This integral has arisen in a pricing formula for an exotic financial option. I have scoured books on integrals, series, and products that might help (e.g., Gradshteyn and Ryzhik; volumes 1-2 of Prudnikov, Brychkov, and Marichev; Ramanujan's integrals), but with limited success. As follows are approaches I have considered, roughly in decreasing order of thoroughness:
- Express $\exp[-Ae^{x}]$ by its Taylor series, swap integration/summation, and evaluate the resulting summand integrals: This leads to a series of normal CDFs with coefficients of the form $\frac{(-A)^n}{n!}e^{Dn^2-Kn}$, $n$ being the index of summation. I've been unable to find formulas simplifying this series, e.g., by expressing the normal CDFs in terms of Kummer $U(a,c,z)$ functions. The only relevant series I located (from NIST) do not sum over $z$, which is of interest here (namely, as the square of the corresponding normal CDF argument).
Alternatively, it would be nice to at least be able to evaluate the following series: $$ \sum\limits_{n=0}^\infty\frac{(-A)^n}{n!}e^{Dn^2-Kn} $$ In this case, I could at least show that the original integral is a countable mixture of normal CDFs (the integral plays the same role as the normal CDF in the Black-Scholes pricing formula).
[EDIT February 14, 2018] Clarification
Explicitly, the first approach solves for the integral as a constant multiple of the following series (see the added parameter specification below for the definition of $\tau$): $$ \sum\limits_{n=0}^\infty\frac{(-A)^n}{n!}e^{Dn^2-Kn}\Phi\left[\frac{2}{\sqrt\tau}(X+K-2Dn)\right] $$ Letting $p(n):=Dn^2-Kn$, then this can be written as follows: $$ \sum\limits_{n=0}^\infty\frac{(-A)^n}{n!}e^{p(n)}\Phi\left[\frac{2}{\sqrt\tau}(X-p'(n))\right] $$ It isn't immediately clear to me whether this accords any simplification.
Use $\exp[-Ae^{x}]=\lim\limits_{m\to\infty}(1-\frac{Ae^{x}}m)^m$, swap this limit and integration, evaluate the integral and then the limit: The only apparent evaluation of this integral is via the Binomial Theorem, which leads to a similar series (of normal CDFs) as above.
- Observe that the integral is the moment generating function (MGF) of a truncated log-normal random variable. Tellambura and Senaratne 1 have had some success in evaluating the non-truncated MGF, and Fact 21.73 of Söderlind 2 indicates how raw moments of truncated log-normal variables may be computed in terms of those of non-truncated variables. However, the only way to merge these two approaches appears to be as above (replacing $\exp[-Ae^{x}]$ by its Taylor series), which results in a similar series of normal CDFs.
- Observe that the integral is the expected value of a truncated double log-normal/'log-log-normal' random variable. The non-truncated distribution has been studied by, for example, Holland and Ahsanullah 3, without much analytical progress.
- Define $I(A,B):=\int\limits_{-\infty}^X\exp[-(Ae^x+Bx+Cx^2)]\mathrm dx$; then $$ I_A(A,B)=-I(A,B-1) $$ Here the subscript denotes partial differentiation. Impose the following boundary and transversality conditions: $$ I(0,B)=\sqrt{\frac\pi C}e^{\frac{B^2}{4C}}\Phi\left(\frac{B+2CX}{\sqrt{2C}}\right)\text{ and }\lim\limits_{B\to-\infty}I(A,B)=\infty\mathbf{1}_{X>0} $$ Here $\mathbf1$ denotes an indicator function, and $\Phi$ the standard normal CDF. Evaluation of the integral is thus equivalent to solving a linear partial differential delay equation. I am not very familiar with such equations.
At any rate, other than theoretical/intellectual curiosity and the potential to contribute meaningfully to an unsolved and non-trivial integral (as far as I can tell), my motivation for working it out, despite its relatively nice existing form, is to continue with further work on my pricing formula; e.g., computing hedging parameters and comparing its accuracy/efficiency to existing approaches. I'd prefer not to find an appropriate simplification later, and have to redo all of this work. Plus, the above attempts have exposed alternate ways to efficiently compute the integral numerically, if need be. Thanks for reading; I'd appreciate any insights, suggestions, and advice.
[EDIT February 14, 2018] Parameter specification:
The above parameters (A, B, C, D, K, and X) are given in terms of the following underlying parameters:
- (Risk-free interest rate) $r\ge0$
- (Option strike price) $k\ge0$
- (Option maturity time) $T\ge0$
- (Time remaining to option expiry) $\tau\in[0,T]$
- (Underlying asset price at time $T-\tau$) $S\ge0$
- (Underlying asset price transformed power mean at time $T-\tau$) $M:=\int\limits_0^{T-\tau}S^\alpha\mathrm{d}t$
- (Underlying asset price volatility) $\sigma\ge0$
- (Option indexing parameter) $\alpha>0$
- (CDF indexing parameter) $b\in\{0,1\}$
- $\zeta\equiv\frac{\sqrt 2}{\alpha\sigma}\left[\ln\frac{\alpha^2\sigma^2M^2}{S^\alpha}-\alpha\left(r-\frac{\sigma^2}2\right)\tau\right]$
In terms of these, compute the following:
- $A\equiv\frac{2\sqrt{{S^\alpha}}}{\alpha\sigma}\ge0$
- $B\equiv1+\frac{2r}{\alpha\sigma^2}-\frac b\alpha+\frac{\sqrt2\zeta}{\alpha\sigma\tau}$
- $C\equiv\frac2{\alpha^2\sigma^2\tau}>0$
- $D\equiv\frac{\alpha^2\sigma^2\tau}8\ge0$
- $K\equiv\frac\alpha4\left(\sqrt2\zeta\sigma+\tau[(\alpha-b)\sigma^2+2r]\right)$
- $X\equiv\ln{\frac{S^{\frac\alpha2}}{\alpha\sigma Tk^\alpha}}$