[ 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 (Settings changed) made by lballabio
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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

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