A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results

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A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results

Belal Baaquie

To cite this version:

Belal Baaquie. A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results. Journal de Physique I, EDP Sciences, 1997, 7 (12), pp.1733-1753. �10.1051/jp1:1997167�.

�jpa-00247484�

(2)

A Path Integral Approach to Option Pricing with Stochastic

Volatility: Some Exact Results

Belal E.

Baaquie (*)

Department of

Physics,

National

University

of

Singapore,

Kent

Ridge, Singapore

119260

(Received

28 October 1996, revised 2 May 1997, accepted 30

July 1997)

PACS.03.65.~w

Quantum

mechanics

PACS.05.40.+j

Fluctuation

phenomena,

random processes, and Brownian motion

PACS.01.90.+g

general

interest

Abstract, The Black~scholes formula for pricing options _on stocks and other securities has been

generalized by

Merton and Garman to the _case when stock

volatility

is stochastic. The

derivation of the price of

a security derivative with stochastic volatility is reviewed starting from the first principles ^of ^finance. The equation ^of Merton and Garman is then recast using ^the

path

integration

technique

of theoretical physics. ^The price of the stock option is shown to be the

analogue

of the

Schr6dinger

wavefuction of quantum ^mechanics ^and ^the exact Hamiltonian and Lagrangian of the system is obtained. The results of Hull and White _are

generalized

to the _case when stock price and

volatility

have _non~zero correlation. Some exact results for pricing stock options for the

general

correlated _case _are derived.

1. Introduction

The

problem

of

pricing

the

European

call

option

has been well-studied

starting

^from ^the

pio~

neering

^work ^of ^Black ^and ^Scholes 11,

2].

^The ^results ^of ^Black ^and ^Scholes were

generalized by Merton,

Garman [3j ^and others for the _case of stochastic

volatility

and

they

derived _a

partial

differential

equation

that the

option price

must

satisfy.

The methods of theoretical

physics

have been

applied

with _some _success to the

problem

of

option pricing by

Bouchaud et al. [4].

Analyzing

the

problem

of

option pricing

^from ^the

point

of view of

physics brings

a whole collection of _new

concepts

to the field of mathematical finance

as well as adds to it _a set of

powerful computational techniques.

This paper is _a continuation of

applying

the

methodology

of

physics,

in

particular,

that of

path integral quantum mechanics,

to the

study

of derivatives.

It should be noted

that,

unlike the paper

by

Bouchaud and Sornette [4] in which the

pricing

of derivatives is obtained

by techniques

which go

beyond

the conventional

approach

in

finance,

the

present

_paper ^is based _on the usual

principles

of finance. In

particular,

_a continuous time random walk is assumed for the

security

and _a risk~free

portfolio

is used to derive the derivative

pricing equation;

these _are reviewed in Section 2. The main focus of this paper is _to

apply

the

computational

tools of

physics

to the field of

finance;

the _more

challenging

task of

radically changing

the

conceptual

framework of finance

using

concepts from

physics

is not

attempted.

(* ^e~mail:

phybeb@nus,edu.sg

@

Les

(ditions

_de

_Physique

₁₉₉₇

(3)

An

explicit analytical

solution for the

pricing

of the

European

call

(or put) option

for the _case of stochastic

volatility

has _so far remained _an unsolved

problem.

Hull and White [5] studied this

problem

and obtained _a series solution for the _case when the correlation of stock

price

and

volatility, namely

_p, is

equal

to zero, I,e. p = 0

(uncorrelated).

Hull and White [5j ^also obtained _an

algorithm

to

numerically

evaluate the

option price using

Monte Carlo methods for the _case of p

~

0.

Extensive numerical studies of

option pricing

for stochastic

volatility

based _on the

algorithm

of Finucane [6j ^have ^been ^carried ^out

by

Mills et al.

[7j.

The main focus and content of this paper is to

study

the

problem

of stochastic

volatility

from the

point

of view of the

Feynman path integral

_[9]. The

Feynman~Kac

formula for the

European

call

option

is well known

[2, 8].

^There ^have ^also ^been some

applications

of

path integrals

in the

study

of

option pricing _[10, iii.

In this _paper the

path integral approach

to

option pricing

is first discussed in its

generality

and then

applied

to the

problem

of stochastic

volatility.

The

advantage

of

recasting

^the

option pricing problem

_as _a

Feynman path integral

is that this allows for _a _new

point

of view and which leads to _new ways of

obtaining

solutions which

are exact,

approximate

as well _as

numerical,

to the the

pricing

of

options.

In Section 2 the differential

equation

of Merton and Garman _{[3] is} derived from the

principles

of finance. In Section 3 this

equation

is recast in the formalism of quantum ^mechanics. ^In Section 4 _a discrete time

path integral expression

is derived for the

option price

with stochastic

volatility

which

generalizes

the

Feynman~Kac

formula. In Section 5 the

path integration

_over

the stochastic stock

price

is

performed explicitly

and _a continuous time

path integral

is then obtained. And

lastly

in Section 6 _some conclusions _are drawn.

2.

Security

Derivative with Stochastic

Volatility

We review the

principles

of finance which

underpin

the

theory

of

security

derivatives, and in

particular

that of the

pricing

of

options.

^We will derive the results not

using

the usual method used in theoretical finance based _on Ito-calculus

(which

_we will review for

completeness),

but

instead from the

Langevin

stochastic differential

equation.

Hence _we start from first

principles.

A

security

is any financial instrument which is traded in the

capitals market;

this could be the stock of _a company, the index of _a stock

market, government

^bonds etc. A

security

derivative is _a financial instrument which is derived from _an

underlying security

^and ^which is also traded in the

capitals

market. The three _most

widely

used derivatives _are

options,

futures and

forwards;

_more

complicated

derivatives _can be constructed out of these _more basic derivatives.

