Conditional Zero-Coupon Bond Prices in the Vasicek Model

Summary

The document raises a question about how to express a zero-coupon bond’s price at a future time when only earlier information is available. It gives the affine Vasicek pricing form, in which the bond price at a given time depends on the short rate at that time, and writes the mean-reverting short-rate solution from an earlier date. The core issue is distinguishing the price conditional on information available at the bond-pricing time from the random future price viewed from an earlier time.

The material is a question rather than a worked answer: it asks whether substituting the stochastic short-rate solution into the affine formula is valid, and requests a formal conditional-expectation justification and the bond’s stochastic differential equation. It therefore frames a useful modeling problem but does not resolve it or supply numerical evidence. Readers would need to derive the conditional distribution and account for the bond’s value dynamics under a specified measure before using the setup for valuation or risk calculations.

Key ideas

• In the Vasicek model, a zero-coupon bond price at time t is expressed as an affine function of the short rate at t.
• The short rate at a future time is random when viewed using information available at an earlier time.
• The document asks how conditional expectations relate the future bond price to earlier information.
• It does not provide the requested derivation or the bond price stochastic differential equation.
• A full treatment needs a specified probability measure and the corresponding rate dynamics.

Tags

# Bond prices at future times under Vasick one-factor model


# Bond prices at future times under Vasick one-factor model












In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time $t$ conditional on the information at this time is

$$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-B(t,T)r(t)} \quad\quad (1)$$ for which $r(t)$ is known since we have information $F_t$ and so this price $P(t,T)$ is deterministic at time $t$ (if I understand it correctly).

But what if we only have information $F_s$ with $s<t$, what is the price of this bond at time $t$ "as seen from time $s$" (which now should be a random variable rather than deterministic)? Do we "just replace" $r(t)$ above with the solution to Vasicek model, i.e. with $$r(t) = r(s)e^{-k(t-s)} + \theta(1-e^{-k(t-s)}) + \sigma\int^t_se^{-k(t-u)}dW(u) ?\quad\quad (2)$$

Is yes, then what is the formal justification for it (in terms of the pricing formula via conditional expectation)?

Putting this another way: what is the SDE for zero-coupon bonds in Vasicek model?

Add 1

I think the use of the same variable $t$ in (1) and (2) causes some confusion (I know it does for me) so let me re-write $(1)$ as $(1^*)$

$$P(s,T) = E[e^{-\int^T_s r(u)du}|F_s] = A(s,T)e^{-B(s,T)r(s)} \quad\quad (1^*)$$

which is the price of the bond at time $s$ given the information $F_s$ up to this time.

What is the price of the bond at time $t>s$ "as seen from" $s$ given the information $F_s$? How is it expressed in terms of the conditional expectation?

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