Modeling Inflation: Pricing Inflation-Linked Products

Inflation eats your money. Slowly, usually, but sometimes fast. If you hold a regular bond, inflation erodes the value of every coupon and the principal repayment. Index-linked bonds solve this by tying payments to an inflation index like the Consumer Price Index (CPI) in the US or the Retail Price Index (RPI) in the UK. Chapter 71 of Wilmott’s book looks at how to model inflation and price these products. The answer turns out to be messier than you might hope.

What Are Inflation-Linked Products?

The simplest one is the inflation-linked bond. Instead of paying a fixed coupon and returning a fixed principal, everything scales with the inflation index I. If inflation goes up 20% over the bond’s life, your principal payment is 20% larger. Your income keeps up with the cost of living.

Then there are inflation caps and floors. A cap pays out when inflation exceeds some level. A floor pays out when it drops below. Imagine a bond that adjusts for inflation each year but with a floor at 3% and a cap at 6%. If actual inflation is 2%, you still get 3%. If it is 8%, you only get 6%. The cap and floor are embedded options on the inflation rate.

Inflation swaps exchange a fixed rate (the market’s guess at average future inflation) for actual inflation over each period. There are also inflation swaptions (options on entering a swap) and barrier contracts that knock in or out based on the inflation level.

The Obvious Approach

The textbook way to price inflation products is straightforward. Treat the inflation index I like a stock price or exchange rate. Write down a stochastic differential equation for I with some drift and volatility. Write another equation for the interest rate r. Done.

Well, almost done. There is a subtlety. Unlike a stock, you cannot trade the inflation index directly. You cannot buy a unit of CPI. This means there is a market price of inflation risk lurking in the equations. Someone has to bear the risk that inflation differs from expectations, and they demand compensation for it.

Wilmott says everyone does it this way, and then immediately suggests something different based on what the data actually show.

What the Data Tell Us

This is where the chapter gets interesting. Wilmott looks at UK RPI data and plots the relative changes R (the instantaneous percentage change in the index). The data, especially since 1960 when there are enough decimal places, do not look like independent random numbers drawn from a normal distribution. They look more like a random walk.

This is an important distinction. If R were white noise, then modeling dI = I * (something * dt + something * dX) would be natural. The index itself would be the random walk, and its returns would be independent period to period. But if R (the rate of inflation) is itself a random walk, then inflation has momentum. High inflation tends to persist. Low inflation tends to persist. The index is integrated twice, not once.

Wilmott suggests writing dR = a(R) dt + b dX as the model for the rate of inflation, putting R at the same level as the interest rate r. Both are stochastic processes. The inflation index I is then just the integral of R, similar to how the price of a bond is determined by integrating the short rate.

Building the Pricing Model

With R modeled as its own stochastic process and r as the interest rate, the pricing equation for inflation-linked products becomes a PDE in r, R, I, and t. That sounds like a four-dimensional problem, but there is a similarity solution. Because the payoff of an inflation-linked bond is proportional to I, the value V(r, R, I, t) can be written as I times H(r, R, t). This reduces the problem to three dimensions.

The equation for H involves the risk-neutral drifts and volatilities of both r and R, their correlation, and the market prices of risk. If you pick linear risk-neutral drifts and constant or square-root volatilities, you can find explicit solutions.

Looking at the Distribution

When Wilmott fits the distribution of monthly R to a normal distribution, he gets the usual result: it does not fit well. There is skew, and the downside appears bounded. Deflation is rare and limited.

A lognormal fit (after shifting R by a constant to make it positive) works much better. The best fit parameters give a shifted lognormal with a floor around negative 1.5% and a mean around 1.9%. This is consistent with the observation that inflation has a hard time going deeply negative. Central banks fight deflation aggressively.

Looking at the volatility of R as a function of the level of R, there is a hint that volatility is higher both for very high inflation and for deflationary periods. But the data are too sparse for firm conclusions. Wilmott settles on constant volatility for simplicity and chooses a(R) to reproduce the observed lognormal steady-state distribution.

Inflation and Interest Rates: The Tangled Relationship

Can we model inflation independently of interest rates? Wilmott plots year-on-year inflation since 1960 alongside interest rates. The relationship is clearly complex and bidirectional. Governments use interest rates to control inflation. Inflation affects spending and saving behavior, which feeds back into interest rates.

Ignoring the high-inflation 1970s (a dangerous but common move), there is a roughly linear relationship: R equals about negative 0.87 plus 0.56 times r. Higher interest rates come with higher inflation, but with a negative intercept that reflects the fact that real interest rates are typically positive.

To capture both the correlation and the deviation from it, Wilmott introduces a residual variable phi representing the difference between actual R and the linear prediction from r. The residual follows its own mean-reverting process. Its steady-state distribution is normal with a standard deviation around 1.1%.

This gives a workable two-factor model: r drives the general level and R = alpha + beta*r + phi captures both the systematic relationship and the random deviations.

The Calibration Problem

Here is where Wilmott gets honest about the limitations of the classical framework. All of that data analysis, the distribution fitting, the correlation estimation, was done on real-world (physical measure) data. But derivative pricing works in the risk-neutral measure where the drifts are different.

This means most of what we estimated gets thrown away. The risk-neutral drift replaces the real-world drift, and only the volatilities survive from the statistical analysis. This is a fundamental tension in quantitative finance: we need data to build models, but the pricing framework ignores much of what the data tell us.

The market price of risk for inflation is the bridge between the two worlds. But it is hard to observe, hard to measure, and unstable over time. You would need to see it in the prices of traded inflation products, but those markets can be thin.

The desire to calibrate, to make theoretical prices match market prices, assumes the market is “correct” in some sense. Wilmott is skeptical of this.

Non-Linear Methods as an Alternative

The chapter ends by pointing toward a different approach. Non-linear pricing methods like the mean-variance model accept that we cannot eliminate inflation risk. Instead of producing a single price, they produce a range: an expected value and a variance. Some prices within that range are more likely than others.

This fits naturally with the Epstein-Wilmott philosophy from the previous chapters. If we cannot perfectly hedge inflation risk (and we cannot, because the inflation index is not traded), then pretending we can through calibration is dishonest. Better to acknowledge the uncertainty and price accordingly.

Wilmott notes that inflation modeling is still in its early stages. Most approaches copy what works for interest rates or credit risk, but inflation has its own quirks. The relationship with interest rates is complex and regime-dependent. The 1970s show that inflation can break out of seemingly stable patterns. And the products being traded are getting more exotic, requiring models that can handle caps, floors, swaptions, and barriers.

Key Takeaways

Three things stand out from this chapter. First, the rate of inflation R behaves more like a random walk than like independent noise. This matters for how you model it. Treating I as a geometric Brownian motion misses the persistence in inflation rates.

Second, inflation and interest rates are deeply connected but the relationship is not stable. Any model that assumes a fixed correlation or a fixed linear relationship is vulnerable to regime changes. The 1970s broke every pattern that worked from 1950 to 1965.

Third, the market price of inflation risk is the elephant in the room. Classical pricing requires it, but nobody can measure it well. Non-linear methods that produce price ranges rather than point estimates may be more honest for inflation products.

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