Interest Rate Modeling Without Probabilities

Every interest rate model you have seen so far in this book assumes some form of random process. Brownian motion, mean reversion, stochastic volatility. They all start with “assume interest rates follow this stochastic differential equation” and then build a pricing framework on top. Chapter 68 of Wilmott’s book throws all of that out the window. No random walks. No probability distributions. No volatility parameters. Just bounds. This is the Epstein-Wilmott model, and it is refreshingly different.

The Problem With Traditional Models

There are two standard ways to think about interest rates. The first is the yield-based approach from Chapter 13, where you assume rates are constant for each product. Simple and useful for basic bonds, but obviously wrong when rates change. The second is the stochastic approach from Chapters 30 and 35, where rates follow some random process driven by one or a few factors.

The stochastic models have a big issue that practitioners know about but textbooks gloss over. If you assume a one-factor model, you are saying that every point on the yield curve is perfectly correlated with every other point. In theory, you could hedge a six-month option on a one-year bond using a ten-year bond. Nobody in their right mind would actually do that. Practitioners use common sense to pick nearby maturities for hedging, but the math says any maturity will do. The model and reality disagree.

On top of that, the parameters of stochastic models (volatilities, correlations, mean reversion speeds) are hard to estimate. And once you estimate them, they change. You calibrate today, and tomorrow your parameters are different. The whole exercise feels fragile.

What If We Just Set Boundaries?

The Epstein-Wilmott approach starts from a much simpler place. Instead of specifying a probability distribution for how interest rates move, we just say two things:

1. The interest rate stays within some range. It cannot go below some floor or above some ceiling.
2. The rate of change of interest rates is bounded. Rates cannot jump by more than a certain amount per unit time.

That is it. No Brownian motion. No drift terms. No volatility. Just upper and lower bounds on where rates can be and how fast they can move. The rate can do whatever it wants within those constraints. It could go up steadily, down steadily, oscillate wildly, or sit still. We make no assumptions about which path it takes.

Given these minimal constraints, we ask: what is the worst that can happen to our portfolio?

The Worst-Case Equation

If you hold a portfolio whose value depends on the short-term rate r and time t, you can use Taylor’s theorem to figure out how the value changes over a small time step. The key difference from the usual setup is that there is no second derivative term in r. Why? Because the process is not Brownian. The rate does not have the locally infinite variation that produces the second-order term.

What we get instead is a first-order nonlinear PDE. The nonlinearity comes from the worst-case analysis. At each moment, we ask: given that the rate can move in any direction within our bounds, which direction is worst for us? Then we assume that direction happens.

The resulting equation has a term that looks at the first derivative of V with respect to r. If that derivative is positive (portfolio value goes up when rates go up), then the worst case is for rates to go down as fast as possible. If the derivative is negative, the worst case is rates going up as fast as possible. The equation picks the worst direction automatically.

To see this concretely, imagine your portfolio could be worth 1.03, 1.02, 1.015, 1.01, or 1.02 after the next time step, depending on what rates do. The worst case is 1.01. So you assume that happens and discount it back. Simple.

The Price Spread Problem

When you solve the worst-case equation for a zero-coupon bond, you get a number that is lower than its “true” value. Solve the best-case equation (just flip the optimization) and you get a number that is higher. Together, you get a spread: the bond is worth somewhere between the worst and best cases.

This sounds like a disadvantage. After all, the market quotes a single price. But think about it: the market itself has a bid-offer spread. Getting a range of prices is actually more honest than pretending you know the exact fair value. The question is whether the spread is narrow enough to be useful.

For a naked zero-coupon bond, the spread is wide. Way too wide to be practical. With a spot rate of 6%, a growth bound of 4% per year, and a range of 3% to 20%, the worst and best case yields diverge dramatically. If this were the whole story, the model would be useless.

Static Hedging Saves the Day

Here is where the model gets clever. You cannot delta hedge in this framework because there is no stochastic process to hedge against. But you can statically hedge. Buy or sell other bonds alongside your target contract and value the whole package together.

Because the equation is nonlinear, the value of a portfolio is not just the sum of individual values. Adding a hedging bond changes which interest rate path is the worst case. The hedge shifts the worst-case scenario to something less extreme.

The example in the chapter is striking. Take a four-year zero-coupon bond and hedge it with bonds of maturities 0.5, 1, 2, 3, 5, 7, and 10 years. Without hedging, the worst-case value is 0.579 and the best-case is 0.878, a spread of 0.299. After optimal static hedging, the worst case improves to 0.730 and the best case drops to 0.758, a spread of just 0.028. That is a factor-of-ten reduction.

Most of the hedging comes from the three-year and five-year bonds, the maturities closest to our target. The ten-year bond is not used at all. This matches the common sense that practitioners already apply: hedge with nearby maturities.

The Yield Envelope

Instead of the traditional yield curve, the Epstein-Wilmott model produces what Wilmott calls the Yield Envelope. For each maturity, you get an upper and lower yield from the best and worst cases. At maturities where a traded bond exists, the envelope collapses to a single point matching the market yield. Between traded maturities, there is a spread.

The yield envelope looks like a “string of sausages,” pinched tight at traded maturities and bulging out between them. The wider the gap between available bonds, the wider the spread. This is intuitive: if you have fewer hedging instruments, you have less certainty about fair value.

This is a fundamentally different way to think about yield curves. Instead of fitting a smooth curve through market points and hoping the interpolation is correct, you acknowledge uncertainty. The envelope tells you exactly how much the yield could vary between observed points, given your assumptions about rate behavior.

Swaps, Caps, and Floors

The model handles more complex products too. For swaps, there are two approaches. The market practice method decomposes the swap into zero-coupon bond cashflows and prices those. The academic method approximates the reference rate with the short-term rate. Both give similar unhedged values, but the decomposition method hedges much more effectively because you are hedging cashflows with cashflows of the same form.

For caps and floors, an interesting asymmetry shows up. Hedging is more effective at reducing the best-case price than at raising the worst-case price. Why? In the worst case, the interest rate moves to make the caplets worthless. The hedge has to fight against that by either redirecting the worst-case path or profiting from the hedging bonds. That is harder than just counteracting positive cashflows in the best case.

Practical Applications

The model has several natural uses. You can identify arbitrage: if a market price falls outside your worst-to-best range, something is mispriced. You can set bid and offer prices: put your bid at the worst case and your offer at the best case, and you are guaranteed not to lose money. You can manage risk: the spread gives you an absolute bound on how much you can lose, not just a “95% confidence” number.

The last point is worth emphasizing. Traditional VaR gives you a maximum loss with some probability. This model gives you the maximum loss, period, as long as rates stay within your bounds. No probability required.

Why This Matters

The Epstein-Wilmott model is a fundamentally different philosophy. Instead of picking a stochastic process and hoping it is right, you set broad constraints and find the worst case. Instead of delta hedging (which requires perfect correlation assumptions), you use static hedging with real traded instruments. Instead of calibrating fragile parameters, you specify simple bounds that are easy to verify against historical data.

The model is nonlinear, which means you have to solve it for the entire portfolio at once. That is computationally more expensive. But the payoff is robustness. Your prices do not break when parameters change because the parameters are just bounds, not precise estimates.

In the next two chapters, Wilmott extends this framework to derivatives pricing and adds features like crash modeling and economic cycles.

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