## Monday, June 16, 2014

### Chapter 6, part 5 of 8: path generators

Welcome back.

This week, back to some content from my book after the screencast of last week (and the announcement of the QuantLib Notebooks series; leave your feedback if you want the transcripts to be published). This post is the fifth in a series that started here and covers chapter 6, that is, the Monte Carlo framework in QuantLib.

Mandatory plug: you can still register for my Introduction to QuantLib Development course (London, September 24th to 26th): three days of lectures and exercises on the architecture of the library.

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### Path generation

Before tackling path generation proper, we need a last basic component: a structure to hold the path itself. Listing 6.9 shows the Path class, which models a random path for a single variable. It contains the TimeGrid instance and the Array holding the node times and values, respectively, and provides methods to access and iterate over them. The method names are usually chosen to follow those of the standard containers, allowing for a more familiar interface; methods with domain-based names such as time are also available.

Listing 6.9: Interface of the Path and MultiPath classes.
    class Path {
public:
Path(const TimeGrid& timeGrid, const Array& values);
bool empty() const;
Size length() const;
Real operator[](Size i) const;
Real at(Size i) const;
Real& operator[](Size i);
Real& at(Size i);
Real front() const;
Real& front();
Real back() const;
Real& back();
Real value(Size i) const;
Real& value(Size i);
Time time(Size i) const;
const TimeGrid& timeGrid() const;
typedef Array::const_iterator iterator;
typedef Array::const_reverse_iterator reverse_iterator;
iterator begin() const;
iterator end() const;
reverse_iterator rbegin() const;
reverse_iterator rend() const;
private:
TimeGrid timeGrid_;
Array values_;
};

class MultiPath {
public:
MultiPath(const std::vector<Path>& multiPath);
// ...more constructors...
Size assetNumber() const;
Size pathSize() const;
const Path& operator[](Size j) const;
const Path& at(Size j) const;
Path& operator[](Size j);
Path& at(Size j);
private:
std::vector<Path> multiPath_;
};

The MultiPath class, also shown in listing 6.9, holds paths for a number of underlying variables. It is basically a glorified container, holding a vector of Path instances and providing a few additional inspectors to uphold the law of Demeter. (For instance, if p is a MultiPath instance, the additional methods allow us to write p.pathSize() instead of p[0].length(). Besides being uglier, the latter might also suggest that p[1].length() could be different.) The simplicity of the implementation is at the expense of some space: by storing a vector of Path instances, we're storing N identical copies of the time grid.

#### Aside: access patterns

The basic inspectors in the Path class, such as operator[] and the iterator interface, return the values Si of the underlying variable rather than the pairs (ti,Si) including the node times. While it could well be expected that p[i] return the whole node, we thought that in the most common cases a user would want just the value, so we optimized such access (it's some kind of Huffman-coding principle, in which the most common operations should require the least typing; the Perl language shines at this, or so I'm told).

In a sad note about reuse (or lack thereof) if we were to add to the Path class a few methods to return whole nodes, we probably wouldn't use std::pair<Time,Real> to hold the data, but an inner Node struct. This is not because we instinctively leap at the most complex implementation—even though it may seem so at times—but because code such as p.node(i).second is not self-describing (wait, is second the time or the value?) The advantage of writing the above as p.node(i).value would offset the cost of defining a Node struct.

What remains to be done for path generation is now simply to connect the dots between the classes that I described in the last few posts. The logic for doing so is implemented in the MultiPathGenerator class template, sketched in listing 6.10; I'll leave it to you to decide whether this makes it an instance of the Factory pattern.

