Differentiation is the central tool of calculus — it formalizes how quantities vary with respect to each other.
In quantitative finance, differentiation measures sensitivity: how a small change in one variable (price, rate, volatility) affects another (portfolio value, risk, or return).
1. Definition
Let $f:\mathbb{R}\to\mathbb{R}$.
The derivative of $f$ at $x=a$ is
if this limit exists. The derivative represents the instantaneous rate of change of $f$ at $x=a$.
If $f’(a)$ exists for every $a$ in an interval, then $f$ is differentiable on that interval.
Geometric interpretation:
The derivative equals the slope of the tangent line to $f(x)$ at $x=a$.
Economic interpretation:
The derivative is the marginal change of output per unit change in input — e.g., marginal profit, marginal cost, marginal return.
2. Basic Differentiation Rules
Let $u(x)$ and $v(x)$ be differentiable functions and $c$ a constant.
- Constant Rule: $(c)’ = 0$
- Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$ for all $n \in \mathbb{R}$
- Exponential Rule: $\frac{d}{dx} e^{x} = e^{x}$, and $\frac{d}{dx} e^{a x} = a e^{a x}$
- Logarithmic Rule: $\frac{d}{dx} \ln(x) = \frac{1}{x}$
- Constant Multiple Rule: $(c u)’ = c u’$
- Sum Rule: $(u+v)’ = u’ + v’$
- Product Rule: $(uv)’ = u’v + uv’$
- Quotient Rule: $\left(\frac{u}{v}\right)’ = \frac{u’v - uv’}{v^2}$
- Chain Rule: $\frac{d}{dx} f(g(x)) = f’(g(x)) \, g’(x)$
Example:
For $f(x) = e^{3x^2}$,
\(f'(x) = e^{3x^2} \cdot 6x.\)
3. Derivatives of Common Functions
| Function $f(x)$ | Derivative $f’(x)$ |
|---|---|
| $x^n$ | $n x^{n-1}$ |
| $e^x$ | $e^x$ |
| $\ln x$ | $1/x$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $a^x$ | $a^x \ln a$ |
| $\log_a x$ | $\frac{1}{x \ln a}$ |
Verification Example:
If $f(x) = x^3 + 2x^2 - 5x + 4$, then
\(f'(x) = 3x^2 + 4x - 5.\)
4. The Chain Rule — Composition of Functions
Suppose $y=f(u)$ and $u=g(x)$, then \(\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx} = f'(g(x))g'(x).\)
Example:
Let $f(x)=\ln(3x^2+2)$. Then
\(f'(x)=\frac{1}{3x^2+2}\cdot6x=\frac{6x}{3x^2+2}.\)
In finance, this principle governs propagation of sensitivities — e.g., the delta of an option whose underlying itself depends on other factors.
5. Higher-Order Derivatives
The second derivative is \(f''(x) = \frac{d}{dx}\left(f'(x)\right).\)
It measures curvature — the rate at which slope itself changes.
If $f’‘(x) > 0$, the function is convex; if $f’‘(x) < 0$, it is concave.
Example:
$f(x)=x^3 \Rightarrow f’‘(x)=6x$ — positive for $x>0$, negative for $x<0$.
Applications:
- Convex payoff functions (options) have positive second derivative (Gamma).
- Portfolio curvature: second derivative of value wrt price or rate measures nonlinearity of exposure.
6. Implicit Differentiation
When $y$ is defined implicitly by $F(x,y)=0$, differentiation uses the chain rule:
\[\frac{dF}{dx} = F_x + F_y \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{F_x}{F_y}.\]Example:
For $x^2 + y^2 = 25$,
$2x + 2y\frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}.$
In Finance:
Used in sensitivities derived from equilibrium or constraint conditions, e.g. yield-curve bootstrapping where yields depend implicitly on discount factors.
7. Logarithmic Differentiation
When differentiating products or powers efficiently:
Take logs on both sides of $y = f(x)$, then differentiate:
\[\ln y = \ln f(x) \quad \Rightarrow \quad \frac{1}{y} \frac{dy}{dx} = \frac{f'(x)}{f(x)}.\]Example:
$y = x^x \Rightarrow \ln y = x \ln x \Rightarrow \frac{1}{y}\frac{dy}{dx} = \ln x + 1 \Rightarrow \frac{dy}{dx} = x^x(\ln x + 1).$
8. Differentiation in Financial Contexts
8.1 Instantaneous Rate of Return
For a continuously compounded return process: \(r_t = \frac{dS_t}{S_t \, dt}.\) The derivative here measures instantaneous proportional growth.
8.2 Yield Curve Slope
If $y(T)$ is the zero-coupon yield for maturity $T$,
the slope $\frac{dy}{dT}$ measures how yields change with maturity (key rate sensitivity).
8.3 Option Greeks
The option price $V(S, \sigma, t)$ yields: \(\Delta = \frac{\partial V}{\partial S}, \quad \Gamma = \frac{\partial^2 V}{\partial S^2}, \quad \nu = \frac{\partial V}{\partial \sigma}.\) All are applications of partial differentiation and rates of change.
8.4 Duration and Convexity
For bond price $P(y)$ as a function of yield $y$: \(D = -\frac{1}{P}\frac{dP}{dy}, \quad C = \frac{1}{P}\frac{d^2 P}{dy^2}.\) Duration and convexity describe first- and second-order sensitivities to rate changes.
9. Differentiability vs Smoothness
- Differentiable ⇒ continuous
- $C^1$ ⇒ continuously differentiable (no kinks)
- $C^\infty$ ⇒ infinitely differentiable (smooth)
Smoothness is essential in models requiring Ito’s lemma or Taylor expansions.
Example:
$|x|$ is not differentiable at 0 (kink), but $x^2$ is $C^\infty$.
10. Summary
- Differentiation formalizes infinitesimal rates of change.
- Fundamental rules (chain, product, quotient) allow decomposition of complex relationships.
- Second and higher derivatives reveal curvature and nonlinear effects.
- In finance, derivatives quantify risk sensitivities — yield slopes, option Greeks, and exposure curvature.