A general Solution to the Bensoussan-Crouhy-Galai Equation appearing in Finance by SolomonAntoniou

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									A General Solution to the Bensoussan-Crouhy-Galai Equation appearing in Finance
Solomon M. Antoniou

SKEMSYS
Scientific Knowledge Engineering and Management Systems
37 oliatsou Street, Corinthos 20100, Greece
solomon_antoniou@yahoo.com

Abstract
We find a general solution to the non-linear PDE introduced by Bensoussan, Crouhy and Galai. This equation models the stochastic equity volatility as a function of the stock price and time. Our solution relies on the exploitation of the underlying Lie symmetries of the given equation. We also determine the classical Lie symmetries of a similar in nature non-linear PDE which models the volatility of the stock price as a function of the total asset value and time.

Keywords: Stochastic equity volatility, Lie Symmetries, Nonlinear PDEs, Exact Solutions.

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1. Introduction
The Black-Scholes option pricing model (ref. [1]) was derived on the assumption of constant volatility. However this assumption does not appear to be true in everyday market transactions (volatility smile, see for example ref. [2]). The volatility dynamics of the stochastic volatility models (or the stochastic volatility models combined with jump diffusion processes, [5]-[9]) appears to be exogenous and cannot be justified on economic grounds. It is the Bensoussan-Crouhy-Galai (BCG) approach introduced some time ago (refs [3]-[4]) that considers the stochastic nature of volatility to be endogenous and comes from the impact of a change in the value of the firm’s assets on the financial leverage. The core of the BCG model relies on a PDE (see equation (11) of [3]) that models the stochastic equity volatility as a function of the stock price and the time. This equation is a nonlinear PDE, where the unknown function is the stochastic equity volatility function given by ∂ ∂ 1 2 2 ∂2 + + (r + 2 ) S =0 S 2 ∂t 2 ∂S ∂S There is a second equation (see equation (12) of [3]) which is also a nonlinear PDE, where the unknown function is the function ~ (V, t ) , V being the total asset (1.1) value and t the time variable. It is given by ~ ∂ ~ 1 2 2 ∂ 2~ ~) V ∂ = 0 (1.2) + + (r + V ∂t 2 ∂V ∂ V2 We have found a general solution to the Bensoussan-Crouhy-Galai equation (1.1), long believed to be as one not having an analytical solution. (S, t ) , S being the stock price and t the time variable. It is

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In solving this equation, we have found the Lie point symmetries of the equation. We then determined two invariants, which allowed us to convert the equation into an ordinary, nonlinear, second order DE. Under two successive transformations, we transformed this last DE into an Emden-Fowler equation, which admits an analytical solution given by the error function erfi( x ) of imaginary argument. We have also determined the Lie point symmetries of the second equation (1.2) for further investigation. Despite the fact that the BCG approach is – to the best of our knowledge – one of the most serious alternative candidates to the Black-Scholes approach, it has not received proper attention. That might come from the non-linear nature of the equations involved and the lack of analytical solutions. A further reformulation of the BCG approach, using the closed-form solution of the above-mentioned equations is under way. The paper is organized as follows: In Section 2 we determine the Lie symmetries of differential equation (1.1). In Section 3 we find a general solution of the BCG equation, taking into account the generators of the Lie symmetry. In Appendix A we prove an auxiliary Theorem that helps to convert an ordinary DE into an Emden-Fowler equation. In Appendix B we solve the Emden-Fowler equation. In Appendix C we give a brief description on how to find the inverse of the error function. In Appendix D we consider another general solution of the equation, which is based on a generator not considered in deriving the general solution in Section 3, using a paraphrasis of the so-called Tanh-Method. In Appendix E we derive the Lie symmetries of the second equation, which can be used for the determination of a general, closedform solution.

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2. Lie Symmetries of the BCG Equation (1.1)
Lie’s method of symmetry is a well-established method initiated by S. Lie (see references [10]-[18]). In principle, this method can be used both for ODEs and PDEs (or systems of these equations), linear or nonlinear, in many areas of science and engineering (and even in finance).

Theorem 1. Equation (1.1) can be transformed to the equation
(2.1) where x = ln S . Proof. We consider the BCG equation (1.1). We make the substitution (2.2) Since S
2 ∂ 2 2

∂ u 1 2 ∂ 2u  1 2  ∂ u + u + r + u  =0 ∂ t 2 ∂ x2  2  ∂ x

x = ln S

∂S

=

∂ 2u ∂x
2

−

∂ ∂u ∂u and S , = ∂S ∂ x ∂x

the BCG equation (1.1) takes the form (2.1), where (S, t ) = u ( x , t ) with x = ln S .