In this paper, ^we

analyze

the

option

of _an

underlying security

S. A

European

call

option

on S is _a financial instrument which

gives

the _owner of the

option

to

buy

_or not to

buy

the

security

at some future time T > t for the strike

price

^{of K.}

At time t

=

T,

when the

option

matures the value of the call

option f(T)

is

clearly given by

~'~ ~~

~

~~~

~~~~

=

gls). lib)

The

problem

of

option pricing

is the

following: given

^the

price

^{of the}

security S(t)

at time t, what should be the

price

of the

option f

at time t _< T?

Clearly f

=

f(t, S(t), K, T).

This is

(4)

a final value

problem

since the final value of

f

at t

= T has been

specified

and its initial value at time t needs to be evaluated.

The

price

of

f

will be determined

by

how the

security Sit)

evolves in time. In theoretical

finance,

it is common to model

S(t)

_as _a random

(stochastic)

variable with its evolution

given by

the stochatic

Langevin equation (also

called _an Ito~wiener

process)

_as

@

=

4Sjtj

+

asRjtj j2aj

where

Rt

is the usual Gaussian white noise with _zero mean; since white noise is assumed to be

independent

for each time

t,

we have the Dirac delta~function correlator

given by

jR~R~,

₌

bit t'j. j2bj

Note

#

is the

expected

return on the

security

S and _a is its

volatility.

White noise

R(t)

has the

following important property.

^If we discretize time t

= n£, then the

probability

distribution function of white noise is

given by

pjR~)

=

~

e-[Rf, _j3a)

2x

For random variable

R)

it _can be shown

using equation (3a)

that

R)

= + random terms of

Oil). (3b)

e

In other

words,

to

leading

order in _e, the random variable

R)

becomes deterministic. This

property

of white noise leads to _a number of

important results,

and goes under the _name of Ito calculus in

probability theory.

As _a _warm-up, consider the _case when _a ₌ constant; ^this ^is ^the famous _case considered

by

Black and Scholes. We have

Q

~~

fit

_{+ E,}

Sit

+

E)) fit, Sit))

dt -o _e

j~~~

or,

using Taylor expansion

But

l~)~

₌

a~S~R)+O(1)

t

=

(a~S~+Oli).

14C)

Hence _we

have,

for _e _~ 0

I

_fit ^~

2"~~~

_fiS2 ^~

~~fiS

^~

"~fiS~'

~~~~

Since

equation (4d)

is of central

importance

for the

theory

of

security

derivatives we also

give

a derivation of it based _on Ito-calculus. Rewrite

equation (2a)

in terms differentials _as dS

=

#Sdt

+ asdz

(5)

where the Wiener process dz

=

Rdt,

with

R(t) being

the Gaussian white noise. Hence from

equation (3b)

(dz)~

=

R)(dt)~

= dt ₊

O(dt~/~)

~~~ ~~~~~

jds)2

=

«2s2dt

₊

ojdt3/2).

From the

equations

for dS and

(dS)~ given

above _we have

d

f

=

)

dt ₊

$dS

₊

~ _(dS)~

+

O(dt~/~)

=

)

+

~a~S~ ^~~

₊

#S$)dt

+

aS$dz

and

using

^dz

/dt

= R _we obtain

equation (4d).

The fundamental idea of Black and Scholes is to form _a

portfolio

such

that, instantaneously,

the

change

of the

portfolio

is

independent

of the white noise R. Consider the

portfolio

x =

f $S _(5a)

I.e. _x is _a

portfolio

in which _an investor sells _an

option f

and

buys

^fi

f/fiS

amount of

security

S. We then have from

equations (4b, 5a)

dx

df fif

dS

I

_dt _fiS _dt

~

~"~~~~~

~~~~

The

portfolio

x has been

adjusted

_so that its

change

is deterministic and hence is free from risk which _comes from the stochastic _nature of _a

security.

This

technique

of

cancelling

the random

fluctuations of _one

security (in

this _case of

f) by

another

security (in

this _case

S)

is called

hedging.

The rate of return _on _x hence must

equal

the risk-free return

given by

the short-term risk-free interest rate _r since otherwise _one could

arbitrage.

Hence

j

= rx

(5c)

which

yields

from

equation (5b)

the famous Black-Scholes [1j

equation

I

^~

~~fiS

^~

2"~ _fiS2 ^'

^~~~~

Note the parameter

#

of

equation (2a)

has

dropped

out of

equation (5d) showing

that _a risk- neutral

portfolio

_x is

independent

of the investors

expectation

as reflected in

#;

_or

equivalently,

the

pricing

of the

security

derivative is based _on _a risk-neutral process which is

independent

of the investors

opinion.

We _now address the _more

complex

_case where the

volatility a(t)

is considered to be _a stochastic

(random)

variable. We then have the

following coupled Langevin equations (following

Hull and White

[5j),

for

a~

=

V,

)

=

#S+aSR (6a)

$

=

~V+fVQ (fib)

(6)

where ~ is the

expected

rate of increase of the variance

V, (

its

volatility

and R and

Q

_are correlated Gaussian white noise with _zero _means and with correlators

lRtRt,)

=

lotot,)

=

bit t')

_16C)

and

jRtot,i

"

Pdjt t') j6d)

with p~ < 1

being

the reflection of the correlation between S and V.

Note that _one _can choose _a different _process from

equation (6)

for stochastic

volatility

_as

has been done

by

various authors [5] but the main result of the derivation is not affected.

The

procedure

for

deriving

^the

equation

^for ^the

price

of _an

option

is _more involved when

volatility

is stochastic. Since

volatility

V is not a traded

security

this _means that _a

portfolio

can

only

have two

instruments, namely f

and S. To

hedge

_away the all fluctuations due to both S and

V,

we need at least three financial instruments to make _a

portfolio

free from the investors

subjective preferences.

Since _we have

only

two instruments from which to form _our

portfolio,

_we will not be able to

completely

avoid the

subjective expectations

of the investors.

Consider the

change

in the

pricing

of the

option

in the _presence of stochastic

volatility; using equations (6a, fib)

_we have from

Taylor expansion

)

=

jfl )( fit

_{+ e,}

V(t

₊

e), Sit

₊

e)) fit, V(t), Sit)))

=

~~

⁺

$ _(~S

+

aSR)

₊

$ _(~V

+

(VQ)

_~~~

^~ ~

_~

^~ ~

