Volatility Band (VB): Option Pricing Bounds

• Volatility Band (VB) is a rigorous, model-independent interval that bounds Black–Scholes implied volatility to ensure no-arbitrage pricing.
• It employs optimization-based variational formulations to derive explicit upper and lower bounds, facilitating numerical inversion and efficient computation.
• VBs are practical tools for model validation, risk management, and super-replication pricing under both static and stochastic volatility conditions.

A volatility band (VB) is a rigorous, model-independent interval enclosing the Black–Scholes implied volatility, defined for arbitrary moneyness and call price. VB encapsulates both theoretical and practical facets: as analytical upper and lower bounds derived from optimized representations of option prices, and as a robust tool for model validation, risk management, and numerical inversion of implied volatility. The concept of volatility bands can also be generalized to settings with uncertain or stochastic bounds on volatility, as in super-replication pricing under volatility-uncertainty (Tehranchi, 2015, &&&1&&&).

1. Theoretical Foundation and Definition

The canonical setting is the normalized Black–Scholes call price function $BS(k,y)$ in terms of log-moneyness $k$ and total volatility $y > 0$:

$$BS(k, y) = \Phi(-k/y + y/2) - e^k \Phi(-k/y - y/2),$$

where $\Phi$ denotes the standard normal CDF. For any arbitrage-free call price $c$ and log-moneyness $k$, the total implied volatility $y = BS^{-1}(k, c)$ solves $BS(k, y) = c$, and the implied volatility is $\sigma^{imp} = y / \sqrt{T}$.

A volatility band $VB(k, c)$ is a deterministic interval $[\sigma_L(k, c), \sigma_U(k, c)]$ (with explicit formulae) such that for all $k$ and all $c$ in the option price’s no-arbitrage range, the true implied volatility satisfies:

$$\sigma_L(k, c) \leq \sigma^{imp}(k, c) \leq \sigma_U(k, c).$$

The construction of VB relies on variational formulations of $BS(k, c)$ as one-dimensional optimization problems, enabling explicit bound derivation (Tehranchi, 2015).

2. Optimization-Based Derivation and Explicit Bounds

The implied total standard deviation $y = BS^{-1}(k, c)$ admits dual variational characterizations. Introduce the parametric functions

$$H_1(d; k, c) = d - \Phi^{-1}(e^{-k} (\Phi(d) - c)), \ H_2(d; k, c) = \Phi^{-1}(c + e^k \Phi(d)) - d.$$

Then, for every $(k, c)$ in the relevant range,

$$BS(k, c) = \inf_{d \in \mathbb{R}} H_1(d; k, c) = \inf_{d \in \mathbb{R}} H_2(d; k, c).$$

Insertion of specific trial values for $d$ yields uniform, explicit upper and lower bounds for $y$, and thus for $\sigma^{imp}$:

• For $k \geq 0$:

$$\sigma_L(k, c) = \frac{-2 \Phi^{-1}\left(\frac{1-c}{2}\right)}{\sqrt{T}}, \quad \sigma_U(k, c) = \frac{-2 \Phi^{-1}\left(\frac{1-c}{1+e^k}\right)}{\sqrt{T}}.$$

• For $k < 0$, the lower bound is replaced by

$$\sigma_L(k, c) = \frac{-2 \Phi^{-1}\left(\frac{1-c}{2e^k}\right)}{\sqrt{T}}.$$

The band $[ \sigma_L(k, c),\; \sigma_U(k, c) ]$ thus traps the true implied volatility for any non-negative, arbitrage-free call price surface. These bounds arise directly from the minimization program, with the upper bound corresponding to a suitable trial $d_2$ in $H_2$ and the lower to a comparison with the ATM case $k=0$ (Tehranchi, 2015).

3. Symmetry Properties and Asymptotic Regimes

The Black–Scholes formula possesses symmetries that enlarge the range of volatility bands:

• Put–Call Symmetry: $BS(k, c) = BS(-k, e^{-k} c + 1 - e^{-k})$. This reduces the study of VBs for $k<0$ to those for $k>0$.
• Volatility Inversion Symmetry: For

$\hat{c} = 1 - \int_0^c \frac{2k}{[BS(k, u)]^2} du,$

one has $BS(k, c) = 2k / BS(k, \hat{c})$, transforming valid bounds for $BS(k, c)$ into new ones for $BS(k, \hat{c})$ and hence for $BS(k, c)$.

