A trinomial tree based method is presented for pricing exotic options. The model is based on a combination of techniques. that is, a tree generation technique and an appropriate backward induction
Trinomial Tree Construction
A trinomial tree based method is presented for pricing exotic options. The model is based on a combination of techniques. that is, a tree generation technique and an appropriate backward induction pricing technique.
Since the volatility parameter in the SDE is of a piecewise constant form, the tree generation techniques may, in some cases, construct trees that are non- recombining. In the worst case, then, the space complexity of the tree generation techniques is proportional to the exponential of the number of time slices in the tree.
Each method includes a technique for constructing, based on the SDE (1), an appropriate tree of discrete prices of the underlying security. Each such technique uses a mathematical result, described below, for ensuring that branching probabilities from each tree node are appropriate (i.e., probabilities, for each node, must be non-negative and sum to one).
By matching mean and variances as described above, and by ensuring that the probabilities sum to one, we obtain the following system of linear equations
Where
and
In this section we present the techniques for generating a tree appropriate for pricing the barrier options described in Section 2. We consider single barrier options first.
Let be a partition of the time interval . Furthermore suppose that the underlying security follows piecewise geometric Brownian motion, in the sense described below, over the interval . Specifically, assume that the underlying security can be modeled as a process, , which, under the risk neutral probability measure, satisfies a stochastic differential equation (SDE) of the form
(1)
where is standard Brownian motion. Here and are deterministic functions of the piecewise constant form
and .
Consider a tree node, , at a time slice, , where ; furthermore, assume that the logarithm of the price of the underlying security at this node is equal to . We assume that node branches into three nodes, at time slice , with respective logarithm of the price of the underlying security of the form and where and .
Here is the value that, among all tree nodes at time , is closest to where and ; furthermore, and are values for the two nodes closest to the node with value . Next we associate with node a discrete random variable, , which takes the values
We seek to determine , above, so that the mean and variance of the discrete random variable match those of the continuous random variable (obtained by solving the SDE (1), with initial condition , for the time interval ).
(2)
for the unknowns . By algebraic manipulation, the linear system of equations above is equivalent to
(3)
(4a)
. (4b)
Notice that while the system of equations above has a unique solution, we have no guarantee that will be non-negative. Next we determine a condition on to ensure that
Recall that the branching rule from node implies that
(5)
Dividing (5) by and substituting for the right hand side of (4b), we have, which we can rewrite as
(6)
for some . Solving (3) and substituting the right hand side of (4b) for , we obtain
(7)
where . Notice that the right hand side of (7) has no dependency on .
Analyzing the right hand side of (7) over the range , we obtain the condition
(8)
on , which ensures that Notice that (8) yields an equivalent condition on , that is,
. (9)
To summarize, for an arbitrary tree node on an arbitrary time slice, appropriate branching probabilities are given as the solution of (2) provided that condition (9) holds. In Appendix B we examine the sensitivity of solutions to (2) with respect to perturbations in the values of and .
Suppose that we have constructed a tree up to time , where . To expand the tree to the next time slice, we first define, at time , an appropriate partition for the logarithm of the underlying security; then, using this partition, we determine the children and associated probabilities of all nodes at time .
Note that, by an appropriate partition for the logarithm of the underlying security at time , we mean a partition such that the inter-node spacing is equal to where is chosen (as in Section 3.1.1) so that branching probabilities are non-negative. Next we describe how to construct such a partition. Then we discuss how to determine the branching and corresponding probabilities for nodes at time .
To define a partition at time (with uniform spacing, , which satisfies the inequality (9)), we first determine whether, for some nodes at time , there is a branch that crosses the barrier at time . This determination is made by checking certain conditions, defined in Appendix A, based on the branching rule (for nodes on the old time slice to nodes on the new time slice) defined. I
f we determine that the barrier will be crossed at time , we generate a partition by placing a node on the actual barrier (i.e., either or ) and all other nodes offset from the barrier by integer multiples of . Otherwise, if the barrier will not be crossed, we define an artificial barrier at time (see Appendix A); we then generate a partition by placing a node exactly on the artificial barrier and all other nodes offset by integer multiples of from this barrier (here the artificial barrier simply acts as a point of reference for generating the partition). We use for a value close to the upper bound, , in the inequality (9).
Once an appropriate partition has been defined at time , we then determine, according to the branching rule presented in Section 3.1.1, the children of each node at time . Suppose that a particular node at time (with value for the logarithm of the underlying security equal to ) branches to nodes at time with values for the logarithm of the underlying security equal to , and , respectively, where . Then, from (2), appropriate branching probabilities are determined by solving the system of linear equations
for the unknowns and . The numerical conditioning of the system of linear equations above should be checked, however, to ensure the accuracy of the computed solution.
The tree construction technique for double barrier options is based on a similar approach as for single barrier options. That is, if the tree has been constructed up to time , an appropriate partition for the underlying security is defined at time ; then branches and associated probabilities are determined for nodes on the old time slice. We describe these techniques next.
Suppose that the tree has been generated up to time , where . If neither barrier is crossed at time , then an artificial barrier is defined, at time , and used (as a point of reference) to generate an appropriate partition. Otherwise, a partition is defined that places nodes on both barriers (see below). Once a partition is defined, branching and corresponding probabilities are determined as above.
