On Trinomial Trees for One-Factor Short Rate Models∗

On Trinomial Trees for One-Factor Short Rate Models∗ Markus Leippoldy Swiss Banking Institute, University of Zurich Zvi Wienerz School of Business Administration, Hebrew University of Jerusalem April 3, 2003 ∗Markus Leippold acknowledges the financial support of the Swiss National Science Foundation (NCCR FINRISK). Zvi Wiener acknowledges the financial support of the Krueger and Rosenberg funds at the He- brew University of Jerusalem. We welcome comments, including references to related papers we inadvertently overlooked. yCorrespondence Information: University of Zurich, ISB, Plattenstr. 14, 8032 Zurich, Switzerland; tel: +41 1-634-2951; fax: +41 1-634-4903; [email protected]. zCorrespondence Information: Hebrew University of Jerusalem, Mount Scopus, Jerusalem, 91905, Israel; tel: +972-2-588-3049; fax: +972-2-588-1341; [email protected]. On Trinomial Trees for One-Factor Short Rate Models ABSTRACT In this article we discuss the implementation of general one-factor short rate models with a trinomial tree. Taking the Hull-White model as a starting point, our contribution is threefold. First, we show how trees can be spanned using a set of general branching processes. Secondly, we improve Hull-White's procedure to calibrate the tree to bond prices by a much more efficient approach. This approach is applicable to a wide range of term structure models. Finally, we show how the tree can be adjusted to the volatility structure. The proposed approach leads to an efficient and flexible construction method for trinomial trees, which can be easily implemented and calibrated to both prices and volatilities. JEL Classification Codes: G13, C6. Key Words: Short Rate Models, Trinomial Trees, Forward Measure. There exist myriads of different term structure models and one might be struck by the variety of different approaches used. In the literature, models are commonly described by categorizing them into equilibrium or arbitrage-free models, short rate or forward rate models, one-factor or multi-factor models and so on. Indeed, most approaches can be fit within another approach and it merely depends on the modeler's taste, which model he wants to work with. As anexample, one can express a forward rate model as a short rate model or vice versa, as long as some technical conditions are fulfilled. However, when it comes to implementation, one has to choose which interest rate he wants to calculate. A widely used practice is then to start with the short rate. In this article, we explore one-factor short rate models and present their implementation using trinomial trees. Our work takes the model of Hull and White (1994) as a starting point. We contribute to the existing literature threefold. First, we show how spanning the tree can be generalized to allow for different alternative branching processes. This not only allows to use alternative branching processes to avoid negative interest rates as in the Hull-White model, but also allows to obtain a \slender" tree at the edges. This can substantially reduce the computational time. Moreover, additional flexibility in defining branching processes becomes important when pricing certain types of exotic options. For barrier options, as an example, a finer grid around the barrier helps to increase the convergence of the numerical tree method. Our second and main contribution is an improvement of Hull-White's procedure to calibrate the tree to bond prices by a computationally much more efficient approach. In particular, when pricing an interest rate derivative, one has first to perform a forward induction to match the tree, and secondly one has to do backward induction to price the instrument. Especially the forward 3 induction can become computationally intensive. With our approach, we are only left with backward induction. Instead of performing a forward induction, we use an approach which builds on the use of the forward measure. Matching is done analytically. An additional advantage of our method is that we can directly match the tree to the discretely discounted forward rates, which is a more natural approach than matching to the instantaneous short rate. We exploit two properties of the forward measure. First, under the forward measure the forward rate is an unbiased estimator of the future interest rate. This simple relationship allows us to match the tree in a straightforward manner and, moreover, the level shift is given in closed-form for a wide range of term structure models. As a prominent example serves the lognormal model of Black and Karasinski (1991). In the Hull-White framework, the tree matching procedure under lognormal short