Deriving the Vasicek Zero-Coupon Bond Pricing PDE

Summary

The document asks how to derive a pricing equation for a zero-coupon bond when the short rate follows a Vasicek process. It defines the bond value as the conditional expected discount factor over the remaining term and gives the short-rate stochastic differential equation, with mean reversion toward a constant level and constant volatility. The proposed route is the Feynman–Kac formula, which links this expectation to a partial differential equation.

Key ideas

• A Vasicek short rate is modeled with mean reversion and constant diffusion volatility.
• A zero-coupon bond price can be expressed as the expected discount factor to maturity.
• Feynman–Kac provides a way to derive a PDE for the bond value as a function of time and the current short rate.
• The document poses the derivation problem but does not provide its solution.

Tags

# Term structure equation in the Vasicek model


# Term structure equation in the Vasicek model












Consider the SDE $$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$ Let $$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r).$$ (F can be interpreted as price of a zero coupon bond with maturity T.)

Use the Feynman-Kac formula to derive a PDE for the function $F(t; r)$.

I wanted to use Ito to obtain formula for $r_t$ and then plug it into Feynman-Kac formula $$E_{t0,x}=(\exp(-\int_{t}^{T}r(s,X_s)ds)\phi(X_t))$$ from the lecture, but I can't derive it. Any help is greatly appreciated

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