Limits and Derivatives

1Introduction

A derivative answers one question: when an input moves a little, how much does an output move, per unit of input? Traders answer it daily without naming it. One TX contract gains NT$200 per index point; a bond with modified duration 9 loses about 9% per percentage point of yield, to first order.

This module builds the mathematics behind such numbers — limits, derivatives and elasticities — and shows where the linear picture breaks.

Sections 2–4 build the tools: limits, continuity, the derivative and its rules. Section 5 reads derivatives as sensitivities: futures delta, duration, rate risk. Section 6 introduces elasticity, which resolves the case.

Section 7 turns derivatives into approximations and shows how they fail. Section 8 computes derivatives numerically, where rounding makes "a smaller step is better" false, and Section 9 does it in Python.

2Limits and Continuity

Where a function is heading

The slope of a chord of $e^x$ from $0$ to $x$ is $(e^x-1)/x$. At $x=0$ the expression is $0/0$ and means nothing. For $x=0.01$ it is 1.005017, and for $x=-0.01$ it is 0.995017: the values head toward 1 from both sides. A limit makes "heads toward" precise, and it does not care what happens at the point.

Limits combine as expected: if $\lim h={\ell }_1$ and $\lim u={\ell }_2$, then $\lim (h\pm u)={\ell }_1\pm {\ell }_2$, $\lim hu={\ell }_1{\ell }_2$, and $\lim h/u={\ell }_1/{\ell }_2$ when ${\ell }_2\ne 0$. These limit laws are stated without proof (Stewart, chapter 2). One more tool does the heavy lifting below.

Continuity

Polynomials, $e^x$ and $\ln x$ (for $x>0$) are continuous. So are sums, products, quotients and compositions of continuous functions, wherever they are defined. Continuity also lets a limit pass inside a function: if ${\lim }_{x\to x_0}u(x)=\ell$ and $h$ is continuous at $\ell$, then ${\lim }_{x\to x_0}h(u(x))=h(\ell )$. All three facts are stated without proof (Stewart, chapter 2).

A function can fail to be continuous in three ways, and each has a financial example.

• Removable discontinuity. $(e^x-1)/x$ is undefined at 0, but its limit exists (it is 1, as shown below). Defining the value at 0 as 1 repairs it.
• Jump. A digital option pays 1 if the index ends above the strike $K$ and 0 otherwise. Its payoff jumps at $K$, where the left and right limits are 0 and 1.
• Infinite discontinuity. The leverage $f=NmF/E$ of a futures account grows without bound as equity approaches zero. For Trader A on 11 June, equity is zero at $F=22,116-600,000/200=19,116$, and $f\to +\infty$ as $F$ falls toward that level.

Continuity is weaker than smoothness. The call payoff $\max (S-K,0)$ is continuous at the strike but has a corner there; Section 3 shows that it has no derivative at $K$.

Two limits that carry the module

By (1.6) of LM1, $e^x={\lim }_{n\to \infty }(1+x/n{)}^n$ for every real $x$. Everything we need about the slope of $e^x$ then follows from Bernoulli's inequality: for every integer $n\ge 1$ and every $x\ge -1$,

Replace $x$ by $x/n$ with $n\ge |x|$: $(1+x/n{)}^n\ge 1+x$. Letting $n\to \infty$ preserves the inequality, so $e^x\ge 1+x$ for every real $x$. Applied to $-x$ it gives $e^{-x}\ge 1-x$, which is positive when $x<1$. Since $e^x{e}^{-x}=1$ by the laws of exponents (LM1),

Subtract 1 and divide by $x$. For $0<x<1$ this gives $1\le (e^x-1)/x\le 1/(1-x)$. For $x<0$ the division flips both inequalities: $1/(1-x)\le (e^x-1)/x\le 1$. In both cases the bounds tend to 1 (Exhibit 2 shows them closing in), so the squeeze theorem yields

A second limit follows by substitution. Let $z=\ln (1+x)$, so that $x=e^z-1$. As $x\to 0$, $z\to 0$ because $\ln$ is continuous at 1, and $z\ne 0$ whenever $x\ne 0$. Then $\ln (1+x)/x=z/(e^z-1)=1/[(e^z-1)/z]$, and the quotient law with (2.2) gives

In financial language, (2.2) says that a continuously compounded rate $x$ produces a simple return $e^x-1$ whose ratio to $x$ tends to 1. Equation (2.3) says the same of a log return and a simple return. Both are "≈" statements about small moves; LM3 measures their error.

Limits at infinity

Limits at infinity describe long horizons, such as the value of an income stream that never stops.

