# Binomial Distribution (binomial model inspired by Merton)

I want to do the following and I hope that anyone of you can help me out.

In short, I have a portfolio consisting of 10 companies, whereas their asset values are driven by a common economic background factor u and the assets in question are correlated. Now I want to calculate the probability that e.g. 3 of 10 companies default simultaneously.

In order to do so, I choose a mixed binomial Merton model. whereas the probability that a given number of companies default is given by:

$$\small \mathbb P[N_m \le n]=\sum_{k=0}^n \binom{m}{k}\int_{-\infty}^\infty N\left(\frac{-(C+\sqrt\rho u)}{\sqrt{1-\rho}}\right)^k \left(1-N\left(\frac{-(C+\sqrt\rho u)}{\sqrt{1-\rho}}\right)\right)^{m-k} \frac1{\sqrt{2\pi}}e^{-\frac{u^2}{2}}\,\mathrm du$$

whereas u is the common economic background factor and a standard normal random variable.

$C_i$ is given by:

$$C_i=\frac{\ln(V_{0,i}/D_i)+\left(r-\frac12 \sigma_i^2\right)T}{\sigma_i \sqrt T}$$

whereas I substituted r with μ

Then I entered the following definitions:

V1 = 100; D1 = 70; μ1 = 0.2; σ1 = 0.1;
V2 = 110; D2 = 75; μ2 = 0.25; σ2 = 0.2;
V3 = 95; D3 = 72; μ3 = 0.1; σ3 = 0.2;
V4 = 120; D4 = 100; μ4 = 0.4; σ4 = 0.3;
V5 = 120; D5 = 110; μ5 = 0.2; σ5 = 0.2;
V6 = 100; D6 = 65; μ6 = 0.25; σ6 = 0.12;
V7 = 99; D7 = 85; μ7 = 0.3; σ7 = 0.05;
V8 = 125; D8 = 76; μ8 = 0.2; σ8 = 0.3;
V9 = 105; D9 = 100; μ9 = 0.23; σ9 = 0.02;
V10 = 100; D10 = 75; μ10 = 0.2 ; σ10 = 0.2;

T = 5; ρ = 0.5; \[ScriptU] = 0.25;

Evaluating Subscript[C, i]

C1 = (Log[V1/D1] + (μ1 - 0.5*σ1^2)*T)/(σ1*Sqrt[T]);

C2 = (Log[V2/D2] + (μ2 - 0.5*σ2^2)*T)/(σ2*Sqrt[T]);

C3 = (Log[V3/D3] + (μ3 - 0.5*σ3^2)*T)/(σ3*Sqrt[T]);

C4 = (Log[V4/D4] + (μ4 - 0.5*σ4^2)*T)/(σ4*Sqrt[T]);

C5 = (Log[V5/D5] + (μ5 - 0.5*σ5^2)*T)/(σ5*Sqrt[T]);

C6 = (Log[V6/D6] + (μ6 - 0.5*σ6^2)*T)/(σ6*Sqrt[T]);

C7 = (Log[V7/D7] + (μ7 - 0.5*σ7^2)*T)/(σ7*Sqrt[T]);

C8 = (Log[V8/D8] + (μ8 - 0.5*σ8^2)*T)/(σ8*Sqrt[T]);

C9 = (Log[V9/D9] + (μ9 - 0.5*σ9^2)*T)/(σ9*Sqrt[T]);

C10 = (Log[V10/D10] + (μ10 - 0.5*σ10^2)*T)/(σ10*Sqrt[T]);


Now I am not sure how to calculate the individual probabilities and add them up.

Would you suggest first to integrate the terms on the right hand side of the integral, or is it possible to enter the whole function in Mathematica considering both the normal distribution and the binomial distribution?

I would be very thankful for any advice.

Thanks.

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Stocks defaulting?? I know the probability of bond defaults can be relatively high... but stocks?? I mean, you don't see a company bankruptcy everyday in the market... –  Rod Jul 11 '13 at 13:57
@ Rod Lm Sorry, let me substitute Stocks by assets so V represents the asset value of a company and D1 etc. the debt level. I will change this in the text above. thenks for reminding me :-) –  Milan Ivica Jul 11 '13 at 14:03
I'm not quite following the equations. C is subscripted by i but there seems no place for that in the main equation. Also [ScriptU] is set to a constant but is also used as the variable of integration. Am I missing something? –  Jim Baldwin May 9 at 17:00