Squeeze Theorem and Bounds

Motivation: why this matters in quant finance

Many results in quantitative finance are proved not by computing a limit directly but by trapping it between two simpler quantities that both converge to the same value. The squeeze theorem (also called the sandwich theorem) formalises this. It is the tool behind convergence proofs for discrete-to-continuous limits (binomial tree → Black-Scholes), for showing that remainder terms in Taylor expansions vanish, and for establishing that quadratic variation sums converge.

Beyond the squeeze theorem itself, the broader skill of bounding — finding upper and lower estimates for quantities you cannot compute exactly — is central to quant finance. No-arbitrage bounds on option prices, worst-case portfolio losses, error bounds on numerical methods, and tail probability estimates all rely on inequalities and bounding arguments rather than exact calculation.

The Squeeze Theorem

Statement

If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $a$ (except possibly at $a$ ), and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$ then $\lim_{x \to a} f(x) = L$ .

The function $f$ is "squeezed" between $g$ and $h$ , which both converge to $L$ , so $f$ must also converge to $L$ . The theorem works equally for $x \to a$ , $x \to \infty$ , and for sequences ( $n \to \infty$ ).

Proof idea

For any $\varepsilon > 0$ , the convergence of $g$ and $h$ gives a $\delta$ such that $|g(x) - L| < \varepsilon$ and $|h(x) - L| < \varepsilon$ for $0 < |x - a| < \delta$ . Then $L - \varepsilon < g(x) \leq f(x) \leq h(x) < L + \varepsilon$ , so $|f(x) - L| < \varepsilon$ .

Key inequalities and bounding tools

Supremum and infimum

The supremum (least upper bound) and infimum (greatest lower bound) of a set

A \subseteq \mathbb{R}

\sup A = \text{smallest number} \geq \text{every element of } A

\inf A = \text{largest number} \leq \text{every element of } A

Unlike $\max$ and $\min$ , the sup and inf always exist (in the extended reals) even when the set does not attain its extreme values.

Finance application: The essential supremum

\text{ess}\sup X

of a random variable is the smallest value

M

such that

\mathbb{P}(X > M) = 0

. It defines the worst-case scenario in an almost-sure sense and is used in robust risk measures, model-free option bounds, and the definition of

L^\infty

spaces.

Limsup and liminf

For a sequence $\{a_n\}$ :

\limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{k \geq n} a_k = \inf_n \sup_{k \geq n} a_k

\liminf_{n \to \infty} a_n = \lim_{n \to \infty} \inf_{k \geq n} a_k = \sup_n \inf_{k \geq n} a_k

The limsup is the largest accumulation point of the sequence; the liminf is the smallest. The sequence converges if and only if $\limsup = \liminf$ , in which case both equal the limit.

Finance application: In the theory of large deviations and extreme value theory, the limsup characterises the asymptotic behaviour of maxima. For instance, the law of the iterated logarithm states that for Brownian motion:

\limsup_{t \to \infty} \frac{W_t}{\sqrt{2t \ln\ln t}} = 1 \quad \text{a.s.}

This gives the precise envelope within which Brownian paths fluctuate — tighter than $\sqrt{t}$ but still unbounded.

Triangle inequality

|a + b| \leq |a| + |b|, \qquad \left|\sum a_i\right| \leq \sum |a_i|

Used constantly in bounding portfolio risk: the absolute P&L of a portfolio is bounded by the sum of absolute P&Ls of its components. This is the basis of the sub-additivity property of coherent risk measures.

Jensen's inequality

If $g$ is convex and $X$ is a random variable with $\mathbb{E}[|X|] < \infty$ :

g(\mathbb{E}[X]) \leq \mathbb{E}[g(X)]

Finance application: Since

e^x

is convex,

e^{\mathbb{E}[X]} \leq \mathbb{E}[e^X]

. This is why the mean of a log-normal is

e^{\mu + \sigma^2/2} > e^{\mu}

— the convexity correction

\sigma^2/2

is the gap between the two sides of Jensen's inequality. This same inequality underlies the

-\sigma^2/2

drift correction in Itô's Lemma applied to

\ln S

Jensen's inequality also gives no-arbitrage bounds: since $\max(S - K, 0)$ is convex in $S$ , the option price satisfies $C \geq \max(\mathbb{E}^{\mathbb{Q}}[S_T] e^{-rT} - Ke^{-rT}, 0) = \max(S_0 - Ke^{-rT}, 0)$ — the familiar lower bound.