3The Derivative

From average to instantaneous rate

The average rate of change of $h$ between $x_0$ and $x_0+\Delta x$ is the difference quotient $[h(x_0+\Delta x)-h(x_0)]/\Delta x$: the slope of the secant through the two points. As $\Delta x$ shrinks, the secant pivots about $(x_0,h(x_0))$. If it settles on a limiting line, that line is the tangent and its slope is the derivative.

A derivative carries units: output units per input unit. That is why one number answers three questions.

• Slope (geometry): the steepness of the tangent line.
• Marginal rate (economics): the extra output from one more unit of input, such as the extra interest from lengthening a loan.
• Sensitivity (risk): the change in a value per unit change in a risk factor. A TX contract has a sensitivity of NT$200 per index point.

Two derivatives from the definition show the mechanics. For $h(x)=x^2$, $[(x+\Delta x{)}^2-x^2]/\Delta x=2x+\Delta x\to 2x$. For $h(x)=1/x$ with $x\ne 0$, $[1/(x+\Delta x)-1/x]/\Delta x=-1/[x(x+\Delta x)]\to -1/x^2$.

The figure below lets you watch the secant become the tangent. Its opening view uses $h(x)=e^x$ at $x_0=0$ with a forward step $\Delta x=0.5$: the secant slope is 1.29744 while the tangent slope is 1. Shrink $\Delta x$ and the error falls in proportion to $\Delta x$; switch to central differences and it falls in proportion to $\Delta x^2$.

Real pricing functions behave the same way. Exhibit 3 applies (2.4) to the price of a 10-year bond with a 2% annual coupon, valued at a 2% yield, where it trades at par. Section 5 derives the exact derivative, $P^{\prime }(2\%)=-898.2585$.

Differentiable implies continuous

If $h$ is differentiable at $x_0$, then $h(x_0+\Delta x)-h(x_0)=\Delta x\cdot \frac{h(x_0+\Delta x)-h(x_0)}{\Delta x}\to 0\cdot h^{\prime }(x_0)=0$, so $h$ is continuous at $x_0$. The converse fails.

The one-sided derivatives $h_{\pm }^{\prime }(x_0)={\lim }_{\Delta x\to 0^{\pm }}[h(x_0+\Delta x)-h(x_0)]/\Delta x$ can exist and differ. For the call payoff $\max (S-K,0)$ the right derivative at $K$ is 1 and the left derivative is 0. The payoff therefore has no derivative at $K$. The digital payoff is not even continuous at $K$, so it cannot be differentiable there.

The mean value theorem

Somewhere, the instantaneous rate equals the average rate. When the bond's yield rises from 2% to 3%, its price falls from 100 to 91.469797, an average slope of −853.0203 per unit of yield. The theorem guarantees a yield between 2% and 3% at which $P^{\prime }$ takes exactly that value.

Two consequences are used repeatedly. If $h^{\prime }>0$ on an interval, $h$ is increasing there. If $|h^{\prime }|\le M$ on $[x_1,x_2]$, then $|h(x_2)-h(x_1)|\le M|x_2-x_1|$, which is how Sections 7 and 8 bound errors.

4Differentiation Rules

Six rules, each proved once from (2.4), differentiate every function built from powers, exponentials and logarithms by sums, products, quotients and compositions.

Linearity and integer powers

From the limit laws, $(\kappa h{)}^{\prime }=\kappa h^{\prime }$ for any constant $\kappa$, and $(h+u{)}^{\prime }=h^{\prime }+u^{\prime }$: differentiation is linear. For a positive integer $j$, the binomial theorem gives $(x+\Delta x{)}^j=x^j+j{x}^{j-1}\Delta x+\Delta x^2\times (\text{a polynomial in }x\text{ and }\Delta x)$. Dividing by $\Delta x$ and letting $\Delta x\to 0$ leaves $\frac{d}{dx}{x}^j=j{x}^{j-1}$.

The exponential and the logarithm

Proof. For $\kappa \ne 0$, the law of exponents gives $[e^{\kappa (x+\Delta x)}-e^{\kappa x}]/\Delta x=\kappa e^{\kappa x}(e^{\kappa \Delta x}-1)/(\kappa \Delta x)$, which tends to $\kappa e^{\kappa x}$ by (2.2) with $\kappa \Delta x$ in place of $x$. For $\kappa =0$ both sides are 0.

For the logarithm, $\ln (x+\Delta x)-\ln x=\ln (1+\Delta x/x)$. With $z=\Delta x/x$,

With $\kappa =1$, the exponential is its own rate of change. Wealth $W(t)=W_0{e}^{gt}$ grows at $W^{\prime }(t)=gW(t)$: a constant proportional rate $g$, the log growth rate of LM1. Its logarithm, $\ln W(t)=\ln W_0+gt$, is a straight line with slope $g$.

