On the Inverse Problem of Dupire’s Equation with Nonlocal Boundary and Integral Conditions

http://www.scirp.org/journal/jmf ISSN Online: 2162-2442

ISSN Print: 2162-2434

DOI: 10.4236/jmf.2017.74051 Nov. 28, 2017 934 Journal of Mathematical Finance

On the Inverse Problem of Dupire’s Equation

with Nonlocal Boundary and Integral

Conditions

Coskun Guler

1

, Volkan Oban

2

1_{Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey} 2_{Department of Mathematical Engineering, Istanbul Technical University, Istanbul, Turkey}

Abstract

In this study, Inverse Problem for Dupire’s Equation with nonlocal boundary and integral conditions is studied. Then, by means of the some transforma-tion, this equation is converted to diffusion equation. The conditions for the existence and uniqueness of a classical solution of the problem under consid-eration are established and continuous dependence of

(

ρ,v

)

on the data is shown. It is emphasized that this problem is well-posed.

Keywords

Mathematical Finance, Dupire’s Formula, Dupire’s Equation, Local Volatility, Diffusion Equation, Inverse Problem, Well-Posedness

1. Introduction

In mathematical finance, Dupire’s formula (local volatility) is expressed in the following form

( )

2

2

1

, 2

V V

rS

t S

S t

S V

S

σ

∂ ∂

+

∂ ∂

=

∂ ∂

The Dupire formula enables us to deduce the volatility function in a local vo-latility model from quoted put and call options in the market. In a local vovo-latility model the asset price model is under a risk-neutral measurement. For the rele-vant formula, reference [1].

Non-homogeneous Dupire’s equation is shown as follows,

How to cite this paper: Guler, C. and Oban, V. (2017) On the Inverse Problem of Dupire’s Equation with Nonlocal Boundary and Integral Conditions. Journal of Ma-thematical Finance, 7, 934-940.

https://doi.org/10.4236/jmf.2017.74051

Received: October 11, 2017 Accepted: November 25, 2017 Published: November 28, 2017

Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

DOI: 10.4236/jmf.2017.74051 935 Journal of Mathematical Finance

( )

2 2

1

, 2

t SS S

V =

σ

⋅S ⋅V −rSV +F S t

The Dupire equation is a forward equation for the call option price V as a function of the strike price S and the time to maturity t. The local volatility of the underlying assets is a deterministic function of assets price and the time t.

(

S tt,

)

σ σ=

Therefore under local volatility model, the stochastic process followed by the stock price is

(

)

dSt=µS ttd +σ S tt, dWt

t

W : The randomness from the stock price: [2].

In the 1970s, when Black-Scholes formula was initially derived, most people were convinced that the volatility of a certain asset given the current circums-tance was a constant number. Then, later on, after the economic crash in 1987, people were starting to doubt the constant volatility assumption. Especially after more and more evidence of volatility smile was collected, people tend to believe that the implied volatilities cannot remain constant during the whole time. They probably have some dependent relationships with some other factors in the op-tion pricing model as well. One of such guesses is that, the implied volatility could be depending on the stock price S(t) and time t. And if we study a model of price processes with a volatility that depends on the stock price S(t) and time t, we can try to explore the inner connection between the implied volatility, and the local volatility. The volatility in such models depends on the σ imply

( )

(

S t t,

)

σ stock price S(t) and time t[3].

In mathematical physics, one of the most famous diffusion equations is a par-tial differenpar-tial equation that describes the flow of heat and is given by

2 2

u u

k

t x

∂ ∂ =

∂ ∂

where u=u x t

( )

, is a function of space x and time t and k is an arbitrary real constant known as the diffusion coefficient. The above is called a heat equation. Black-Scholes PDE, even though explicitly not a heat equation, can be trans-formed into a heat equation like the above with suitable transformation of va-riables. To be exact, the heat equation is not the most generalized form of diffu-sion equation that arises in physics, but we shall currently limit our discusdiffu-sion to the heat equation because that will suffice for our understanding of dynamics of financial derivatives.

