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How to solve the stochastic differential equation of the Vasicek model for the analysis of credit risk? I search in the article "The Distribution of loan portfolio value" (Vasicek) but he doesn't solve the equation.

$$dA_i=\mu_i A_i dt+\sigma_i A_i dx_i$$

The solution is:

$$log A_i(T)=log A_i+\mu_i T-\frac{1}{2}\sigma_i^2T+\sigma_i\sqrt{T}X_i$$

The probability of default of the i-th loan is then

$$p_i=P[A_i(T)<B_i]=P[X_i<c_i]=N(c_i)$$

where

$$c_i=\frac{log B_i-log A_i-\mu_i T+\frac{1}{2}\sigma_i^2T}{\sigma_i\sqrt{T}}$$

and N is the cumulative normal distribution function.

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  • $\begingroup$ The first equation is just the geometric brownian, and you can find its solution here: math.stackexchange.com/questions/873704/… $\endgroup$ – Magic is in the chain Apr 21 at 0:17
  • $\begingroup$ And to compute the probability, you will need to substitute the solution you have for A_i, and then isolate X, on the left hand side. $\endgroup$ – Magic is in the chain Apr 21 at 0:21
  • $\begingroup$ @Magicisinthechain I went to the link you sent and tried to solve the differential equation from the beginning. T is the maturation time of the credit. The problem is that if we pass everything to the logarithm, in the formula that I wrote in the question appears $\sqrt{T}$ in $\sigma_i\sqrt{T}X_i$ and in the post that you sent me does not appear that term. Do you know how to get that missing term? $\endgroup$ – Carmen González Apr 21 at 2:15
  • $\begingroup$ It will be Brownian motion in the solution. As BM has mean zero and variance T (standard deviation equal to square root of T), you can write it as sqrt T times standard normal for simulation purpose. $\endgroup$ – Magic is in the chain Apr 21 at 14:19

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