In quantitative finance, many quantities depend on multiple continuous variables — asset prices, interest rates, and volatility states.
Multivariable integration generalizes one-dimensional integration to compute accumulated effects over multidimensional domains, such as joint probability distributions and multi-asset payoffs.
1. Double and Multiple Integrals
For a function $f(x,y)$ defined on a rectangular region $R = [a,b]\times[c,d]$, the double integral is defined as
\[\iint_R f(x,y)\,dx\,dy = \lim_{m,n\to\infty} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_i^*, y_j^*)\,\Delta x\,\Delta y,\]provided the limit exists and is finite.
This represents the volume under the surface $z=f(x,y)$ over region $R$.
Similarly, a triple integral \(\iiint_V f(x,y,z)\,dx\,dy\,dz\) computes the hyper-volume over a 3D region $V$.
2. Iterated Integrals and Fubini’s Theorem
If $f$ is continuous on $R$, then the double integral equals the iterated integral:
\[\iint_R f(x,y)\,dx\,dy = \int_c^d \int_a^b f(x,y)\,dx\,dy = \int_a^b \int_c^d f(x,y)\,dy\,dx.\]Example: \(\iint_{[0,1]^2} (x+y)\,dx\,dy = \int_0^1 \left[\frac{x^2}{2} + xy\right]_0^1 dy = \int_0^1 \left(\frac{1}{2} + y\right) dy = \frac{1}{2} + \frac{1}{2} = 1.\)
Application:
In finance, this structure appears in joint expectations where two or more random variables interact (e.g., multi-asset options, correlation products).
3. Integration Over Non-Rectangular Regions
When the domain $R$ is bounded by curves or surfaces, limits of integration depend on $x$ or $y$.
Example:
Region bounded by $y=0$, $y=x$, and $x=1$:
\(\iint_R y\,dx\,dy = \int_0^1 \int_0^x y\,dy\,dx = \int_0^1 \frac{x^2}{2}\,dx = \frac{1}{6}.\)
4. Change of Variables and Jacobians
Often it is convenient to transform variables $(x,y)$ to new variables $(u,v)$ for simpler integration.
If $(x,y)$ are smooth functions of $(u,v)$:
then
\[dx\,dy = \left|\frac{\partial(x,y)}{\partial(u,v)}\right|\,du\,dv,\]where the Jacobian determinant is
\[\frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}.\]Example (Polar Coordinates):
Let $x = r\cos\theta$, $y = r\sin\theta$.
Then
\(\frac{\partial(x,y)}{\partial(r,\theta)} =
\begin{vmatrix}
\cos\theta & -r\sin\theta \\
\sin\theta & r\cos\theta
\end{vmatrix} = r.\)
Thus $dx\,dy = r\,dr\,d\theta$.
Illustration: \(\iint_R f(x,y)\,dx\,dy = \int_{\theta_1}^{\theta_2} \int_{r_1}^{r_2} f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta.\)
This formula is used extensively when integrating over circular or elliptical regions, such as correlation ellipses or bivariate normal densities.
5. Integration of Joint Probability Densities
If $(X,Y)$ has joint pdf $p(x,y)$, the probability over region $A$ is
\[P((X,Y)\in A) = \iint_A p(x,y)\,dx\,dy.\]Marginal densities are obtained by integrating out the other variable:
\[p_X(x) = \int_{-\infty}^\infty p(x,y)\,dy, \quad p_Y(y) = \int_{-\infty}^\infty p(x,y)\,dx.\]Example:
If $(X,Y)$ are jointly normal with correlation $\rho$,
\(p(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}
\exp\!\left(-\frac{x^2 - 2\rho xy + y^2}{2(1-\rho^2)}\right).\)
Then
\(p_X(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}, \quad p_Y(y)=\frac{1}{\sqrt{2\pi}}e^{-y^2/2}.\)
In Finance:
This is the basis for pricing basket options, spread options, and correlation derivatives.
6. Expectation and Covariance via Multivariable Integrals
Expected value of a function $g(X,Y)$ is \(E[g(X,Y)] = \iint_{\mathbb{R}^2} g(x,y)p(x,y)\,dx\,dy.\)
Covariance is computed as \(Cov(X,Y) = \iint_{\mathbb{R}^2} (x - E[X])(y - E[Y])p(x,y)\,dx\,dy.\)
Example:
For jointly normal $(X,Y)$, $Cov(X,Y) = \rho$.
7. Application: Bivariate Normal Transformation
Let $(U,V)\sim N(0,I)$ be independent.
Define correlated variables:
\(X = U, \quad Y = \rho U + \sqrt{1-\rho^2}\,V.\)
Then $(X,Y)$ has joint pdf with correlation $\rho$.
The Jacobian determinant of the linear transformation is 1, hence densities transform directly.
This change of variables is essential in Monte Carlo simulation of correlated random variables for portfolio or VaR modeling.
8. Multivariate Integration in Pricing
A European option on two assets with payoff $f(S_1,S_2)$ has risk-neutral price:
\[V_0 = e^{-rT}\iint_0^\infty f(S_1,S_2)p(S_1,S_2)\,dS_1\,dS_2.\]Example:
Basket call with weights $w_1, w_2$ and strike $K$:
Analytic solutions are rare; Monte Carlo integration is used to approximate this double integral.
9. Monte Carlo and Numerical Multivariate Integration
For $n$-dimensional integrals, direct quadrature becomes impractical as $n$ grows.
Monte Carlo replaces the integral by an expectation estimated through random sampling:
Properties:
- Error $\mathcal{O}(1/\sqrt{N})$, independent of dimension.
- Applicable for any distribution shape.
- Used in pricing, risk aggregation, and exposure simulation.
10. Change of Variables in Probability Integrals
If $Y=g(X)$ and $X$ has pdf $p_X(x)$, then \(p_Y(y) = p_X(x)\left|\frac{dx}{dy}\right|.\)
Example:
If $X$ is uniform on $[0,1]$ and $Y=-\ln X$,
then $p_Y(y)=e^{-y}$ for $y>0$, i.e., an exponential distribution.
This concept generalizes to multivariate transformations through the absolute Jacobian determinant.
In Finance:
Used for transforming between return spaces (log vs. arithmetic), yield–discount mappings, or volatility parameterizations.
11. Summary
- Multiple integrals extend one-dimensional area to multidimensional volume.
- The Jacobian corrects for distortion under variable transformations.
- In probability, integration computes expectations, covariances, and prices.
- Monte Carlo integration generalizes to high-dimensional financial problems.
- These tools underpin simulation, copula modeling, and multi-asset derivative pricing.