We can think of a call option (on a financial asset) being governed by an equ-ation similar to the above, where u=u S t

( )

, will be the call option price as a function of a spatial variable, S the asset price (here, x=S) and time, t. The diffusion coefficient, k can be thought as the volatility of the asset price.

Formerly studies related to inverse problem for parabolic equations can be examined in [4]-[12].

DOI: 10.4236/jmf.2017.74051 936 Journal of Mathematical Finance (i.e., Black-Scholes-Merton Equation, Backward Kolmogorov Equation, and Du-pire’s Equation, etc.)

In this study, we take into consideration the following equation

( ) ( )

2 2

1

, 2

t SS S

V =

σ

⋅S ⋅V −rSV +p t ⋅F S t , (1.1)

( )

{

, :1 , 0

}

.

T

D = S t < <S e < <t T with the initial condition,

( )

(

,

)

( )

;

( )

ln

( )

; 1

V ω S O =φ S ω S = S < <S e (1.2)

The periodic boundary condition

( )

1,

( )

, , s

( )

, 0; 0

V t =V e t V e t = ≤ ≤t T (1.3) And the over determination condition

( )

( )

1

, d eV S t

S E t

S =

∫

(1.4)

The problem of finding a pair

{

p t V S t

( ) ( )

, ,

}

in (1.1)-(1.4) is called inverse problem for Dupire’s Equation.

2. Converting the Dupire’s Equation to the Diffusion

Equation

( ) ( )

2 2

1

, 2

t SS S

V =

σ

⋅S ⋅V −rSV +p t ⋅F S t

Under the transformations;

2

2 e ;x

S t T τ

σ

= = −

(

)

2

ln ; 2

x S τ σ T t

⇒ = = −

( )

( )

2

( )

2

(

)

2

, , e , ln ,

2 x

v x

τ

V S t V T

τ

v S

σ

T t

σ

 

 

= ⇒ _ − _= _ − _

   

(

)

t x t t

V v x vτ τ

⇒ = ⋅ + ⋅

2

1 2 t

V =

σ

⋅v_τ

⇒ −

Differentiation yields

(

)

2

1 1

;

S x SS xx x

V v V v v

S S

⇒ = = −

( ) ( )

2 2

1

, 2

t SS S

V

σ

S V rSV p t F S t

⇒ = ⋅ ⋅ − + ⋅

2 2

2 2

2 2

1 1 1

e

2 2

x

v v v v

r S pF

x S x

S x

σ

σ

τ

 

∂ ∂ ∂ ∂

⇒ − − _ − _+ ⋅ ⋅ +

∂ _∂ ∂ _ ∂

( ) ( )

2

2

,

xx x x

r

v_τ v v v ρ τ f xτ

σ

DOI: 10.4236/jmf.2017.74051 937 Journal of Mathematical Finance

( ) ( )

2 2 1 , xx x r

vτ v _σ v ρ τ f xτ

⇒ = +_ − _ + ⋅

 

When 2

2 r

σ = ⋅ taken, we get,

( ) ( )

,

xx

v_τ v ρ τ f xτ

⇒ = +

( )

2

{

( )

}

, : 0 1, 0 , : 0 1, 0

2

Dτ xτ x T σ t T Dτ xτ x τ T

 

= < < < ⋅ < < ⇒ = < < < <

 

( ) ( )

,

xx

vτ =v +ρ τ f xτ (2.1)

( )

, 0 Ψ

( )

, 0 1

v x = x ≤ ≤x (2.2)

( ) ( ) ( )

0, 1, , x 1, 0; 0

v t =v t v t = ≤ ≤τ T (2.3)

( )

( )

1

0v x,τ dx=E τ , 0≤ ≤τ T

∫

(2.4)

3. Existence and Uniqueness of the Solution of the

Transformed Inverse Problem [13]

There are the following assumptions on the data of problem (2.1)-(2.4). (v1) E

( )

τ

∈C1

[ ]

0,T

(v2) Ψ

( )

x ∈C4

[ ]

0,1 ;

1) Ψ 0

( )

=Ψ 1 ,Ψ 1 0,Ψ 0

( ) ( )

′ = ′′

( )

=Ψ 1 ,′′

( )

2) 1

( )

( )