rates requires the use of a root search algorithm1, which needs to be applied in every time slice of the tree. In our approach however, the level shift can be determined analytically for a large class of term structure models. The second property we are exploiting is the fact that in discrete time the one-period forward measure equals the risk-neutral measure. Hence, the backward induction to determine derivative prices remains the same as in the standard trinomial tree. Finally, we show how the tree can be adjusted to the volatility structure in such a manner that our approach to match the initial term structure is still applicable. This is done by exploiting the flexibility embedded in the trinomial tree model. 4 1. Basic Assumptions and Notation We assume the market to be complete and arbitrage opportunities are absent. This is essen- tially equivalent to the assumption that there exists a probability measure P on (Ω; ) such F that for every security S without intermediate payments on a time interval [0; T ] its discounted price process is a P-martingale. When discounting occurs by taking the money market account, P is termed the risk-neutral probability measure. Then, T − rsds St = E e t ST t ; (1) R j F with r the short rate process2 and E ( ) the conditional expectation under the measure P. s · j Ft To implement a trinomial tree approximation for the interest rate r, we assume that trading takes place at discrete time steps. The market is complete in the sense that there exists for every time t 0 a zero bond with the respective maturity. Imposing absence of arbitrage, ≥ there exists a unique risk-neutral pricing measure P (in discrete time) such that every claim discounted by the money market account is a martingale under P. Further, for every time t the state-space is assumed to be finite. The state space is described by the one-dimensional process Xt with X0 = x. We assume that X follows a time-homogeneous Markov process. Interest rates and bond prices can be expressed as functions of X, i.e. we assume P (Xt; t; T ) to be the time-t pricing functional for a zero bond in state Xt, which pays $1 at the maturity date T . The entire term structure can be captured by the strictly positive function P (Xt; t; T ). We further require the zero bond to satisfy the conditions P (Xt; t; t) = 1 and lim P (Xt; t; T ) = 0 as T for all X and t. To abbreviate notation, bond prices observed from the initial term ! 1 t 5 structure, P (x; 0; T ), are denoted by P ?(T ). We next discuss the different steps to construct a tree approximation of the short rate process. 2. State Variable Tree The standard procedure to construct a tree approximation of the short rate process is to start spanning a tree for the state variable X on an equidistant tree. The process X is assumed to follow a time-homogeneous stochastic differential equation (SDE). Then, a trinomial tree can be used to provide a discrete-time and discrete-space Markov approximation for X. In the extended Vasicek model of Hull and White (1994) the transitions in the tree for the standard branching process are defined the following way. With πi+1;j+1 they denote the probability by which the process moves upward from node (i; j) to node (i + 1; j + 1) within one time interval. Probabilities are assumed to be independent of state and time. In Hull and White (1994) it is assumed that the transition from a node (i; j) to nodes (i 1; j + 1); (i; j + f − 1); (i + 1; j + 1) are equidistant both on the time as well as on the space axis. Furthermore, g it is assumed that the nodes (i; j) and (i; j + 1) remain on the same vertical level. Obviously, there are many degrees of freedom when building a trinomial tree. In general however, the more degrees of freedom, the less stable will the tree be when it comes to pricing derivatives. Therefore, one has to be careful, as in most cases flexibility comes at the price of less stability. Hull and White (1994) span the tree for X in such a way that the conditional expectation and conditional variance of the process X are matched exactly for all i and j. Instead of matching the variance as in Hull and White (1994), we match the second moment. This gives 6 1 (3;3) - 1PP 1 (2;3) π2 PP PP PP (1;1) - PP-Pq 1PP 1PP 1 (1;3) π1 PP PP PP PP PP PP π0 (0;1)- PPPq- PPPq- (0;0)PP PP 1PP 1 (0;3) PP PP PP PπP−1 PP PP PP PP PP P(−P1;1) PP PP PqP -PqP 1-Pq (−1;3) PP PP PP PP PP PP PP PP π−2 PPqP PPq- (−2;3) PP PP PP PP PPq (−3;3) Figure 1. Trinomial Tree. The tree starts at note (0; 0). At each node there is a threesome πi−1; πi; πi+1 evolving to the neighbor nodes (i 1; j), (i; j), and (i + 1; j) respectively.

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