Fixed Income Term Sheets: Real Product Examples

Theory is nice. But at some point you have to price actual products that real people are trading. Chapter 38 of Wilmott’s book takes two interesting fixed-income contracts and walks through how to price them from scratch. No hand waving. Just the math, the logic, and even the code.

These two products, the Chooser Range Note and the Index Amortizing Rate Swap, show how the tools we have been building throughout the book get applied to real-world deals.

The Chooser Range Note

A vanilla range note pays you based on how many days the reference rate (usually LIBOR) stays inside a specified band. Simple enough. But the Chooser Range Note (CRN) takes it up a notch.

In a CRN, the band is NOT predetermined. The contract holder gets to choose the range at the start of each period. In the term sheet Wilmott shows, there are four periods, so the holder makes four decisions during the life of the contract.

And these are not simple yes/no decisions like “do I exercise?” Each decision involves picking a mid point $M$ for the range, which is a continuous variable. You could pick any number. So the space of possible strategies is infinite.

Sounds impossible to price, right? Actually, it is not that bad if you think about it correctly.

The Pricing Logic

The key insight is that you price from the hedger’s perspective. The writer of the contract must sell it for enough money to cover the worst case, meaning the holder picks the range that maximizes the contract value.

Here is the approach for the first leg:

1. Pick a value for the mid point $M$
2. Price a standard (non-chooser) range note with that fixed mid point
3. Repeat for many different values of $M$
4. Plot the contract value as a function of $M$
5. Find the $M^$ that gives the maximum value $V^$

That maximum value is what you must charge for the first leg. If you charge less, the holder could pick $M^*$ and you would lose money. If they pick something else, you profit from the difference.

Why the Writer Wins

Here is where it gets interesting. The writer prices in a risk-neutral world because they are hedging. The holder makes decisions based on their real-world view of where rates are going.

The forward rate curve is typically upward sloping, containing a risk premium. But actual short-term rates rarely follow the forward curve that dramatically. So the price-maximizing ranges (based on the risk-neutral forward curve) are usually different from the ranges the holder actually picks (based on their real-world view).

This gap between risk-neutral optimal and real-world chosen ranges creates a windfall profit for the writer. Same concept as with American options: the writer sells for the worst-case price, but the holder rarely exercises optimally.

Later Legs Are Trickier

The first leg is straightforward because you know today’s spot rate. For the second, third, and fourth legs, you have to think ahead.

For the second leg, you need the function $V^f(r, M, t_1)$, the value of a non-chooser range note starting at time $t_1$ for different starting rates $r$ and mid points $M$. Maximize over $M$ to get the optimal value as a function of $r$ alone. That function then becomes the terminal condition for a pricing PDE or simulation from now until $t_1$.

You repeat this backward through each leg. Each optimization over $M$ collapses one dimension, and then you roll back through time to find today’s value.

What is clever about this contract: despite having embedded decisions (like an American option), it can be priced either by PDE methods or by Monte Carlo. That is unusual. Most contracts with decisions strongly prefer the PDE approach.

The Index Amortizing Rate Swap

The second product Wilmott examines is the Index Amortizing Rate Swap (IARS), which we first saw described in Chapter 32. Here we get the actual math and Visual Basic code.

In a plain interest rate swap, you exchange fixed-rate payments for floating-rate payments on a constant notional principal. In an IARS, the principal is not constant. It amortizes (shrinks) according to a schedule that depends on the current interest rate level.

Think of it this way: when rates are low, the principal amortizes faster. When rates are high, it stays put.

Why Does This Exist?

The IARS was invented to mimic the prepayment behavior of mortgage-backed securities. When rates drop, homeowners refinance, returning principal to investors early. The IARS captures this feature synthetically.

The Math

The value of the swap is $V(r, P, t)$ where $r$ is the spot rate, $P$ is the current principal level, and $t$ is time. The principal $P$ is not stochastic. It is deterministic but jumps at each reset date.

Between reset dates, $P$ is constant, so the governing equation is the standard one-factor bond pricing equation:

$$\frac{\partial V}{\partial t} + \frac{1}{2} w^2 \frac{\partial^2 V}{\partial r^2} + (u - \lambda w) \frac{\partial V}{\partial r} - rV = 0$$

where $u - \lambda w$ is the risk-adjusted drift and $w$ is the volatility of the spot rate.

At each reset date $t_i$, two things happen simultaneously:

1. Interest exchange. The swap value jumps by $(r - r_f)P$, where $r_f$ is the fixed rate
2. Principal amortization. The principal changes from $P$ to $g(r)P$, where $g(r)$ is the amortizing schedule

The jump condition combines both:

$$V(r, P, t_i^-) = V(r, g(r)P, t_i^+) + (r - r_f)P$$

At maturity, there is one final interest exchange:

$$V(r, P, T) = (r - r_f)P$$

The Similarity Reduction

Because the amortizing schedule is linear in $P$, the problem has a nice simplification. You can write $V(r, P, t) = PH(r, t)$ and solve a two-dimensional problem for $H$ instead of the three-dimensional problem for $V$. This cuts computation time significantly.

The jump condition for $H$ becomes:

$$H(r, t_i^-) = g(r)H(r, t_i^+) + r - r_f$$

And the final condition is simply:

$$H(r, T) = r - r_f$$

The Amortizing Schedule

In the code Wilmott shows, the amortizing schedule is a simple piecewise linear function. If the rate $r$ is below 10%, the principal amortizes proportionally ($g(r) = 10r$, so at $r = 0$ everything amortizes, at $r = 0.1$ nothing does). Above 10%, no amortization occurs ($g(r) = 1$).

In practice, the schedule would come from the term sheet as a table mapping rate levels to amortization percentages. The code handles this through a separate function that can be modified for different deal structures.

The Numerical Solution

The code uses an explicit finite-difference scheme. The interest rate is discretized on a grid, and the equation is stepped backward in time from maturity. At each reset date, the jump condition is applied to the entire grid.

The boundary conditions are set so that the solution behaves reasonably at very low rates (where the drift dominates) and very high rates (where a linear extrapolation is used).

What These Products Teach Us

Both products illustrate important principles from the book:

Decisions create value asymmetry. The chooser range note shows how embedded decisions benefit the holder but can be managed by the writer through worst-case pricing. This is the same principle behind American option pricing.

State variables matter. The IARS needs the principal $P$ as an extra variable. Recognizing that $P$ is deterministic (not random) and exploiting the linearity of the amortizing schedule reduces the computational problem dramatically.

Theory meets practice. Both products require the full toolkit: stochastic interest rate models, PDE methods, boundary and jump conditions, and careful numerical implementation. This is what quantitative finance looks like in the real world.

You will fully appreciate the code once you have read about numerical methods later in the book. But even now, the logic should be clear. Define the problem, write the equations, handle the special conditions at reset dates, solve numerically, done.

This is part of a series covering “Paul Wilmott on Quantitative Finance”. Next up: Merton Model: Your Company’s Equity Is Just an Option.

Previous: HJM and BGM Models: Forward Rate Modeling.