# Yield Curve Modelling

Economic theories of the term structure of interest rates have been part of economic thought since around the start of the twentieth century. Irving Fisher, the American economist whose eponymous separation theorem we discussed earlier, was one of the early theorists on what determines the shape of the interest rate yield curve. In 1896 he developed what became known as the Expectations Hypothesis. This says that long-term interest rates simply reflect the market’s expectations for the future path of the short-term interest rate.^{[1]} Under this hypothesis, the expected return on risk-free bonds is the same for all terms of bond.

The Keynesian revolution in macroeconomics that emerged in the 1930s as a response to the Great Depression generated, amongst other things, new ideas in the theory of interest rates. One such idea was the Liquidity Preference Hypothesis. This was mainly developed by the British economists John Maynard Keynes and John Hicks as part of broader economic models of the behaviour of saving, investment and money.^{[2]} The Liquidity Preference Hypothesis said that the expected future path for the short-term interest rate was not the only factor that would impact on the prices of long-term riskfree bonds: a risk premium would need to be offered to investors to induce them to bear the additional price volatility associated with holding longer- term bonds. Under this hypothesis, the risk premium offered by longer-term bonds would always be positive and would increase with the bond term.

A third economic theory of the determination of the risk-free yield curve was developed in US academia in the 1950s and 1960s, most notably by Modigliani and Sutch (yes, *that* Modigliani).^{[3]} This theory, which became known as the Market Segmentation Hypothesis or Preferred Habitat Hypothesis, could be viewed as a generalisation of the Liquidity Preference Hypothesis. It argued that different investors had different investment horizons and this horizon determined investors’ bond maturity preferences. Investors would require a risk premium to induce them to hold bonds with terms different to their maturity preference. The Liquidity Preference Hypothesis can be viewed as a special case of this theory where all investors have an instantaneously short investment horizon. Under the Market Segmentation Hypothesis, the term premiums embedded in risk-free bond prices could be positive or negative at different points on the term structure, depending on investors’ investment horizons and the supply of bonds.

In a world of deterministic interest rates, a simple arbitrage argument shows that the Expectations Hypothesis must hold true. So theories of how yield curves are determined are really about how interest rate *uncertainty *impacts on bond prices. This is a complex and quantitative problem to tackle rigorously—to make meaningful progress, a mathematical description of the nature of interest rate uncertainty is inevitably required; and, intuitively, any theory must recognise that risk-free bonds of similar durations are close substitutes for each other and must be priced in a consistent way that does not create any trivial free lunches (i.e. the bond prices must be arbitrage-free). These two ingredients—a stochastic process for an economic variable and its use in the determination of arbitrage-free asset prices—were the keys to Black—Scholes—Merton’s option pricing breakthrough. So it is unsurprising to find that the breakthrough in how interest rate uncertainty impacts on bond prices was directly inspired by the Black-Scholes dynamic replication and noarbitrage argument.

This breakthrough was achieved in a seminal paper by Oldrich Vasicek that was published in 1977.^{45} Vasicek was a Czech probability theorist who migrated to the USA in the late 1960s and began a career in investment banking in California. In his role as a researcher at a bank, he attended seminars by Fisher Black and Myron Scholes in the early 1970s where they presented their early ideas on option pricing. Vasicek saw that a risk-free bond of a given term could be viewed as a form of derivative security where the short-term interest rate was the underlying asset. This opened up the possibility of applying the dynamic hedging ideas of Black-Scholes to find a formula for the arbitrage- free behaviour of the entire yield curve.

