Spring is well underway, and my young colleagues’ fancy lightly turns to the dating apps on their phones. I’m married with children, so I finished my series on the new finite-difference framework instead. The collected series forms chapter 8 of Implementing QuantLib, whose main content is now finished. I’m still missing about half of the appendix, after which I’ll have the first complete version. But of course, Leanpub being Leanpub, you can buy the book now and get further versions for free when I update it. Neat, huh?
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Schemes and solvers
As in the old framework, schemes provide a discretization of the time derivative in the equation we want to solve; you can go back to an earlier post for more details. The new framework provides reimplementations of schemes used in the old one, such as explicit and implicit Euler, as well as a number of new ones I won’t describe; as usual, you can look them up in the library.
Like in the old framework, they have no common base class; they all
courteously agree to implement
with the same semantics described in the section I just quoted, but
there’s no inheritance relationship holding them to any contract.
This is a bit of a problem, given that the new framework usually
eschews generic programming and templates in favor of more
object-oriented techniques; the absence of a base class, of course,
prevents us from writing non-template code taking an instance of a
The way the framework works around this is to define the
FdmSchemeDesc class, shown in the following listing. Its instances
can be passed around instead of the scheme they describe, and contain
the type of the chosen scheme and the parameters that might be needed
to instantiate it. The available types are enumerated in the class
declaration, and yes, that’s not optimal. I’ll get back to this
Besides the enumeration, the class declares a bunch of static methods returning a few pre-built schemes with sensible parameters; if you want a custom one instead, you can build an instance with any old values.
Note also that this class and the others described in this section are
not templates; they’re written to work specifically with the
FdmLinearOpComposite class. My guess is that their authors
learned the lesson from the old framework, which was written for a
generic operator and turned out to be only ever used with the
Now, if the type enumeration seemed a bit icky to you, you might want
to brace yourself for the next listing and the
Its constructor collects and stores
an operator, a set of boundary and step conditions, and a scheme
description. They are used in the
rollback method, which puts
them together in a finite-difference model, takes an array of initial
values, and runs the model between the two given times
to in the given number of steps, possibly with a few initial
damping steps. And, as you can see, the implementation of
rollback is a big switch on type.
Was that a gasp? Yes, I know. Both the
rollback method need to be changed if we want to add a
new scheme, which is a violation of the open-closed principle. It’s
not like the authors of the code were sloppy, though. Let’s say we go
ahead and inherit all schemes from a base class, so we can use a
polymorphic pattern such as Strategy to select one. There are two
forces at play here. First, if the user must be able to choose it,
the scheme instance would have to be passed to the engine from
outside. Second, the operator and the boundary conditions, which the
scheme requires, are created inside the engine, because having the
user write code to create them would cause duplication, and thus the
scheme must be created inside the engine, too. Tricky, isn’t it?
Two possible solutions come to mind. One is to use some kind of
factory: the scheme might have, say, a
clone method that returns
another instance with the same type and parameters and with the passed
boundary and step conditions added (we could also have factory
classes, but that would mean a parallel hierarchy, which is never
good). Another is to change the interface of the classes we’ve seen,
so that the scheme doesn’t contain the conditions; since backward
compatibility prevents us from changing the
class, this requires writing a new model class, too. Both would have
been possible, and the maintainer who reviewed and merged the
framework when it was contributed should have seen the problem and
should have suggested one of them. That maintainer was me, in case
you’re curious. Live and learn, I guess.
And after the
FdmBackwardSolver class, we’re on the home
stretch; the remaining classes add a couple of layers that make it
more convenient to use the solver, but not a lot of new logic. To the
more observant among you: yes, in this section I abandoned my initial
intent to write a top-down description of the components. I think it
makes more sense this way.
FdmSolverDesc structure shown in
the listing that follows groups together the arguments needed
to initialize a solver so that we can pass them around together. The
operator is not included; that’s because, as you’ll see in a minute,
we’ll define higher-level solvers that define the operator and can be
reused for different instruments by passing them different
Fdm1DimSolver class sketched in
the next listing takes care of some of the plumbing
around a solver.
Its constructor takes a solver description, a scheme description and an operator, stores them, and prepares a number of other data members for the calculations. In particular, it uses the layout in the solver description to allocate the correct size for the initial values, and the calculator to fill them; also, it extracts from the mesher the values of the underlying variable on the chosen grid.
The calculations are implemented inside the
method (yes, the class inherits from
LazyObject, which puzzles
me; I’ll come back to this). As I mentioned, it’s mostly plumbing:
the code makes a copy of the initial values, creates an instance of
FdmBackwardSolver, uses it to roll back the values, stores the
results, and wraps them in an interpolation. Methods such as
interpolateAt make the results available to client code; other
inspectors, not shown, give access to the first and second derivative
of the interpolation, as well as the time derivative of the result.
Going back one step: I gave the matter some thought, but I don’t see
the need for inheriting from
LazyObject here. For one thing, the
Fdm1DimSolver doesn’t register with anything, which
kind of defeats the purpose of the pattern (registration might happen
in derived classes, but a quick search in the library shows that
Fdm1DimSolver doesn’t have any). Moreover, we’ll see that solvers
are normally created by an engine during calculation, used right
there, and discarded afterwards. All in all, we could have put the
whole calculation in the constructor, or even turned the class into a
function returning some kind of result structure containing the
interpolation and some inspectors to access it.
I should mention that
Fdm1DimSolver has siblings: the framework
also defines the
classes, which perform the same calculations and produce a 2-D or 3-D
interpolation, respectively. The interface is almost the same, with
some difference due to the higher dimensions; for instance, the
interpolateAt method takes as arguments
in the 2-D case and
z in the 3-D case.
Finally, specific solvers are tasked with creating an operator and
using it with one of the generic solvers I described. It’s the case
FdmBlackScholesSolver class, shown in
the next listing.
Besides the usual descriptions and a
few other parameters for the model, it takes a handle to a
Black-Scholes process that it stores and with which it registers for
notifications; again, this would make more sense if engines kept the
solver around instead of destroying it. Its
performCalculations method creates an operator based on the
process and uses it to build a 1-D solver, which is then stored.
Inspectors such as
calculate first, which triggers the construction of the 1-D
solver, and then delegate to the corresponding methods of the stored
solver, which in turn trigger the rollback calculation inside it. The
inspectors also take care of the transformation between the underlying
value, used in the interface, and its logarithm, used in the process.
As I mentioned earlier, the solver takes the solver description from the outside and thus can be used for different engines, provided that the underlying of the option follows the same dynamics. In the listing in this post, you can see it used for a vanilla put or call option; by changing the boundary conditions, it can be used, and in fact it is used, for barrier options; and by changing the initial conditions, it might be used for digital options or other kinds of instruments. You can see how this design is more flexible than the old, inheritance-based one; or at least, that it is much simpler to get flexibility.
Multi-dimensional solvers, which I won’t describe, work in the same
way. If you’re interested, you can look at the
class for a 2-D example and at the
class for a 3-D one. And, as the saying goes, there’s plenty more
where those come from.
With solvers, we finally have all the pieces we need to build our engines—and round up this chapter, I might add. Looking back at the listing of the engine we took as example, we can now see how a finite-difference engine works. It takes the scheme and the other required parameters at construction, and then performs its calculations by building the required mesher, the initial, step and boundary conditions, and finally the solver from which the results can be extracted.