~~~V ~ ~

~

~~~~~~~'

_~~~~

Using

R)

=

Q)

= +

Oil), Rtot

=

~+O(1) _(7b)

we

have,

similar to

equation (4a),

for _e _~ 0

~

~~~~

^~

~~~~

^~

^"~~~~~

^~

^~~

~

~~~~~~S~

+aS

~~

R +

eV$Q. ^(7c)

Define the deterministic and stochastic terms

by

~~

m

fw

₊

foiR

₊

fo2Q. (7d)

dt

Consider _a

portfolio

which consists of

91It)

amount of

option f

^and 92

It)

amount of stock

S,

that is

7rit)

₌

°ilt)fit)

₊

°21t)Sit). _18a)

Then

~~~

^~

~~~

~

~' ~~~~

(7)

We consider _a

self-replicating portfolio

in which _no cash flows in _or out and the

quantities 91It)

and 92

It)

_are

adjusted accordingly.

It then follows that

)

=

91$

+ 92

$ _(8c)

~~~~~

dx

~

~~

~

jo~s

+

la

R +

02Q)hf

+

°25"~'

_~~~~

~~ ⁱ ^~

We choose the

portfolio

such

that,

in matrix notation

which

yields

j~

=

91fw

+

4925. j9a)

Note that all the random terms from the rhs of

equation (8d)

have been removed

by

_our

choice of

portfolio

which satisfies the constraint

given

in

equation (8e).

Since

dx/dt

is _now

deterministic,

the absence of

arbitrage requires

)

= rx

(9b)

or, from

equations (8d, 9a),

(w r)91f

+

ii r)925

= 0.

(9c)

We hence have from

equations (8e, 9c)

that

equation (9c)

above _can be satisfied in

general by making

the

following

choice _w and

#, namely

w r =

]ioi

+

]2a2 (9d)

1-r

=

]ia. (9e)

Eliminating ]i yields using equation (9e) yields

from

equation (9d)

fw

_r

f

=

~

_~

foi

+

]2fo2

"

and hence from

equations (7c, 7d)

we obtain the Merton and Garman [3]

equation given by

Note the fact that both

#

and

]i

have been eliminated from

(10a)

since _we could

hedge

for the fluctuations of S

by including

it in the

portfolio

_x. In order words the

subjective

view of the investor visa vis

security

S has been removed.

However,

the _appearance of

]2

in

equation

(10a)

reflects the view of the investor of what he

expects

for the

volatility

of S. Since V is

(8)

not

traded,

the investor can't execute a

perfect hedge against

it _as _was the case for constant

volatility

and hence the

pricing

of the

option given

in

equation (10a)

has _a

subjective

factor in it.

It has been

argued by

Hull and white [5] that for _a

large

class of

problems,

_we _can consider

]2

to be _a constant which in effect

simply

redefines _~ to be ~

]2.

Hence _we have

Equation

(10) ^bove is

an

_equation _for _any security _derivative _with

stochastic

volatility.

Whether

it

is an

European

option

_or an _Asian

option

or

an

American

option

^is

the

^undary that are

imposed

on

f;

in

3.

Black-Scholes-Schr6dinger Equation

for

Option

Price

We recast the Merton and Garman

equation given

in

equation (10)

in terms of the formalism

of

quantum mechanics;

in

particular

_we derive the Hamiltonian for

equation (10)

which is the

generator

of infinitesimal translations in time. Since both S and V _are random variables which

can never take

negative values,

_we define _new variables _x and _y

S

= e~ -oo<x<oo

(11a)

V

= eY _oo _< y ^< oo.

(11b)

From

equation (10b),

_we then obtain the

Black-Scholes-Schrodinger equation

^for ^the

price

of

security

derivative _as

~~

_~

_~~~

_~~~~~

where

f

is the

Schr6dinger

wave function. The "Hamiltonian' H is _a differential operator

given by

~y

fi2

1 fi

fi2 j2

fi2 fi

~

2 fix2 ^~

2~~

^~~_fix ^~~~~~~

fixfiy

2

fiy2

^~

2~~

^~~

_By

^~~~~~

From

equation (12a)

_we have the formal solution

f

= e~~~+~~

f (13)

where

/

_is _fixed

_by

_the

_boundary

_condition. _Note _from

_equation ₍₁₃₎

_that

option price f

_seems

to be unstable

being represented by

_a

growing exponential; however,

_as will be _seen later since the

boundary

condition for

f

is

given

at final time

T, equation (13)

will be converted to _a

decaying exponential

in terms of

remaining

time _T

= T t.

Let

p(x,

_y,

T[x', y')

be the conditional

probability that, given security price

^x' and

volatility y'

at time _T

=

0,

^it ^will ^have a value of _x and

volatility

_y at time T. We have the

boundary

condition at _T

= 0

given by

Dirac

delta-functions, namely Plx,

_v,

01x', v')

=

dlx' x)dlv' v). l14)

The derivative

price

is then

given,

for 0 < t <

T, by

the

Feynman-Kac

formula

[5,

8] as +on

f(t;

_x,

y)

₌

^e~~~~~)~ dx'p(x,

_y, T

t[x')g(~') (Isa)

~o~

(9)

where

~~

p(z,

_y, T

t[x')

=

/ _dy'p(x,

y, T

t[x', y'). (lsb)

-~

Note that

integration

over

y'

is

decoupled

