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### Trees and tree-based lattices

In order to use the assets I described in the previous post, we need a lattice. The first question we have to face is at the very beginning of its design: should the tree itself be the lattice, or should we have a separate lattice class using a tree?

Even if the first alternative can be argued (a tree can very well be seen as a kind of lattice, so an is-a relationship would be justified), the library currently implements the second. I’m afraid the reasons for the choice are lost in time; one might be that having a class for the lattice and one for the tree allows us to separate different concerns. The tree has the responsibility for maintaining the structure and the probability transitions between nodes; the lattice uses the tree structure to roll the asset back and adds some additional features such as discounting.

#### The Tree class template

As I just wrote, a tree provides information about its structure. It can be described by the number of nodes at each level (each level corresponding to a different time), the nodes that can be reached from each other node at the previous level, and the probabilities for each such transition.

The interface chosen to represent this information (shown in the listing below) was an index-based one. Nodes were not represented as instances of a Node class or structure; instead, the tree was described—not implemented, mind you—as a series of vectors of increasing size, and the nodes were identified by their indexes into the vectors. This gives more flexibility in implementing different trees; one tree class might store the actual vectors of nodes, while another might just calculate the required indexes as needed. In the library, trinomial trees are implemented in the first way and binomial trees in the second. I’ll describe both later on.

The methods shown in the interface fully describe the tree structure. The columns method returns the number of levels in the tree; evidently, the original designer visualized the time axis as going from left to right, with the tree growing in the same direction. If we were to change the interface, I’d probably go for a more neutral name, such as size.

The size name was actually taken for another method, which returns the number of nodes at the i-th level of the tree; the underlying method takes two indexes i and j and returns the value of the variable underlying the tree at the j-th node of the i-th level.

The remaining methods deal with branching. The branches method returns the number of branches for each node (2 for binomial trees, 3 for trinomial ones and so on). It takes no arguments, which means that we’re assuming such a number to be constant; we’ll have no trees with a variable number of branches. The descendant identifies the nodes towards which a given node is branching: it takes the index i of the level, the index j of the node and the index of a branch, and returns the index of the corresponding node on level i+1 as exemplified in the following figure. Finally, the probability method takes the same arguments and returns the probability to follow a given branch.

The original implementation actually sported a base Tree class as shown in the listing above. It worked, but it committed what amounts to a mortal sin in numerical code: it declared as virtual some methods (such as descendant or probability) that were meant to be called thousands of times inside tight nested loops. (If you’re wondering why this is bad, see for instance [1] for a short explanation—along with a lot more information on using C++ for numerical work.) After a while, we started to notice that small glaciers were disappearing in the time it took us to calibrate a short-rate model.

Eventually, this led to a reimplementation. We used the Curiously Recurring Template Pattern (CRTP) to replace run-time with compile-time polymorphism (see the aside below if you need a refresher) which led to the Tree class template shown in the next listing. The virtual methods are gone, and only the non-virtual columns method remains; the machinery for the pattern is provided by the CuriouslyRecurringTemplate class template, which avoids repetition of boilerplate code by implementing the const and non-const versions of the impl method used in the pattern.

This was not a redesign; the existing class hierarchy was left unchanged. Switching to CRTP yielded slightly more complex code, as you’ll see when I describe binomial and trinomial trees. However, it paid off in speed: with the new implementation, we saw the performance increase up to 10x. Seeing the results, I didn’t stop to ask myself many question (yes, I’m the guilty party here) and I missed an opportunity to simplify the code. With a cooler head, I might have noticed that the base tree classes don’t call methods defined in derived classes, so I could have simplified the code further by dropping CRTP for simple templates. (If a new branch of the hierarchy were to need CRTP, it could be added locally to that branch.) Yet another thing to remember when we decide to redesign parts of the library.

#### Aside: curiouser and curiouser.

The Curiously Recurring Template Pattern (CRTP) was named and popularized by James Coplien in 1995 [2] but it predates his description. It is a way to achieve a static version of the Template Method pattern (see [3], of course), in which a base class implements common behavior and provides hooks for customization by calling methods that will be implemented in derived classes.

The idea is the following:

Now if we instantiate a Derived and call its foo method, it will in turn call the correct bar implementation: that is, the one defined in Derived (note that Base doesn’t declare bar at all).

We can see how this work by working out the expansion made by the compiler. The Derived class doesn’t define a foo method, so it inherits it from the its base class; that is, Base<Derived>. The foo method calls impl, which casts *this to a reference to the template argument T, i.e., Derived itself. We called the method on a Derived instance to begin with, so the cast succeeds and we’re left with a reference to our object that has the correct type. At this point, the compiler resolves the call to bar on the reference to Derived::bar and thus the correct method is called. All this happens at compile time, and enables the compiler to inline the call to bar g and perform any optimization it sees fit.

Why go through all this, and not just call bar from foo instead? Well, it just wouldn’t work. If we didn’t declare bar into Base, the compiler wouldn’t know what method we’re talking about; and if we did declare it, the compiler would always resolve the call to the Base version of the method. CRTP gives us a way out: we can define the method in the derived class and use the template argument to pass its type to the base class so that the static cast can close the circle.

#### Binomial trees

As I mentioned earlier, a tree might implement its required interface by storing actual nodes or by just calculating their indexes on demand. The BinomialTree class template, shown in the listing below, takes the second approach. In a way, the nodes are the indexes; they don’t exist as separate entities.

The class template models a rather strict subset of the possible binomial trees, that is, recombining trees with constant drift and diffusion for the underlying variable and with constant time steps. (The library contains, in its experimental’’ folder, an extension of this class to non-constant parameters; you’re welcome to have a look at it and send us feedback. For illustration purposes, though, I’ll stick to the constant version here.)

