A Stochastic Differential Equation Model

A Stochastic Differential Equation Model

N. Hema1 & Dr. A. Jeyalakshmi2 1Research Scholar,SCSVMV, Kanchipuram, India. 2Prof & Dean of Sciences, SCSVMV, Kanchipuram.

Abstract :In this paper, we propose a stochastic differential equation model where the underlying stochastic process is a jump- diffusion process.The stochastic differential equation is represented as a Partial Integro Differential Equation(PIDE) using the Fokker Planck equation. The solution of the PIDE is obtained by the method of finite differences. The consistency, the convergence of the solution and the stability of the finite difference scheme are discussed. The model is applied to forecast the daily price changes in a commodity derivative. The observed values are compared graphically with the values expected from the proposed model.

Keywords: jump-diffusion process, partial integro differential equation, stability of the finite difference scheme, consistency and convergence of a numerical solution, commodity derivative.

Introduction

There are two reasons for studying a SDE. One motivation is that, many physical phenomena can be modelled as random processes (e.g. thermal motion). When such a process enters a physical system, we get a SDE model. The second reason is that, in statistical modelling, unknown forces are modelled as random processes. This again leads to a SDE. Unknown forces are often found to occur in modelling financial quantities like asset price, interest rate, options and other derivatives. In this paper, we present a SDE model and apply it for pricing a commodity derivative. In recent years the complexity of the models used has increased and in turn has led to complicated equations. Of particular interest is a type of differential equation containing an integral term. Such equations are called Partial- Integro Differential Equations. I. Florescu and Mariani[3]studied these type of problems and proved the existence of a solution under a general hypothesis about the integral term. R.Cont suggested a finite difference scheme with discretization of the integral term[9] .

In this paper, we discuss a particular type of PIDE . The paper is organised as follows: In sections 1 and 2 we give a review of a SDe with jump diffusions and the finite difference scheme for second order partial derivatives. In section3, we recall the eigenvalues and spectral value of a tri diagonal matrix. Also we recall the definitions of consistency, convergence and stability of the solution obtained by the numerical scheme. Finally in section 4, the proposed model and the assumptions made are described. We prove the consistency, the uniqueness of the solution to the system of equations under consideration and also obtain the conditions of stability of the finite difference scheme.The proposed model is tested with a set of real time data. The observed and the estimated values are compared graphically.

Section 1: A SDE and jump- diffusion process:

Generally a SDE is in the form dX(t) = (t, X(t)) dt + (t, X(t)) dW(t), where W denotes a Wiener process. This equation should be interpreted as an integral equation as

X(t) X(0) = (t, X(t)) dt + (t, X(t)) dW(t), X(0) being the initial value of X.Sometimes in a SDE, the stochastic process may consist of three components namely: a drift term, a diffusion term and a jump term. The jump term accounts for abnormal changes in value.The jumps occur at random times and are usually assumed to follow Poisson distribution. The magnitudes of the jumps may be assumed to follow Poisson or other distributions like beta, exponential, log normal or Pareto[1],[9].There is a direct relationship between a stochastic differential equation and a boundary value problem for a parabolic partial differential equation (diffusion equation) given by the Fokker Planck equation:

If X(t) is a jump diffusion process satisfying the SDE ,

dXt = f(X(t),t)dt + g(Xt,t)dWt + h(X(t),t,Q)dP(Xt ,t) then the expectation v(x,t) = E(u(X(T)|X(t)=x) is the solution to the PIDE ,

0 = +f + 1 2 2 + + , , ,

2 2

where f ,g,h are continuously differentiable functions, Q is the random variable denoting the jump amplitude mark, P is the jump process, W(t) is the Weiner process , (q) is the density of Q and T is the terminal time. Florescu and Mariani studied these types of equations and proved the existence of a solution under a general hypothesis on the integral term. To solve such kind of equations, a finite difference scheme with discretization of the partial derivative terms and integral term is suggested by R.Cont.

Section 2: The finite difference scheme

Let C0 be the space of continuously differentiable functions with a norm .

Let v C0([0,T]xR) be continuously differentiable as many times as required for t [0,T] and x R . The first and second order partial derivatives of v with respect to x are discretized as follows:

For (tn,xi) [0,T] x R ,

= +1 ;

2 = +1 2+ 1 .

2 i 2

Moreover an integral of the form + , , , , ()dq is approximated

=

=

to a sum 2

1

1() (+ ) using trapezoidal quadrature formula after replacing the

limits of integration by a suitable bounded interval say [ ,]. By the introduction of a uniform grid on [0,T] x [,] and discretization of the derivative as well as the integral terms, the PIDE may be reduced to a system of linear equations in the form Ax = B.We apply this discretization process for the PIDE considered in this paper.

