M.2.2 Single-Variable Calculus

Basic Prerequisites: 01 · From Zero (what a derivative is).

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1. Intuition & Practical Objective

Single-variable calculus is the reference implementation of everything that follows. It gives you four tools, and each one has a direct finance job:

1. Limits - make sense of “instantaneous” quantities: a continuously-compounded return is the limit of periodic compounding, and the price of a bond as maturity $\to 0$ is a limit.
2. Derivatives - local sensitivity and the first-order optimality condition (Fermat): “flat $\Rightarrow$ optimum.”
3. Taylor polynomials - replace a messy function by a polynomial you can compute with. This is the expansion behind delta–gamma–theta P&L, duration–convexity, and every “local model” on a desk.
4. Integration - turn a density into a probability and a payoff into a price, via the Fundamental Theorem of Calculus.

The one mental model: Taylor’s theorem says every smooth function is a polynomial plus a controlled error. Stop at degree 1 and you get the tangent (delta). Go to degree 2 and you get curvature (gamma). The remainder term is the honest part - it tells you the size of the error you are eating by truncating, and it is exactly the term a trader is exposed to when the market moves more than “a little.”

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2. Mathematical Ground Truth & Derivations

Limits and continuity. ${\lim }_{x\to c}f(x)=L$ means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ close enough to $c$. $f$ is continuous at $c$ if ${\lim }_{x\to c}f(x)=f(c)$. The algebra of limits (sums, products, compositions of continuous functions are continuous) lets one compute almost everything without $ϵ$–$\delta$ arguments. L’Hôpital’s rule handles the indeterminate $0/0$ form:

$$\text{if }f(c)=g(c)=0\text{ and }{g}^{\prime }(c)\ne 0,\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\frac{f^{\prime }(c)}{g^{\prime }(c)},\text{(iterated if the ratio is still }0/0\text{).}$$

Bernstein’s note deliberately defines limits via continuity (convergence at a point is “the redefined function is continuous there”), which makes L’Hôpital a corollary of the mean value theorem rather than a separate axiom.

Derivative rules. From the limit definition: $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$, $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$, $(f/g{)}^{\prime }=(f^{\prime }g-f{g}^{\prime })/g^2$, $(f\circ g{)}^{\prime }=(f^{\prime }\circ g)g^{\prime }$, and $(x^n{)}^{\prime }=n{x}^{n-1}$.

The mean value theorem (MVT). If $f$ is continuous on $[b,c]$ and differentiable on $(b,c)$, there is $x\in (b,c)$ with

$$f(c)-f(b)=f^{\prime }(x)(c-b).$$

Its corollaries drive optimisation and numerics: $f^{\prime }\equiv 0$ on an interval $\Rightarrow f$ constant; and the secant slope of a finite-difference scheme equals the true derivative at some intermediate point, which is what makes truncation-error bounds possible.

Fermat, Rolle and the second-order test.

• Fermat: an interior extremum of a differentiable $f$ has $f^{\prime }(x^{\ast })=0$.
• Rolle: $f(b)=f(c)$ $\Rightarrow$ some interior point has $f^{\prime }=0$.
• Second-order sufficiency: if $f^{\prime }(x^{\ast })=0$ and $f^{\prime \prime }(x^{\ast })>0$, then $x^{\ast }$ is a strict local minimum ($f^{\prime \prime }<0$ $\Rightarrow$ maximum). If $f^{\prime \prime }(x^{\ast })=0$ the test is inconclusive ($x^3$, $x^4$ at $0$).

Taylor’s theorem with remainder. For $f$ $(n+1)$-times differentiable on an interval containing $a$ and $x$, there is $z$ between them with

$$f(x)=\underset{P_n(x)\text{Taylor polynomial}}{\underbrace{\sum _{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k}}+\underset{R_n(x)\text{remainder}}{\underbrace{\frac{f^{(n+1)}(z)}{(n+1)!}(x-a{)}^{n+1}}}.$$

The Lagrange form of the remainder (Stewart §11.10–11.11; Spivak Part III) is the key to error control: it says the error is “the next term, with the derivative evaluated at some unknown point $z$.” For $e^x$ about $0$, $R_n(x)=e^z{x}^{n+1}/(n+1)!$, so on $[0,1]$ the error is bounded by $e/(n+1)!$ - a provable rate.