Cauchy-Schwarz and Chebyshev

Cauchy-Schwarz:

|\mathbb{E}[XY]| \leq \sqrt{\mathbb{E}[X^2]\,\mathbb{E}[Y^2]}

. Used in bounding covariances and correlations.

Chebyshev's inequality:

\mathbb{P}(|X - \mu| \geq k\sigma) \leq 1/k^2

. A model-free tail bound — it holds for any distribution with finite variance, giving a worst-case probability for large deviations. It is weaker than Gaussian tail bounds but applies more broadly.

Examples and applications

Example 1: proving $\lim_{x \to 0} x^2 \sin(1/x) = 0$

The function $\sin(1/x)$ oscillates wildly near $x = 0$ , but:

-x^2 \leq x^2 \sin(1/x) \leq x^2

Both $-x^2 \to 0$ and $x^2 \to 0$ as $x \to 0$ , so by the squeeze theorem, $x^2\sin(1/x) \to 0$ .

Analogy in finance: When a pricing error term oscillates but is bounded by a quantity that shrinks (e.g., the discretisation error in a binomial tree is bounded by

O(1/n)

regardless of the specific path), the squeeze theorem proves convergence even though the error may not be monotone.

Example 2: no-arbitrage bounds on option prices

For a European call with strike $K$ , maturity $T$ :

\max(S_0 - Ke^{-rT}, 0) \leq C \leq S_0

The lower bound follows from put-call parity and the non-negativity of the put. The upper bound follows because the call payoff $(S_T - K)^+ \leq S_T$ , so $C = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T - K)^+] \leq e^{-rT}\mathbb{E}^{\mathbb{Q}}[S_T] = S_0$ .

These are model-free bounds — they hold regardless of the stock price dynamics. Any model that produces a price outside these bounds admits arbitrage. The bounds squeeze the option price into a known interval without computing it exactly.

Example 3: quadratic variation convergence

The quadratic variation of Brownian motion is proved by showing:

\mathbb{E}\left[\left(\sum_{i}(\Delta W_i)^2 - T\right)^2\right] = \sum_i \text{Var}((\Delta W_i)^2) = 2\sum_i (\Delta t_i)^2 \leq 2\|\Delta\| \cdot T \to 0

The variance of the sum is bounded above by $2\|\Delta\|T$ , which tends to zero as the mesh $\|\Delta\| \to 0$ . This is a bounding argument that establishes $L^2$ convergence without computing the sum exactly.

Example 4: bounding the Taylor remainder

The Lagrange remainder of the Taylor expansion of

f

at order

N

satisfies:

|R_N(h)| \leq \frac{M_{N+1}}{(N+1)!}|h|^{N+1}

where $M_{N+1} = \sup |f^{(N+1)}(\xi)|$ over the relevant interval. This bound shows the remainder is $O(h^{N+1})$ and vanishes as $h \to 0$ . In the Itô expansion, this bounding argument proves that third- and higher-order terms vanish as $\Delta t \to 0$ , justifying the truncation at second order.

Common confusions and pitfalls

"The squeeze theorem requires $g \leq f \leq h$ everywhere." No — only near the limit point. The inequality can fail far from

a

; what matters is that it holds in some neighbourhood.

Confusing sup with max. The supremum may not be attained:

\sup\{1 - 1/n : n \geq 1\} = 1

, but no element of the set equals 1. The Extreme Value Theorem guarantees that continuous functions on closed bounded intervals attain their sup, but in general, sup and max are different.

Applying Jensen's inequality backwards. Jensen gives

g(\mathbb{E}[X]) \leq \mathbb{E}[g(X)]

for convex

g

and

g(\mathbb{E}[X]) \geq \mathbb{E}[g(X)]

for concave

g

. Misidentifying the convexity direction flips the inequality.

Where this goes next

The squeeze theorem and bounding tools connect to:

Limits: The squeeze theorem is a limit theorem — it establishes limits indirectly via bounds.
Taylor Series: Remainder bounds are bounding arguments that justify truncation.
Convergence of Sequences and Series: Comparison tests for series convergence are squeeze-theorem-style arguments applied to partial sums.
Brownian Motion: Quadratic variation convergence and the law of the iterated logarithm use bounding and limsup arguments.
Improper Integrals: Comparison tests for integral convergence are the continuous analogues of the squeeze theorem.

References

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals (6th ed.). Thomson Brooks/Cole. Ch. 2 Section 2.3 (Limit Laws and Squeeze Theorem), Ch. 11 Sections 11.3-11.4 (Integral and Comparison Tests).