Section3: Stability and uniqueness of solution of a tri diagonal system of equations

Let Ax = B be a system of equations where A is a tri diagonal matrix of order n having each of its entry on the main diagonal as a, the sub diagonal entries are all c and super diagonal

entries as b.The eigen values of this matrix are k = a + 2 cos+1 for k = 1,2,3.n provided bc > 0.

The spectral radius of A is (A) = max and the criterion for the stability of the scheme is (A) < 1.

A matrix of order n x n is said to be diagonally dominant if the sum (in modulus) of all the off-diagonal elements in any row is less than the modulus of the diagonal element in that row. Consistency and Convergence of the solution

A finite difference scheme is said to be consistent with the PIDE it represents, if for any sufficiently smooth solution u of this equation, the truncation error of the scheme , tends uniformly to zero as (t,x)0. Moreover, if there exist C1 and C2 independent of t and

x such that | | C1 (t)h +C2 (x)k then the order of convergence is (h,k).

The Lax-Ritchmeyer equivalence theorem:

Given a properly posed linear initial or boundary value problem, if a linear finite difference approximation to it is consistent and stable , then the scheme is convergent.

A problem is properly posed if:

(i) the solution exists, (ii) the solution is unique and (iii) the solution depends continuously on the initial data[2].

Section 4:The Proposed Model

1. Assumptions for the model:

   Let X(t) be a stochastic process and f, g, p, p be continuously differentiable functions of X and t. Let Q be the random variable denoting the jumps.

   Define Q+ = max(Q,0) and = max(-Q,0).Then Q+ and are respectively the positive and negative parts of Q. Moreover, Q = Q+ – . We shall assume Pareto distribution for Q+ and

   .The density fuctions of Q+ and are :

   (q ) =

   (q ) =

   +

   (+ ) +1

   (q ) =

   (q ) =

   –

   ( ) +1

   , q+ , > 0. ( , are the parameters of the distribution of Q+)

   , q- , > 0. ( , are the parameters of the distribution of )

   The expectations of these two random variables are given by :

   E(Q+) =

   1

   = and E() =

   1

   = .

   We shall assume and are different from 1.The parameters , , and of the distributions may be estimated from historical data. Let P+ and be the densities of Q+ and respectively.

   We represent the process as X = Z + Q+ – where Z is normal in [, ] , Q+ and are Pareto variables in the intervals [,] and [,] respectively..Now we construct the stochastic differential equation in the form

   dX(t) = f(X(t),t)dt + g(X(t),t)dW(t) + pdP+1[,] + pd1[, ] ——— (4.1) where 1 is the indicator function.

2. The PIDE

   Let v(x,t) be continuously differentiable in x and t as many times as required while bounded at . Then the conditional expectation of the process V(X(t),t) namely v(x,t) = E[V(X(t)|X(t0)=x0] is the solution to the PIDE

   0 =

   + f

   + g 2

   + +

   , , + , , (+)d+ +

   2 1

   2 , , , , ()d ———(4.2)

   The domains of Q+ and Q are truncated to the bounded intervals [,] and [,] by choosing and suitably.

   In equation (4.2), + and denote the upward and downward jumps respectively in X. 1 and 2 are the respective density functions.Let P1, P2, and N1, N2 be the parameters of the distributions of + and respectively. The numbers , are suitably chosen as the bounds of + and such that < N2 (=) and P2(=) < .

   2

   2

   Let D = f + 2 2 , I = = I1 I2 and L = D + I

   Now equation 4.1 takes the form + Lv = 0. ————(4.3)

   We shall include the initial conditi v 0 = x

   and the notations

   on 0 0

   1

   2

   = + 1 , , + , , ,

   = + 2 , , , , .

3. The explicit implicit scheme for equation (4.3):

   Let the period of time under consideration be T. Introduce a uniform grid on [0,T] x [, ] as

   tn = nt, for n = 0,1,2,..N-1; xi = + kx for k = 1,2M with t = and x = .

   Let {vkn} be the solution of the numerical scheme to be defined. Let

   K1+

   and K2+

   be real

   numbers such that [P2,] is contained in the interval [(K1+ -½)x, (K2+ +½)x] and K1- , K2- be such that [ ,N2] is contained in [(K1- -½)x, (K2-+½)x] .

   We use an explicit finite difference approximation for the differential operator D and an implicit difference approximation for the integral operator I as in R.Cont[10 ].

   k

   k

   The derivatives are discretized as ( )

   2

   2

   +1

   =

   =

   1

   1

   and ( 2

   )

   )

   2 k

   2 k

   = +12 +1. ()

   Also (J (v))

   =

   (+)dq+ +

   (+) (

   )

   2

   2

   1 k

   1 1

   =+

   1

   +

   (J (v))

   =

   ()dq-

   () (

   )

   2 k 2

   2

   2

   +

   2 = 1

   Now we replace equation (4.3) by the system of equations:

   +1

   +Dvkn+1 +J1vkn J2vkn = 0 , for k = 1,2.M ;, vkn+1 = 0, k > M ,

   n=

   -1 and v 0 = x .