Vasicek first specified a general mathematical structure for the uncertainty in future interest rate behaviour by defining a stochastic modelling framework for the short-term interest rate. He assumed that the instantaneous short-term interest rate (which we will refer to as the short rate) followed a continuoustime stochastic process that had a few key properties: it was continuous (the short rate could not jump); it was Markovian (its future value depended only on its current value and not on previous values); and it had a single source of randomness (what would now be called a single-factor model). He then supposed that the yield on a risk-free bond of any term could be expressed as the expectation for the short rate over the term of the bond plus a risk premium component. Of course, if there are no restrictions on the how the risk premium component behaves as a function of the term of the bonds, this statement is just a tautology—the risk premium at any point in the term structure is merely the difference between the market bond yield and the expected short rate path over the bond term. One of Vasicek’s key results was to show that in order for the yield curve to be arbitrage-free, there were well-specified constraints on what form the risk premium component could take across the term structure of risk-free bonds.

The assumption that the short rate process had a single source of randomness (and that the risk premium component of bond prices did not have any additional source of randomness) implied that all variation in bond prices of all terms must be driven by the same single source of randomness. This, in turn, implied that all risk-free bond returns would be instantaneously *perfectly correlated* (though bond expected returns and volatilities would vary with term). This perfect correlation implied that some combination of any two bonds of different maturities could be held that would (instantaneously) produce a certain return (this would involve being long one bond of a given term, and short another bond of another given term). In other words, the short-term interest account could be *replicated* by a dynamic combination of any two risk-free bonds. Like Black-Scholes, Vasicek argued that as this dynamic portfolio was instantaneously riskless, its rate of return must be the short rate in order to avoid arbitrage.

This argument was very general—for *any* two bonds of two different terms, portfolio weights could be found such that the hedge portfolio would be instantaneously riskless. The relative portfolio weights would be a function of the relative sizes of the bond volatilities at the two different bond terms, and these in turn would be determined by the short rate’s stochastic process. Vasicek had not yet considered a specific form of stochastic process—the dynamic replication argument held for any short rate process that satisfied his general modelling assumptions. From this construction, and in very similar style to Black-Scholes, Vasicek used the dynamic hedge portfolio argument to derive a partial differential equation for the bond price of any term t, which was a function of the current short rate, its drift and variance processes and the risk premium assumed to apply to the bond of term t.

This analysis uncovered the key arbitrage-free property of risk-free bond prices in Vasicek’s framework. It showed that every risk-free bond’s instantaneous expected return in excess of the short rate must vary in linear proportion to the bond price’s instantaneous volatility. So at any given moment, a ratio of the excess expected return to volatility existed that was constant across all terms of risk-free bond. The risk premium component of all bond yields was determined by this ratio, which Vasicek called the ‘market price of risk’.

Mathematically, this market price of risk could be written as:

where p(t, T) is the instantaneous drift rate at time t of the price of a bond maturing at time T; r(t) is the short rate at time t, and o(t, T) is the instantaneous volatility at time t of the price of a bond maturing at time T. In Vasicek’s framework, no-arbitrage implied that q(t), the market price of risk at time t, did not vary as a function of the bond term T.

In this setting, the risk premium component could be positive or negative, and it could vary in size over time, but at any given instant, it could only take one sign across all points on the yield curve and it varied across the yield curve in this prescribed linear way. If a bond had a risk premium different to that implied by the market price of risk and the bond’s volatility, it would be possible to generate arbitrage profits by constructing a riskless portfolio that generated returns in excess of the short rate.

This restriction on the term structure of the risk premium had significant implications for the economic theories of the term structure. The Expectations Hypothesis (market price of risk is zero) and the Liquidity Premium Hypothesis (market price of risk is always positive) could both be accommodated by Vasicek’s no-arbitrage relationship, but it placed significant limits on the broader Preferred Habitat Hypothesis. In essence, Vasicek had shown that investors’ ability to reconstruct bond maturities through dynamic portfolios of bonds of other maturities meant that there was a limit to the way in which maturity preferences could impact on the relative differences in arbitrage-free bond prices. For example, in Vasicek’s model, if the ten-year bond offered a negative risk premium and the 20-year bond offered a positive risk premium, investors could perfectly replicate the ten-year bond pay-off through a dynamic combination of the 20-year bond and cash that would cost less than the ten-year bond price.

