User:Mgalashin/Dothan model

The Dothan model is a model of mathematical finance, discribing the dynamics of short interest rates. It belongs to one factor model class, i.e. assumes interest rates to be dependent on previous period rate and random term only. The major merit of the model is it is the only lognormal model with explicit formula for discount bond price. The model was introduced by L. Uri Dothan in 1978.

Details

Initially, L.U. Dothan considered a model following stochastic differential equation



$$ dr_t =\sigma r_t \, dW_t$$

However, there is little difference in techniques if we assume a non-zero drift term:


 * $$ dr_t = \alpha r_t \, dt + \sigma r_t \,  dW_t$$

where The process is geometric Brownian motion.
 * $$ r_t $$ is the instanteneous interest rate process
 * $$ \alpha $$ is a drift term, the direction of long term change in interest rates
 * $$ \sigma $$ is instantaneous volatility of the interest rate, measure of amplitude of randomness implied by the model
 * $$ W_t $$ is a Wiener process under risk-neutral measure

Discussion

Expression, mean, variance

The analitical expression is obtained solving the SDE:
 * $$ r_t = r_0 e^{\alpha t - \frac {\sigma^2}{2} + \sigma W_t}$$

Taking mean:
 * $$ E[r_t] = r_0 e^{\alpha t}$$

and varinance:
 * $$ Var[r_t] = {r_0}^2 e^{2\alpha t} (e^{\sigma^2t} - 1)$$

Hence long term mean is either convergent to zero when $$a \=< 0$$ or divergent when $$a > 0$$