   0,1,2.N 0 0

4. Consistency of the system of equations Proposition 1:

   The system of equations (4.4) is consistent.

   2

   2

   .ie. for all v ( [0,T] x [,] ) and all (tn,xk) [0,T] x [,],

   (

   (

   +1

   n+1

   n n

   +Dvk

   +J1vk

   J2vk

   ) (

   + Lv)(tn, xk) k (, ) 0 as (, ) 0.

   Also there exist C1 and C2 such that | k | C1t + C2x .

   Proof :

   Using Taylors expansion up to the second order,

   +1

   2

   1 2

   –

   2

   2 =

   2

   2 t 0 as (, ) 0

   Dvkn+1 Dv(tn,xk) = Dvkn+1 Dv(tn+1,xk) – tDv( ,xk) , where (tn,tn+1)

   +1

   tDv( ,xk) +f(tn+1 ,xk)(

   ) f(tn+1 ,xk)

   +

   2 (+1 2+1 + +1) – 2 (t

   ,x ) ()2

   2()2

   +1

   1

   2

   n+1 k

   2 2

   2 2

   tDv( ,x ) + 1 f(t

   ,x ) () + f(t

   ,x ) x

   k

   n k 2 2

   n k

   + 2

   (+1-+1) + ( +1 -+1)- 2(t

   ,x ) ()2

   2()2

   +1

   1

   2

   n+1 k

   0 as (t,x) 0 .

   2

   2

   J v n I v(t ,x )= [ +

   [ tn, x

   + + (tn, xk)]

   (+)-

   (+)dq+ ]

   1 k 1 n k

   =+

   1

   1

   +

   k

   ( +1)

   1 1 1

   + + + + +

   = 2

   2 [ tn, x

   +

   ( ) v(t ,x +q ) (q )]dq ,

   1

   1

   2

   2

   =+

   ( 1)

   k

   1

   n k 1

   since v

   +

   (+)

   (tn, xk)

   (+)dq+

   2

   2

   1

   1

   k =+

   n

   1

   +

   1

   ( +1) +

   + +

   J1vk

   I1v(tn,xk)=[ 2 + 2

   ],

   2

   2

   = 1 ( 1)

   2

   2

   where u(tn,xi+) = v(tn,xi+)1 ().

   k

   k

   So J1v n I1v(tn,xk) (-)

   ( +1)

   +

   ( 1)

   2

   2

   (-) ( K2+ – K +)x

   Similarly ,

   1

   0 as (t,x) 0 .

   J2vkn I2v(tn,xk) ( ) ) (K2- -K1-)x ,where w(tn,xi+) = v(tn,xi+)2 ()

   0 as (t,x) 0 .

   Hence the error term k satisfies

   | | 12 t +[ (-) ( K + – K +) +| |( ) ) (K – -K -)]x

   k 2 2

   2 1

   2 1

   = C1t +C2x, where C1 and C2 are independent of t and x .

5. Stability of the finite difference scheme

+1

+1

2 +

2 +

The system of equations (III) may be written in the form,

+1

+ f +1 +1

2 +1 +1 +1

k

k

+ +1 1 = J2v n J1vkn , k = 0,1,.N,

2

k

k

vk0 = x0, v n+1 = 0, for k > N.

or equivalently, c t vk-1n+1 + (1 a t)vkn+1 +b t vk+1n+1 = vkn + (J2v n J1v n) t , k =

+

+

0,1,2, .N wherea =

2

2 , b =

+

2

k

2 and c =

2

k

2 , vk

n+1

= 0,

2

for k > N. It can be seen that a = b + c .

x may be chosen to satisfy b > 0. Obvoiusly c 0. Hence we have a 0 .

Thus the system of equations is of the matrix form AX = B

1 at bt 0

ct 1 at bt

0 c 1 at

where A =

0 0 0

0 0 0

ct 1 at bt

0 ct 1 at

It is seen that A is a tridiagonal matrix.

Proposition 2:

The solution of the system of equations (4.5) is unique and stable if t < Proof:

1 .

2

Let t <

1 .

2

1 1

Then 1 a t > 2 whereas (b+c) t = a t < and the system has a unique solution.

2. Thus the matrix A is diagonally dominant

To prove the tability of the finite difference scheme we shall verify that the spectral radius of the matrix A is less than unity.

