Introduction to Term Structure Lattice Models

In this lesson, we learned about fixed income derivative pricing in lattice models.

Fixed income markets are enormous, even bigger than equity markets.  In Q3, 2012 fixed income markets were worth $35.3 trillion, while equity markets were worth $26 trillion.
In this lesson we will focus on interest rate and bond derivatives using binomial lattice models.

Fixed income models are more complex than security models.  The problem is that we need to model the entire term-structure of the interest rates (i.e. how the spot interest rate changes throughout time, not just the price of the bonds that carry the interest rates).   The short rate, r(t) is the variable of interest in fixed income models.  It is the risk-free interest rate between periods t and t+1.  It is a random process before it becomes known at time t.

In modeling, we will specify risk-neutral probabilities for each r(t), not the true probabilities of r(t).  This is different from q, (1-q), p, and (1-p) in the binomial model.  In lattice models we will, price fixed income derivatives in such a way that guarantees not arbitrage, then match the prices of liquid securities via a calibration procedure.

The following is an example of a binomial model for the short-rate interest rate:

We will take zero coupon bond (zcb) prices to be our basic securities.  The notation of r(i,j) refers to the short rate r at time i and at state j.  The short rate r(i,j) refers to the short-rate at node N(i,j).  We will use the form Z(i,j,k) to refer to a zcb for time i, state j and matures at time k.  We would like specify a binomial model by setting up all the Z(i,j,k)'s at each node.

We will also let Z(i,j) to be the date i, state j, price of a non-coupon paying security.  This basically means that Z(i,j) is the price of a zcb at particular time i and state j.  We will use risk-neutral pricing to price every security such that:

where q(u) and q(d) are the risk-neutral probabilities of an up-move and a down-move, respectively.  Since they are probabilities, they are between 0 and 1 and sum to 1.  There can be no arbitrage when we price using this formula because it is not possible for Z(i+1, j+1) and Z(i+1,j) to be greater than or equal to zero (see the above formula) AND  have Z(i,j) be less than zero.  Since this is not possible, there must be no arbitrage.


Next we looked at if the security pays a "coupon", or more loosely, pays an intermediate cash flow between time purchased and maturity.  This is similar to a stock paying a dividend.
If the security pays coupon C(i+1,j) at date i+1 and state j, then

There can also not be any arbitrage in this case either.

Similar to what was said earlier, Z(i+1,j+1) + C(i+1, j+1) ≥ 0 and Z(i+1,j) + C(i+1,j) ≥ 0 , so Z(i,j) cannot be less than 0.  This shows that there cannot be Type A arbitrage.

There cannot be Type B arbitrage either.  Type B arbitrage states that V(0) ≤ 0, but V(1) > 0.  If Z(i+1,j+1) + C(i+1, j+1) ≥ 0 and Z(i+1,j) + C(i+1,j) > 0, then Z(i,j) cannot be less than or equal to 0.

Finally, it is common practice to set q(u) = q(d) = 0.5 and we will in fact assume this in future lessons.