Why a Zero-Coupon Bond Reaches Par at Maturity in HJM

Summary

The document asks how a zero-coupon bond price can follow an Ito process before maturity and still equal one at its maturity date. It frames the puzzle using the HJM bond-price dynamics, in which the price changes through drift and volatility, and asks whether a risk-neutral zero drift leaves any mechanism to enforce the terminal value.

The text presents the question but supplies no answer or derivation. The key distinction needed to resolve it is not developed: the stochastic differential equation is for times before maturity, while the bond price at maturity is set by its payoff, and admissible volatility behavior near maturity must be consistent with that boundary condition. As a result, this is a useful conceptual prompt about term-structure modeling, but not a complete explanation or evidence for a particular HJM specification.

Key ideas

• A zero-coupon bond's maturity value is a terminal payoff condition.
• The document questions how that condition fits with stochastic price dynamics before maturity.
• Under a risk-neutral measure, a zero drift in the stated dynamics raises a boundary-condition question.
• The text offers no resolution or model-specific conditions on volatility near maturity.

Tags

# How can a bond price that follows an Ito process possably have value 1 at maturity?


# How can a bond price that follows an Ito process possably have value 1 at maturity?












Consider the HJM model for instance. According to Wikipedia, the price $P(t,T)$ of a ZCB at time $t$ with maturity time $T$ is of the form

$$ {\displaystyle {\frac {dP(t,T)}{P(t,T)}}=\mu\left(t,T\right)dt-{{\sigma }}(t,T)dW_{t}}. $$

How can this possibly force $P(T,T)=1$? It's said that $\mu\left(t,T\right)$ is an adapted process (wrt to $t$), so it may depend on $P(t,T)$. I would think it must, in order to steer the process back to 1 at $T$.

But then under the risk-neutral measure $\mathbb{Q}$, of course $\mu\left(t,T\right)=0$ , then I can't possibly see how $P(T,T)=1$ can hold. We would need $\sigma(t,T)\to 0$ as $t\to T$, but that would seems to slow the volatility to 0 freezing the process at the value just before time $T$ possibly not 1. I don't see how it can possibly be steered back to 1 without a drift term.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.