The eigen values of the tri diagonal system are given by

k = 1- a t + 2 cos+1 , k = 0,1,2,3.N.

The absolute value of the maximum among the eigen values is | 1- a t + 2 | Now < 1 requires | 1- a t + 2 | < 1.

i.e, -1 < 1- a t + 2 < 1

i.e, 0 < (a -2 ) t < 2

2

or <

2

<

<

1 2

1

Hence the condition t <

2 is sufficient for stability of the solution.

2

It should be noted that b 0 requires

2

+2 2 0. This is true if f 0.

If f < 0, we shall select <

2 .

Also it can be easily checked that 1 < 1 < 2 . For the sake of simplicity, the

2 2

stability condition is stated in the form given in Proposition 2.

Section5: Application of the SDE model to estimate the expected variation in the price of a commodity derivative.

A derivative is a financial insruent that gives the owner a certain payment depending on the value of the underlying asset or delivery of the asset according to the agreement between the buyer and seller. A futures contract is an agreement between two parties to buy or sell a certain asset for a certain price at a future date. Trading is done through exchanges and the underlying asset may be stocks, bonds, interest rates, exchange rates or commodities.

We consider the data from MCX (Multi-Commodity Exchange at Mumbai) for all wheat futures contracts from April 2011 having expiry as October 2011. In India, the marketing season for the rabi (or winter) wheat crop begins in the month of April every year. So data is considered at the beginning of the marketing season. We assume f(X(t),t) = m cos(2 ),

where m is the arithmetic mean of the percentage changes in the derivative price obtained

from historical data and R is the demand during the period T. We shall assume R is constant throughout the period. We have chosen f as a periodic function because the demand of an agricultural commodity is seasonal, as far as a commodity exchange is concerned. Let the function g = s2/2, s being the standard deviation of the data selected. Based on the test data the parameters of the distributions of + and may be estimated by the use of any one of the statistical techniques available. For the data under consideration the test for goodness of fit for Pareto distribution has been performed. Both Anderson Darling test and Smironov test accept the fit. The parameters of the distributions of + and are denoted by P1, P2 and N1,N2 respectively.

The empirical parameters are found to be P1=0.341, P2=0.016, N1=0.308, N2=0.016. Also m=o.o43819 and s = 0.56341. The limit specified by MCX for the commodity wheat is 3% either way. This means that the price variation in wheat cannot exceed 3% or fall below it. So we shall take the values of and as -3 and 3 respectively. We take =1. The condition < 1/2a now requires > 2s-m. Hence is taken suitably. Let N=30 and M=5.

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

actual estimated

1 4 7 101316192225283134374043464952555861

Plots showing the observed percentage changes in the futures price and the estimated values. The test data consists of all the wheat futures contracts at MCX for a period of fifty days, beginning 11th April 2011and having expiry month October 2011.

Conclusion

The estimated changes and the actual changes (in percentage relative to the closing price on the first day of the period under consideration) are plotted . It is found that the estimated values agree moderately with the actual values.A MATLAB code has been developed to obtain the solution from the finite difference scheme. Further work will focus on improving the choice of input parameters .

The MATLAB code is given below.

function f=price(mean,s,alpha,delta,delx,N1,N2,P1,P2,T,R,xo,N,M) K1plus=round((P1/delx)+0.5);

K2plus=round((delta/delx)-0.5); K1min=round((N1/delx)+0.5); K2min=round((-alpha/delx)-0.5); c=(s^2)/(2*(delx^2));

for n=2:N

for i=2:M

b=c+(mean*(cos(2*pi*R*(n-1))/T)/delx); a=b+c;

A(i)=1-a*n/T;

I=i-1;

B(I)=b*n/T; C(I)=c*n/T; IV(i)=xo;

end

X=diag(A)+diag(B,1)+diag(C,-1); NV=inv(X)*IV';

for i=2:M

RHS1(i)=NV(i);

J1=0;

J2=0;

for j=K1plus:K2plus l=i+j; qplus=P2*(l-2); if (l <= M)

phi1=P1*(P2^P1)/(P2^(P1+1)); J1=J1+(phi1*(NV(l)-NV(i)));

else

J1=J1+0;

end

end

for k=K1min:K2min m=i+k; qmin=N2*(m-2); if(m <= M)

phi2=N1*(N2^N1)/(N2^(N1+1)); J2=J2+(phi2*(NV(m)-NV(i)));

else

J2=J2+0;

end

end

RHS2(i)=(n/T)*(J2-J1)*NV(i);

end

price=inv(X)*(RHS2'+RHS1') end

end

The command to execute the code and the input parameters are given below:

>> price(0.043819,0.56341,-3,3,1.2 ,0.308,0.016,0.341,0.016,30,5917808.2,0.17628,30,5)

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