Products and quotients

Proof. Add and subtract $u(x)h(x+\Delta x)$:

The chain rule

Most pricing functions are nested: a bond price depends on discount factors, which depend on the yield. The chain rule differentiates compositions.

The intuitive argument writes $\Delta h/\Delta x=(\Delta h/\Delta u)(\Delta u/\Delta x)$ and lets $\Delta x\to 0$. It is correct whenever $\Delta u\ne 0$ for small $\Delta x$, which can fail if $u$ oscillates. The Deep Dive gives a proof without that gap.

The chain rule completes the toolkit.

• Real powers. For $x>0$, $x^{\alpha }=e^{\alpha \ln x}$, so $\frac{d}{dx}{x}^{\alpha }=e^{\alpha \ln x}\cdot \alpha /x=\alpha x^{\alpha -1}$ for any real $\alpha$.
• Other bases. For a base $\beta >0$, ${\beta }^x=e^{x\ln \beta }$, so $\frac{d}{dx}{\beta }^x={\beta }^x\ln \beta$.
• Discount factors. $\frac{d}{dy}(1+y{)}^{-t}=-t(1+y{)}^{-t-1}$ and $\frac{d}{dr}{e}^{-rt}=-t{e}^{-rt}$.
• Logarithmic derivative. For $h>0$, $\frac{d}{dx}\ln h(x)=h^{\prime }(x)/h(x)$: the relative rate of change, the basis of elasticities in Section 6.

Exhibit 4 collects the rules with the places they appear in this book.

5Derivatives as Sensitivities

In risk management a derivative is a sensitivity: the change in a value per unit change in one risk factor, other factors held fixed. This section computes three that desks quote daily: futures delta, bond duration and the rate sensitivity of a present value.

Delta of a futures position

An account holding $N$ contracts opened at $F_0$ with equity $E_0$ has equity $E(F)=E_0+Nm(F-F_0)$ after marking to market. Its derivative is the position's delta:

Delta per index point converts to delta per 1% move by multiplying by $F/100$. At $F=22,344$ a 1% move is 223.44 points, worth NT$44,688 to one TX contract. The quantity $NmF$ is the position's notional, or exposure, and "NT$ per 1%" is notional ÷ 100.

Duration: the derivative of a bond price

A bond with annual cash flows $C{F}_t$ at $t=1,\dots ,n$ is priced at yield $y$ as $P(y)={\sum }_{t}C{F}_t(1+y{)}^{-t}$. By linearity and the discount-factor rule,

As a sensitivity, (2.6) says $\Delta P/P\approx -D_{\text{mod}}\Delta y$, with an error of order $\Delta y^2$ (Section 7). The price fall for a one-basis-point rise in yield, $-P^{\prime }(y)\times 0.0001$, is the bond's DV01. LM3 adds the second-order term, convexity.

Rate sensitivity of a present value

For a single payment due in $T$ years, the sensitivity depends on the compounding convention. At a continuously compounded rate $r$, $PV=e^{-rT}$ per unit of payment and $dPV/dr=-T\cdot PV$: the relative sensitivity is exactly $T$. At an annually compounded rate $y$, $PV=(1+y{)}^{-T}$ and $dPV/dy=-\frac{T}{1+y}PV$. For NT$1,000,000 due in 5 years at 2%, the sensitivities are −452.42 and −443.99 per bp, on present values of 904,837.42 and 905,730.81.

The same reasoning applies to the index futures fair value $F=S{e}^{(r_f-q)\tau }$, which LM6 derives by no-arbitrage. Holding $r_f$, $q$ and $\tau$ fixed, $dF/dS=e^{(r_f-q)\tau }$, close to 1.

Holding $S$, $r_f$ and $\tau$ fixed, $dF/dq=-\tau F$. For a three-month contract ($\tau =0.25$) at $F=22,000$, a rise of 0.1 percentage point in the expected dividend yield lowers fair value by 5.50 points. A derivative that holds the other inputs fixed is a partial derivative, treated properly in LM7.

6Elasticity and Leverage

A unit-free derivative

A delta of NT$200 per point cannot be compared with a duration of 9 years, and neither says what happens to an account in percent. Elasticity removes units by comparing relative changes.

The last form follows from the chain rule. Put $z=\ln x$, so $x=e^z$ and $\ln h=\ln h(e^z)$. Then $\frac{d\ln h}{dz}=\frac{h^{\prime }(e^z)e^z}{h(e^z)}=\frac{x{h}^{\prime }(x)}{h(x)}$.