from the function

g(x')

due to

equation ii).

It follows from

equation (14)

that

fit;

_x,

y) given

in

(15a)

satisfies the

boundary

condition

given

in

equation ii),

_i-e-

flT,

_x, _Y)

=

gl~). l15C)

We rewrite

equations (15a, 15b)

in the notation of quantum mechanics. The function

fix)

can be

thought

of _as _an infinite dimensional vector

f)

of _a function _space with

components fix)

=

(xi f)

and

f*(x)

=

fix) (where

^* stands for

complex conjugation).

For _-oo < _x < _oo the bra vector

(xi

is the basis of the function _space and the ket _vector

ix)

^is its dual with

normalization

(T[X')

_#

d(T T'). (16)

The scalar

product

of two functions is

given by

lflg)

₊

/)

dx

f*lx)glx)

=

lfll )~ dxlxllxlllg)

and

yields

the

completeness equation

1=

f~ _dxjxjjxj j17a)

where I is the

identity operator

on function _space. For the _case of stochastic stock

price

^and stochastic

volatility,

_we have

~

I =

~c~ dxdy ix, y)(x, vi (17b)

where

ix, y)

m

ix)

li

Iv).

From

equation (13)

_we

have,

in Dirac's notation

f, t)

=

e~+jf. 0)

and

boundary

condition

given

in

equation (16) yields If, T)

=

e~~~~~~ If, ₀₎

=

lg).

Hence

if, t)

=

e-<T-t)(H+~) _ig) _i~~~~

or, more

explicitly

fit,T,v)

=

lT,vlf,tl

=

e~lT, vle~lg) l18b)

or,

using completeness equation (17b) fit,

_~,

y)

=

e-~(T-t) j ~j _dx'dy'jx, _yje-(T-t)H ix', y'jgjx') j19a)

(10)

which

yields

for

remaining

time _T

= T

t,

from

equation (15b) Pix, v,Tlx', v')

=

ix, vle~~~lx', u'). i19b)

Note

remaining

time _runs

backwards,

I.e. when _T

=

0,

_we have t

= T and when _T

=

T,

real

time t

=

0;

as mentioned

earlier, expressed

in terms of

remaining

time _T the derivative

price

is

given by

_a

decaying exponential

_as

given

in

equation (19b).

Before

tackling

the _more

complicated

case of

option pricing

with stochastic

volatility

I

apply

the formalism

developed

_so far to the

simpler

_case of constant

volatility;

the detailed derivation is

given

ⁱⁿ

Appendix

A. The well known results of Black and Scholes _are _seen to emerge

quite

naturally

in this formalism.

4. The Discrete-Time

Feynman

Path

Integral

The

Feynman path integral

_[9] is _a formulation of quantum ^mechanics ^which ^is ^based on

functional

integration.

In

particular,

the

Feynman path integral provides

_a functional inte-

gral

realization of the conditional

probability

_p(o~, _y, _T[o~',

y').

To obtain the

path integral

_we

discretize time _T into N

points

with

spacing

_e

=

TIN. Then,

from

(19b), p(x,

_y,

T[x', g')

= lim

ix, _vi ^e~ ^e~ ix', g'). (20)

N~O~ ~

N-times

Hence

forth,

_we will

always

_assume the N _~ _oo limit.

Inserting

the

completeness equation

for _x and _g,

namely equation (17b), (N I)

times in

equation (20) ^yields

p(x,

_y,

T[x', g')

=

j dxijyi jj(xi, yi[e~~~[xi-i, vi-1) (21a)

i=1~

i=1

with

boundary

conditions

To =

T',

_yo

=

v' (21b)

~N ~ ~, UN ~ Y.

(~lC)

We show in the

Appendix

B that for the H

given by equation (12b)

_we have the

Feynman

relation

given

in

(B.5, B.9)

1

~-y~/2

~~~'~~~~

~~~~~'~ 2xejfi@~~~~

^~~~~~

where the

'Lagrangian'L

is

given

in

equation (B.11),

for dx~ e xi x~-i and

dj

e j g~-i

as

~~~~

~2 ~~~

~ ~

~~~~

For _e _~

0,

we

have,

_as

expected

i~i, v~le~~~ lx~-i, vi-i)

=

dix~

_xi-i

)div~ vi-i)

₊

°iE) ₁₂₃₎

and the

prefactors

to the

exponential

on the rhs of

(22a)

_ensure the correct limit.

(11)

We form the 'action' S

by

N

S

= e

~ _L(I)

₊

_O(e). ₍₂₄₎

1=1

Note S is

quadratic

in _~~ and non-linear in the j variables.

Hence,

from

equations (20-22, 24)

we have the

path integral

Pi~,

_v,

Tl~')

=

)~ _{dg' pj~,}

y,

Tjx', g') j25a)

= lim

/DXDYe~ ^(25b)

N~C~

where,

for _e

=

TIN

le-YN/2 _jj~ _/°' ^dxi

^~~~~~~

125C)

~~

2xe(1 p~)

~=i ^-°~ ~'~~~~ ~~~

and

l~~ § _~c~

ill

^~~~~~

Note _we have included _an extra

duo

=

dg' integration

ⁱⁿ ^DY

given

in

(25d)

due to the

integra-

tion _over

y'

in

equation (25a).

Equation (25)

is the

path integral

for stochastic stock

price

with stochastic

volatility

in its full

generality.

To evaluate

expressions

such _as the correlation of

Sit)

with

V(t'),

the

path

integral

in

equation (25)

has to be used.

Since the action S is

quadratic

in the _x~, the

path integral

_over the _x~ variables in

equation (25)

can be done

exactly.

The details _are

given

in

Appendix

C.

From

equation (C.6)

_we obtain

~so+si

j26a)

