Previous Topic

Next Topic

Black-Derman-Toy (BDT)

The standard BDT model is constructed algorithmically to be consistent with both the existing term structure of zero-coupon yields, and (optionally) the term structure of yield volatilities. That is, the main aim of the tree building procedure is to derive a binomial representation for the level of the short-term interest rate such that zero-coupon bond prices computed from the tree are exactly equal to the set of zero coupon prices that are directly observable in the market. If desired, the model can also be constructed so that the implied interest rate distribution at each time step matches an observed interest rate volatility curve.

A general method for constructing binomial trees for the short rate is described fully in Hull and White (1996), Rebonato (1998), and Jamshidian (1991). Clewlow and Strickland (1998) present a comprehensive summary of the Jamshidian construction method, and that reference is the main source of the discussion that follows.

Consider constructing a binomial tree for the short rate that spans a period between time 0 and time T. The entire period can be split into an arbitrary number of i = 1 to N periods such that the length of each period is equal to Dt = T / N. With two discrete outcomes possible from every node, there are a total of i + 1 nodes at each of the i 'th timesteps. This general binomial representation is shown in the following figure, where each node is indexed as (i,j).

The number of states at each timestep is indexed by j. At the initial branch in the tree (i = 0), there is only 1 state and that is indexed as j = 0. At the end of the first period, the short rate has either increased to state j = 1 or decreased to state j = -1. In general, the rate at any arbitrary node i, j (that is not at the extreme top or bottom of the tree) will either move to a node indexed as i + 1, j + 1 (after an up-jump) or i + 1, j - 1 (after an down-jump). The extreme top and bottom nodes are indexed as N, N and N, -N, respectively.

As at time 0, we assume that both the term structure of zero coupon rates and rate volatilities are known. These curves are typically defined for a discrete number of points, and some interpolation scheme will be used to determine rates for maturity dates that are not explicitly included in the input data. Assuming that the zero rate for maturity i is defined as ZR(i ), then a discount function can be defined as:

 

Equation Template

where,

P(0,i ) = the time-0 price of a zero-coupon bond with unit face value that matures at time i

Dt = the length (in years) of the timestep assumed in the tree construction

Note that this representation of the discount function assumes that the input zero rates are defined with a compounding frequency that matches the chosen length of the timesteps used in the construction of the tree.

The variable of interest at each node in the tree is the short-term interest rate that will apply over the next time interval, r i, j. This rate is known at the beginning of each period and defines the current one period discount rate:

 

Equation Template

where,

DFi,j = the time i discount factor for state j

Dt = the length (in years) of the timestep assumed in the tree construction

With this background, we can now consider the general intuition behind the tree building process set out by Jamshidian (1991). The technique is referred to as a forward induction process, and involves working through each successive branch in the tree searching for the collection of short-term interest rates which will return a zero-coupon bond price that is consistent with the observed bond price. Further, if the model is also fitted to observed yield volatilities then the chosen set of rates must be consistent with both yields and volatilities.

Jamshidian shows that the level of the short rate at the i'th step in the tree is characterised by:

 

Equation Template

where,

U(i ) = the median of the lognormal distribution for r at step i

s(i ) = the short-rate volatility at step i

z(i ) = the value of a standard Wiener process

Clewlow and Strickland (1998) show that for each step i in the binomial tree, the expression in the above equation can be operationalized to give an expression for the short rate at each node in the tree. That is,

 

Equation Template

The remainder of the tree building process is then related to determining the appropriate values for both U(i ) and s(i ) such that the resulting tree is arbitrage-free. That is achieved in part using the concept of state prices (also referred to as Arrow-Debreu prices), which are defined as the time zero value of a pure security that provides a payoff of $1 if node (i, j) is reached, and zero otherwise:

 

Equation Template

where,

Equation Template the time zero state price for node (i, j)

PV = present value operator

Zero-coupon bond prices can also be defined in terms of the entire vector of state prices at step i :

 

Equation Template

State prices can therefore be interpreted as the discounted probability of reaching any node in the tree, and they represent the basic building blocks that can be used to value any financial claim through time. Further, given that the state price at node (0, 0) is equal to 1 by definition and by assuming that the risk-neutral probabilities of an up-jump and down-jump are both equal to 1/2, then we can construct the state prices at all successive nodes 'within' the tree using the following relation:

 

Equation Template

The state prices for the top and bottom nodes in the tree at each step i are given by:

 

Equation Template

Consider first just fitting a BDT tree to a set of known zero-bond prices. By assuming a constant volatility (s(i ) = s), the expression in equation Equation Template (from above) for the short rate is only a time-varying function of U(i ). At each step i in the tree, we have computed all j state prices and are attempting to determine the no-arbitrage short rate at all j nodes. We do so by searching for the rates that satisfy the requirement that:

 

Equation Template

That is, given that the current value of a zero-coupon bond maturing at time (i + 1) . Dt is known, we can use a technique such as Newton-Raphson to solve for the only unknown in the above equation, U(i ).

Although the BDT model was initially derived specifically using the binomial lattice framework, several authors have subsequently shown that the model implies the following stochastic differential equations for the evolution of the short rate through time. These processes can be directly compared to the equivalent expressions for the Hull-White and Black-Karasinski models discussed in the following sections. The original specification of the model, fit to both the term structure of zero rates and the term structure of volatilities is:

 

Equation Template

where,

r = the instantaneous short rate at time t

q(t) = value of the underlying asset at option expiry

s(t) = time dependent short rate volatility

When the BDT model is restricted to fit the initial term structure only, the expression in the above equation reduces to:

 

Equation Template

where,

s = the constant short rate volatility

 

Return to www.derivativepricing.com website

Copyright 2013 Hedgebook Ltd.