Taylor Series

Motivation: why this matters in quant finance

The Taylor series is the tool that turns differentiation from a local operation (the slope at a point) into a global approximation (the entire function, rebuilt from its derivatives at a single point). In quantitative finance, this bridge is used in three essential ways:

It is the starting point of Itô's Lemma. When you want to find $dv(S, t)$ — how the option price changes as the stock price and time evolve — you begin with a Taylor expansion of $v$ around the current point. In ordinary calculus, you keep the first-order terms and drop everything else. In stochastic calculus, the Brownian motion increment $\Delta W \sim \sqrt{\Delta t}$ makes the second-order term survive, and that is Itô's Lemma. The Taylor series is therefore the direct ancestor of every stochastic differential equation in quant finance.
It provides local polynomial approximations for pricing and risk. The Greeks — delta, gamma, theta, and the rest — are the coefficients of a Taylor expansion of the option price around the current market state. The P&L of a delta-hedged portfolio over a small time interval is approximated by:

\Delta\text{P\&L} \approx \Theta\,\Delta t + \frac{1}{2}\Gamma\,(\Delta S)^2 + \cdots

This is a second-order Taylor expansion in

\Delta S

and

\Delta t

. The approximation is accurate for small moves and breaks down for large moves — knowing when to truncate and why the higher-order terms matter is the essence of understanding Taylor series in a quant context.

It connects discrete and continuous formulations. The Taylor expansion of $e^x = 1 + x + x^2/2! + \cdots$ is what turns discrete compounding into continuous compounding; the expansion of $\ln(1 + r) \approx r - r^2/2 + \cdots$ is what converts simple returns to log-returns. These are not just algebraic tricks — they are Taylor series applied to specific functions, and knowing the error term tells you when the approximation is safe.

Definition and setup

Single-variable Taylor series

Let

f

be a function that is infinitely differentiable (smooth) at a point

a

. The Taylor series of

f

about

a

is:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots

where

f^{(n)}(a)

is the

n

th derivative of

f

evaluated at

a

, and

n! = n \times (n-1) \times \cdots \times 1

is the factorial.

When

a = 0

, this is called the Maclaurin series:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\,x^n

Notation convention: Throughout this page and the rest of the vault, we use

h = x - a

for the displacement from the expansion point. The Taylor series then reads:

f(a + h) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}\,h^n

Taylor polynomials (finite truncation)

The $N$ th-order Taylor polynomial is the partial sum:

T_N(x) = \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n

This is a polynomial of degree at most $N$ that agrees with $f$ at $a$ in value and in all derivatives up to order $N$ . The first few:

Order	Polynomial	Name
0	$T_0 = f(a)$	Constant approximation
1	$T_1 = f(a) + f'(a)\,h$	Linear (tangent line)
2	$T_2 = f(a) + f'(a)\,h + \frac{1}{2}f''(a)\,h^2$	Quadratic

In quant finance, the second-order polynomial is by far the most important because it is the truncation that Itô's Lemma uses. The first-order polynomial is the ordinary chain rule; the second-order polynomial is Itô's correction.

The remainder (error term)

The difference between

f

and its Taylor polynomial is the remainder

R_N

f(a + h) = T_N(a + h) + R_N(a + h)

Taylor's theorem (Lagrange form): there exists some

\xi

between

a

and

a + h

such that:

R_N = \frac{f^{(N+1)}(\xi)}{(N+1)!}\,h^{N+1}

The remainder is $O(h^{N+1})$ , meaning it shrinks faster than $h^N$ as $h \to 0$ . This is why truncation works: for small $h$ , the error from dropping order- $(N+1)$ terms is negligibly small compared to the terms you kept.

Why this matters for Itô's Lemma: In the deterministic case,

h = \Delta x = O(\Delta t)

, so the second-order term

\frac{1}{2}f''h^2 = O(\Delta t^2)

vanishes as

\Delta t \to 0

. But when

h = \Delta W = O(\sqrt{\Delta t})

, the second-order term is

O(\Delta t)

— it survives. The third-order term is

O(\Delta t^{3/2})

and vanishes. This is why the Itô expansion truncates at exactly second order: the

\sqrt{\Delta t}

scaling of Brownian increments promotes one extra term from "negligible" to "leading order."