To complete his analysis of this general arbitrage-free bond pricing framework, Vasicek then noted that the bond pricing equation that he derived through the dynamic hedging process was equal to the bond price implied by the expected short rate when the market price of risk is assumed to be zero (though this was not a central object of his analysis, it was essentially an incidental by-product of his ‘real-world’ modelling presentation). In the same way that Black-Scholes-Merton showed that the risk-neutral stock process determined the value of any option on the stock, Vasicek had shown that the risk-neutral short rate process determined the arbitrage-free price of risk-free bonds of all terms.

Having established the above framework and its general properties, Vasicek then proceeded to illustrate it with the use of a simple example: he assumed that the short rate followed a mean-reverting process with a normally distributed volatility process (sometimes referred to as an Orstein-Uhlenbeck process):

He also assumed the market price of risk was some constant *q* (i.e. it did not vary over time). These model dynamics were never meant to be taken as a recommendation of a realistic description of interest rates, but rather to provide a simple and tractable illustration of Vasicek’s pricing framework. Nonetheless, this short rate model is universally referred to as the ‘Vasicek model’. With this modelling specification, the arbitrage-free bond pricing equation could be solved to provide an analytical formula for the bond price as a function of term. And the expected return and volatility of every risk-free bond were well- defined functions of the parameters of the short rate process and the market price of risk (as well as the bond’s term).

Vasicek’s demonstration that the Black-Scholes-Merton idea of arbitrage- free pricing by dynamic replication could be used to price the risk-free term structure was the permanent foundation for a new branch of quantitative finance. A vast amount of yield curve modelling research followed in the next two decades that further developed Vasicek’s big idea. One of the most significant developments to arise in the years following the Vasicek paper came from a pair of papers published by Cox, Ingersoll and Ross (CIR) in 1985.^{46} These papers were far less accessible than Vasicek’s paper. Vasicek’s paper had been very focused in its ambitions: he showed that, under certain assumptions, a given stochastic process for the short rate together with a market price of risk could determine the arbitrage-free pricing of risk-free bonds of all terms. Cox, Ingersoll and Ross broadened the setting. Rather than specifying a framework where the stochastic process for the short rate is specified exogenously, they developed a broader stochastic model of the wider economy (production possibilities, wealth, individual’s utility) and all the assets that traded in it, and then derived the short rate stochastic process that was implied by the specified economy system.

Before specifically considering interest rate behaviour, their first paper derived the by now ubiquitous risk-neutral valuation property for their economic system. It is interesting to note that Vasicek did not focus the presentation of his bond pricing mathematics in a risk-neutral setting. He used a ‘real-world’ probability measure, and showed that the bond pricing equation could use these real-world probabilities together with the market price of risk assumption to obtain the bond price (and this could be presented as a risk- neutral valuation where the market price of risk transforms the real-world probability measure into a risk-neutral one). After Vasicek, the biggest application of the arbitrage-free yield curve modelling that he pioneered was increasingly in interest rate derivatives pricing and hedging rather than in macroeconomic descriptions of how interest rate uncertainty impacted on yield curve behaviour. These applications were most efficiently addressed by working in risk-neutral probability measures, and the use of real-world probabilities increasingly disappeared from the yield curve modelling literature. Cox, Ingersoll and Ross’s grand economic framework seamlessly moved between real-world probabilities and risk-neutral probabilities.

The CIR paper, ‘A Theory of the Term Structure of Interest Rates’, broke new ground beyond Vasicek in a number of important ways. Like Vasicek, they derived the arbitrage-free bond pricing equation via a stated stochastic process for the short rate. Their short rate model (below) assumed that the volatility of the short rate moved in proportion to the square root of the short rate:

This was a slightly more sophisticated and realistic model of interest rate behaviour than that used by Vasicek. Most notably, it precluded the possibility of negative interest rates (this has traditionally been seen as a desirable feature of interest rate models, though that view has been subject to some revision in the 2010s!). Like Vasicek, they derived an analytical solution for the risk-free bond pricing equation implied by the short rate process. Cox, Ingersoll and Ross also went a step further and considered how the same valuation framework could be used to value bond derivatives and, as an illustrative example, derived the arbitrage-free pricing equation for a call option on a risk-free bond. This was an early illustration of the potential applicability of stochastic yield curve models to pricing and hedging interest rate derivatives. A huge expansion in the types and volumes of such securities occurred in the final two decades of the twentieth century. This was facilitated by the advanced mathematics of these models and motivated further advancements in the theory and its implementation.

