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