Key Taylor expansions

The following expansions are used throughout quant finance. All are Maclaurin series ( $a = 0$ ) unless stated otherwise, valid for the indicated range of $x$ .

Exponential

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} \qquad \text{for all } x \in \mathbb{R}

This converges everywhere and converges rapidly. It is the most important series in quantitative finance because the exponential appears in discount factors (

e^{-rT}

), stock price models (

S_T = S_0 e^{(\mu - \sigma^2/2)T + \sigma W_T}

), and the moment generating function of the normal distribution.

Finance application — small-rate approximation: For small

r\Delta t

e^{-r\Delta t} \approx 1 - r\Delta t + \frac{(r\Delta t)^2}{2} - \cdots \approx 1 - r\Delta t

This first-order approximation is used throughout fixed income when time steps are small. The error is $O((r\Delta t)^2)$ , which is negligible for daily or even monthly steps when $r$ is moderate.

Natural logarithm

\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\,x^n \qquad \text{for } -1 < x \leq 1

Finance application — log-returns vs simple returns: If the simple return over a period is

R = \Delta S / S

(a small number), then the log-return is:

\ln(1 + R) \approx R - \frac{R^2}{2}

For small returns (e.g., daily equity moves of ~1%), $R^2/2$ is tiny and log-returns $\approx$ simple returns. For large returns (e.g., a 50% crash, $R = -0.5$ ), the difference is significant: $\ln(0.5) = -0.693 \neq -0.5$ . The Taylor expansion tells you exactly when the approximation breaks down and by how much.

Geometric series

\frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots = \sum_{n=0}^{\infty} x^n \qquad \text{for } |x| < 1

Finance application — present value of a perpetuity: The PV of an infinite stream of payments

c

discounted at rate

r

per period is:

\text{PV} = \sum_{n=1}^{\infty} \frac{c}{(1+r)^n} = \frac{c}{r}

This is the geometric series with $x = 1/(1+r)$ , summed and rearranged. The convergence condition $|x| < 1$ corresponds to $r > 0$ — the discount rate must be positive for the PV to be finite.

Power function (binomial series)

(1 + x)^\alpha = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^n = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^2 + \cdots \qquad \text{for } |x| < 1

where $\binom{\alpha}{n} = \frac{\alpha(\alpha-1)\cdots(\alpha - n + 1)}{n!}$ is the generalised binomial coefficient. When $\alpha$ is a positive integer, the series terminates (it becomes the binomial theorem). For non-integer $\alpha$ , the series is infinite.

Finance application — duration and convexity: The price of a bond as a function of yield

y

can be expanded around a base yield

y_0

P(y_0 + \Delta y) \approx P(y_0) + P'(y_0)\,\Delta y + \frac{1}{2}P''(y_0)\,(\Delta y)^2

The first-order term is (negative) duration

\times

price

\times \Delta y

; the second-order term is convexity

\times

price

\times (\Delta y)^2 / 2

. This is a second-order Taylor expansion of the bond price around the current yield — the same structure that appears in the Greeks expansion for options.

Multivariable Taylor expansion

Two-variable expansion

For a function $f(x, y)$ expanded around $(a, b)$ , with $h = x - a$ and $k = y - b$ :

f(a + h, \, b + k) = f(a, b) + f_x h + f_y k + \frac{1}{2}\left(f_{xx} h^2 + 2f_{xy} hk + f_{yy} k^2\right) + \cdots

where all partial derivatives are evaluated at

(a, b)

. The first-order terms give the total differential

df = f_x\,dx + f_y\,dy

(the multivariable chain rule); the second-order terms add the curvature corrections.