0Ψ x dx=E 0

∫

(v3) f x

( )

,τ ∈C

( )

Δ ;T f x

( )

,τ ∈C4

[ ]

0,1 ,∀ ∈τ

[ ]

0,T ; 1) f

( )

0,τ = f

( ) ( )

1,τ ,fx 1,τ =0,fxx

( )

0,τ = fxx

( )

1,τ ;

2) 1

( )

[ ]

0 f x,τ dx≠ ∀ ∈0, τ 0,T

∫

Then the so-called inverse problem (2.1)-(2.4) has a unique solution. Proof:

Take into consideration the following function system on [0,1]. Riesz bases in L2

[ ]

0,1

( )

( )

[

]

( ) (

)

[

]

0 x 2, 2n1 x 4 cos 2πnx , 2n x 4 1 x sin 2πnx n, 1, 2,

χ = χ ₋ = χ = − = _{ (3.1)}

( )

( )

[

]

( )

[

]

0 , 2n1 cos 2π , 2n sin 2π , 1, 2,

y x =x y ₋ x =x nx y x = nx n=  (3.2) It is easy to verify that each systems (3.1) and (3.2) are biorthogonal on [0,1]. Applying the standard procedure of the Fourier method, we get the following presentation for the solution (2.1)-(2.3) for arbitrary ρ

( )

t ∈C

[ ]

0,T :

( )

( ) ( )

( )

( )

( )

( ) ( )

( ) ( )

( )

( )

( )

( )

( )

( ) ( )

( )

( ) ( )(

)

( ) ( ) 2 2 2 2 2 2π

0 ₀ 0 0 ₁ 2 2

2π

2 2

1 0

2

2 1 2 2 1

1

2π

2 1 2 1

1 0

2π 2

1 0

, Ψ d Ψ e

e d

(Ψ 4 Ψ )

e d

4π e d

n t n n n n n n n n

n n n

n n n n n n n n

v x f x x

f x

n e x

f x

n f

τ

τ _{τ δ}

π τ

τ _{τ δ}

τ _{τ δ}

τ ρ δ δ δ χ χ

ρ δ δ δ χ

π τ χ

ρ δ δ δ χ

ρ δ δ τ δ δ

∞ − = ∞ − − = ∞ − − − = ∞ − − − − = ∞ − − =     =_ + _ + _{ }     + _{ }     + _ − _     + _{ }   − −

∑

∫

∑ ∫

∑

∑ ∫

∑

_

∫

_χ2n−1

( )

x

DOI: 10.4236/jmf.2017.74051 938 Journal of Mathematical Finance where 1

( ) ( )

0 d

k x yn x x

Ψ = Ψ

∫

_and

( )

1

( ) ( )

0 , d

n n

f τ =

∫

f xτ y x x, n=0,1, 2, The assumption Ψ

( )

0 = Ψ

( )

1 , Ψ′

( )

1 =0, f

( )

0,τ = f

( )

1,τ , fx

( )

1,τ =0 are consistent conditions for the presentation (3.3) of the solution v x t

( )

, to be valid. Besides, under the smoothness assumptions _Ψ

( )

4

[ ]

0,1 x ∈C ,

( )

,

( )

ΔT

f xτ ∈C and f x

( )

,

τ

∈C4

[ ]

0,1 , ∀ ∈τ

[ ]

0,T , the series (3.3) and its x- partial derivative (

x

∂

∂ ) uniformly convergent in ΔT because their majorizing sums are absolutely convergent. So, their sums v x

( )

,τ and vx

( )

x,τ are con- tinuous in ΔT. Additionally, the τ -partial derivative (

t ∂

∂ ) and xx-second

or-der partial or-derivative ( 22

x

∂

∂ ) series are uniformly convergent for

τ ε

≥ >0 (ε

is an arbitrary constant-positive number). Thus,

( )

2,1

( )

1,0

( )

, ΔT ΔT

v xτ ∈C C

and satisfies conditions (2.1)-(2.3). Additionally, v xt

( )

,τ is continuous in ΔT since the majorizing sum of τ -partial derivative (

t

∂

∂ ) series is absolutely

con-vergent according to the conditions Ψ′′

( )

