Week 4 : Numerical Simulation of Stochastic Differential Equations 1

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Week 4 Numerical Simulation of Stochastic
Differential Equations 1
The Euler Maruyama Method
This lecture is based on the following two
articles
1.) Higham, D. An Algorithmic Introduction to
Numerical Simulation of SDE. SIAM Review, Vol 43,
No.3 2.) Higham, D. Kloeden, P. Maple and
Matlab for SDE in Finance
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A scalar autonomous SDE is an object of the
following type
A solution is a stochastic process X(t),
satisfying
SDEs play a prominent role in a range of
applications, including biology, chemistry,
epidemiology, mechanics, microelectronics and of
course finance.
One important SDE is the one for the stock price
in the Black-Scholes model
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Sometimes the coefficients in the SDE are non
autonomous, i.e. depend explicitly on time
Sometimes the SDE is multidimensional, in this
case a(t,x) is a vector valued function, b(t,x)
is a matrix valued function and W is a
multidimensional Brownian motion. The solution X
is then a vector valued stochastic process.
If b(t,x) is constant zero, then we have an
ordinary differential equation.
For the SDE from the previous slide, describing
the evolution of a stock in the Black-Scholes
model, there exists an explicit solution ( see
CTF ).
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For most SDEs occurring in practical
applications however, this is not the case The
following SDE
This SDE plays an important role in stochastic
volatility models and in interest rate theory. It
does in general not have an explicit solution.
We therefore have to discuss numerical
approximations of the solution of a given SDE.
Let us first recapture some of the basics already
discussed in Continuous Time Finance
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Average of u(W(t)) and five individual paths are
displayed in the following figure
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What should the average be ? Think about that !
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Example We know already that
Numerically
Results itoerr 0.0158, straterr 0.0186
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Ito formula
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The Euler Maruyama method for solving a SDE
The approximation in order to obtain this scheme
is the following
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And similar for the dt term.
The main question now is, how good is this
approximation ?
Do we have convergence and in what sense ?
Theoretical results about convergence are in
general available when the coefficient functions
are Lipschitz and satisfy a linear growth
condition. We speak about this in CTF.
Here we assume these conditions, see however also
the important remarks about convergence at the
end.
Before we study convergence in detail, we have to
say what we actually mean by convergence.
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Two modes of convergence are of importance for us

1.) strong convergence
2.) weak convergence
A numerical scheme is said to
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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.
Obviously The higher the convergence order, the
better the method.
Strong convergence is a path wise property
To see this note the Markov inequlity
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Weak convergence is basically a convergence of
the distribution.
If a term like E(h(X(T))) has to be computed via
Monte Carlo simulation, then the weak convergence
concept is enough. This is for example the case
standard European options.
If the payoff of an option involves properties of
the path, such as Asian or Barrier options, then
strong convergence is the right concept.
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Let us reconsider the following SDE
The coefficients are given by
We know that the exact solution is given by
Lets see how good the Euler Maruyama method
works
The following Matlab program implements the Euler
Maruyama method for the SDE above. The step size
for the EM method is taken to be 4 times the step
size for the BM.
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We can now vary the step size in order to get
some idea about the approximation error
If we approximate the inequality for the strong
convergence error from above and assume equlity ,
then by taking logarithms we obtain for the
strong convergence error
Log(strong error EM) log(KT) ½ Delta
This relationship can indeed be observed in the
following figure
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A similar analysis carried out for the weak
convergence order leads to the following figure
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Example Application in Finance
The following system of two SDEs represents a
basic version of a stochastic volatility model.
X1t represents the price of a stock, X2t
volatility, which is itself a stochastic process,
or more precisely the solution of a SDE.
The following figure shows simulated paths for
the stock price using different step sizes
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We want to compute the price of a European call
(X1t-1)
where the parameters are chosen to be lambda
0.05, sigma0 0.8 and the interestrate is assumed
to be r.05.
The following Matlab code does this, using the EM
method and Monte Carlo.
The result is Price 0.3402.
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For this option type the EM method is appropriate
since the payoff only depends on the distribution
of X1t. In the case of a path dependent option,
like an Asian option, better results can be
obtained by using a method with a higher strong
convergence order.
We will discuss the Milstein method, which has
strong convergence order 1 in detail in the next
lecture.