The expansion that becomes Itô's Lemma

This is the central connection in the vault. Let $v(S, t)$ be a twice-differentiable function (the option price), and let $S$ change by $\Delta S$ and $t$ by $\Delta t$ . The second-order Taylor expansion is:

\Delta v \approx v_t\,\Delta t + v_S\,\Delta S + \frac{1}{2}v_{SS}\,(\Delta S)^2 + v_{St}\,\Delta S\,\Delta t + \frac{1}{2}v_{tt}\,(\Delta t)^2

Now substitute the stock price dynamics. In the Black-Scholes model,

\Delta S = \mu S\,\Delta t + \sigma S\,\Delta W

. The key step is determining which terms survive as

\Delta t \to 0

Term	Order	Survives?
$v_t\,\Delta t$	$O(\Delta t)$	Yes
$v_S\,\Delta S$	$O(\sqrt{\Delta t})$	Yes
$\frac{1}{2}v_{SS}\,(\Delta S)^2$	$O(\Delta t)$ via $(\Delta W)^2 = O(\Delta t)$	Yes
$v_{St}\,\Delta S\,\Delta t$	$O(\Delta t^{3/2})$	No
$\frac{1}{2}v_{tt}\,(\Delta t)^2$	$O(\Delta t^2)$	No

The third row is the Itô correction. In ordinary calculus,

(\Delta S)^2 = O(\Delta t^2)

and the entire second-order block vanishes. But with Brownian motion,

(\Delta W)^2 = O(\Delta t)

, so

(\Delta S)^2 = \sigma^2 S^2 (\Delta W)^2 + \cdots = \sigma^2 S^2 \Delta t + \cdots

, and the

v_{SS}

term survives. The result, after passing to the limit, is Itô's Lemma:

dv = v_t\,dt + v_S\,dS + \frac{1}{2}v_{SS}\,\sigma^2 S^2\,dt

This is a Taylor expansion truncated at second order — but truncated at a different place than in ordinary calculus, because the

\sqrt{\Delta t}

scaling of Brownian increments changes which terms are "leading order."

The Black-Scholes derivation uses exactly this expansion: the

v_t\,dt

term is theta, the

v_S\,dS

term is delta times the stock move, and the

\frac{1}{2}v_{SS}\sigma^2 S^2\,dt

term is the gamma correction. Setting up a hedged portfolio that cancels the

dW_t

term and requiring the riskless remainder to earn the risk-free rate

r

gives the Black-Scholes PDE.

Convergence

Radius of convergence

A Taylor series

\sum c_n (x - a)^n

converges for

|x - a| < R

and diverges for

|x - a| > R

, where

R

is the radius of convergence. At

|x - a| = R

, behaviour must be checked case by case.

For the key series:

Function	Centre	Radius of convergence
$e^x$	0	$\infty$
$\ln(1+x)$	0	1
$\frac{1}{1-x}$	0	1
$(1+x)^\alpha$	0	1
$\sin x$ , $\cos x$	0	$\infty$

The exponential and trigonometric series converge everywhere; the logarithm and geometric series converge only for $|x| < 1$ (with possible endpoint convergence).

Finance implication: The approximation

\ln(1 + R) \approx R

is a Taylor series with radius of convergence 1. For

R > 1

(a return exceeding 100%) or

R \leq -1

(a total loss), the series diverges and the approximation is meaningless. In practice, daily returns are far inside the radius, but cumulative returns over long horizons may not be.

Analytic functions

A function is analytic at

a

if its Taylor series converges to

f(x)

in some neighbourhood of

a

. Not all infinitely differentiable functions are analytic — the classic counterexample is

f(x) = e^{-1/x^2}

(with

f(0) = 0

), which has all derivatives zero at

x = 0

but is not identically zero. In quant finance, virtually all pricing functions encountered in practice (Black-Scholes, bond pricing, etc.) are analytic where they are smooth, so this subtlety rarely causes problems.