Listing 6.10: Sketch of the MultiPathGenerator class template.
    template <class GSG>
class MultiPathGenerator {
public:
typedef Sample<MultiPath> sample_type;
MultiPathGenerator(const shared_ptr<StochasticProcess>&,
const TimeGrid&,
GSG generator,
bool brownianBridge = false);
const sample_type& next() const { return next(false); }
const sample_type& antithetic() const { return next(true); }
private:
const sample_type& next(bool antithetic) const;
bool brownianBridge_;
shared_ptr<StochasticProcess> process_;
GSG generator_;
mutable sample_type next_;
};

template <class GSG>
const typename MultiPathGenerator<GSG>::sample_type&
MultiPathGenerator<GSG>::next(bool antithetic) const {
if (brownianBridge_) {
QL_FAIL("Brownian bridge not supported");
} else {
typedef typename GSG::sample_type sequence_type;
const sequence_type& sequence_ =
antithetic ? generator_.lastSequence()
: generator_.nextSequence();
Size m = process_->size(), n = process_->factors();
MultiPath& path = next_.value;
Array asset = process_->initialValues();
for (Size j=0; j<m; j++)
path[j].front() = asset[j];
Array temp(n);
next_.weight = sequence_.weight;

TimeGrid timeGrid = path[0].timeGrid();
Time t, dt;
for (Size i = 1; i < path.pathSize(); i++) {
Size offset = (i-1)*n;
t = timeGrid[i-1];
dt = timeGrid.dt(i-1);
if (antithetic)
std::transform(sequence_.value.begin()+offset,
sequence_.value.begin()+offset+n,
temp.begin(),
std::negate<Real>());
else
std::copy(sequence_.value.begin()+offset,
sequence_.value.begin()+offset+n,
temp.begin());

asset = process_->evolve(t, asset, dt, temp);
for (Size j=0; j<m; j++)
path[j][i] = asset[j];
}
return next_;
}
}

Its constructor takes and stores the stochastic process followed by the variable to be modeled; the time grid to be used for generating the path nodes; a generator of random Gaussian sequences, which must be of the right dimension for the job (that is, at each draw it must return N × M numbers for N factors and M steps); and a boolean flag to specify whether or not it should use Brownian bridging, which at this time cannot be set to true and with which we're unfortunately stuck for backward compatibility.

The interface of the class provides a next method, returning a new random path (well, multipath, but cut me some slack here); and an antithetic method, returning the antithetic path of the one last returned by next. (Unfortunately, the correctness of the antithetic method depends on its being called at the correct time, that is, after a call to next; but I can't think of any acceptable way out of this.) Both methods forward to a private method next(bool), which implements the actual logic. If the brownianBridge flag was set to true, the method bails out by raising an exception since the functionality is not yet implemented. Following the general principles of C++, which suggest to fail as early as possible, we should probably move the check to the constructor (or better yet, implement the thing; send me a patch if you do). As it is now, one can build a path generator and have the constructor succeed, only to find out later that the instance is unusable.

If Brownian bridging is not required, the method gets to work. First, it retrieves the random sequence it needs: a new one if we asked for a new path, or the latest sequence we used if we asked for an antithetic path. Second, it performs some setup. It gets a reference to the path to be built (which is a data member, not a temporary; more on this later); retrieves the array of the initial values of the variables and copies them at the beginning of the path; creates an array to hold the subsets of the random variables that will be used at each step; and retrieves the time grid to be used. Finally, it puts the pieces together. At each step, it takes as starting point the values at the previous one (it uses the same array where it stored the initial values, so the first step is covered, too); it retrieves from the time grid the current time t and the interval dt over which to evolve the variables; it copies the subset of the random variables that it needs (the first N for the first step, the second N for the second step, and so on, N being the number of factors; also, it negates them if it must generate an antithetic path); and to close the circle, it calls the evolve method of the process and copies the returned values at the correct places in the paths.

As I mentioned, the returned path is stored as a data member of the generator instance. This saves quite a few allocations and copies; but of course it also makes it impossible to share a path generator between threads, since they would race for the same storage. However, that's just the last nail in the coffin. Multithreading was pretty much out of the picture as soon as we stored the Gaussian number generator as a data member; the path generator can only be as thread-safe as its RNG—that is, not much, since most (if not all) of them store state. A much easier strategy to distribute work among threads would be to use different MultiPathGenerator instances, each with its own RNG. The problem remains of creating RNGs returning non-overlapping sequences; for algorithms that allow it (such as Sobol's) one can tell each instance to skip ahead a given number of draws.

