Exact Vasicek Short-Rate Simulation at a Future Horizon

Summary

The document explains how to simulate a Vasicek short rate at a future time without generating every intermediate value. Starting from the model’s mean-reverting stochastic differential equation, it applies an integrating factor and integrates over the full time horizon. Because the stochastic integral has a deterministic integrand, the terminal rate is normally distributed, with a mean that moves the initial rate toward the long-run level and a variance determined by the model’s volatility, mean-reversion speed, and horizon.

This gives a direct one-step simulation method for the rate at the chosen horizon. The discussion does not explain how to construct an entire yield curve from the simulated short rate, despite that being part of the original question. It also does not address parameter estimation, calibration, or whether the Vasicek model fits observed rates; the result depends on the stated model assumptions.

Key ideas

• The Vasicek model permits direct simulation of the short rate at a future horizon without generating intermediate path values.
• An integrating factor yields the terminal rate as a deterministic mean component plus a stochastic integral.
• The terminal rate is normally distributed because the stochastic integral has a deterministic integrand.
• The terminal mean reflects mean reversion toward the model’s long-run level.
• The answer does not describe how to derive a full yield curve from the simulated rate.

Tags

# Timesteps in Vasicek model


# Timesteps in Vasicek model












When simulating stocks one can easily use GBM with only one random variable per simulation to create a new stock price in say 5 years, you don't need to create the whole asset paths if you don't need that.

Now I wonder if that is also the case for the Vasicek model. Can I use the Vasicek short rate model with only one random variable per simulation to create a new short rate in 5 years (without constructing the whole path to the short rate in 5 years?).

If so, how do you go from the new simulated short rate to the whole new yield curve?

Thanks.

## Answer by user9403 (score 3)

https://quant.stackexchange.com/a/22895

Yes you can! Any SDE that has an analytic solution can be simulated exactly. The vasicek model has dynamics $dr=a(b-r)dt+\sigma dW_t$. By Ito's lemma, $$d\left(e^{at}r\right)=e^{at}\left(a(b-r)dt+\sigma dW_t\right) +a e^{at} r dt$$ Simplifying, $$d\left(e^{at}r\right)=e^{at} ab +e^{at}\sigma dW_t$$ Integrating, $$e^{aT} r_T=r_0+b(e^{aT}-1)+\sigma \int_0 ^ T e^{at} dW_t$$ Solving for $r_T$, $$r_T=r_0 e^{-aT} +b(1-e^{-aT})+\sigma \int_0 ^ T e^{-a(T-t)} dW_t $$ Since the Ito integrand is deterministic, the distribution of the Ito integral is normal with mean zero and variance $$\sigma^2\int_0 ^ T e^{-2a(T-t)} dt =\frac{\sigma^2}{2a}\left(1-e^{-2aT}\right) $$ The distribution of $r_T$ is thus normal with expectation $$r_0 e^{-aT} +b(1-e^{-aT})$$

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