## Monday, June 2, 2014

### Chapter 6, part 4 of 8: more processes

Hello everybody.

This is the fourth in a series of posts that cover chapter 6 of my book (that is, the Monte Carlo framework) and started here. This week, a couple more short examples.

In other news, you can still register for my Introduction to QuantLib Development course; I went on and on about it already, so I'll leave it at that.

Follow me on Twitter if you want to be notified of new posts, or add me to your circles, or subscribe via RSS: the widgets for that are in the sidebar, at the top right of the page. Also, make sure to check my Training page.

### Example: more processes

In the last post I exposed all the flaws in out implementation of the Black-Scholes process. For an example of a well-behaved process, you can look at listing 6.7 instead, which shows the OrnsteinUhlenbeckProcess class.

Listing 6.7: Implementation of the OrnsteinUhlenbeckProcess class.
    class OrnsteinUhlenbeckProcess : public StochasticProcess1D {
public:
OrnsteinUhlenbeckProcess(Real speed,
Volatility vol,
Real x0 = 0.0,
Real level = 0.0);
Real x0() const;
... // other inspectors
Real drift(Time, Real x) const  {
return speed_ * (level_ - x);
}
Real diffusion(Time, Real) const  {
return volatility_;
}
Real expectation(Time t0, Real x0, Time dt) const  {
return level_ + (x0 - level_) * std::exp(-speed_*dt);
}
Real stdDeviation(Time t0, Real x0, Time dt) const  {
return std::sqrt(variance(t,x0,dt));
}
Real variance(Time t0, Real x0, Time dt) const {
if (speed_ < std::sqrt(QL_EPSILON)) {
return volatility_*volatility_*dt;
} else {
return 0.5*volatility_*volatility_/speed_*
(1.0 - std::exp(-2.0*speed_*dt));
}
}
private:
Real x0_, speed_, level_;
Volatility volatility_;
};

The Ornstein-Uhlenbeck process is a simple one, whose feature of interest here is that its mean-reverting drift term $$\theta(\mu - x)$$ and its constant diffusion term $$\sigma$$ can be integrated exactly. Therefore, besides the mandatory drift and diffusion methods, the class also overrides the expectation and stdDeviation methods so that they implement the formulas for their exact results. The variance method (in terms of which stdDeviation is implemented) has two branches in order to prevent numerical instabilities; for small $$\theta$$, the formula for the variance is replaced by its limit for $$\theta \to 0$$.

Finally, for an example of a multi-dimensional process, we'll have a look at the StochasticProcessArray class, sketched in listing 6.8.

Listing 6.8: Partial implementation of the StochasticProcessArray class.
    class StochasticProcessArray : public StochasticProcess {
public:
StochasticProcessArray(
const std::vector<shared_ptr<StochasticProcess1D> >& ps,
const Matrix& correlation)
: processes_(ps), sqrtCorrelation_(pseudoSqrt(correlation)) {
for (Size i=0; i<processes_.size(); i++)
registerWith(processes_[i]);
}
// ...
Disposable<Array> drift(Time t, const Array& x) const {
Array tmp(size());
for (Size i=0; i<size(); ++i)
tmp[i] = processes_[i]->drift(t, x[i]);
return tmp;
}
Disposable<Matrix> diffusion(Time t, const Array& x) const {
Matrix tmp = sqrtCorrelation_;
for (Size i=0; i<size(); ++i) {
Real sigma = processes_[i]->diffusion(t, x[i]);
std::transform(tmp.row_begin(i), tmp.row_end(i),
tmp.row_begin(i),
bind2nd(multiplies<Real>(),sigma));
}
return tmp;
}
Disposable<Array> expectation(Time t0, const Array& x0,
Time dt) const;
Disposable<Matrix> stdDeviation(Time t0, const Array& x0,
Time dt) const;
Disposable<Array> evolve(Time t0, const Array& x0,
Time dt, const Array& dw) const {
const Array dz = sqrtCorrelation_ * dw;
Array tmp(size());
for (Size i=0; i<size(); ++i)
tmp[i] = processes_[i]->evolve(t0, x0[i], dt, dz[i]);
return tmp;
}
private:
std::vector<shared_ptr<StochasticProcess1D> > processes_;
Matrix sqrtCorrelation_;
};

This class doesn't model a specific process, but rather the composition in a single entity of N correlated one-dimensional processes. I'll use it here to show how correlation information can be included in a process.

Its constructor takes the vector of 1-D processes for the underlyings and their correlation matrix. The processes are stored in the corresponding data member, whereas the correlation is not: instead, the process precomputes and stores its square root. (That would be its matricial square root; that is, $$\sqrt{A} = B$$ if $$B B^T = A$$.) The constructor also registers with each process, in order to forward any notifications they might send.

Most other methods, such as initialValues, drift, or expectation simply loop over the stored processes, calling their corresponding methods and collecting the results in an array. The diffusion method also loops over the processes, but combines the results with the correlation: it multiplies each row of its square root by the diffusion term of the corresponding process, and returns the results (if you multiply it by its transposed, you'll find the familiar terms $$\sigma_i \rho_{ij} \sigma_j$$ of the covariance matrix). The stdDeviation method does the same, but using the standard deviation of the underlying processes which also include the passed $$\Delta t$$.

This leaves us with the evolve method. If we knew that all the processes behaved reasonably (i.e., by adding the calculated variations to the values of their variables) we might just inherit the default implementation which takes the results of the expectation and stdDeviation methods, multiplies the latter by the array of random variates, and adds the two terms. However, we don't have such guarantee, and the method needs to be implemented in a different way. First, it deals with the correlation by multiplying the Gaussian variates by its square root, thus obtaining an array of correlated random variates. Then, it calls the evolve method of each one-dimensional process with the respective arguments and collects the results.