Finally, Cox, Ingersoll and Ross also highlighted the limitations of the single-factor models that had been exclusively considered by Vasicek and themselves up to this point. The single-factor model structure had severe limitations—most obviously, it implied that the entire term structure was perfectly correlated. From a derivatives pricing perspective, it would prove difficult to recover the market prices of derivatives that depended on both bond price volatilities and correlations with such models. From a risk management perspective, these models implied that a ten-year liability could be perfectly hedged by dynamically rebalancing long and short positions in a three-year and two-year risk-free bonds. These limitations had been recognised by Brennan and Schwartz in the immediate aftermath of the Vasicek paper, and they published a specific two-factor arbitrage-free interest rate model in 1979.^{[4]} Cox, Ingersoll and Ross developed a more general multifactor modelling setting within the arbitrage-free pricing framework first pioneered by Vasicek. You may recall that the perfect correlation of risk-free bond returns was an important step in Vasicek’s bond pricing derivation. Cox, Ingersoll and Ross showed that Vasicek’s dynamic replication argument could still work in this multi-factor, decorrelated yield curve world—more factors just meant more risk-free bond holdings of different maturities were required to construct the dynamically riskless portfolio.

These three themes beyond Vasicek that Cox, Ingersoll and Ross developed—more realistic volatility behaviour for the short rate; the application of the model to interest rate derivative pricing; and multi-factor short rate modelling—were themes that were further developed by academics and practitioners over the following few years. Hull and White,^{[5]} Black, Derman and Toy^{[6]} and Black-Karasinski^{[7]} were important contributions that pursued these themes in an extremely active research programme that was at least partly motivated by the huge growth in the trading of interest rate derivatives in securities markets.

A notable departure from these threads of development arose in a paper by Heath, Jarrow and Morton (HJM) published in 1992.^{[8]} The HJM approach reconsidered how to describe the stochastic evolution of the yield curve. As we have seen, short rate models explicitly specified a stochastic process for the short-term interest rate, and then deduced the arbitrage-free pricing of all risk-free bonds from that process. So one explicit equation (the short rate process) determined the behaviour of every point on the yield curve. This was very parsimonious, but it also restricted modelling flexibility and meant that there were limited degrees of freedom to fit to calibration targets such as volatilities of different points of the yield curve (or to simultaneously fit to the prices of many different types of interest rate derivatives).

Instead of only explicitly specifying a stochastic process for the short rate, HJM suggested explicitly specifying a stochastic process for every point on the yield curve. This general framework provided almost unlimited freedom in specifying the volatility and correlation structure of the different points on the yield curve. This flexibility was very powerful in the context of interest rate derivative pricing (which was the intended purpose of their framework). HJM then found the no-arbitrage conditions that specified the risk-neutral drifts that must be generated by each forward rate of the yield curve in order to avoid arbitrage.

Their paper was closely related to the quantitative work of the Harrison option pricing papers discussed above. HJM showed that their interest rate framework could be embedded as the interest rate modelling piece of the broader mathematical economic framework of Harrison and Pliska. They were therefore able to make use of the ‘equivalent martingale measure’ concept of Harrison and Pliska to determine the drift processes for any forward rate on the yield curve. Where Vasicek’s yield curve modelling was inspired by Black-Scholes, the modelling framework developed by HJM was inspired by Harrison and Pliska: with this paper, yield curve modelling had fully caught up with and become an integral part of option pricing theory.