Theorem 2. The generators of the Lie Symmetries of the equation (2.1) are given
by (2.3) X1 = ∂ ∂t ∂ ∂ 1 ∂ +t − u ∂x ∂t 2 ∂u ∂ ∂ + e −(x − r t) u ∂x ∂u

(2.4)

X2 = r t

(2.5)

X3 = − e −(x − r t) X4 = ∂ ∂x

(2.6)

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Proof. Let (2.7)

( x , t , u ( 2) ) be defined by 1  1  ( x , t , u ( 2) ) = u t + u 2 u xx +  r + u 2  u x 2  2  ∂ ∂ ∂ + (x, t, u ) + (x, t, u ) ∂x ∂t ∂u

Introduce the vector field X (the generator of the symmetries) by (2.8) X = (x, t, u )

The second prolongation of the vector field is defined in our case by the equation ∂ ∂ ∂ ∂ ∂ + t + xx + xt + tt pr ( 2) X = X + x ∂ux ∂ut ∂ u xx ∂ u xt ∂ u tt Invariance of the equation under the symmetry of the vector field V is implemented by the equation pr ( 2) X [ ( x , t , u ( 2) )] = 0 , as long as

( x , t , u ( 2) ) = 0 . We have pr ( 2) X [ ( x , t , u ( 2) )] = 0 1   1   ⇔ pr ( 2) X u t + u 2 u xx +  r + u 2  u x  = 0 2  2    or (2.9) [ u u xx + u u x ] + 1  1  + x  r + u 2  + t + xx u 2 = 0 2  2  The coefficients
x t x

,

t

and

xx

are given by

= x + ( u − x )u x − u u 2 − x u t − u u x u t x = t − t u x + ( u − t )u t − u u x u t − u u 2 t = xx + (2 xu − xx )u x − xx u t + ( uu − 2 xu ) u 2 − x
− 2 xu u x u t − uu u 3 − uu u 2 u t + ( u − 2 x ) u xx − x x

xx

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− 2 x u xt − 3 u u x u xx − u u t u xx − 2 u u x u xt For the functions , and we have = (x, t, u )
x

= (t) ,

= (x, t )

Thus, using the previous expressions for (2.10)

,

t

and

xx

, equation (2.9) becomes

 1  [ u u xx + u u x ] +  r + u 2  [ x + ( u − x )u x ] +  2  + t − t u x + ( u − t )u t + 1 + u 2 { xx + (2 xu − xx )u x + 2 + uu u 2 + ( u − 2 x )u xx } = 0 x

1  1  We substitute u t by − u 2 u xx −  r + u 2  u x into (2.10): 2  2  [ u u xx + u u x ] +  1  +  r + u 2  [ x + ( u − x )u x ] + t − t u x +  2   1  1   + ( u − t )  − u 2 u xx −  r + u 2  u x  +  2    2 1 + u 2 { xx + (2 xu − xx )u x + 2 + uu u 2 + ( u − 2 x ) u xx } = 0 x which can be written in the equivalent form (2.11) 1  1 2     r + u  x + t + u 2 xx  + 2  2     1  +  u + r + u2 ( u − x )− t −  2  

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  1  1 − ( u − t ) r + u 2  + u 2 (2 xu − xx )  u x +  2  2  1 1 + [ u − ( u − t ) u 2 + ( u − 2 x )u 2 ] u xx + 2 2 1 + u 2 uu u 2 = 0 x 2 All the coefficients of the partial derivatives of the function u are set equal to zero. We then have a system of partial differential equations: • Coefficient of the zero-th order derivative of u: (2.12) 1 2  1 2  r + u  x + t + u xx = 0 2  2 

• Coefficient of u x (2.13)  1  u + r + u2 ( u − x )− t −  2   1  1 − ( u − t ) r + u 2  + u 2 (2 xu − xx ) = 0  2  2 • Coefficient of u xx : (2.14) 1 1 u − ( u − t )u 2 + ( u − 2 x )u 2 = 0 2 2

• Coefficient of u 2 : x (2.15) 1 2 u uu = 0 2
uu = 0 , which means that

We now have to solve the system of equations (2.12)-(2.15). From (2.15) we get to u: (2.16) ( x, t, u ) = (x, t ) ⋅ u + ( x, t ) is a linear function with respect

From (2.14), after canceling opposite terms and dividing by u, we get the equation

