Francois Dubois & Tony Lelievre (2005)
Numerical results for Rogers & Shi PDE
Numerical results for Rogers & Shi PDE
Numerical results for Rogers & Shi PDE
Change of variable Francois Dubois & Tony Lelievre (2005)
• Rogers & Shi’s PDE gives poor results, especially when the volatility is small. These poor results are due to the fact when x is close to zero, the advective term is larger than the diffusion term.
• Change of variable
• This PDE has been obtained by Vecer (2001) by using some financial arguments
)1()1(),(
0)/(2
)/(
),(),(
/
22
xxxTq
qTtyrqTty
q
xtfyt q
Tty x
yyyt
The reason why we chose this change of variable
• The PDE (2) :
xxxTf
frxT
fx
f xxxt
)(),(
01
2
22
)1()1(),(
0)/(2
)/(
1
),(),(
/
22
xxxTq
qTtyrqTty
q
fT
f q
f q，fq
xtfytq
Ttyx
yyyt
xtt
xxyyxy
The reason why we chose this change of variable
• In order to fit the term
we solve the ODE
• So that we chose
xfT
1
T
t x
Tdt
dx
1
),(),(
1/
xtfytq
fT
fqT tyx xtt
Change of variable
• This approach can be generalized. For example, in order to completely get rid of the advective term, we solve the ODE:
• Change of variable
)1(1
),(
02
11,),(
2)(2
)(
zrT
zTh
hezh
zerT
tfzt h
zztTr
t
tTr
Trx
dt
dx 1
11 )( tTrzerT
x