~~~'~'~~~'~

~~'/2xe(1 p~) j~$1

_~~~

where the first term in

equation (22b) gives

So

-

i ^~i~

_{+ ~}

II

^~

_126b)

and

Si

is the result of the DX

path integration given by equation (C.6b)

_as

Si

=

~

(x

^~' + e

~(r

_eY~

2(1 P~)E Li=1~~~ i~

_~

(

~~~/~ ldYl +

~l~ )ll~' _126~)

1=1

Extensive numerical studies of the

pricing

of

European

call

option

^has been carried out in

[13]

based _on

equations (26a-c).

(12)

5. Continuous Time

Feynman

Path

Integral

We first derive the continuum limit of the Black-Scholes constant

volatility

_case before

analyzing

the _more

complex

_case of stochatic

volatility.

The

DXBS-Path intjgral

^for ^the Black-Scholes _case, from

equation (A.5b),

has _a _measure

DXBS

₌

~)~~~ fldxi

_which _is

essentially

the _measure for the flat _space

llt~;

we can

2xea

~_~

hence take the N _~ _oo for DX and obtain _a well defined continuous-time

path integral

_[9].

From

equation (A.5a)

_we obtain

taking

the continuum limit of _e _~ 0

with

boundary

conditions

x(0)

=

x' and

x(T)

= ~. We _see from above that the Black-Scholes

case of constant

volatility corresponds

to the evolution of _a free

quantum

mechanical

particle

with _mass

given by lla~.

This

yields

the Black-Scholes result

pBs(x,T[~')

=

(x[e~~~BS[x')

=

DXBSe~BS

The exact result for _pBs

lx, T[x')

is

given

ⁱⁿ

equation (A.4).

Equation (26) provides

_a

mathematically rigorous

basis for

taking

the N _~ _oo limit for the _case of stochastic

volatility.

On

exactly performing

the

f DX-path integral,

the

remaining

fDY-path integral

has _a _measure DY

=

~

~]~/~fl~~dyi which, just

as in the Black-

2xe(

^~"

Scholes _case, is

essentially

the _measure for the flat _space

llt~ (unlike

the nonlinear

expression

in

Eq. (25c));

_we _can hence take the N _~ _oo for DY and for

So

+

Si

and obtain _a well defined

continuous-time

path integral.

~f~Y~

~ ~

/ _dte~~~~

~ T'~'~~~

~~~

~~ take the linfit Of _E _~

°'

_"~~

~~~~'

_~~~ ~

~=i °

Hence,

from

(26)

we have

S

=

So

+

Si (27a)

So

₌

-j/~dti(j+~-(12)2 127b)

~~

211 p2)w ~~ ~'

_~

~~

~

~~ ~~~~~

+

~~(e~1°~/~ e~/~)

^~

(~ /

dt

e~~~~/~)~ (27c)

( (

~

with

boundary

value

y(T)

= y

(27d)

and

~~~'~'~~~'~

~~

~s

2x(1 p2)Tw

^~~~~

Note

taking

the continuum limit is

possible only

after the discrete

f

DX

path integration

has been

performed

since the nonlinear _measure

given by (25c)

does not have _a finite continuum

(13)

limit. All correlation functions

(S(t)V(t'))

_can be obtained from the continuum limit of the discrete correlators

(e~«e~").

For the _case of p =

0,

to _recover the Black-Scholes

formula,

set _w

=

a~,

where _a

=

e~/~

_is

the

volatility

at time t

= 0. As has been noted

by

Hull and White [5], for the _case _{of p}

= 0

the conditional

probability depends

on stochastic

volatility y(t) only through

the combination

w ₌

/~ ^e~~~~/~dt.

^For p

~ 0,

we see from

equation (27c)

that

p(x, y,T[x')

_now

depends

_on

T o

w as well _as _on _u

=

/~ ^e~~~~/~dt

and

e~~°~/~

We hence have from

(27, 28)