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+ We substitute

1 tu − xu = 0 2

in the previous equation by the expression (2.16), to obtain ⋅u + + 1 tu − xu = 0 2

It follows that (2.17) and (2.18) + 1 t − x =0 2 by (2.16), we get =0

From equation (2.12), and substituting

 1 2 r + u ( x ⋅ u + x ) + ( t ⋅ u + t ) +  2  1 + u 2 ( xx ⋅ u + xx ) = 0 2 which can be written in equivalent form 1 r ( x + t ) + (r x + t ) u + ( x + xx ) u 2 + 2 1 + ( x + xx ) u 3 = 0 2 From the previous equation we get (since (2.19) (2.20) r x + t =0
x + xx = 0

= 0)

From (2.13), canceling opposite terms and substituting  1  u ( ⋅ u + ) + r + u2 ( t − x )− t +  2  1 + u 2 (2 x − xx ) = 0 2 The previous equation can be written in equivalent form

by (2.16), we get

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{r ( t − x ) − t } + u + 1 1 + { + ( t − x ) + (2 x − xx )} u 2 = 0 2 2 from which we get (2.21) and (2.22) 1 1 + ( t − x ) + (2 x − xx ) = 0 2 2 r( t − x )− t = 0

Since, (as it follows from (2.18)),
x =

+

1 t and 2

xx = x

equation (2.22) can be written as 1 1  1 +  t − − t  + (2 x − x ) = 0 2 2  2 which is equivalent to (2.23)
x+

=−

1 t 2

The previous equation has the general solution (2.24) (x, t ) = − 1 ′( t ) + g ( t ) e − x 2

We can prove that the solution in (2.24) is compatible with (2.20). After substitution of (2.24), equation (2.19) takes the form − r g( t ) e − x − which is equivalent to − 1 ′′( t ) + { g ′( t ) − r g( t )}e − x = 0 2 1 ′′( t ) + g ′( t ) e − x = 0 2

From this equation we get

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(2.25) and (2.26)

−

1 ′′( t ) = 0 2

g ′( t ) − r g ( t ) = 0 ( t ) = a1 + a 2 t

Equation (2.25) conveys that (2.27) and (2.26) that (2.28) g( t ) = a 3 e r t

We therefore get from (2.24) that (2.29) 1 ( x, t ) = − a 2 + a 3 e − ( x − r t ) 2

From (2.16), (2.17) and (2.29), we determine ( x , t , u ) : (2.30) 1 (x, t, u ) = − a 2 u + a 3 e − ( x − r t ) u 2

We can now find x from (2.18) :
x = a3 e −(x − r t)

Integrating with respect to x we get (2.31) = −a 3 e − ( x − r t ) + f ( t )

Using (2.21), (2.27) and (2.31) we can determine the unknown function f ( t ) . The differential equation we get for the function f ( t ) is f ′( t ) = r a 2 Integrating with respect to t, we obtain (2.32) f (t) = a 2 r t + a 4

The function ( x , t ) can be derived from (2.31): (2.33) (x, t ) = a 2 r t − a 3 e − ( x − r t ) + a 4

The vector X, the generator of the symmetries, can be expressed as

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X = (a 2 r t − a 3 e − ( x − r t ) + a 4 )

∂ ∂ + (a 1 + a 2 t ) + ∂x ∂t

 1  ∂ +  − a 2 + a 3 e −(x − r t)  u  2  ∂u The previous expression can be written as X = a1  ∂ ∂ ∂ 1 ∂  + + a2  r t +t − u  ∂x ∂t ∂t 2 ∂u     ∂ ∂  ∂  + a4 + a 3  − e −(x − r t) + e −(x − r t) u   ∂x ∂u  ∂x  From the previous expression we get the generators of the symmetries given by (2.3)-(2.6).

3. A General Solution of the BCG Equation (1.1)
After determining the symmetries of the BCG equation, we are now in a position to find its general solution.

3.1 Some preliminary Theorems
We state and prove the following

Theorem 3. Equation (2.1) takes the form
yy + y =1

under the substitutions u (x, t ) = y = x−rt Proof. We consider a linear combination of the generators X1 , X 2 and X 4 : (3.1) where and X=
1+ 2+ 4

r ( y) 1+ r t

are constants to be determined.

