Deriving Vasicek Zero-Coupon Bond Prices from Short-Rate Dynamics

Summary

The document asks how to solve the bond pricing equation under a Vasicek short-rate model, using an exponential-affine price form for a zero-coupon bond. It proposes substituting that form into the pricing equation to derive differential equations for its time-dependent coefficients, then using them to obtain the bond price. The stated inputs include a spot rate, process volatility, and a repeated parameter value, though the parameter notation is ambiguous.

The answer sketches an alternative derivation from the short-rate dynamics: solve the linear stochastic differential equation with an integrating factor, integrate the short rate over the bond’s life, and rearrange the stochastic integral. It then uses the normal distribution of that integral and the expectation of an exponential normal variable to price the bond. The reply outlines the calculation rather than carrying it through, so it does not give the coefficient solutions or a numerical bond price; the original question’s parameter inconsistency also limits direct application.

Key ideas

• An exponential-affine price assumption can reduce the bond pricing equation to equations for time-dependent coefficients.
• The Vasicek short-rate process can be solved using an integrating factor.
• A zero-coupon bond price can be expressed as the expectation of the exponential of the integrated short rate.
• The stochastic integral can be characterized using its normal distribution and variance.
• The response gives a derivation outline rather than a completed symbolic or numerical solution.

Tags

# Deriving solution to bond pricing equation


# Deriving solution to bond pricing equation












Consider the Vasicek model for the spot model:

$$ 𝑑𝑟 = (𝛼 − 𝛾𝑟)𝑑𝑡 + √𝛽𝑑𝑊 $$

Suppose $𝛾 = 0.1, 𝛾 = 0.1$, and the volatility of the process is 0.02. The spot rate is 10%.

Assume the form of solution to the BPE (Bond Pricing Equation) is $Z(t, T; r) = e^{A(t, T) - rB(t,T)}$ and derive equations for $A$ and $B$. Solve the equation and obtain the form of $Z(t, T; r)$, hence price the zero coupon bond.

How do I go about this problem? Do I need to substitute Z into the BPE and get the partial derivatives from there?

## Answer by achirikhin (score 1)

https://quant.stackexchange.com/a/79277

It is very easy to do it by direct calculation; it is a bit easier for the dynamics

$dr = a(b-r)dt + \sigma dW_t$

- Integrate it using the integrating multiple $e^{at}$ and moving $-are^{at}dt$ to the left side

- Compute $\int_0^T {r dt}$, for which you will need to change order of integration, to end up with $\int_0^T{ ... dW_t}$. By Ito isometry, this will be a normal variable with known mean and variance.

- Take expectation of the exponential of that normal variable

Everyone has to do this calculation once :) Good luck!

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