T o

CC ~Si(U,V,W)

~~~'~'~~~'~

~~°~~~~

~~~~'~'~°~

~~~~~

where

and

gin,

u,

w)

is ^given

^the ^ath

^integral

gin,

,

w)

T

o~ T

_o~

From

^quation _(29c)_we_see _that g(u, u, w) is

the

probability density for u,

u,

and

w.

ntegral for

gin,

_u, w) is

nonlinear and

not

be

_exactly. We see that

p(x,

_{y, T[x')}

is

the

eightedverage of the integrand e~i~">~>~~ ^/ ^x(1 ^-

p2)Tw

_with

espect

to

g(u, u, Note gin, u, w)

has

the

remarkable

roperty

that

it _is independent of p. _ollowing

Hulland

White [5] _one

can

xpand

the

_ntegrand

in

_(29a) in _an nfinite owereries in _u, u, w and

reduce

the evaluation

(u"w'°uP)

_m

/

c~ _dudw

u"w'°uPg(u,

_u,

w) (30a)

o

=

/ _DY[

^~

_/~

e~~~~/~dt]"

^~

/~ e~~~~dt]~°eP~~°)/~e~° (30b)

T o T

o

The

path integral

for

(u"w~°uP)

_can be

performed exactly.

Rewrite

(30b)

_as

iU"W'~U~)

₌

~nlm j~dti ^dtndtn+i ^dtn+m ^{zu, y,p)} ^130c)

~/~~~~

zj~,

_y,_p~

~

j

Dy

expi j~

^dt

J it) vit)ieP~i°~/~e~° i~~~~

and from

equations (30b, 30c)

we have

~ ⁿ ^n+m

j(t)

=

~ _bit _t~)

₊

~ _bit _t~)

₊ ^~

_bit) (31b)

~

i=i i=n+1

~

n+m

m

~ _a~d(t _t~)

₊

~d(t). _(31c)

1=1

~

(14)

The

path integral

for

Z(j,

_g,

p)

is evaluated

exactly

in

Appendix

C and

yields

~£

a~Y n+m _~

(u"w~uP)

= UP

~~~ fl _dt~ e~ (32a)

~

~=1

where from

equations (D.20a,

D.20b.

31c)

we

have,

after _some

simplifications

F

=

i~ j12) ~f

a~t~

+12 f

a~aj91t~ tj)tj

+

F' 132b)

=1 ij-1

with

~'

~~ ~

~

~ ~~

~

~~~~~

Note that in

obtaining equation (32)

_we have used that

9(0)

=

1/2,

_as

given

in

equation (D.16)

and reflects the

identity f) dtd(t _T)

=

1/2.

All the t~

integrations

in

(32)

can be

performed exactly

since the exponent ^is linear in the

l's.

The

expression (32)

for

(u"w'~uP) generalizes

the result of Hull and White [5j as we have _an exact

expression

for all the moments of _u, _u and

w as well _as for their cross-correlators.

We

explicitly

evaluate the first few _moments

using equation (32)

~Y ~

(w)

=

dte"~

T

~

p~

~) (~~)

~ ~e

~T

where V

=

e~;

~~2Y

^~ ^ti _~

j~2j

~~ ~~

~p(ti+t2)ei

t2

~ fi

/

1

/

2

T 0 0

2v2 ~(2p+t~)~

_eM~ 1

@ jj2

+

~)(2~

+

j2) ~(~

+

j2)

^~

~(2~

+

12)

^~~~~

We evaluate

(w~)

for the _case of _~

= 0: _we have from

equation (32)

3Y ~ t t

j~3j

~

~~

j

~~~

j

^~

~~~

j

^~

_dt~ ~i~(t2+2t3)

~3

0 0 0

v3

~ ~

3(6T2

^~~~~

~ ~~~ ~ ~ ~ ~

~~~~~'

~~~~

Equations (33-35

_agree

exactly

with the results stated

(without derivation)

in Hull and White [5]. ^It ^is

reassuring

to see that _two _very different formalisms agree

exactly

and this increases

ones confidence in the

path integral approach.

We

compute

a few moments that have not been

given

in Hull and White [5]. ^We further have

(recall

_a

=

e~/2)

~Y/2

~

jq~) = dt

~~H/2-(

^/8)t

T o

2°

[~(M/2-t~/8)~ ij ^j36a)

T(~ (2/4)

(15)

j~2j

_~

^2~~' /~

~~~

~~

~~~

~p(ti+t2)/2~-t~(ti-t2)/8 _j~~~)

T~

0 0

4v

elm/2-(2/8)~ ~~(p/2-(2/4)~

_l

T~

l~~

+

t~/4)l~ t~/8) l~

₊

t~/4)l~ t~/2) l~ t~/2)l~ t~/8)

^~~~~

and

la8tly,

^for _ai ₌

1/2,

_a2

= in

equation (31c),

_we have

~~3Y/2

~ ti

(uw)

~

= ~

dti dt2 e"~~i/~+~~~e~

^~~i~~~~~/~

T

4~v

e(3p/2+3(2/8)~ ~(p/2-(2/8)~

_i

T~

3j~1+ j~/2)j~l+ j~ /4)

_j~l +

j~/2)j~l~ j~ /4)

^~ _3j~1

j4/16) ^l~~~

All the other moments

(u"w'~uP)

_can

8imilarly

be

evaluated,

which in turn

yields

an infinite series solution for

p(x,

_y,

T[x')

in

equation (29a).

6. Conclusion

The

complete

information

regarding

the

dynamics

of how stock

price Sit)

and its

volatility V(t) evolve,

their cross-correlators _as well _as their fluctuations is

given by

the discrete time

path integral given

in

equation (25).

In

particular,

to

numerically study

correlators such _as

(e~«e~")

_we need to start from the

path integral given

in

equation (25).

However,

if _one is interested

solely

in the

price

^of the

option,

_one needs to

only

determine

p(x,

_y,

T[x', y')

and in this _case the

simplified

discrete-time

path integral

obtained in

equation (26)

^should ^be used _as the

starting point

for numerical studies. A detailed numerical simulation

based _on the results of this _paper has been carried out in reference

[13];

ⁱⁿ

particular,

it has been shown that

implied volatility

smile and frown _can be obtained for different values of the

correlation

parameter

_p as well _as for

varying

values of

Sit).

In summary, we reformulated the

option pricing problem

in the

language

of

quantum

me-

chanics. We then obtained _a number of _exact results for the

option price

^of a

security

derivative

using

^the

path integral

method. Stochastic

volatility

introduces _a

high

order of

nonlinearity

in the

option pricing problem

and needs to be studied

using approximate

and numerical tech-

niques.

The continuous time

path integral given

in

equation (29c)

is _a nonlinear

path integral

which has _no known exact solution. Various

techniques

could be used to obtain _new

approximations

for

g(u,

_u,

w)

^relevant ^for

special applications.

Numerical

algorithms

based _on the

path integral provide

_a wide range of _new numerical

algorithms

which go

beyond

the usual binomial tree and its

generalizations.

The

comparative efficiency

of the different

algorithms

is

being

studied. The

Feynman path integral

has been

numerically

studied

using

_many

algorithms

and which could prove useful in

addressing problems

in finance.

Acknowledgments

1would like _to thank Dr. Lawrence

Ma,

^Dr. Toh Clioon

Peng

and Dr. Ariff Mohamed for

having

^introduced ^this

subject

to me as well _as Mr. L-C- Kwek for many fruitful discussions.

(16)

Appendix

A

Black-Scholes Formula for Constant

Volatility

For the _case of constant

volatility

the Black-Scholes

equation

is

given by

the

following

_[2]

Making

the

change

of variable _as in

(2)

S=e~,

-oo<x<oo

yields

~~

_~