Examples and applications

Example 1: Greeks as Taylor coefficients

The P&L of an option position over a small time step $\Delta t$ with stock move $\Delta S$ is, to second order:

\Delta v \approx \underbrace{v_S}_{\Delta}\,\Delta S + \underbrace{v_t}_{\Theta}\,\Delta t + \frac{1}{2}\underbrace{v_{SS}}_{\Gamma}\,(\Delta S)^2

This is a Taylor expansion with the derivatives named as Greeks. Each Greek is a Taylor coefficient:

Delta ( $\Delta = v_S$ ): the coefficient of $\Delta S$ . First-order sensitivity.
Theta ( $\Theta = v_t$ ): the coefficient of $\Delta t$ . Time decay.
Gamma ( $\Gamma = v_{SS}$ ): the coefficient of $(\Delta S)^2 / 2$ . Curvature correction.
Higher-order Greeks (speed $= v_{SSS}$ , charm $= v_{St}$ , etc.) are coefficients of higher-order Taylor terms.

A delta-hedged portfolio eliminates the $\Delta S$ term, leaving:

\Delta\text{P\&L}_{\text{hedged}} \approx \Theta\,\Delta t + \frac{1}{2}\Gamma\,(\Delta S)^2

This is the theta-gamma trade-off: a delta-hedged option earns theta (time decay) and pays gamma (cost of convexity), or vice versa. For a long call position,

\Theta < 0

and

\Gamma > 0

: you lose from time decay but gain from large moves. The Taylor expansion makes this decomposition precise.

Example 2: duration and convexity of a bond

Let $P(y)$ be the price of a bond as a function of yield $y$ . Expand around the current yield $y_0$ :

P(y_0 + \Delta y) \approx P(y_0) + P'(y_0)\,\Delta y + \frac{1}{2}P''(y_0)\,(\Delta y)^2

Define:

\text{Duration} = D = -\frac{P'(y_0)}{P(y_0)}, \qquad \text{Convexity} = C = \frac{P''(y_0)}{P(y_0)}

Then the percentage price change is:

\frac{\Delta P}{P} \approx -D\,\Delta y + \frac{1}{2}C\,(\Delta y)^2

Duration gives the first-order sensitivity (analogous to delta); convexity gives the second-order correction (analogous to gamma). For small yield changes, duration dominates. For large yield changes, the convexity term matters — it always helps (convexity is positive for vanilla bonds), which is why bond traders say "convexity is your friend."

This is structurally identical to the options Greeks expansion. Both are second-order Taylor expansions of a pricing function around the current state.

Example 3: why $\ln(1 + R) \approx R - R^2/2$ matters

If a stock returns $R = 5\%$ in simple terms, the log-return is:

\ln(1.05) = 0.05 - \frac{0.05^2}{2} + \frac{0.05^3}{3} - \cdots = 0.04879\ldots

The first-order approximation ( $\ln(1+R) \approx R = 0.05$ ) overestimates by about 0.12%. The second-order approximation ( $R - R^2/2 = 0.05 - 0.00125 = 0.04875$ ) is accurate to 0.004%.

Now consider a crash: $R = -30\%$ .

\ln(0.70) = -0.3567\ldots

First-order: $-0.30$ (error: $-5.67$ percentage points). Second-order: $-0.30 - 0.045 = -0.345$ (error: $-1.17$ pp). Third-order: $-0.30 - 0.045 - 0.009 = -0.354$ (error: $-0.27$ pp).

The Taylor expansion converges, but slowly for large

|R|

. In risk management, this means linear approximations (VaR based on delta alone) significantly underestimate tail risk. Including the gamma (second-order) term captures the curvature and gives a more accurate loss estimate — exactly the same logic as the Itô correction.

