Vasicek Bond Price Dynamics

Article Quant Q&A · Author: PTQuoc

Summary

The document explains how the bond price dynamics in the Vasicek short-rate model follow from the model’s closed-form bond price. That price is written as a time-dependent factor multiplied by an exponential involving the current short rate and the function B(t,T). Applying Itô’s lemma to this expression produces a stochastic return term whose volatility is proportional to the short-rate volatility and B(t,T). The factor appears because the bond price’s sensitivity to the short rate depends on time to maturity.

Under the risk-neutral measure, the bond’s expected instantaneous return must equal the short rate. Imposing that condition determines the drift and yields the stated price dynamics. The response also notes that Q is conventional notation for a pricing measure, while P often denotes the physical probability measure. The explanation sketches the derivation but omits intermediate algebra and does not address the separate modeling choices involved in specifying physical-measure dynamics or estimating parameters from market data.

Key ideas

The Vasicek bond price depends exponentially on the current short rate through B(t,T).
Applying Itô’s lemma to the bond price produces a diffusion term scaled by B(t,T).
Risk-neutral pricing requires a traded bond’s expected instantaneous return to match the short rate.
Q is common notation for a risk-neutral measure, while P often denotes the physical measure.

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Fixed income Derivatives pricing Statistics

Full text

# Bond-price dynamics in the Vasicek model


# Bond-price dynamics in the Vasicek model












Hello I am studying about interest rate modeling

There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable to replicate which is:

$dP(t,T) = r(t)P(t,t)dt - \sigma B(t,T)P(t,T)dW(t)$

This equation on 2nd page (or page 18th according to document paging). It locates about 1/3 page top down. Anyone understand how we get this one? What border me is why there is $B(t,T)$ appear. I tried but unable to obtain that result.

Besides, the side question is why in interest rate stochastics process it is always express under risk neutral $\mathbb{Q}$ why a traditional stock price S is often expressed in $\mathbb{P}$

Thank you so much

## Answer by Magic is in the chain (score 3)

https://quant.stackexchange.com/a/46477

You know the bond price formula takes this form:

$P \left( t, T \right)= A \left( t, T \right) e^{ -r_{t} B \left(t, T \right) }$

Now apply Ito's lemma, so you will get after some manipulation:

$\frac{dP}{P}= \left(\frac{1}{A} \frac {\partial A}{\partial t} -r \frac {\partial B}{\partial t} - \kappa \theta B + \kappa r B+ \frac{1}{2} B^2 {\sigma}^2\right) dt - \sigma B d w_{t}$

The expected return under the risk neutral measure must be r, so you can set the drift term equal to r, and you get your equation.

Don't think it is necessary to represent risk neutral measure by Q, but Q has become sort of conventions. Probably originated from the fact that people use P to represent probability, and then use the next letter Q, to denote the next measure one want to talk about.

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

Deriving Vasicek Bond Price Dynamics with Itô’s Lemma

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