We therefore consider the generator

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(3.2)

X = ( + t)

∂ ∂ 1 ∂ + ( r t + 1) − u ∂t ∂x 2 ∂u

Another linear combination of the generators might be possible. Any other choice depends on the boundary or/and initial conditions. We can now determine the invariants corresponding to X given by (3.2). For this purpose, we are going to solve the system (3.3) dt dx = = + t rt + 1 du 1 − u 2

The equation

dt dx = can be written as + t rt + 1  1− r ⋅ dx =  r +  t+    dt  

and by integration, x = r t + k ⋅ ln( t + ) + C1 where k = (3.4) 1− r . Therefore, one invariant of the equation is y = x − r t − k ⋅ ln( t + ) dt = + t du we get 1 − u 2

Integrating the equation

1 ln u = − ln( + t ) + C 2 or ln u = ln[( + t ) −1 / 2 A] , or 2 u = ( + t ) −1 / 2 A where A is a constant. Therefore, another invariant is given by (3.5) where = ( y) . = + tu

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Using (3.4) and (3.5), we find ut = −  1 k ( t + ) −3 / 2 −  r +  2 t+    ( t + ) −1 / 2 y  

u x = ( t + ) −1 / 2 y u xx = ( t + ) −1 / 2 yy Thus, under the transformations (3.4) and (3.5), the non-linear partial differential equation ∂ u 1 2 ∂ 2u  1 2  ∂ u + u + r + u  =0 ∂ t 2 ∂ x2  2  ∂ x becomes the ordinary second order (nonlinear) differential equation − 1 2 −k
y+

1 2 1 2 yy + y =0 2 2

Under the choice k = 0 , i.e. (3.6)
yy + y =1

1 = , the previous equation takes the form r

where we have taken

= 1 (any other choice might be possible). y = ln ,

Theorem 4. Under the transformation
(3.7)

equation (3.6) can be transformed to the equation (3.8)
2

+2

=

1

Proof. Introducing the substitution (3.7), since
y =

and

yy =

2

+

equation (3.6) takes the form (3.8).

Theorem 5. Equation (3.8) admits a similarity transformation
(3.9) = , = −1 w , w = w ( )

13

and takes the final form (3.10) w = 1 w

which is an Emden-Fowler equation. Proof. The proof is in Appendix A.

Theorem 6.
(I) The general solution of equation (3.10) is given by (3.11) = ±∫ dw ln w + 2A
2

+B

where A and B are arbitrary constants. (II) The integral appearing in (3.11), can be expressed in terms of the erfi( x ) , (3.12) = ± e−A  erfi   2  2 A + ln w 2  +B  2 

the error function of complex argument (Appendix C). Proof. (I) Appendix B. (II) Under the substitution
2 ln w 2 + 2 A = 2 z , i.e. w = e − A e z ,

the integral in (3.11) takes the form
2 = ± 2 e − A ∫ e z dz + B

from which, taking into account (C.3) and readjustment of the constants A and B, we arrive at (3.12). Note. The “Integrator” by Wolfram Research (integrals.wolfram.com) gives the following result for the integral in (3.11): e−a  log( x 2 ) x 2 erfi  a +  2 2  x    

∫

1 2a + log( x )
2

dx =

14

which, for x > 0 , is essentially the result we have found in (3.12). The following Theorem provides the general solution of the equation (3.10).

Theorem 7. The function w ≡ w ( ) , which is a solution to (3.10), is given by
(3.13) where (3.14) =± 2 e− ( − ) w ( ) = exp{[(erfi) −1 ( )]2 − }

The function (erf i) −1 ( x ) is the inverse error function of complex argument. Proof. From (3.12) we get (3.15) where erfi ( A + ln w ) = is given by (3.14). Inverting (3.15) we find A + ln w = (erfi) −1 ( ) Solving the previous equation with respect to w, we find w ( ) given by (3.13).

3.2 A General Solution
Let us now put everything together.We see that the final equation depends on the variable variables. We have that = (Theorem 5), = e y (Theorem 4) and y = x − r t (Theorem 3) . Therefore we have to find the relation between and the original

where x = ln S . Therefore = e y = e x − r t = e ln S − r t or (3.16) = S ⋅ e−r t (S, t ) = u ( x , t ) (Theorem 1),

Concerning the unknown function, we see that where u ( x , t ) =

1 r ( y) (Theorem 3) and ( y) = w ( ) (Theorem 4). 1+ r t

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Function w ( ) is given by w ( ) = exp{[(erfi) −1 ( )]2 − } (Theorem 7). Therefore ( y) = 1 S⋅e
−r t

exp{[(erfi) −1 ( )]2 − } , or

(3.17)

er t ( y) = exp {[(erfi) −1 ( )]2 − } S

We thus arrive at the following

Theorem 8. The general solution of equation (1.1) is given by
(3.18) (S, t ) = = S ⋅ e − rt and r er t exp{[(erfi) −1 ( )]2 − } 1+ r t S =± e− ( − ) .

where

2

This is the main result of our paper.

