Wednesday, June 15, 2011

Solving the Modified OU PDE and Calculate Mean Reversion rate

Calculating theta from the hazard rate curve:
We believe that the hazard rate follows the Ornstein-Uhlenbeck process
dh=α(θ-h)dt+σhdW
iven an initial term structure of hazard rate, we can find the value for θ by first calculating the closed form solution for the above pde and then eliminating the stochastic term from the solution.
We follow the integrating factor approch for finding the close form solution for the pde.
If we have a PDE of the form dX=F(t,X)dx+σXdW ,Initial Condition=X0
We can find the Integrating factor as
IF=exp(-∫sigma^tdw〖σdw+1/2∫sigma^tdw〖σ^2 dw〗〗)
Hence d(IF*X)=IF*F(t,X)dt
d(IF*X)/dt=IF*F(t,X)

h_t=h_0*exp⁡(-kt+σW_t )+α□(θ/k)*(1+exp(-k*t)
differentiating ht wrt time

∂h/∂t=h_0*exp⁡(-kt+σW_t )*(-k)+ α□(θ/k)*(exp(-k*t)*(-k)
∂h/∂t=(-k)(h_t- α□(θ/k))

θ=(∂h/∂t+k*h_t )/α
As we see our theta is dependent over the rate of change of h w.r.t to time and on the level of h as well. So our next task is to generate the different values ofθt, corresponding to the values of alpha and sigma.