0 = Ψ′′

( )

1 and fxx

( )

0,τ = fxx

( )

1,τ in ΔT. [13]

Equation (2.4) could be differentiated according to (v1) to obtain;

( )

( )

1

0vτ x,τ dx=E′ τ

∫

(3.4) Besides, under the consistency assumption 1

( )

( )

0Ψ x dx=E 0

∫

the formulas (3.3)-(3.4) result the following Volterra integral equation (Second kind):

( )

( )

₀t

(

,

) ( )

d ,

[ ]

0,

P τ =F τ +

∫

K τ ζ ρ ζ ζ τ∈ T (3.5)

where

( )

( )

( )

( )

( )

2

2π 2 1 0 1 2

8π e

2 1

2

π

n n n

n n

E n

F

f f

n

τ

τ τ

τ τ

∞ − =

∞ =

′ + ⋅ Ψ

=

+

∑

∑

(3.6)

( )

( )

( ) ( )

( )

( )

2

2π 2

1

2 1

4π e

,

1 1

π

n n

n

n n

n f K

f F

n τ ς

τ τ ς

τ τ

∞ − −

= ∞

=

⋅ =

+

∑

∑

(3.7)

Consider that the dominator in (3.6) and (3.7) is never equal to zero, since the assumption 1

( )

0f x,τ dx≠0

∫

, ∀ ∈τ

[ ]

0,T . In the light of the assumptions (v1)-(v3), the function F

( )

τ and the kernel K

( )

τ ς, are continuous functions

on

[ ]

0,T and

[ ] [ ]

0,T × 0,T , respectively. So, we get a unique function ρ

( )

t which is continuous on

[ ]

0,T and together with the solution of the so-called problem (2.1)-(2.3) given by Fourier series (3.3), forms the unique solution of the inverse problem (2.1)-(2.4) [13].

4. Continuous Dependence of (

P

,

v

) on the Data

DOI: 10.4236/jmf.2017.74051 939 Journal of Mathematical Finance the assumptions (v1)-(v3) and let

( )

2,0_Δ 0

T

C

f ≤M , Ψ C2_{[ ]}0,1 ≤M1, E _C1[ ]_0,1 ≤M2, 0<M3≤min( )x t,∈ΔT f x

( )

,τ

for some positive constants M ii, =0,1, 2, 3.

Then, the solution pair

{ }

P v, of the so-called problem (2-1)-(2.4) depends on the data in Λ _{for small T.}

Proof: ϕ=

{

f,Ψ,E

}

and ϕ=

{

f,Ψ,E

}

be two data in Λ. Let’s

demon-strate

( ) [ ]

2,0_ΔT 2[0,1] 10,1

C C C

f E

ϕ = + Ψ + and

( )

[ ]

2

2,0_{Δ [0,1] 1}

0,1

T C

C _C

f E

ϕ

= + Ψ +

Let

{ }

P v, and

{ }

P v, be solutions of the inverse problem (2.1)-(2.4) cor-responding data

ϕ

and

ϕ

, respectively. It is apparent from (3.5)-(3.7) there are positive constants M ii, =4, 5 such that

[ ]0, 4 ([ ] [ ]0, 0, ) 5 [ ]0, 4 5

, ,

1

C T C T T C T

M

f M K M P

TM

×

≤ ≤ ≤

− , where

0 4 2 1 5

3

2 2

,

6 6

M

M M M M

M

= + = _.

It infers from (3.6)-(3.7) that

[ ]_0, 6 2,0

( )

_Δ 7 2[ ]_0,1 8 _{1[ ]}

0,

T

C T C C _{C T}

F−F ≤M F−F +M Ψ − Ψ +M E−E

[ ] [ ]

(_{0, 0,} ) 9 2,0( )_Δ

T

C T T C

K K M F F

×

− ≤ −

where

6 2 1 2 7 2 0

4 4

1 4 2 2 1 4 2

2 ,

3 3

6 6 6

M M M M M

M M

      

= _ + _ +_ + _{ } = _ + _

     

 

8 2 0 9 2 0

4 4

1 2 1 8 4

2

3

6 , 6

M M M M

M M

   