3.3 Inverting the Error Function
We now have to find an analytical expression for the inverse of the error function of imaginary argument. In Appendix C we prove the relation (C.10): (3.19) (erfi) −1 ( ) = − i (erf ) −1 ( i )

where (erf ) −1 ( x ) is the inverse error function given by the series expansion (C.4) and implemented in many Symbolic Languages, like “Mathematica”. Using (C.4) and (C.10), we can prove that (erfi) −1 ( ) has the series expansion (C.11).

Appendix A. Proof of Theorem 5.
We try a transformation of the form (A.1) = , = w

16

under which equation (3.8) is invariant. Under this transformation we find that the partial derivatives are transformed to 2 and
2

=

2

−1

1

w+

2

1

w

 ( − 1) 1   = +  − 1          
−1 1

−2

1

w+
2

1  11  2 + +  − 1 2    

w +

1
2

w

We therefore find that under the transformation (A.1) the differential equation (3.8) takes the form  ( − 1) 1   +  + 1      
1  11  2 + +  + 1 2     −2 1

w+
1

    

−1

w +

+

1
2

2

w

= − w −1

The coefficients of w and w of the previous equation vanish simultaneously under the choice (A.2) = 1, = −1 = −1 w

Equation (3.8) therefore under the transformation = , takes the form (A.3) w = w −1

17

This is an Emden-Fowler (type of) equation.

Appendix B
In this appendix we shall solve the Emden-Fowler equation (3.10). Consider the equation d2y d x2 = 1 y dy we get the equation dx

Multiplying the previous equation by 2

2

dy d 2 y dy 1 ⋅ =2 ⋅ dx d x 2 dx y d  dy  dy 1   =2 ⋅ dx  dx  dx y
2 2

which can be written as

dy  dy  Integrating with respect to x we find   = 2 ∫ from which we get y  dx   dy  2   = ln y + 2 A , where A is an arbitrary constant.  dx  Therefore we have dy = ± ln y 2 + 2 A dx and by a further integration with respect to x, ±∫ dy ln y + 2A
2 2

+B=x

where B is another arbitrary constant. Note 1. Another method of finding the general solution is by the substitution dy d2y dp = p and therefore = p⋅ . 2 dx dy dx

18

The original equation can then be converted into p dp = integrated, etc.

dy which can be y

Note 2. The general form of the Emden-Fowler equation d2y dx
2

= a x m yn

can be solved for a variety of choices for m and n using Lie Symmetry Analysis.

Appendix C
The error function is denoted by erf ( x ) and it is defined by the integral (C.1) erf (z) = 2
z

0

∫e

−x2

dx

The error function of imaginary argument is denoted by erfi( x ) and it is defined by (C.2) erfi( x ) = −i erf (i x )

We can prove, using (C.2) and (C.1), that (C.3) erfi(z) = 2
z

0

∫

e x dx

2

which means that erfi( x ) is a real-valued function. The inverse of the error function is a real-valued function given by (C.4) (C.5) (C.6) erf
−1

 ck   (x) = ∑ x   k = 0 2k + 1  2 

∞

2 k +1

c0 = 1 ck = c m ⋅ c k −1− m m = 0 ( m + 1)(2m + 1)
k −1

∑

19

and is being implemented in many of the well-known Symbolic Languages like “Mathematica”. The first few terms of the previous expansion are given by erf
−1 2 3 4   x + x 3 + 7 x 5 + 127 x 7 + 4369 (x) = x9 + 2  12 480 40320 5806080 

   

In order to determine the function w ( ) we have to invert the erfi( x ) function. Let us explain briefly how to invert the erfi( x ) function. Suppose we want to invert the function (C.7) erfi( x ) = z Since, according to (C.2), erfi( x ) = −i erf (i x ) , equation (C.7) can be written as − i erf (i x ) = z and therefore erf (i x ) = i z , from which we get i x = erf −1 (i z) or (C.8) x = − i erf −1 (i z)

Inverting directly (C.7), we have (C.9) x = (erfi) −1 ( z) (erfi) −1 (z) = − i erf −1 (i z)

Comparing (C.8) and (C.9), we get the formula (C.10)

Using the series expansion (C.4), we get from (C.8)  ck   x = −i ∑ iz    k = 0 2k + 1  2
∞ 2 k +1

Since − i ⋅ i 2k +1 = i 2k + 4 = i 2k = (−1) k , the previous relation can be written as (C.11)   (erfi) (z) = ∑ z   k = 0 2k + 1  2 
−1 ∞ ( −1) k c  k 2 k +1

Details about the error function and its inverse can be found for example in References [19]-[21].