Asymptotic analysis of the uniform bounds yields leading-order behavior for extreme strikes, maturities, and prices. For example:

• As $c\rightarrow 1$ (long-maturity, in-the-money regime):

$$BS(k, c) = \sqrt{-8 \ln(1-c)} + O\left(\frac{\ln[-\ln(1-c)]}{\sqrt{-\ln(1-c)}}\right)$$

• As $c \to 0$ (short-maturity, far out-of-the-money), and for right/left wings, analogous expansions follow directly by inserting bounds into limits (Tehranchi, 2015).

4. Numerical Implementation and Computational Aspects

The explicit construction of the volatility band enables efficient numerical procedures. For any model or calibrated price curve $(k, T) \mapsto c$, corresponding VBs can be computed by applying the closed-form formulas. These bands provide initial bracketing intervals for root-finding (e.g., bisection or Newton–Raphson) in numerical inversion for implied volatility, ensuring global convergence.

Numerical illustrations confirm that the bands $[y_L, y_U]$ frequently provide sharp enclosures, especially in the wings and for extreme maturities. However, in the at-the-money (ATM) region, the band may be relatively wide (conservative), and tightening it remains an open challenge (Tehranchi, 2015).

5. Applications in Model Validation, Risk Management, and Super-Replication

Volatility bands possess several practical applications:

• Model Validation: For any parametric or nonparametric model yielding theoretical prices $c_{model}(k, T)$, the test $\sigma_{model}(k, T) \in VB(k, c_{model})$ can immediately detect misspecification by identifying price-implied volatility values outside the theoretically admissible band.
• Risk Management: VB supplies robust, worst-case bounds for implied volatility under arbitrary shape of the call price surface, essential for stress testing derivative portfolios.
• Numerical Inversion: The deterministic bands localize the true implied volatility, enabling guaranteed convergence when numerically inverting for $\sigma^{imp}$.
• Super-Replication under Volatility Uncertainty: In an extended framework where volatility itself lies within stochastic bands—i.e., a process bounded by $L_t$, $U_t$ where $L_t = d\sqrt{Z_t}$, $U_t = u\sqrt{Z_t}$ for a slow mean-reverting $Z_t$—the concept of volatility bands appears in the optimal control formulation of the super-replication price. The link is via a Hamilton–Jacobi–Bellman (HJB) PDE, whose nonlinear control term reflects stochastic bounds (Fouque et al., 2017).

6. Extensions: Stochastic Bands, Computational Reduction, and Open Problems

The volatility band methodology extends beyond the static, model-independent Black–Scholes setting.

• Uncertain Volatility Model with Stochastic Bands: In the framework where volatility bounds are themselves stochastic, e.g. $L_t = d\sqrt{Z_t}$ and $U_t = u\sqrt{Z_t}$ with $Z_t$ a CIR process, the robust option price is the supremum over all volatilities in $[L_t, U_t]$. The super-replication price satisfies a two-dimensional, fully nonlinear HJB–Black–Scholes–Barenblatt PDE. Using regular perturbation (expansion in small $\delta$, the time-scale parameter for $Z_t$), the computational complexity reduces dramatically—only two families of one-dimensional PDEs must be solved, as opposed to a full two-dimensional PDE (Fouque et al., 2017). The bands themselves now reflect market-observed stochasticity, e.g. via the VIX index.
• Open Problems: While VBs are tight in the wings and for extreme maturities, the ATM region remains a domain where band width may be overly conservative. Further, extensions to local-volatility or general pricing kernels, as well as fully multidimensional (strike, maturity) analogues that preserve arbitrage conditions, are not yet systematically explored (Tehranchi, 2015).

7. Summary Table: Volatility Band Formulas for $k \geq 0$

Quantity	Formula	Domain
Lower bound $\sigma_L$	$-2\Phi^{-1}\left(\frac{1-c}{2}\right)/\sqrt{T}$	$k \geq 0$, $c \in [(1-e^k)^+,1)$
Upper bound $\sigma_U$	$-2\Phi^{-1}\left(\frac{1-c}{1+e^k}\right)/\sqrt{T}$	$k \geq 0$, $c \in [(1-e^k)^+,1)$
Implied volatility $\sigma^{imp}$	Satisfies $\sigma_L \leq \sigma^{imp} \leq \sigma_U$	$k \geq 0$, $c \in [(1-e^k)^+,1)$

These formulas and their counterparts for $k<0$ provide the explicit volatility bands essential to all outlined practical and theoretical applications (Tehranchi, 2015). In stochastic and uncertain volatility environments, the concept is further enriched and retains computational and theoretical tractability (Fouque et al., 2017).