[ quantlib-Bugs-2637105 ] error in method expectation of BlackScholesMerton process

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Bugs item #2637105, was opened at 2009-02-25 15:16
Message generated for change (Comment added) made by heckl
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Category: None
Group: None
Status: Open
Resolution: Accepted
Priority: 5
Private: No
Submitted By: Michael Heckl (heckl)
Assigned to: Luigi Ballabio (lballabio)
Summary: error in method expectation of BlackScholesMerton process

Initial Comment:
The method "expectation(Time t0, Real x0, Time dt)" does wrong calculations. I tried it for a BlackScholesMerton Process and the result is wrong. After following up the code i found out that the method calculates:

return x0 * std::exp(dx);


where  dx comes from

Real EulerDiscretization::drift(const StochasticProcess1D& process,
                                    Time t0, Real x0, Time dt) const {
        return process.drift(t0, x0)*dt;
    }

And the process drift from
Real GeneralizedBlackScholesProcess::drift(Time t, Real x) const {
        Real sigma = diffusion(t,x);
        // we could be more anticipatory if we know the right dt
        // for which the drift will be used
        Time t1 = t + 0.0001;
        return riskFreeRate_->forwardRate(t,t1,Continuous,NoFrequency,true)
             - dividendYield_->forwardRate(t,t1,Continuous,NoFrequency,true)
             - 0.5 * sigma * sigma;
    }


This is simply wrong. What is wrong is that you didnt consider the Jenson inequality. Since exp is a strictly convex function we have:

let X_t be the BlackScholes Merton Process and S_t be ln(X_t) the related log-process, then
exp{E[S_t]}=exp{E[ln(X_t)]}<E[exp{ln(X_t)}]=E[X_t]

But the method expectation calculates the left side which is always to small. You can easily illustrate it if you choose the BlackScholesMerton Process in a way that the drift is 0. Check out the attached code for clarification.

Greetings
Michael

----------------------------------------------------------------------

>Comment By: Michael Heckl (heckl)
Date: 2009-03-02 15:22

Message:
No, sorry, regrettably i dont have an easy patch to fix this bug.

I was trying a few things but in the end it turned out more difficult to
fix this bug then you would expect considering it only takes a little
change in the mathematical formula.

It is not possible to simply adjust the method:
- StochasticProcess1D::expectation(Time t0, Real x0, Time dt)
or the method
- Real GeneralizedBlackScholesProcess::apply(Real x0, Real dx)

those methods do the math. But the problem is that they are both used by
the method:
- Real ExtendedBlackScholesMertonProcess::evolve

which is responsible to do generate the pathes. So a change in one of the
upper methods which calculate the expected value immediatly affects the
"evolve" method and therefore we get wrong pathes.

imho the easiest way would be to make a new method and change the names to
make it more intuitional (i.e. make it more obvious what method calculates
what values; does the value belong to the process or the log-process; what
does the value describe?). The new method could look something like:

Real StochasticProcess1D::expectationOfX(Time t0, Real x0, Time dt) const
{
        Real sigma = diffusion(t0,x0);
        Real mue = drift(t0,x0);
        Real dx = (mue+0.5*sigma*sigma)*dt;
        return apply(x0, dx);
    }

----------------------------------------------------------------------

Comment By: Luigi Ballabio (lballabio)
Date: 2009-02-27 12:18

Message:
You're right.  The problem comes from the process being born as an exp
wrapper for the underlying log process.  The expectation doesn't work.  Do
you have a patch that makes it return the correct value?

----------------------------------------------------------------------

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