Even with these constraints, there’s quite a few different possibilities for the construction of the tree, so most of the implementation is left to derived classes. This class acts as a base and provides the common facilities; first of all, a branches enumeration whose value is defined to be 2 and which replaces the corresponding method in the old Tree class. Nowadays, we’d use a static const Size data member for the purpose; the enumeration is a historical artifact from a time when compilers were less standard-compliant.

The class constructor takes a one-dimensional stochastic process providing the dynamics of the underlying, the end time of the tree (with the start time assumed to be $$0$$, another implicit constraint), and the number of time steps. It calculates and stores a few simple quantities: the initial value of the underlying, obtained by calling the x0 method of the process; the size of a time step, obtained by dividing the total time by the number of steps; and the amount of underlying drift per step, once again provided by the process (since the parameters are constant, the drift is calculated at $$t=0$$). The class declares corresponding data members for each of these quantities.

Unfortunately, there’s no way to check that the given process is actually one with constant parameters. On the one hand, the library doesn’t declare a specific type for that kind of process; it would be orthogonal to the declaration of different process classes, and it would lead to multiple inheritance—the bad kind—if one wanted to define, say, a constant-parameter Black-Scholes process (it would have to be both a constant-parameter process and a Black-Scholes process). Therefore, we can’t enforce the property by restricting the type to be passed to the constructor. On the other hand, we can’t write run-time tests for that: we might check a few sample points, but there’s no guarantee that the parameters don’t change elsewhere.

One option that remains is to document the behavior and trust the user not to abuse the class. Another (which we won’t pursue now for backward compatibility, but might be for a future version of the library) would be for the constructor to forgo the process and take the constant parameters instead. However, this would give the user the duty to extract the parameters from the process at each constructor invocation. As usual, we’ll have to find some kind of balance.

The two last methods define the structure of the tree. The size method returns the number of nodes at the $$i$$-th level; the code assumes that there’s a single node (the root of the tree) at level $$0$$, which gives $$i+1$$ nodes at level $$i$$ because of recombination. This seems reasonable enough, until we realize that we also assumed that the tree starts at $$t=0$$. Put together, these two assumptions prevent us from implementing techniques such as, say, having three nodes at $$t=0$$ in order to calculate Delta and Gamma by finite differences. Luckily enough, this problem might be overcome without losing backward compatibility: if one were to try implementing it, the technique could be enabled by adding additional constructors to the tree, thus allowing to relax the assumption about the start time.

Finally, the descendant method describes the links between the nodes. Due to the simple recombining structure of the tree, the node at index $$0$$ on one level connects to the two nodes at index $$0$$ and $$1$$ on the next level, the node at index $$1$$ to those at index $$1$$ and $$2$$, and in general the node at index $$i$$ to those at index $$i$$ and $$i+1$$. Since the chosen branch is passed as an index ($$0$$ or $$1$$ for a binomial tree) the method just has to add the index of the node to that of the branch to return the correct result. Neither index is range-checked, since the method will be called almost always inside a loop; checks would not only be costly (which is possible, but I haven’t measured the effect, so I’ll leave it at that) but actually redundant, since they will already be performed while setting up the loop.

The implementation of the BinomialTree class template misses the methods that map the dynamics of the underlying to the tree nodes: that is, the underlying and probability methods. They are left to derived classes, since there are several ways in which they can be written. In fact, there are families of ways: the two class templates in the following listing are meant to be base classes for two such families.

The first, EqualProbabilitiesBinomialTree, can be used for those trees in which from each node there’s the same probability to go to either branch. This obviously fixes the implementation of the probability method, which identically returns 1/2, but also sets constraints on the underlying method. Equal probability to go along each branch means that the spine of the tree (that is, the center nodes of each level) is the expected place to be; and in turn, this means that such nodes must be placed at the forward value of the underlying for the corresponding time (see the figure below for an illustration of the idea). The class defines an underlying method that implements the resulting formula for the logarithm of the underlying (you can tell that the whole thing was written with the Black-Scholes process in mind). A data member up_ is defined and used to hold the half-displacement from the forward value per each move away from the center, represented by $$\Delta x$$ in the figure; however, the constructor of the class does not give it a value, leaving this small freedom to derived classes.

The second template, EqualJumpsBinomialTree, is used for those trees in which going to either branch implies an equal amount of movement in opposite directions for the logarithm of the underlying (again, see the figure below); since one of the directions goes with the drift and the other against it, the probabilities are different. The template defines the underlying method accordingly, leaving to derived classes the task of filling the value of the dx_ data member; and it also provides an implementation for the probability method that returns one of two values (pu_ or pd_) which must also be specified by derived classes.

Finally, the three classes in the following listing are examples of actual binomial trees, and as such are no longer class templates. The first implements the Jarrow-Rudd tree; it inherits from EqualProbabilitiesBinomialTree (note the application of CRTP) and its constructor sets the up_ data member to the required value, namely, the standard deviation of the underlying distribution after one time step. The second implements the Cox-Ross-Rubinstein tree, inherits from EqualJumpsBinomialTree, and its constructor calculates the required probabilities as well as the displacement. The third one implements the Tian tree, and provides an example of a class that doesn’t belong to either family; as such, it inherits from BinomialTree directly and provides all required methods.

#### Bibliography

[1] T. Veldhuizen, Techniques for Scientific C++. Indiana University Computer Science Technical Report TR542.
[2] J.O. Coplien, A Curiously Recurring Template Pattern. In S.B. Lippman, editor, C++ Gems. Cambridge University Press, 1996.
[3] E. Gamma, R. Helm, R. Johnson and J. Vlissides, Design Patterns: Element of Reusable Object-Oriented Software. Addison-Wesley, 1995.