# Hull White Analytic Closed-Form Solution¶

## Overview¶

In financial mathematics, the Hull-White model is a model of future interest rates and is an extension the Vasicek model.

Its an no-arbitrage model that is able to fit todays term structure of interest rates.

It assumes that the short-term rate is normally distributed and subject to mean reversion.

The stochastic differential equation describing Hull-White is:

$\delta{r} = [\theta(t) - ar]\delta{t} + \sigma\delta{z}$

These input parameters are:

$$\delta r$$ - is the change in the short-term interest rate over a small interval

$$\theta (t)$$ - is a function of time determining the average direction in which r moves (derived from yield curve)

$$a$$ - the mean reversion

$$r$$ - the short-term interest rate

$$\delta t$$ - a small change in time

$$\sigma$$ - the volatility

$$\delta z$$ - is a Wiener (Random) process