Solving the Vasicek model for reversion to the mean of interest rates.

Reminder: Ito Lemma: IfdX = a(X, t)dt + b(X, t)dW

Then

dg(X, t) =(

agx +1

2b2gxx + gt

)

dt + bgxdW .

The Vasicek model isdX = α(r − X)dt + sdW

Look at g(X, t) = eαt(X − r). From Ito:

dg = (α(r − X)eαt + αeαt(X − r))dt + seαtdW = seαtdW .

Integrating, we have

eαt(X(t) − r) − (X(0) − r) = s

∫

t

0eαudW (u)

= s

(

eαtW (t) − α

∫

t

0W (u)eαudu

)

.

To get the last line we have used Ito again, with “g” equal to eαtW (and X = W ).Rearranging gives

X(t) = r + (X(0) − r)e−αt + s

(

W (t) − α

∫

t

0W (u)e−α(t−u)du

)

.

Notes:

1. There is a problem in this model that X can become negative.

2. Everyone not in finance calls the process the “Ohrnstein-Uhlenbeck” process — it hasmany applications outside finance.

1