#### Aside: stepping on one's own toes.

Unfortunately, there's quite a bit of memory allocation going on during multi-path generation. The obvious instances (the two Arrays being created to store the current variable values and the current subset of the random numbers) could be avoided by storing them as data members; but on the one hand, we would have yet more mutable members, which is something we should use sparingly; and on the other hand, the allocation of the two arrays only happens once in a whole path generation, so it might not make a lot of difference.

Instead, the worst offender might be the underlying process, which at each step creates an Array instance to be returned from its evolve method. How can we fix this? In this case, I wouldn't return a reference to a data member if I can help it: the StochasticProcess class might just work in a multithreaded environment (well, if market data don't change during calculation, or if at least the process is kept frozen) and I'd rather not ruin it. Another possibility might be that the evolve method take a reference to the output array as an argument; but then, we (and for we, I mean whoever will implement a stochastic process) would have to be very carefully about aliasing.

The sad fact is that the natural way to write the evolve method would usually be something like (in pseudocode, and in the case of, say, two variables)
    void evolve(const Array& x, Time t, Time dt,
const Array& w, Array& y) {
y[0] = f(x[0], x[1], t, dt, w[0], w[1]);
y[1] = g(x[0], x[1], t, dt, w[0], w[1]);
}

where x is the input array and y is the output. However, it would be just as natural for a user to write
    p->evolve(x,t,dt,w,x);

(that is, to pass the same x as both input and output) meaning to replace the old values of the variables with the new ones.

Well, you see the catch. When the first instruction in evolve writes into y[0], it actually writes into x[0]. The second instruction then calls g with a value of x[0] which is not the intended one. Hilarity ensues.

Fixing this requires either the developer to make each process safe from aliasing, by writing
    a = f(...); b = g(...); y[0] = a; y[1] = b;

or something to this effect; or the user to be mindful of the risk of aliasing, and never to pass the same array as both input and output. Either solution requires more constant care than I credit myself with.

So, where does this leave us? For the time being, the current interface might be an acceptable compromise. One future possibility is that we take away the allocation cost at the origin, by using some kind of allocation pool inside the Array class. As usual, we're open to suggestions.

To close the section, I'll mention that the library also provides a PathGenerator class for 1-D stochastic processes. It has some obvious differences (it calls the StochasticProcess1D interface and it creates Paths instead of MultiPaths; also, it implements Brownian bridging, which is a lot easier in the 1-D case) but other than that, it has the same logic as MultiPathGenerator and doesn't warrant being described in detail here.

#### Pricing on a path

There's not much to say on the PathPricer class template, shown in listing 6.11. It defines the interface that must be implemented by a path pricer in order to be used in a Monte Carlo model: an operator() taking a path and returning some kind of result. It is basically a function interface: in fact, if we were to write the framework now instead of ten years ago, I'd just use boost::function and dispense with PathPricer altogether.

Listing 6.11: Interface of the PathPricer class template.
    template<class PathType, class ValueType=Real>
class PathPricer : public unary_function<PathType,ValueType> {
public:
virtual ~PathPricer() {}
virtual ValueType operator()(const PathType& path) const=0;
};

The template arguments generalize on both the type of the argument, which can be a Path, a MultiPath, or a user-defined type if needed (for instance, when pricing instruments with an early-termination feature, one might want to save time by using a path class that generates nodes lazily and only if they are actually used); and the type of the result, which defaults of a simple Real (usually the value of the instrument) but might be an array of results, or some kind of structure. On the one hand, one might like to have the choice: an Array would be more appropriate for returning a set of homogeneous results, such as the amounts of a series of coupons, while a structure with named fields would be better suited for inhomogeneous results such as the value and the several Greeks of an instrument. On the other hand, the results will need to be combined, and therefore they'll need some basic algebra; the Array class already provides it, which makes it ready to use, whereas a results structure would need to define some operators in order to play well with the facilities described in the next section.