Observations spaced at intervals Δt are taken on the short rate. The i th observation is ri (0 ⩽

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Observations spaced at intervals Δt are taken on the short...

Observations spaced at intervals $\Delta t$ are taken on the short rate. The $i$ th observation is $r_i$ $(0 \leqslant i \leqslant m)$. Show that the maximum-likelihood estimates of $a, b^*$, and $\sigma$ in Vasicek's model are given by maximizing $$ \sum_{i=1}^m\left(-\ln \left(\sigma^2 \Delta t\right)-\frac{\left[r_i-r_{i-1}-a\left(b^*-r_{i-1}\right) \Delta t\right]^2}{\sigma^2 \Delta t}\right) $$

Observations spaced at intervals $\Delta t$ are taken on the short rate. The $i$ th observation is $r_i$ $(0 \leqslant i \leqslant m)$. Show that the maximum-likelihood estimates of $a, b^*$, and $\sigma$ in Vasicek's model are given by maximizing
$$
\sum_{i=1}^m\left(-\ln \left(\sigma^2 \Delta t\right)-\frac{\left[r_i-r_{i-1}-a\left(b^*-r_{i-1}\right) \Delta t\right]^2}{\sigma^2 \Delta t}\right)
$$

Key Concepts

Transition Density Function

The transition density function describes the probability of transitioning from one state to the next in the discretized setting. For the Vasicek model, the increments are normally distributed with a mean that reflects the drift (mean-reverting behavior) and a variance that scales with the time interval and the diffusion coefficient. This function is central to setting up the likelihood used in MLE for parameter estimation.

Discretization of Stochastic Differential Equations

Since continuous-time models like the Vasicek model are observed at discrete intervals in practice, discretization methods (such as the Euler–Maruyama scheme) are used to approximate the continuous-time process. This results in a discrete-time model where the increments can be treated as conditionally normally distributed, facilitating the construction of a likelihood function based on observed data.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a model by maximizing the likelihood function, which measures the probability of the observed data given particular parameter values. In the context of the Vasicek model, MLE is applied to derive the estimates of the speed of mean reversion, the long-term mean, and the volatility by using the transition probabilities of the discretized process.

Vasicek Model

The Vasicek model is a continuous?time model for interest rate dynamics, described by a stochastic differential equation (SDE) that captures mean reversion—i.e., the tendency of the rate to move towards a long?term average level. This model is a classic application of the Ornstein–Uhlenbeck process and is fundamental for modeling short rates in finance.

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