~~~

with the Black-Scholes Hamiltonian

given by

To evaluate the

price

of the

European

call

option

with constant

volatility

_we have the

Feynman-

Kac formula

fit, x)

~

=

e~ dx'(x[e~~~BS [x')g(x').

To compute _pBs

lx, T[x')

₌

(x[e~~~BS ix'),

where

remaining

time is _T

= T

t,

we use the

Hamiltonian

approach

to illustrate another

powerful technique

of quantum ^mechanics. ^We go to the "momentum" basis in which

HBS

is

diagonal.

The Fourier transform of the

ix)

basis to

'momentum

space'

is

given by

(XIX')

₌

diT x')

=

)( ^e~~~~~~'~

=

)~ )ixlP)lPlx')

which

yields

for momentum space basis

[p)

the

completeness equation

1

=

)~ _)lP)lPl _iA.3)

and the scalar

product

l~lfl)

" ~~~~i

lPl~)

_" ^~~~~~

From

(A.2)

and the

equation

above _we have the matrix elements of H

given by

Using

and

the

~ ,

ove

j°~

dP

^H

')

(x[e~~

~(T) " _{_~2x~~~~}

~

~~~~~~~~~

=

~~~

~~~~~

~( j ^~

j

^' ⁺ ^j l~'~~

(17)

The result above is the Black-Scholes distribution. Recall x'

=

log(S(T)),

_x ₌

log(S(t))

and

T = T

t;

the stock

price

evolves

randomly

from its

given

value of

Sit)

at time t to a whole range of values for

SIT)

at time T.

Equation (A.4)

above states that

log(S(T))

has _a normal distribution with _mean

equal

to

log(S(t))

+ IT

a~ /2)(T t)

and variance of

a~(T t)

_as is

expected

for the Black-Scholes _case with constant

volatility

_[2].

In

general

for _a _more

complicated (nonlinear)

Hamiltonian such _as the _one

given

in

equation (12b)

for stochastic

volatility

it is not

possible

to

exactly diagonalize

H and then be able to

exactly

evaluate the matrix elements of

e~~~

_The

Feynman path integral

is _an efficient

theoretical tool for

analysing

such nonlinear

Hamiltonians,

and this is the _reason for

using

the

path integral

formalism.

Appendix

B The

Lagrangian

The

Lagrangian

L is central to the

path integral

formulation of quantum mechanics. We first find

LBS

for the Black-Scholes _case of constant

volatility

before

tackling

the _more

complicated

case. For infinitesimal time _e it is

given by Feynman's

formula

j~j~-eHBsj~')

=

Nj~)~eLBs

where

N(e)

is _a normalization constant. Since the formula

(A.4)

is exact, we

have,

for

dx = x x'

~~~ ~2

~ ~

~~

_~~

'~

and with

~~~~ ~7

~~

~~~

We _now

analyze

the _case of stochastic

volatility.

From

equation (12b),

the Hamiltonian is

given by

'~

^~~

~~''

~'~

'

~~~ ^~~

~'~ ~'

_~~"

_~'~

=

/~' ^~PX ~PY~ipx(x-x')~~py (y-y')~-eH(x,y,px,py) j~ ~~)

_~

2x 2x

and from

(AA)

the matrix elements of the Hamiltonian is

given by

Toperform the

the matrix

~

jpeY/2 ~

(18)

Note

det M

=

(~e~(I _p~) _(B.5)

and

~

(~ II p2

~~p~

~/~

~~

^~~~

'

Let

A=x-~'+er- ~e~ _(B.7)

2 and

, e ~

B=y-y +q-j(. jB.8)

From

(B.2, B.3),

_on

performing

the Gaussian

integrations,

_we have the

Feynman

relation

'~

~~~~"

~'~ 2xe~~

_~~

where from

(B.2, B.4, B.6)

~

e2(l~-

p2)

^~~

~~~

_~

2 _~~

~

~ ~~~~~~

~

'~

and

finally

from

(B.7, B.8),

for dx

= ~ x' and

by

₌ _y

y',

_we have the

negative

definite

Lagrangian given by

---_

~~

Y

~

2~2~~~~~((

^~

^j~T

¹ ^~ ^p ^~

~~~ ~~~ ~

~~

2~ j~~~~()+~-jj2)j2 ^j ^~~~~

Appendix

C

The

f

DX-Path

Integration

The

path integration

_over the

x(t)-variables

in

equation (19)

_can be done

exactly,

and the derivation is

given

below [9]. ^Let

Q

=

DXe~~ (C.1a)

where from

(22b)

the

x-dependent

term of the

Lagrangian

is

given by

Let

~~ ^~

~ ~~~~~~

~~

~~~'

'~

(C.2b)

~ ^~

~~ ~ ~~ _~ ~ ~~~)2

~~~~

SX

"

2E(1 P~) 1~

^~

with

boundary

values

given by

To =

x',

TN = x.