= _ + _ = _ + _

   

From (3.5), we get that

[ ]0, [ ]0, 5 [ ]0, 4 ([ ] [ ]0, 0, )

5

1

C T

C T C T C T T

M

P P F F TM P P T K K

TM ×

− ≤ − + − + −

−

(

1−TM5

)

P−P _{C[ ]0,T} ≤M10 ϕ ϕ− , where

4 9

10 6 7 8

5

max , ,

1 M M

M M T M M

TM

 

= _ + _

−

 

(

TM5<1

)

for small. Eventually, we get,

[ ]0, 11

C T M

ρ ρ− ≤ ϕ ϕ− _, 10 11

5

1 M M

TM

=

− .

In a similar way, we get the difference v v− from (3.3)

( )_ΔT 12

C

v−v ≤M v−v [13].

5. Conclusion

Boun-DOI: 10.4236/jmf.2017.74051 940 Journal of Mathematical Finance dary and Integral Conditions has been examined. In the theoretical, the condi-tions for the existence, uniqueness and continuous dependence on the data of the problem have been established. Then it is shown that the problem is well-posed problem (in the sense of Hadamard).

References

[1] Davis, M.H.A. The Dupire Formula.

http://wwwf.imperial.ac.uk/~mdavis/FDM11/DUPIRE_FORMULA.PDF

[2] Wu, J. Local Volatility and Stochastic Volatility.

https://www.quantconnect.com/tutorials/introduction-options-stochastic-volatility/

[3] Ye, H. (2011) A Comparison of Local Volatility and Implied Volatility. Uppsala University, U.U.D.M. Project Report 2011:12.

[4] Kanca, F. and Ismailov, M.I. (2012) The Inverse Problem of Finding the Time-Dependent Diffusion Coefficient of the Heat Equation from Integral Overde-termination Data. Inverse Problems in Science and Engineering, 20, 463-476.

https://doi.org/10.1080/17415977.2011.629093

[5] Cannon, J.R. (1968) Determination of an Unknown Heat Source from Over Speci-fied Boundary Data. Society for Industrial and Applied Mathematics Journal on Numerical Analysis, 5, 275-286.https://doi.org/10.1137/0705024

[6] Cannon, J.R., Lin, Y. and Wang, S. (1991) Determination of a Control Parameter in a Parabolic Partial Differential Equation.The ANZIAM Journal, 33, 149-163.

https://doi.org/10.1017/S0334270000006962

[7] Dou, F.-F., Fu, C.-L. and Yang, F. (2009) Identifying an Unknown Source Term in a Heat Equation. Inverse Problems in Science and Engineering, 17, 901-913.

https://doi.org/10.1080/17415970902916870

[8] Farcas, A. and Lesnic, D. (2006) The Boundary-Element Method for the Determina-tion of a Heat Source Dependent on One Variable. Journal of Engineering Mathe-matics, 54, 375-388.https://doi.org/10.1007/s10665-005-9023-0

[9] Ivanchov, M.I. (1998) The Inverse Problem of Determining the Heat Source Power for a Parabolic Equation under Arbitrary Boundary Conditions. Journal of Mathe-matical Sciences, 88, 432-436.https://doi.org/10.1007/BF02365265

[10] Ionkin, N.I. (1977) Solution of a Boundary-Value Problem in Heat Conduction with a Non-Classical Boundary Condition. Differential Equations, 13, 204-211.

[11] Johansson, T. and Lesnic, D. (2007) Determination of a Space Wise Dependent Heat Source. Journal of Computational and Applied Mathematics, 209, 66-80.

https://doi.org/10.1016/j.cam.2006.10.026

[12] Vigak, V. (1994) Construction of a solution of the Heat-Conduction Problem with Integral Conditions. Doklady Akademii Nauk Ukrainskoj SSR Serija A, 8, 57-60. [13] Ismailov, M.I., Kanca, F. and Lesnic, D. (2011) Determination of a Time-Dependent

Heat Source under Nonlocal Boundary and Integral Overdetermination Conditions. Applied Mathematics and Computation, 218,4138-4146.