# How to calculate the integral of Normal CDF and Normal PDF?

I'm trying to find $$\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$$ where

$$A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^2}},\ t=b-c$$.

This integral arrives from trying the find $$E[\max\{X,Y,a\}]$$, where $$X\sim N(b,B^2)$$, $$Y\sim N(c,\frac{1}{A^2})$$, independent. I've tried differentiating w.r.t. $$A$$ but after solving the integral, I can't integrate back w.r.t. $$A$$. This is trying to follow the strategy from How can I calculate $$\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$$.

Is there a closed form solution for this?

I realise that it is similar to rainbow options where they calculate $$\max\{X,Y,k\}$$ but $$X$$ and $$Y$$ follow lognormal distribution. The closed formed solution for that involves the bivariate normal distribution.

You have $$tA + ABx$$, while the answer you linked is differentiating with $$\frac{x-\mu}{\sigma}$$.

The answer you linked provides the solution (when integratingn over $$-\infty, \infty$$)

$$\int_{-\infty}^\infty \Phi \left( \frac{x-\mu}{\sigma} \right) \phi (x) dx = \Phi \left( \frac{-\mu}{\sqrt{1+\sigma^2}} \right)$$

It seems to me you can just transform your integral into that form and apply the answer they provided.

Write

$$tA + ABx = ABx - (-tA) = \frac{x - (-tA)(AB)^{-1}}{(AB)^{-1}} = \frac{x-\mu}{\sigma}$$

Where

$$\mu = (-tA)(AB)^{-1}$$

$$\sigma = (AB)^{-1} = \frac{1}{AB}$$

Then you now have an integral of the form used in that question. So following the logic of that answer, the integral should be

$$\Phi \left( \frac{tA(AB)^{-1}}{\sqrt{1+(AB)^{-2}}} \right) = \Phi \left( \frac{tA}{\sqrt{(AB)^2+1}} \right)$$

Now, in your example you have a lower bound that is not $$-\infty$$. So, using the transformation above, you should be able to replicate the steps taken in that answer (or similar, as their are multiple on this site) with your lower bound, provided a closed form solution exists.

Perhaps you can run some basic simulations in R or Python to verify this is correct.

• Hey! Thanks for the reply! I have a few questions regarding your solution. Are you allowed to differentiate with respect to $\mu$ if $\mu$ is a function of $\sigma$? I'm not sure if that is possible or correct. Since you are treating $\sigma$ as a constant. Also, the solution provided seems to be integrating from $-\infty$ to $\infty$ which is not as the same as mine. – icecream_sandwich07 Oct 13 '18 at 10:24
• The integral the OP is asking about has a different lower limit than the integral in the cited answer whose method is being applied here. – Dilip Sarwate Oct 13 '18 at 11:44
• Good catch @DilipSarwate the linked answer had excluded the bounds so it confused me – Xiaomi Oct 13 '18 at 11:47