1 The Mean Field of the Sun Leif Svalgaard Stanford University Sept. 2, 2011.
Solving the Vasicek model for reversion to the mean of...
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Solving the Vasicek model for reversion to the mean of interest rates. Reminder: Ito Lemma: If dX = a(X, t)dt + b(X, t)dW Then dg (X, t)= ag x + 1 2 b 2 g xx + g t dt + bg x dW . The Vasicek model is dX = α(r - X )dt + sdW Look at g (X, t)= e αt (X - r). From Ito: dg =(α(r - X )e αt + αe αt (X - r))dt + se αt dW = se αt dW . Integrating, we have e αt (X (t) - r) - (X (0) - r) = s t 0 e αu dW (u) = s e αt W (t) - α t 0 W (u)e αu du . To get the last line we have used Ito again, with “g ” equal to e αt W (and X = W ). Rearranging gives X (t)= r +(X (0) - r)e -αt + s W (t) - α t 0 W (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 has many applications outside finance. 1
Transcript of Solving the Vasicek model for reversion to the mean of...
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.
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