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Fixed Income and Interest-Rate Models / Short-Rate Models

Expectations, Volatility, and the Shape of the Term Structure

Once you have calibrated a short-rate model and built its risk-neutral tree, you price every bond and derivative by backwards induction. But what is the model actually assuming about the economy? Why does the risk-neutral probability differ from the true probability? And why do two models with different parameter choices — one with high expected rates, one with a large risk premium and low expected rates — produce the same prices?
This lesson answers those questions by decomposing what determines yield into three distinct forces: the expected path of the short rate, the convexity correction from Jensen's inequality, and the risk premium for bearing duration risk. The previous lesson showed how to price on a tree; this lesson explains what economic content the risk-neutral probabilities encode. The companion lesson on expectations and convexity in the term structure covers the static decomposition; this lesson derives the yield equation dynamically from Itô's lemma and shows why only the risk-neutral drift — not its decomposition into expectations and risk premium — matters for pricing.
Learning objectives. By the end of this note you should be able to:
  • compute the convexity correction from Jensen's inequality for a two-state short-rate tree and explain why it always lowers long-rate spot yields;
  • state how higher volatility increases the convexity effect and how the convexity effect grows with maturity;
  • define the risk premium λ\lambda per year of modified duration and show how it is implemented by shifting the risk-neutral drift upward;
  • state the yield decomposition equation y=r+D E[dy/dt]+λD−12Cσ2y = r + D\,\mathbb{E}[dy/dt] + \lambda D - \tfrac{1}{2}C\sigma^2 and identify which term each force corresponds to;
  • explain observational equivalence: why the term structure cannot distinguish a high-expectations/low-risk-premium world from a low-expectations/high-risk-premium world;
  • explain why arbitrage-free models (Ho-Lee) require a time-dependent drift while equilibrium models (Vasicek, CIR) do not necessarily fit the initial term structure.

In this lesson

    1. Expectations alone: a certainty benchmark
    1. Convexity: Jensen's inequality lowers long yields
    1. Risk premium: compensating for duration risk
    1. The yield decomposition equation
    1. Observational equivalence
    1. Equilibrium versus arbitrage-free models

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