User:Mgalashin/Dothan model

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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

${\displaystyle dr_{t}=\sigma r_{t}\,dW_{t}}$

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

${\displaystyle dr_{t}=\alpha r_{t}\,dt+\sigma r_{t}\,dW_{t}}$

where

• ${\displaystyle r_{t}}$ is the instanteneous interest rate process
• ${\displaystyle \alpha }$ is a drift term, the direction of long term change in interest rates

• ${\displaystyle \sigma }$ is instantaneous volatility of the interest rate, measure of amplitude of randomness implied by the model
• ${\displaystyle W_{t}}$ is a Wiener process under risk-neutral measure

The process is geometric Brownian motion.

Discussion

Expression, mean, variance

The analitical expression is obtained solving the SDE:

${\displaystyle r_{t}=r_{0}e^{\alpha t-{\frac {\sigma ^{2}}{2}}+\sigma W_{t}}}$

Taking mean:

${\displaystyle E[r_{t}]=r_{0}e^{\alpha t}}$

and varinance:

${\displaystyle Var[r_{t}]={r_{0}}^{2}e^{2\alpha t}(e^{\sigma ^{2}t}-1)}$

Hence long term mean is either convergent to zero when Failed to parse (syntax error): {\displaystyle a \=< 0} or divergent when ${\displaystyle a>0}$