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Appendix D
In trying to find a general solution of the BCG equation (2.2), we have used a linear combination of the generators X1 , X 2 and X 4 . It is obvious that the generator X 3 corresponds to the so-called Tanh-method of traveling-wave solutions (see for example [22] and references therein). In this case we introduce a new variable by = x − r t . Using the fact that u t = − r U ′( ) , u x = U ′( ) and u xx = U ′′( ) equation (2.2) transforms into 1  1  − r U ′( ) + U 2 U ′′ +  r + U 2  U ′ = 0 2  2  which is equivalent to U ′′ + U ′ = 0 and has a general solution U = C1 + C 2 e − This solution is not of any practical use (see also [3]).

Appendix E. Lie Symmetries of the BCG Equation (1.2)
We consider the BCG equation (1.2) given by ∂ ~ 1 2 2 ∂ 2~ ∂~ + + (r + ~) V =0 V ∂t 2 ∂V ∂ V2 where ~ ≡ ~(V, t ) . We make the substitution x = ln V . Since V2 ∂ 2~ ∂ V2 = ∂ 2u ∂ x2 − ∂u ∂x and V ∂~ ∂u = ∂V ∂x

the BCG equation takes the form ∂ u 1 2 ∂ 2u  1 2 ∂u + + r − + u =0 ∂t 2 ∂x ∂ x2  2 where ~(V, t ) = u ( x, t ) with x = ln V . The Lie Symmetries of the previous equation are determined next.

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Let

( x , t , u ( 2) ) be given by ( x , t , u ( 2) ) = u t + 1 2  1 2  + uux u xx +  r − 2  2 

Introduce the vector field X (the generator of the symmetries) by (E.1) X = (x, t, u ) ∂ ∂ ∂ + (x, t, u ) + (x, t, u ) ∂x ∂t ∂u

The second prolongation is defined in our case by ∂ ∂ ∂ ∂ ∂ + t + xx + xt + tt pr ( 2) X = X + x ∂ux ∂ut ∂ u xx ∂ u xt ∂ u tt We have now to implement the equation pr ( 2) X [ ( x , t , u ( 2) )] = 0 . We have pr ( 2) X [ ( x , t , u ( 2) )] = 0 1 2   1 2   + uux  = 0 pr ( 2) X u t + u xx +  r − 2  2     1 2  ⇔ [ ux ]+ xr − + u +  2  1 2 + t + xx =0 2 For the functions , and we have = (x, t, u )
x

= (t) ,

= (x, t ) ,

Thus, using the same expressions for get that  1 2  + u + [ ux ]+ xr −  2 

t

and

xx

as in the equation (1.1), we

1 2 + t + xx =0 2

or

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(E.2)

 1 2  + u  [ x + ( u − x )u x ] + [ ux ]+ r −  2  + t − t u x + ( u − t )u t + + 1 2 { xx + (2 xu − xx )u x + 2 + uu u 2 + ( u − 2 x )u xx } = 0 x

We substitute u t by −

1 2  1 2  + u  u x into (E.2): u xx −  r − 2  2 

 1 2  + u  [ x + ( u − x )u x ] + [ ux ]+ r −  2   1 2  1 2   + t − t u x + ( u − t ) − + uux  + u xx −  r −  2    2 + 1 2 { xx + ( 2 xu − xx )u x + uu u 2 + ( u − 2 x )u xx } = 0 x 2

which can be written in the equivalent form (E.3) 1 2  1 2   + u x + t +  r − xx  + 2   2   +   1 2  + r − + u( u − x )− t −  2 

  1 2  1 2 − ( u − t ) r − + u + (2 xu − xx ) u x +  2  2  1 1 + [ − ( u − t ) 2 + ( u − 2 x ) 2 ] u xx + 2 2 + 1 2 2 uu u x = 0 2

Equate all the coefficients of the partial derivatives of the function u to zero. We then have a system of partial differential equations:

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• Coefficient of the zero-th order derivative of u: (E.4) 1 2  1 2  + u x + t + r − xx = 0 2  2 