Example 4: the Itô correction as a Taylor remainder

Apply the Taylor expansion to $f(S) = \ln S$ around the current stock price, with $\Delta S = \mu S\,\Delta t + \sigma S\,\Delta W$ :

\Delta(\ln S) \approx \frac{1}{S}\,\Delta S + \frac{1}{2}\left(-\frac{1}{S^2}\right)(\Delta S)^2 = \frac{\Delta S}{S} - \frac{1}{2}\frac{(\Delta S)^2}{S^2}

Now $(\Delta S)^2 \approx \sigma^2 S^2\,\Delta t$ (keeping only the leading term from $(dW)^2 = dt$ ):

\Delta(\ln S) \approx \frac{\Delta S}{S} - \frac{1}{2}\sigma^2\,\Delta t

Passing to the limit:

d(\ln S) = \frac{dS}{S} - \frac{1}{2}\sigma^2\,dt = \mu\,dt + \sigma\,dW - \frac{1}{2}\sigma^2\,dt = \left(\mu - \frac{1}{2}\sigma^2\right)dt + \sigma\,dW

This is the Itô's Lemma result for

\ln S

under geometric Brownian motion. The

-\frac{1}{2}\sigma^2

correction is the second-order Taylor term that ordinary calculus discards but stochastic calculus keeps. This correction determines that log-returns are normally distributed with drift

\mu - \sigma^2/2

(not

\mu

), and it is what makes the stock price log-normally distributed rather than normally distributed.

Common confusions and pitfalls

"The Taylor series always converges to $f$ ." Not necessarily. The series may diverge outside its radius of convergence, or it may converge to a different function (as with non-analytic functions). In quant finance, the functions you encounter are typically analytic where smooth, so this is rarely an issue in practice — but it explains why Taylor-based approximations fail for large perturbations.

"I can always truncate at first order." Only if the second-order term is negligible. In ordinary calculus with smooth functions,

(\Delta x)^2 = O(\Delta t^2) \to 0

and first order suffices. With Brownian motion,

(\Delta W)^2 = O(\Delta t)

and the second-order term is the same order as the first-order terms — you must keep it. This is the entire content of Itô's Lemma: the truncation point shifts from first to second order.

"Higher-order Greeks are always negligible." For small moves and short time steps, yes. But for large moves (gap risk, stress scenarios), third- and higher-order Greeks can dominate. The "speed" (

v_{SSS}

), "colour" (

v_{SSt}

), and other higher-order sensitivities are the third-order Taylor coefficients. Risk managers who rely exclusively on delta-gamma approximations (second-order Taylor) may underestimate losses from extreme moves.

"Duration fully captures interest rate risk." Duration is a first-order Taylor approximation. For large yield changes, the convexity correction (second-order) is material. For very large changes (e.g., a 300bp shock), even convexity may be insufficient and higher-order terms matter. The Taylor expansion makes the error structure explicit.

Confusing the expansion variable with the function argument. In the Itô expansion, you expand

v(S, t)

with respect to

\Delta S

and

\Delta t

, not with respect to

S

and

t

themselves. The expansion point is the current value

(S_t, t)

, and the displacement is the increment

(\Delta S, \Delta t)

. Mixing these up leads to incorrect expressions.

Where this goes next

The Taylor series is the bridge between three parts of the vault:

From limits and differentiation: The Taylor polynomial is built from derivatives, which are limits. The chain rule, product rule, and quotient rule compute the derivatives that become Taylor coefficients.
To Itô's Lemma: The multivariable Taylor expansion of $v(S, t)$ , truncated at second order with the Brownian scaling $(\Delta W)^2 = \Delta t$ , is Itô's Lemma. The entire stochastic calculus correction is a consequence of the Taylor remainder not vanishing at second order when the increment is driven by Brownian motion.
To the Black-Scholes derivation: The derivation begins with the Taylor expansion $dv = v_t\,dt + v_S\,dS + \frac{1}{2}v_{SS}(dS)^2$ , substitutes the GBM dynamics, constructs a hedged portfolio, and obtains the PDE. Every step traces back to the Taylor series on this page.

The Black-Scholes paper itself rests on this chain: the Taylor expansion produces the Itô differential, the Itô differential enables hedging, and hedging produces the no-arbitrage price. The Taylor series is where the chain begins.

References

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals (6th ed.). Thomson Brooks/Cole. Ch. 11 Sections 11.10-11.11 (Taylor and Maclaurin Series) for Taylor polynomials, remainders, and convergence.