Because elasticities are derivatives of logarithms, the rules of Section 4 become rules of addition and multiplication. For positive functions $u$ and $h$:

• power: if $h$ is proportional to $x^{\alpha }$, then ${\eta }_{h,x}=\alpha$;
• product and quotient: ${\eta }_{uh}={\eta }_u+{\eta }_h$ and ${\eta }_{u/h}={\eta }_u-{\eta }_h$;
• chain: the elasticity of $h(u(x))$ with respect to $x$ is ${\eta }_{h,u}\cdot {\eta }_{u,x}$;
• sum: ${\eta }_{u+h}=\frac{u}{u+h}{\eta }_u+\frac{h}{u+h}{\eta }_h$, a value-weighted average.

The first three follow from $\ln (uh)=\ln u+\ln h$, $\ln x^{\alpha }=\alpha \ln x$ and (2.5). For the sum, ${\eta }_{u+h}=\frac{x(u^{\prime }+h^{\prime })}{u+h}=\frac{u}{u+h}\cdot \frac{x{u}^{\prime }}{u}+\frac{h}{u+h}\cdot \frac{x{h}^{\prime }}{h}$.

The sum rule is why portfolio sensitivities aggregate by value weights. A portfolio 60% in an index fund (elasticity 1) and 40% in cash (elasticity 0) has elasticity 0.6. Exhibit 6 gives each rule a financial reading.

The elasticity of a futures account is its leverage

An account holding $N$ contracts has equity $E(F)=E_0+Nm(F-F_0)$, so $E^{\prime }(F)=Nm$. By (2.7),

The index enters through the chain rule. With the cost-of-carry price $F=S{e}^{(r_f-q)\tau }$ and $r_f$, $q$, $\tau$ fixed, ${\eta }_{F,S}=S{e}^{(r_f-q)\tau }/F=1$. Therefore ${\eta }_{E,S}={\eta }_{E,F}{\eta }_{F,S}=f$.

This covers margin purchases. Buy 1,000 shares of 0050 at an illustrative NT$60 with NT$24,000 of your own and NT$36,000 borrowed: $f=60,000/24,000=2.50$, so a 4% fall removes 10% of equity.

The exchange's margins cap leverage. TAIFEX's table effective 12 August 2026 requires an initial margin of NT$701,000 and a maintenance margin of NT$538,000 per TX contract. The MTX and TMF figures, NT$175,250 and NT$35,050 initial, are exactly 1/4 and 1/20 of TX, in proportion to the multipliers.

TAIFEX recalculates margin levels after each regular session and may adjust them when the change reaches a set threshold. That is why this module uses an illustrative 2025 figure rather than a quoted one. Scaled by TAIFEX's ratio 538,000/701,000, the illustrative initial margin of NT$350,000 corresponds to an illustrative maintenance margin of about NT$268,600.

Leverage moves: its elasticity is $1-f$

Leverage is a ratio, so its elasticity follows from the quotient rule for elasticities and (2.8):

A 1% rise in the futures price cuts Trader A's leverage of 7.3720 by about 6.4%, to a first-order 6.9023. The exact value is 6.9345. On 11 June itself, a 1.0309% rise took A from 7.3720 to 6.9219, against a linear prediction of 6.8877.

Over months the drift is large. Exhibit 7 follows both accounts from the close of 10 June to 31 October 2025, holding one near-month contract and rolling the day before each expiry. TAIEX rose 26.94% and the futures position gained 6,824 points, or NT$1,364,800. A's leverage fell from 7.37 to 2.89 and B's from 4.91 to 2.51.

Despite falling leverage, A's equity rose 227.47% and B's 151.64%. A fixed position's profit is linear in index points. Its return on the initial equity is therefore $f_0$ times the points gained divided by $F_0$: $7.3720\times 6,824/22,116=227.47\%$. Multiplying $f_0$ by TAIEX's return would have predicted only $7.3720\times 26.94\%\approx 198.6\%$.

The futures also gained 832.79 more points than the index (6,824 against 5,991.21). The rolls added a net 510 points: +565 from the June, July and August rolls into cheaper contracts, −55 from the September and October rolls. The other 322.79 points came from the basis, which moved from a 126.14-point discount on 10 June to a 196.65-point premium on 31 October. LM6 explains this basis.

7Differentials and Linear Approximation

The differential

A derivative turns into an approximation by running the definition backwards. If $\Delta x$ is small, $h(x_0+\Delta x)-h(x_0)$ is close to $h^{\prime }(x_0)\Delta x$.