Integration is the inverse operation of differentiation.
Where differentiation measures instantaneous change, integration measures total accumulation.
In finance, integration underlies pricing by expectation, continuous compounding, and risk aggregation — computing areas, totals, or probabilities over continuous domains.
1. The Concept of Integral
Given a continuous function $f:[a,b]\to\mathbb{R}$, the definite integral of $f$ over $[a,b]$ is the limit of Riemann sums:
\[\int_a^b f(x)\,dx = \lim_{n \to \infty}\sum_{i=1}^{n} f(x_i^*) \,\Delta x_i,\]where $\Delta x_i = \frac{b-a}{n}$ and $x_i^* \in [x_{i-1},x_i]$.
This represents the area under the curve $y=f(x)$ from $x=a$ to $x=b$.
If the limit exists and is finite, $f$ is Riemann integrable on $[a,b]$.
Geometric interpretation:
- For $f(x)\ge0$, the integral is the exact area under $f$.
- For $f(x)$ of mixed sign, areas below the axis subtract from the total.
2. Indefinite Integral and Antiderivative
The indefinite integral (antiderivative) of $f$ is any function $F$ such that $F’(x)=f(x)$:
\[\int f(x)\,dx = F(x) + C,\]where $C$ is an arbitrary constant.
Example:
$\int 3x^2\,dx = x^3 + C$.
Fundamental Theorem of Calculus:
If $f$ is continuous on $[a,b]$ and $F’(x)=f(x)$, then
\(\int_a^b f(x)\,dx = F(b) - F(a).\)
Differentiation and integration are inverse processes.
3. Linearity and Basic Rules
For integrable functions $f,g$ and constant $c$:
- $\int (f + g)\,dx = \int f\,dx + \int g\,dx$
- $\int c f\,dx = c \int f\,dx$
- $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$
- If $a<b<c$, $\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx$
Example: \(\int_0^2 (3x^2 - 2x)\,dx = [x^3 - x^2]_0^2 = 8 - 4 = 4.\)
4. Standard Integrals
| Function | $\displaystyle \int f(x)\,dx$ |
|---|---|
| $x^n$, $n\neq -1$ | $\dfrac{x^{n+1}}{n+1}+C$ |
| $1/x$ | $\ln|x|+C$ |
| $e^x$ | $e^x+C$ |
| $a^x$ | $\dfrac{a^x}{\ln a}+C$ |
| $\sin x$ | $-\cos x+C$ |
| $\cos x$ | $\sin x+C$ |
| $\sec^2 x$ | $\tan x+C$ |
| $\frac{1}{1+x^2}$ | $\tan^{-1}x+C$ |
These form the building blocks for analytic integration.
5. Integration Techniques
5.1 Substitution (Change of Variable)
If $u=g(x)$, then \(\int f(g(x))g'(x)\,dx = \int f(u)\,du.\)
Example: \(\int e^{3x}\,dx = \frac{1}{3}e^{3x}+C.\)
5.2 Integration by Parts
If $u=u(x)$ and $v=v(x)$ are differentiable, \(\int u\,dv = uv - \int v\,du.\)
Example: \(\int x e^x dx = x e^x - \int e^x dx = e^x(x - 1) + C.\)
5.3 Partial Fractions
For rational functions, decompose into simpler fractions and integrate term by term.
Example: \(\int \frac{1}{x(x+1)} dx = \int \left( \frac{1}{x} - \frac{1}{x+1} \right) dx = \ln\left|\frac{x}{x+1}\right| + C.\)
5.4 Improper Integrals
When limits are infinite or integrand has singularity, use limit definition:
\[\int_a^\infty f(x)\,dx = \lim_{b\to\infty} \int_a^b f(x)\,dx.\]Example: \(\int_1^\infty \frac{1}{x^2}dx = \left[-\frac{1}{x}\right]_1^\infty = 1.\)
6. Applications in Finance
6.1 Continuous Compounding
The continuously compounded value of principal $P_0$ at rate $r$ over $[0,T]$ is \(P_T = P_0 e^{\int_0^T r(t)\,dt}.\)
If $r(t)$ is constant, $\int_0^T r\,dt = rT$, giving $P_T = P_0 e^{rT}$.
6.2 Expected Value and Pricing
If $X$ is a continuous random variable with pdf $p(x)$, then \(E[X] = \int_{-\infty}^{\infty} x p(x)\,dx.\)
In risk-neutral pricing: \(V_0 = e^{-rT}\int_0^\infty f(S_T)p(S_T)\,dS_T.\)
Thus, valuation becomes integration of discounted payoff weighted by probability density.
6.3 Continuous Portfolio Returns
Instantaneous return rate: \(r_t = \frac{dS_t}{S_t dt} \quad \Rightarrow \quad \ln\frac{S_T}{S_0} = \int_0^T r_t\,dt.\)
6.4 Total Risk (Variance Integration)
Variance of a continuous variable: \(Var(X) = \int_{-\infty}^\infty (x - E[X])^2 p(x)\,dx.\)
This integration measures dispersion around the mean — core to risk quantification.
7. Connection to Differentiation
If $F(x)=\int_a^x f(t)\,dt$, then by the fundamental theorem, \(\frac{dF}{dx}=f(x).\)
Conversely, $\int f’(x)\,dx = f(x) + C$.
Differentiation decomposes accumulation; integration reconstructs it.
8. Numerical Integration
When analytic evaluation is impossible, approximate via discrete summation.
Riemann approximation: \(\int_a^b f(x)\,dx \approx \sum_{i=1}^n f(x_i)\Delta x.\)
Trapezoidal rule: \(\int_a^b f(x)\,dx \approx \frac{\Delta x}{2}\left(f(a)+2\sum_{i=1}^{n-1}f(x_i)+f(b)\right).\)
Monte Carlo integration: \(\int_a^b f(x)\,dx \approx (b-a)\frac{1}{N}\sum_{i=1}^N f(U_i), \quad U_i \sim \text{Uniform}(a,b).\)
In finance, Monte Carlo integration generalizes to expectation over high-dimensional stochastic spaces.
9. Improper and Infinite Domain Integrals in Probability
For densities $p(x)$ defined over $\mathbb{R}$, we require: \(\int_{-\infty}^\infty p(x)\,dx = 1.\)
Example:
For the standard normal density
\(p(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2},\)
we have
\(\int_{-\infty}^\infty p(x)\,dx = 1.\)
Many pricing integrals (e.g., Black–Scholes expectation) are Gaussian-weighted improper integrals.
10. Summary
- Integration accumulates infinitesimal contributions.
- The definite integral computes total change or area; the indefinite integral reverses differentiation.
- Financial applications include continuous compounding, expected values, and risk aggregation.
- In probabilistic pricing, every valuation integral is an area under a probability density.
- Numerical integration (trapezoidal, Monte Carlo) bridges analytic calculus with computational finance.