Vasicek Zero-Coupon Bond Pricing Under the Physical Measure

Summary

The document asks whether the Vasicek short-rate model can produce a zero-coupon bond price under the physical, or real-world, probability measure. It gives the familiar affine bond-price form under the risk-neutral measure, including its coefficient formulas, then specifies a physical-measure short-rate process with mean reversion, constant volatility, and a market price of risk term.

The question highlights the distinction between modeling rate dynamics under the physical measure and valuing bonds for arbitrage-free pricing. It does not provide a derivation, answer, or evidence that the proposed physical-measure process directly yields a market price. It also leaves the risk premium’s form and assumptions unexplored. The material is therefore useful as a setup for studying measure changes and bond valuation, but not as a complete pricing method; a pricing application would need to clarify the relationship between the physical and risk-neutral dynamics.

Key ideas

• The document contrasts Vasicek short-rate dynamics under physical and risk-neutral measures.
• It states an affine zero-coupon bond price form for the risk-neutral model.
• The physical-measure process includes a risk premium term in its drift.
• The document poses the pricing question but does not derive or establish a physical-measure valuation formula.
• Market valuation requires clarity about the measure used and the assumptions linking the two dynamics.

Tags

# Zero coupon price using Vasiceks model under the Real-world P measure model


# Zero coupon price using Vasiceks model under the Real-world P measure model












I'm wondering if there is a way to work out the formula for the price of the zero-coupon bond using the Vasicek's model (P measure). I have tried to find reference on it but could not, I don't know if it is possible.

I know that under the Q measure, the zero-coupon bond price would be

$P(t,T)= A(t,T)e^{r(t)B(t,T)}$

where

$A(t,T)=exp\{(b - \frac{\sigma^2}{2a^2})(B(t, T)-T+t)-\frac{\sigma^2}{4a}B^2(t,T)\}$

$B(t,T)=\frac{1-e^{-a(T-t)}}{a}$

I have a P-measure model $dr_t=(a(b-r_t)-\lambda\sigma)dt+\sigma dW^\mathbb{P}_t$, where the $\lambda$ is the risk premium. I don't know if it is possible to do it or that I'm just heading the wrong way and should instead work with the Q-measure ZCB formula.

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