Calculus is the mathematical foundation of continuous change.
In quantitative finance, it underpins derivative pricing, risk management, and dynamic optimization — from computing sensitivities (Greeks) to modeling stochastic processes in continuous time.
1. The Concept of Limit
Let $f: \mathbb{R} \to \mathbb{R}$.
We say that $f(x)$ approaches $L$ as $x$ approaches $a$ if, for every $\epsilon > 0$, there exists $\delta > 0$ such that
This formal $\epsilon$–$\delta$ definition allows calculus to treat continuous motion precisely.
Example:
For $f(x) = 3x + 2$,
\(\lim_{x \to 1} f(x) = 5.\)
Remark:
Limits describe continuity, differentiability, and integral accumulation — all subsequent operations are built upon this notion.
2. Continuity
A function $f(x)$ is continuous at $x=a$ if:
- $f(a)$ is defined,
- $\lim_{x \to a} f(x)$ exists, and
- $\lim_{x \to a} f(x) = f(a)$.
Equivalently, small perturbations in $x$ produce small changes in $f(x)$.
Example:
Polynomial and exponential functions are continuous everywhere; step or piecewise functions may not be.
In Finance:
Continuity allows infinitesimal modeling — asset prices $S_t$ are modeled as continuous semimartingales, even though real prices move discretely.
3. The Derivative
The derivative of $f$ at point $x=a$ is defined by the limit:
\[f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}.\]It measures the instantaneous rate of change or the slope of the tangent line to $f$ at $x=a$.
Existence:
If the above limit exists (finite), $f$ is differentiable at $a$.
Example:
$f(x) = x^2 \Rightarrow f’(x) = 2x.$
4. Higher-Order Derivatives
Repeated differentiation yields higher-order derivatives:
\[f^{(n)}(x) = \frac{d^n f(x)}{dx^n}.\]These measure curvature and higher-order sensitivities — analogous to Gamma, Vomma, and higher Greeks in option pricing.
Example:
For $f(x) = e^{ax}$,
\(f^{(n)}(x) = a^n e^{ax}.\)
5. Differentiation Rules
Let $u(x)$ and $v(x)$ be differentiable functions.
- Linearity: $(au + bv)’ = au’ + bv’$
- Product Rule: $(uv)’ = u’v + uv’$
- Quotient Rule: $\left(\frac{u}{v}\right)’ = \frac{u’v - uv’}{v^2}$
- Chain Rule: $\frac{d}{dx} f(g(x)) = f’(g(x)) \cdot g’(x)$
Example (Chain Rule):
If $f(x) = e^{3x^2}$,
then $f’(x) = e^{3x^2} \cdot 6x.$
6. Differentiability and Continuity
Every differentiable function is continuous,
but continuity does not imply differentiability.
Counterexample:
$f(x) = |x|$ is continuous at $x=0$ but not differentiable there:
\(\lim_{h\to 0^+}\frac{|h|-0}{h}=1,\quad
\lim_{h\to 0^-}\frac{|h|-0}{h}=-1.\)
7. The Mean Value Theorem (MVT)
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$,
then there exists $c \in (a,b)$ such that
Interpretation:
There exists a point where the instantaneous slope equals the average rate of change.
In Finance:
The theorem underlies linear approximations like Delta hedging — local linearization of payoff with respect to the underlying.
8. The Integral — The Inverse of Differentiation
Given $f(x)$ continuous on $[a,b]$,
the definite integral is the limit of Riemann sums:
It represents the accumulated area under $f(x)$ from $a$ to $b$.
Fundamental Theorem of Calculus:
If $F’(x) = f(x)$, then \(\int_a^b f(x)\,dx = F(b) - F(a).\)
Thus differentiation and integration are inverse operations.
9. Applications in Quantitative Finance
- Instantaneous returns:
$r_t = \frac{dS_t}{S_t\,dt}$ → continuous compounding. - Sensitivity analysis:
Greeks = partial derivatives of option price $V(S, \sigma, t, \dots)$. - Optimization:
Portfolio weights solved via derivative-based first-order conditions. - Continuous-time models:
Ito calculus extends these limits to stochastic processes.
10. Key Takeaways
- Limits formalize continuity and smoothness.
- Derivatives quantify infinitesimal changes.
- Integrals accumulate effects over continuous domains.
- The interplay between the two forms the foundation of continuous-time finance.
This groundwork supports more advanced topics such as partial differentiation, multivariable integration, and stochastic calculus.
Next, Differentiation and Rates of Change.
Some differentiation formulas for reference:
| Function | Derivative |
|---|---|
| $c$ (constant) | $0$ |
| $x^n$ | $nx^{n-1}$ |
| $e^{x}$ | $e^{x}$ |
| $e^{cx}$ | $ce^{cx}$ |
| $\ln(x)$ | $\frac{1}{x}$ |
| $\sin(x)$ | $\cos(x)$ |
| $\cos(x)$ | $-\sin(x)$ |
| $\tan(x)$ | $\sec^2(x)$ |
| $u(x)v(x)$ | $u’(x)v(x) + u(x)v’(x)$ |
| $\frac{u(x)}{v(x)}$ | $\frac{u’(x)v(x) - u(x)v’(x)}{v^2(x)}$ |
| $f(g(x))$ | $f’(g(x)) \cdot g’(x)$ |
| $f^{(n)}(x)$ | $\frac{d^n f(x)}{dx^n}$ |
Derivative Rules Summary:
| Derivatives Rule | Description |
|---|---|
| Sum | $\frac{d}{dx}[u + v] = \frac{du}{dx} + \frac{dv}{dx}$ |
| Product | $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$ |
| Quotient | $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ |
| Chain | $\frac{d}{dx}f(g(x)) = f’(g(x)) \cdot g’(x)$ |
| Power | $\frac{d}{dx}[u^n] = nu^{n-1}\frac{du}{dx}$ |
| Inverse | $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$ |