Financial Mathematics

Appendix: Some Useful Integral Identities and Symmetry Properties of Normal Random Variables

Throughout all the formulas below, a, b, c, A, B, C are any real constants and X is a normal random variable with mean µ ∊ ℝ and standard deviation σ > 0, i.e., X ~ Norm(µ, σ2) with PDF ${\varphi }_{\mu ,\sigma }\left(x\right)\equiv \frac{{\text{e}}^{-{\left(x-\mu \right)}^2/2{\sigma }^2}}{\sigma \sqrt{2\pi }},-\infty <x<\infty$. The functions $\mathcal{N}$(x) and $\mathcal{N}$2(x, y; ρ) are the standard normal univariate and bivariate cumulative distribution functions, respectively.

$\text{E}\left[{\text{e}}^{BX}{\mathbb{I}}_{\left\{X>A\right\}}\right]\equiv {\displaystyle {\int }_A^{\infty }{\text{e}}^{Bx}{\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x={\text{e}}^{\mu B+\frac{1}{2}{\sigma }^2{B}^2}\mathcal{N}}\left(\sigma B+\frac{\mu -A}{\sigma }\right)\text{ (A.1)}$

$\text{E}\left[{\text{e}}^{BX}{\mathbb{I}}_{\left\{X<A\right\}}\right]\equiv {\displaystyle {\int }_{-\infty }^A{\text{e}}^{Bx}{\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x={\text{e}}^{\mu B+\frac{1}{2}{\sigma }^2{B}^2}\mathcal{N}}\left(-\sigma B+\frac{A-\mu }{\sigma }\right)\text{ (A.2)}$

$\text{E}\left[\mathcal{N}\left(AX+C\right)\right]\equiv {\displaystyle {\int }_{-\infty }^{\infty }\mathcal{N}\left(Ax+C\right){\varphi }_{\mu ,\sigma }\left(x\right)\text{d}x=\mathcal{N}}\left(\frac{\mu A+C}{\sqrt{1+{\sigma }^2{A}^2}}\right)\text{ (A.3)}$

$\begin{array}{c}\text{E}\left[{\text{e}}^{BX}\mathcal{N}\left(AX+C\right)\right]\equiv {\displaystyle {\int }_{-\infty }^{\infty }{\text{e}}^{Bx}\mathcal{N}\left(Ax+C\right){\varphi }_{\mu ,\sigma }\left(x\right)}\text{d}x\\ ={\text{e}}^{\mu B+\frac{1}{2}{\sigma }^2{B}^2}\mathcal{N}\left(\frac{\mu A+C+{\sigma }^{2}AB}{\sqrt{1+{\sigma }^2{A}^2}}\right)\end{array}\text{ (A.A)}$

${\displaystyle {\int }_0^{\infty }\mathcal{N}\left(Ax+B\right){\text{e}}^x\text{d}x=-\mathcal{N}\left(B\right)+{\text{e}}^{\left(1-2AB\right)/2A^2}\mathcal{N}\left(\frac{1-AB}{|A|}\right)}(\text{for}$