How Short-Rate Models Determine Zero-Coupon Bond Prices

Summary

The document explains how a stochastic short-rate model can support interest-rate derivative pricing. Under risk-neutral valuation, the price of a default-free zero-coupon bond is the conditional expectation of the discount factor accumulated from the present to maturity. In this framework, a specified process for the short rate determines the bond price; it is not necessary for the instantaneous rate at one moment alone to determine every maturity’s price.

Models such as Vasicek, Hull–White, and CIR prescribe short-rate dynamics, often through a stochastic differential equation. Given those dynamics and the risk-neutral setup, bond prices and related instruments, including bond options and swaptions, can be valued, sometimes analytically. A constant deterministic rate provides a simple illustration, producing exponential discounting over the time to maturity. The explanation assumes no default risk and relies on risk-neutral pricing with the bank account as numeraire. It does not discuss calibration, model risk, or how competing short-rate specifications affect observed market prices.

Key ideas

• A default-free zero-coupon bond is valued as the risk-neutral expectation of its accumulated discount factor.
• The short-rate process over the full period, rather than only its current value, determines the model’s bond price.
• Short-rate models specify stochastic dynamics that can be used to price bonds and interest-rate derivatives.
• A constant deterministic rate yields exponential discounting over the bond’s remaining term.
• The explanation assumes no default risk and does not address calibration or model selection.

Tags

# If short rates $r(t)$ do not determine the bond prices $P(t, T)$, then what is the basis for short rate models?


# If short rates $r(t)$ do not determine the bond prices $P(t, T)$, then what is the basis for short rate models?












The question title says it all: We know that in general, specifying the short rate $r(t)$ does not specify the bond prices $P(t, T)$. So how can a model for short rates—for example the Vasicek model—be powerful enough to price interest rate derivatives?

## Answer by Kevin (score 2, accepted)

https://quant.stackexchange.com/a/47550

Let $r(s)$ be the process of a short rate. Then, by risk neutral pricing, $$ P(t,T) = \mathbb{E}^\mathbb{Q}\left[ \exp\left( -\int_t^T r(s)\mathrm{d}s\right) \Bigg| \mathcal{F}_t\right].$$ Thus, the zero-coupon bond is determined completely by the short rate process. Here, $P(t,T)$ denotes the time $t$ price of a zero-coupon bond maturing at time $T$. You just take the risk-neutral expectation of the discounted payoff. The payoff is $1$ for almost all states of the world $\omega\in\Omega$ (assuming no default risk). Thus, the price of the bond is the conditional expectation of the discount factor. The risk-neutral measure $\mathbb{Q}$ uses a bank account $(B_t)$ as numeraire with $\mathrm{d}B_t=r(t)B_t\mathrm{d}t$.

Short rate models (such as Vasicek, Hull-White, CIR, etc.) specify a stochastic model for $r(s)$, typically a (perhaps multidimensional) SDE and then, you can find (sometimes analytical) prices for bonds, bond options, swaptions etc.

The easiest case is a deterministic and constant short rate $r(s)\equiv r$. Then, $$P(t,T)=e^{-r(T-t)}$$ and clearly the short rate $r$ gives you the bond price.

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