• Coefficient of u x (E.5)  1 2  + r − + u( u − x )− t −  2   1 2  1 2 − ( u − t ) r − + u + (2 xu − xx ) = 0  2  2 • Coefficient of u xx : (E.6) 1 1 − ( u − t) 2 + ( u −2 x) 2 = 0 2 2

• Coefficient of u 2 : x (E.7) 1 2 uu = 0 2
uu = 0 , which means that

We are to solve the system of equations (E.4)-(E.7) From (E.7) we get to u: (E.8) ( x, t, u ) = (x, t ) ⋅ u + ( x, t ) is a linear function with respect

From (E.6), after canceling opposite terms and simplifying, we get the equation (E.9)
x =

1 ′( t ) 2

which has the general solution (E.10) (x, t ) = 1 ′( t ) ⋅ x + ( t ) 2

From the previous equation we get (E.11) (E.12)
t =

1 ′′( t ) ⋅ x + ′( t ) 2

xx = 0

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From equation (E.5), after canceling opposite terms and making the substitutions (E.8)-(E.12), we get the equation (E.13) ( ⋅u + ) + − 1 1 2  ′( t )  r − + u −  2  2 

1 ′′( t ) ⋅ x − ′( t ) + 2 x = 0 2

The previous equation can be written as  1 ′  ( t ) u +  + 2    +  + 1 1 2 1  2 ′( t )  r −   − ′′( t ) ⋅ x − ′( t ) + x=0 2  2  2 

From the previous equation we derive the two equations (E.14) (E.15) + 1 ′( t ) = 0 2 + 1 1 2 1 2 ′( t )  r −   − ′′( t ) ⋅ x − ′( t ) + x =0 2 2  2  by (E.8), we get

From equation (E.4), and substituting

 1 2  + u( x ⋅ u + x ) + ( t ⋅ u + t ) + r −  2  + 1 2 ( xx ⋅ u + xx ) = 0 2

which can be written in equivalent form 1 2  1 2 r −  x+ t+ xx + 2  2   1 2  +  r −  x+  2 
x+ t+

1 2  xx  u + 2 

xu

2

=0

From the previous equation we get

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(E.16)

1 2  1 2 r −  x+ t+ xx = 0 2  2   1 2 r −  x+  2 
x =0 x+ t+

(E.17) (E.18)

1 2 xx = 0 2

From (E.14) we get (E.19) (x, t ) = − 1 ′( t ) 2

which is compatible with (E.18). Since xx = 0 , we get from (E.17) that (E.20)
x + t =0

and therefore, using (E.19) and the above equation, we find (E.21)
x =

1 2

′′( t )

and so integrating the previous equation we obtain (E.22) (x, t ) = 1 x ⋅ ′′( t ) + ( t ) 2

From (E.16), using (E.21) and the fact (as it follows from (E.22))
t =

1 x ⋅ ′′′( t ) + ′( t ) 2

we derive the equation (E.23)  1 2 1 r −   2 2 ′′( t ) + 1 x ⋅ ′′′( t ) + ′( t ) = 0 2

The previous equation holds for any x if (E.24) and (E.25) ′′′( t ) = 0  1 2  1 ′′ ( t ) + ′( t ) = 0 r −   2 2

Finally, equation (E.15) can be written, using the previous expressions,

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 1  1  1 2  1 ′′ x ⋅ ′′( t ) + ( t )  + ′( t )  r − ( t ) ⋅ x − ′( t ) = 0 −  2 2  2  2   which is equivalent to the equation (E.26) (t) + 1 1 2 ′( t )  r −   − ′( t ) = 0 2  2 

Equation (E.25) gives (E.27) ( t ) = a1 + a 2 t + a 3 t 2

We can now determine ( t ) from equation (E.24). In fact, using (E.27) we find ′( t ) = − and by integration (E.28) 1 1 2 (t ) = −  r −  a 3t + a 4  2  1  1 2 r −  2a 3 2  2 

Finally we can determine ( t ) from equation (E.26). We find, using (E.27) and (E.28) 1 2  1  1 1 2 ′( t ) = −  r −   a 3t + a 4  +  r −  ( a 2 + 2a 3 t )  2    2 2  and by integration, (E.29) 1 1 2 (t) = a 4 t +  r −  a 2t + a5 2 2 

From (E.19) we can determine ( x , t ) : (E.30) 1 ( x, t ) = − (a 2 + 2a 3 t ) 2

and from (E.22) the function ( x , t ) (E.31) ( x, t ) = 1 x ⋅ a3 − 1 1 2 r −  a 3t + a 4  2 

