ARCH Variance in a Discrete Vasicek Rate Model

Summary

The document poses a modeling question about combining mean reversion in interest rates with ARCH-style conditional variance. It starts from the continuous-time Vasicek process, whose solution can be used to derive the rate’s expectation and variance, then contrasts it with discrete equations in which the rate moves toward a long-run level and innovations have a variance updated from a lagged squared shock.

The author asks whether this discrete specification can be expressed in continuous time and solved in closed form. No answer or derivation is provided, so the document does not establish a solution or particular moment formulas. Its useful content is the distinction between a familiar continuous-time mean-reverting diffusion and a discrete conditional-heteroskedastic model, and the recognition that ARCH variance dynamics complicate direct use of the standard Vasicek solution. Any application would need a precise time-step convention and assumptions about the shock distribution and parameter constraints.

Key ideas

• The Vasicek model describes a rate that reverts toward a long-run level while receiving stochastic shocks.
• The discrete model makes innovation variance depend on a lagged squared shock.
• The document asks whether the combined specification has a continuous-time representation and closed-form moments.
• No solution is supplied, so the expected rate and variance remain unresolved in the source.

Tags

# ARCH-Vasicek model solution


# ARCH-Vasicek model solution












I understand how we can obtain the solution of Vasicek model $dr_t=\alpha(\mu-r_t)dt+\sigma dW_t$: $$ r_t=r_0e^{-\alpha t}+\mu(1-e^{-\alpha t})+\sigma\int_0^te^{-\alpha(t-s)dW_{s}} $$ This easily allows me to find $\mathbb{E}[r_t]$ and $Var[r_t]$.

However, I am stuck with following ARCH-Vasicek discrete equations. What I'm trying to do is write the model in continuous form and find its closed-form solution to get the expectation and variance of $r_t$.

$$ \Delta r_t=\alpha(\mu-r_{t-1})dt+\epsilon_t \\ \epsilon_t = \sqrt{\sigma^2_t}u_t, u_t \to iid(0,1)\\ \sigma^2_t=\omega+b_1\epsilon^2_{t-1} $$ I'm not very good at stochastic calculus and I suspect there may not be a closed-from solution. Anyway, any help would be appreciated.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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