The finance reading: with $a=S_t$ and $x=S_t+\Delta S$,

$$V(S_t+\Delta S)=V+\Delta \Delta S+\frac{1}{2}\Gamma (\Delta S{)}^2+\frac{1}{6}\text{Speed}(\Delta S{)}^3+\cdots ,$$

i.e. the delta–gamma–theta expansion is a Taylor expansion of the pricing function, and truncating at order 1 leaves the gamma term as the remainder you are exposed to.

Integration and the FTC. The definite integral ${\int }_b^{c}f$ is the limit of Riemann sums $\sum f({\xi }_i)\Delta x_i$. Linearity and positivity are immediate; the Fundamental Theorem of Calculus is

$${\int }_b^c{f}^{\prime }(x)dx=f(c)-f(b).$$

Under the Kurzweil–Henstock (gauge) definition used in Bernstein’s note, every derivative is integrable, so the FTC holds with no extra technical hypotheses - which is why $\mathbb{E}[f(X)]=\int fdF$ always makes sense for the densities finance uses.

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3. Computational Implementation - Taylor convergence, its radius, and the FTC

Stdlib only. Three experiments: (A) truncation error of the Taylor polynomial of $e^x$ shrinking by factorially more digits; (B) the radius of convergence - the same series at $x=0.5$ (inside) versus $x=2$ (outside); (C) Riemann sums converging to an integral at rate $O(1/n)$.

Read the tables. (A) The $e^x$ error drops by a factor of ~$1/n$ each extra term - factorial convergence, hitting machine precision (2.2e-15) by $n=16$. This is why “add one more Taylor term” is the cheapest accuracy in all of numerics. (B) The same series evaluated at $x=2$ blows up to $-8.4\times 10^{27}$: a Taylor series is only trustworthy inside its radius of convergence ($|x|<1$ here). Calibrating an expansion too far from the expansion point is a classic modelling error. (C) The Riemann error falls by exactly $10\times$ each time $n$ grows by $10\times$ - the $O(1/n)$ law, made visible.

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4. Failure Modes & First-Principles Breakdowns

1. Truncating a Taylor series is a choice with a price, given by the remainder. Delta hedging keeps only the first term; the dropped $\frac{1}{2}\Gamma (\Delta S{)}^2$ is the whole hedging risk. The Lagrange remainder tells you its size - never expand without asking what you threw away.
2. The radius of convergence is finite and bites. As experiment (B) shows, the expansion of $\log (1+x)$ is exactly wrong outside $|x|<1$, not merely inaccurate. Local models (delta, duration) are only valid locally; a large move invalidates the expansion, it does not just degrade it.
3. Fermat is not sufficient. $f^{\prime }(x^{\ast })=0$ at $x^{\ast }=0$ for $x^3$ (a saddle), $x^4$ (a min) and $-x^4$ (a max). Always run the $f^{\prime \prime }$ test (and if $f^{\prime \prime }=0$, higher order) before declaring an optimum.
4. Differentiability can fail at exactly the interesting point. $\max (S-K,0)$ has no derivative at $S=K$; its second derivative is a Dirac mass there. Inconsistent-looking Greeks and FDM ringing around the strike are the numerical shadow of this kink.
5. Integration by quadrature is cursed by dimension. A left Riemann sum converges at $O(1/n)$ here, the trapezoid at $O(1/n^2)$ - but in $d$ dimensions a tensor-product rule costs $O(n^{-2/d})$, which is why high-dimensional integrals in finance are done by Monte Carlo ($O(n^{-1/2})$ in every dimension, Glasserman §1.1) rather than by quadrature.
6. A 0/0 limit is not automatically “differentiate top and bottom.” L’Hôpital requires the hypotheses ($f(c)=g(c)=0$, differentiability, $g^{\prime }\ne 0$); on non-indeterminate forms it silently returns nonsense.

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5. References

• Bernstein, D. J.: Calculus for Mathematicians (1997 draft)
• Spivak, Michael: Calculus (4th ed.)
• Stewart, Clegg & Watson: Calculus: Early Transcendentals (9th ed., 2020)
• Simon & Blume: Mathematics for Economists

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Next: 03 · Multivariable Calculus (the same ideas in ${\mathbb{R}}^n$)
• Applied: The Greeks & Dynamic Hedging (Taylor = delta–gamma) · Finite-Difference Methods (MVT-driven truncation error)
• Base: Linear Algebra & Matrices