Using (E.30) and (E.31) we can determine the function ( x, t ) from (E.8):

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(E.32)

1 1 1 1 2 ( x , t ) = − (a 2 + 2a 3 t ) u + x ⋅ a 3 −  r −  a 3t + a 4 2  2 

The function ( x , t ) can be derived from (E.10), using (E.27) and (E.29): (E.33) 1 1 1 2 ( x , t ) = − (a 2 + 2a 3 t ) x + a 4 t +  r −  a 2t + a5 2 2 2  1 1 2  1  ∂ + X = − (a 2 + 2a 3 t ) x + a 4 t +  r − a 2 t + a5  2 2   2 ∂x + (a 1 + a 2 t + a 3 t 2 ) ∂ + ∂t

The vector X, the generator of the symmetries, can be expressed as

1 1 1 2  1  ∂ + − (a 2 + 2a 3 t ) u + x ⋅ a 3 −  r − a3 t + a 4   2   2 ∂u The previous expression can be written as X = a1 ∂ + ∂t  1 ∂ 1 ∂  1 1 2  ∂ + a 2  − x +  r − + t − u + t ∂t 2 ∂u 2 2  ∂x  2
 ∂ ∂  1 1 1 2  ∂  + a 3 − t x + t2 + − tu + x − r − t + ∂x ∂t  2   ∂u 

+ a4

∂ ∂ + a5 ∂u ∂x

Therefore, the generators of the symmetries are: X1 = ∂ ∂ ∂ ∂ , X 2 = (x − t) − 2t + u ∂x ∂t ∂u ∂t ∂ ∂ ∂ + t2 + (− t u + x − t ) ∂x ∂t ∂u

X3 = − t x

28

X4 = where

∂ ∂ , X5 = ∂u ∂x 1 2 2

=r−

The above generators can be used in finding a solution of the BCG equation (1.2) along the lines of Section 3, once the initial conditions are given.

References
[1] F. Black and M. Scholes: “The pricing of options and corporate liabilities”. Journal of Political Economy 81 (1973) 637-654 [2] T. Bollerslev, Y. Chou Ray and K. F. Kroner: “ARCH modeling in finance: a review of the theory and empirical evidence”. Journal of Econometrics 52 (1992) 5-59 [3] A. Bensoussan, M. Crouhy and D. Galai : “Stochastic equity volatility related to the leverage effect. I. Equity volatility behaviour”. Applied Mathematical Finance 1(1994) 63-85 [4] A. Bensoussan, M. Crouhy and D. Galai : “Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants”.Applied Mathematical Finance 2(1995) 43-59 [5] S. Heston: “A closed-form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”. Review of Financial Studies 6(1993) 327-343 [6] J-P. Fouque, G. Papanicolaou and K. R. Sircar: “Derivatives in Financial Markets with Stochastic Volatility”, CUP 2000. [7] A. Lewis: “Option Valuation under Stochastic Volatility with Mathematica Code”, Finance Press 2000. [8] D. Bates: “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in PHLX Deutsch Mark Options” Review of Financial Studies 9 (1996) 69-107

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S. Kou: “A jump diffusion model for option pricing” Management Science 48 (2002) 1086-1101

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[13]

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N. H. Ibragimov: “Elementary Lie Group Analysis and Ordinary Differential Equations” Wiley, New York

[15]

L. Dresner: “Applications of Lie’s Theory of Ordinary and Partial Differential Equations” IOP Publishing, 1999

[16]

R. L. Anderson and N. H. Ibragimov: “Lie-Bäcklund Transformations in Applications ” SIAM, Philadelphia 1979

[17]

N. H. Ibragimov (ed.): “CRC Handbook of Lie Group Analysis of Differential Equations” Vol.1 1994, Vol.2 1995, Vol.3 1996 CRC Press, Boca Raton, FL.

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W. Hereman: “Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations” Euromath Bulletin 2, No.1 Fall 1993

[19] M. Abramowitz and I.A. Stegun: “Handbook of Mathematical Functions” Dover, 1972 [20] Wikipedia: “Error Function”. Available online en.wikipedia.org/wiki/Error_function [21] L. Carlitz: “The Inverse of the error function” Pacific J. Math 13(1963) 459-470

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W. Hereman and W. Malfliet: “The Tanh method: A Tool to Solve Nonlinear Partial Differential Equations with Symbolic Software” Institute of Mines report

Acknowledgements.
I would like to thank Prof. A. Bensoussan for kindly reading a preliminary draft of this report.

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