Brownian Bridge™

Setup

Market Context

The implied vol surface $\sigma_{\mathrm{imp}}(K, T)$ is the complete language of the options market. Every option price, hedge ratio, and P&L scenario is rooted in this surface. Calibrating it accurately and consistently is a core task on any derivatives desk.

A vol surface must satisfy two properties to be financially meaningful:

Arbitrage-free: no calendar spread, butterfly, or call spread arbitrage is implied.
Smooth: the surface must be differentiable enough to produce well-defined local vols via the Dupire formula, and to compute stable vega ladders.

Fitting a surface point-by-point (interpolating market quotes directly) satisfies neither: it typically produces arbitrage-violating kinks and is unstable off-grid. A parametric smile model is used instead.

Conventions

Throughout this module:

Total implied variance: $w(k, T) = \sigma_{\mathrm{imp}}^2(k, T) \cdot T$ , where $k = \ln(K/F)$ is the log-moneyness and $F = S e^{(r-q)T}$ is the forward.
Working with $w$ instead of $\sigma$ simplifies arbitrage conditions (they become inequalities on $\partial w / \partial k$ and $\partial^2 w / \partial k^2$ ).
A vol slice at fixed $T$ is an implied vol smile. A collection of smiles across maturities is the vol surface.

Theory: The Raw SVI Parametrisation

Motivation: Derman-Kani and the Wings

At extreme log-moneyness $|k| \to \infty$ , the implied vol smile must grow roughly linearly in $|k|$ . This follows from Lee's moment formula: if the $p$ -th moment of the stock price is finite under $\mathbb{Q}$ , the right tail of $\sigma_{\mathrm{imp}}$ grows at most as $\sqrt{|k|/T}$ times a function of $p$ . More precisely:

$\limsup_{k \to +\infty} \frac{\sigma_{\mathrm{imp}}^2(k, T) \cdot T}{|k|} \leq \psi(\tilde{p}),$

where $\tilde{p}$ is the critical moment. Any smile that grows faster than $\sqrt{|k|}$ in total variance violates moment bounds and implies arbitrage. The SVI form is designed to reproduce this linear-in- $|k$ wing behaviour exactly.

Raw SVI

Gatheral (2004) proposed the Stochastic Volatility Inspired (SVI) parametrisation. For a fixed maturity $T$ , the total implied variance as a function of log-moneyness $k$ is:

$w(k) = a + b\!\left[\rho(k - m) + \sqrt{(k-m)^2 + \sigma^2}\right],$

where the five raw SVI parameters $(a, b, \rho, m, \sigma)$ satisfy:

$a \in \mathbb{R}$ : overall level of total variance (vertical shift).
$b \geq 0$ : slope of the wings; controls how fast variance grows with $|k - m|$ .
$\rho \in (-1, 1)$ : correlation-like parameter; controls the asymmetry between left and right wings (skew).
$m \in \mathbb{R}$ : location of the ATM vertex (horizontal shift); usually $\approx 0$ .
$\sigma > 0$ : smoothness of the vertex; larger $\sigma$ gives a wider, rounder bottom.

Geometric Interpretation

The graph of $w(k)$ is a rotated hyperbola with:

Asymptotes: as $k \to +\infty$ , $w(k) \approx a + b(1+\rho)(k-m)$ (right wing slope $b(1+\rho)$ ); as $k \to -\infty$ , $w(k) \approx a + b(\rho-1)(k-m)$ (left wing slope $b(\rho-1) < 0$ , so the wing rises).
Vertex: at $k = m$ , $w(m) = a + b\sigma$ . This is the minimum of the smile (for the call side) when $\rho = 0$ .
ATM vol: $\sigma_{\mathrm{ATM}} = \sqrt{w(0)/T}$ . Since $m \approx 0$ in typical calibrations, $w(0) \approx a + b\sigma$ .

Note: the left wing slope $b(\rho - 1)$ is always negative (wings rise on both sides) since $b \geq 0$ and $\rho < 1$ . The right wing slope $b(1 + \rho)$ is always non-negative. The asymmetry between wings is controlled by $\rho$ : negative $\rho$ (typical for equities) steepens the left wing and flattens the right, consistent with the negative equity skew.

Arbitrage-Free Conditions

A slice $w(k)$ is free of static arbitrage if and only if:

Condition 1: No Butterfly Arbitrage

Butterfly arbitrage is absent if and only if the function

$g(k) = \left(1 - \frac{k\, w'(k)}{2w(k)}\right)^2 - \frac{(w'(k))^2}{4}\!\left(\frac{1}{w(k)} + \frac{1}{4}\right) + \frac{w''(k)}{2} \geq 0 \quad \text{for all } k,$

where $w' = dw/dk$ and $w'' = d^2w/dk^2$ . This condition (Gatheral 2004, following Dupire) ensures the local variance implied by the Dupire formula is non-negative:

$\sigma_{\mathrm{loc}}^2(k, T) = \frac{\partial w / \partial T}{g(k)}.$

If $g(k) < 0$ for some $k$ , the local variance is negative — an immediate arbitrage.

For the raw SVI form, $g(k) \geq 0$ is not automatic: it must be checked or enforced by constraining the parameters.

Necessary condition (Lee's bound): A weaker, necessary condition is:

$0 \leq w'(k) \leq \frac{4}{k^2}\left(1 + \sqrt{1 - k w'(k)/2}\right) \quad \text{for all } k \neq 0.$

A sufficient condition for no butterfly arbitrage that is easier to check analytically: for the raw SVI form, $b(1 + |\rho|) \leq 4/T$ (Roper 2010).

Condition 2: No Calendar Spread Arbitrage

The total implied variance must be non-decreasing in maturity for each fixed log-moneyness:

$w(k, T_1) \leq w(k, T_2) \quad \text{for all } k, \quad T_1 \leq T_2.$

If this fails at any point, a calendar spread (long far-dated call, short near-dated call at the same log-moneyness) is riskless arbitrage.

Condition 3: No Call Spread Arbitrage

Total variance must satisfy:

$w'(k) > -2 \quad \text{for all } k.$

(Equivalent to the call price being non-increasing in strike.)

Jump-Wings Parametrisation

The raw SVI parameters $(a, b, \rho, m, \sigma)$ are poorly conditioned for optimisation: small changes in $m$ and $a$ can produce large changes in the smile shape, and the parameters are correlated. The jump-wings (JW) parametrisation (Gatheral and Jacquier 2014) reparametrises in terms of quantities with direct financial meaning: