Week 5 : Numerical Simulation of Stochastic Differential Equations 2

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Week 5 Numerical Simulation of Stochastic
Differential Equations 2
Reminder of last week
We were looking for numerical methods how to
simulate/approximate a solution.
The idea was as follows
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The obvious generalization if the Euler method to
the case of an SDE is the Euler-Maruyama method
Based on the following approximation
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Question Convergence and if yes, in what sense ?
Two modes of convergence are of importance for us

1.) strong convergence
2.) weak convergence For each polynomial g
Under the appropriate conditions on the
coefficient functions, one can show that the
Euler Maruyama method has strong convergence
order ½ and weak convergence order 1.
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We use our numerical schemes in combination with
Monte Carlo method in order to compute prices for
options etc.
The higher the convergence order, the better the
method, but
Strong convergence is the decisive factor if the
payoff of the options depends on path wise
properties.
Weak convergence is the decisive factor if the
payoff of the options depends only on the
distribution of the asset prices at maturity.
In order to obtain better results for the first
case, we need a method which has strong
convergence order higher than the Euler-Maruyama
method.
This leads us to the Milstein method.
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Convergence of Euler-Maruyama and Milstein
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with
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In the general case the double integral has to be
computed numerically, using either Euler-Maruyama
or Milstein.
One can show, that this does not affect the
strong order of convergence.
Example A stochastic volatility model
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MIL Milstein method on 3D finance SDE SDE
is dX(1) X(1) X(2) dW(1)
X(1)_0 1 dX(2) -(X(2)-X(3))
dt 0.3 X(2) dW(2) X(2)_0 0.1
dX(3) (X(2)-X(3)) dt
X(3)_0 0.1 Discretized Brownian path
over 0,1 has delta 2(-18). Milstein
timestep is Delta sqrt(delta). Substeps for
double integral are of size delta.clfrandn('sta
te',1)T 1 Delta 2(-9) delta Delta2
L T/Delta K Delta/deltaX1
zeros(1,L1) X2 zeros(1,L1) X3
zeros(1,L1)Y2 0X1(1) 1X2(1)
0.1X3(1) 0.1for j 1L Y1 0 Winc1
0 Winc2 0 for k 1K dW1
sqrt(delta)randn dW2
sqrt(delta)randn Y1 Y1 Y2dW1
Y2 Y2 dW2 Winc1 Winc1 dW1
Winc2 Winc2 dW2 end X1(j1)
X1(j) X1(j)X2(j)Winc1 ...
X1(j)(X2(j)2)0.5(Winc12 - Delta) ...
0.3X1(j)X2(j)Y1 X2(j1)
X2(j) - (X2(j) - X3(j))Delta 0.3X2(j)Winc2
... 0.9X2(j)0.5(Winc22 -
Delta) X3(j1) X3(j) (X2(j) -
X3(j))Deltaendplot(0DeltaT,X1,'r-'), hold
onplot(0DeltaT,X2,'bl--')plot(0DeltaT,X3
,'b-.')legend('X1','X2','X3')xlabel('t','Font
Size',16), ylabel('X','FontSize',16)
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