(C.2c)

(19)

We make the

change

of variables

1

x~ = z~ e

~

cj,

dx~

=

dz~,

I

=

1, 2,.

,

N 1

(C.3a)

j=1

with

boundary

values

zo =

x' (C.3b)

N

zN = x + e

~

cj.

(C.3c)

j=1

We hence have from

(C.2, C.3)

s~

=

~ ^~

~-t

~ ~

~)2 j~ ~~)

2e(1 p~)

~

~ ~

and,

from

(25c, C.3)

e~~~/~ fl

/~~'

^~~~ ^~

^~~~ ^e~z ^(~'~~~

Q 2xeji p2)

~~i -~J

~

All the _zi

integrations

can be

performed exactly;

_one starts from the

boundary by

first inte-

grating

_over _zi, and then _over _z2, and

finally

_over _zN-i Consider the _zi

integration.

We have

/~ l~2)~~~~ e(l~

p2)

_~°~~~~

~Y2/2

~Yi _{+ eY2}

~~~~

2e(1 p2

~Yi + eY2 ~~~

~°~' '~~

Repeating

this

procedure (N I)

times

yields

e~i (C.6a)

~ tie(1 _p2) _£$1

_~~

where

~~

2e(1

~~

£$1

_~~

^~~~

^~~~~

1

,

( j2 (C.6b)

~~

_~ ~ ~ ~~

~£(~

~~

~

~~

~ ~

For the _case of constant

volatility (

= 0

= p and

ef

=

a2

= constant for all

I; equation (C.6)

then reduces to the Black-Scholes

given

in

equation (A.3).

(20)

Appendix

D

Generating

Function

Z(j,

_y,

p)

~~/~ ~~~~~~~~

zu, y,p~ j

Dy

expi j~

^dt

Jit)vit)ieP~i°~/~e~°. _i~.~~~

from

(27b)

_we have

~° (2 ~

~~~

~

~ _~ ~~~~

'~

with

boundary

condition

y(T)

= y,

jD.1c)

We first evaluate

Z(j,

_y,

y') given by

the

path integralin (D.1a)

but with the

boundary

condition

y(°)

"

y',y(T)

" y

(D.2)

and from

equation (D.1)

Z(J, y,P)

~

/ _dy'Z(j,

y,

y')ePY'/~ (D.3)

Define _new

path integration

variables

z(t) by

Z(t)

=

y(t) y'

₊

_IV' y). (D.4)

Note the _new variables

z(t)

due to

(D.2)

have

boundary

conditions

zjo)

₌ o

t

zjy). jD.5)

~~~~~

zjj,

_v,

v')

=

e~ /

_D~

_~ _

~~

with

~ ~(2

~~'

~ ~~ ~ ~~~~~~ ~

~' ~

~~_~_~~~

~'

T

~

~~ _~~

~~~

'

and

~~ ~

~~ ~~~~~~~~

~2 ~

~~~

~

~~'

'

To

perform

the

path integral

_over

z(t),

_note that from

boundary

condition

(D.5)

_we have the Fourer sine

expansion

on

z(t)

=

~ sin(xnt/T)zn. (D.9)

n=1

From

(D.8, D.9)

_we have

g~2 ^°~ ^°~ ^~

Sz

= -j

~j _n~z(

₊

~j[

dt

j(t) sin(xnt/T)]zn. (D,10) 4T(

~~~ ~_~

(21)

The

path integration

_over

z(t)

factorizes into

infinitely

_many Gaussian

integrations

over the

zn's

and _we obtain

/DZe~z

⁼

^C' ~ ^dzidz2dz3. ^dzo~e~z ^(D.11)

=

C(T)e~~ (D.12)

where

W

=

) / ~

_dt

dt'j(t)D(t, t')j(t') (D.13)

=

~~

/~

^dt

_/~ ^{dt'j it)} (T t)t'j it') (D.14)

T o o

since

,

f

I

~j~j~~~/~)~in(xnt'/T) ~~~~~

D(t,t

~2

~=i

=

fl [t'9(t t')

₊

t9(t' t)

^~~ _].

(D.16a)

where the

step

function is defined

by

Ii

^t ^> ⁰

flit)

=

1/2

t

= 0.

(D.16b)

0 t _< 0

The normalization function

C(T)

_can be evaluated

by

first

considering

the discrete and finite version of the

DZ-path integral along

the lines discussed in Section 3. The result is

given

_[9]

by

~~~~ ~fi _~~

~~~

Collecting

_our results _we have

~wu+w zlJ, u',u)

=

~ _xi

_T

^lD.18)

and

performing

the

y'

Gaussian

integration

in

(A.24) finally yields Z(j,

_y,

p)

=

e~ (D.19)

with

F

= y dt

j(t)

₊

(~L

(~) dt(T t) j it)

₊

(~

dt

j(t) _(T t) ^dt' j it')

₊

F' (D.20a)

~~

₂

~

where

~

F'= ~yp

₊

~(~p / _dt(T _t)j(t)

+

~pT(~L ~(~)

+

~p~(~T. (D.20b)

2 2

o 2 2 8