Integral Calculus and Differential Equations › Integration Techniques for Finance

The following examples use several integration techniques. They are useful when a model lacks a convenient antiderivative. We begin with substitutions and products before turning to numerical integration.

Integration by substitution reverses the chain rule of differentiation, allowing us to simplify integrals by changing variables. The core insight is that a complex integrand might become simple under the right transformation. If $g(t)$ is a differentiable function and $f$ is continuous, then:

$$\int f(g(t)) \cdot g'(t) \, dt = \int f(u) \, du$$

where $u = g(t)$.

The power of substitution lies in recognizing when an integrand contains a function and its derivative: the signature of the chain rule applied in reverse. When we spot this pattern, we can simplify the integral by treating the inner function as a new variable.

Consider pricing a bond with continuous coupon payments where the coupon rate decays exponentially. This structure might arise when modeling a company whose cash flows are expected to decline over time. If the coupon rate at time $t$ is $C(t) = C_0 e^{-\alpha t}$ and the discount factor is $D(t) = e^{-rt}$, we need to compute the present value of this decaying stream of payments:

$$PV = \int_0^T C_0 e^{-\alpha t} \cdot e^{-rt} \, dt$$

where:

• $C_0$: initial coupon cash-flow rate at $t=0$, in currency per year
• $\alpha$: decay rate of the coupon (how fast it decreases)
• $r$: risk-free discount rate
• $T$: maturity time

The present value of coupons from $0$ to $T$ is:

$$PV = \int_0^T C_0 e^{-\alpha t} \cdot e^{-rt} \, dt = C_0 \int_0^T e^{-(r+\alpha)t} \, dt$$

Here we used $e^a \cdot e^b=e^{a+b}$ to combine the exponentials. When $r+\alpha\ne0$, substitution with $u=-(r+\alpha)t$ gives the first branch below. At $r+\alpha=0$, the integrand is the constant $C_0$:

$$PV=\begin{cases}\dfrac{C_0}{r+\alpha}\left(1-e^{-(r+\alpha)T}\right),&r+\alpha\ne0,\\[6pt]C_0T,&r+\alpha=0.\end{cases}$$

For the usual nonnegative discount and decay rates, $r+\alpha$ combines the two forces that reduce future coupon value. For $C_0>0$, the finite perpetuity limit $C_0/(r+\alpha)$ exists only when $r+\alpha>0$. At equality the value grows linearly as $C_0T$; when $r+\alpha<0$, it grows exponentially rather than converging. A zero stream has value zero for every rate.

Integration by parts reverses the product rule, enabling us to integrate products of functions by transferring the derivative from one factor to another. This technique is especially valuable when one function becomes simpler upon differentiation while the other has a known antiderivative. For continuously differentiable functions $u(t)$ and $v(t)$:

$$\int u \, dv = uv - \int v \, du$$

where:

• $u$: a function chosen to simplify when differentiated
• $v$: an antiderivative of the factor represented by $dv$, so $dv=v'(t)\,dt$
• $dv$: the differential of $v$
• $du$: the differential of $u$

The strategy is to choose $u$ and $dv$ so that $\int v \, du$ is simpler than the original integral. A useful mnemonic for choosing $u$ is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), which suggests the order of preference for what to differentiate.

This technique is important when integrating products, especially in option pricing where payoffs multiply probability densities. When computing expected values, we often encounter integrals of the form $\int x \cdot f(x) \, dx$, where $f(x)$ is some probability density.

For an option-pricing application, let $F$ and $p$ be the CDF and density of a nonnegative terminal price $S_T$. Provided the expectation is finite and $(s-K)[1-F(s)]\to0$ as $s\to\infty$, integration by parts turns a call's expected payoff into a tail-probability integral. Choose

$$u=s-K,\qquad dv=p(s)\,ds,\qquad du=ds,\qquad v=F(s)-1.$$

Then the boundary term is explicit:

$$\begin{aligned} E[(S_T-K)^+] &=\int_K^\infty (s-K)p(s)\,ds\\ &=\left[(s-K)(F(s)-1)\right]_K^\infty +\int_K^\infty [1-F(s)]\,ds\\ &=\int_K^\infty [1-F(s)]\,ds. \end{aligned}$$

Consider computing the expected payoff of a call option under a uniform distribution (a simplified model for illustration). If the stock price $S_T$ at expiration is uniformly distributed on $[80, 120]$ with strike $K = 100$, the expected payoff is:

$$E[\max(S_T - K, 0)] = \int_{100}^{120} (S_T - 100) \cdot \frac{1}{40} \, dS_T$$

Direct integration gives the same value as integrating the uniform survival function $(120-s)/40$ from 100 to 120:

$$= \frac{1}{40} \left[ \frac{(S_T - 100)^2}{2} \right]_{100}^{120} = \frac{1}{40} \cdot \frac{400}{2} = 5$$

The Gaussian integral is fundamental to quantitative finance because normal distributions appear throughout risk modeling and option pricing. Some mean-variance analyses assume normal arithmetic returns; the mean-variance criterion itself uses means and covariances rather than requiring normality. Under Black-Scholes, continuously compounded (log) returns are normal and terminal prices are lognormal; other option models need not use normal returns. In each normal-model setting, we encounter integrals involving the exponential of a quadratic. The basic Gaussian integral states:

$$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$$

For this particular unnormalized Gaussian, the area is $\sqrt{\pi}$; changing its height or width changes that area. One derivation squares the integral and integrates $e^{-(x^2+y^2)}$ over the plane. Polar coordinates give $2\pi\int_0^\infty \rho e^{-\rho^2}\,d\rho=\pi$, so the positive original integral is $\sqrt{\pi}$.

More generally, for the standard normal density:

$$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, dx = 1$$

This integral equals 1 because it represents the total probability under the standard normal density. A probability density, when one exists, integrates to 1. Discrete probability masses are normalized by summation; more generally, a probability measure assigns total mass 1 to its sample space. The factor $1/\sqrt{2\pi}$ is precisely the normalization constant that ensures this property.

The cumulative distribution function of the standard normal:

$$\Phi(z) = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \, dt$$

where:

• $\Phi(z)$: cumulative distribution function (CDF) of the standard normal distribution
• $z$: upper limit of integration, representing the number of standard deviations
• $t$: dummy variable of integration
• $\frac{1}{\sqrt{2\pi}} e^{-t^2/2}$: the standard normal probability density function

has no expression in elementary functions. It can be written with the special function $\operatorname{erf}$ as $\Phi(z)=\tfrac12[1+\operatorname{erf}(z/\sqrt2)]$ and is evaluated numerically in practice. It appears directly in the Black-Scholes formula for option pricing, where call and put prices are expressed in terms of $\Phi(d_1)$ and $\Phi(d_2)$.