Implied Vol Surfaces and Smile Dynamics

Setup

Market Context

Black-Scholes assigns a single constant volatility $\sigma$ to all options on the same underlying. In practice, if you invert the Black-Scholes formula using market prices, the resulting implied volatility varies across strikes and maturities:

$\hat{\sigma}(K, T) = \mathrm{BS}^{-1}(C_{\mathrm{mkt}}(K, T)),$

where $\mathrm{BS}^{-1}$ denotes the unique positive root of $\mathrm{BS}(S, K, r, T, \sigma) = C_{\mathrm{mkt}}(K, T)$.

The function $(K, T) \mapsto \hat{\sigma}(K, T)$ is the implied volatility surface. It is not a model. It is a quotation convention — a compact re-parametrisation of market call prices that separates the structural input (the model) from the market data.

Understanding its shape, arbitrage constraints, and dynamics is prerequisite to calibration of any stochastic volatility model.

Notation and Conventions

Throughout this module:

• $F = S e^{(r-q)(T-t)}$ denotes the forward price ($q$ = dividend yield, set to zero if not stated).
• Moneyness is often expressed as the log-forward moneyness $k = \ln(K/F)$.
• Implied variance is $w(k, T) = \hat{\sigma}^2(K, T) \cdot T$ (total variance; $T$ measured from today).
• Rates are continuously compounded. All vols are annualised.

Arbitrage-Free Conditions on the Surface

Not every surface $\hat{\sigma}(K, T)$ is admissible. Three no-arbitrage conditions must hold, corresponding to three types of static arbitrage:

Call Spread Monotonicity

For fixed $T$, the call price must be non-increasing in $K$:

$\frac{\partial C}{\partial K}(K, T) \leq 0.$

A violation means you can buy the $K_1$-strike call, sell the $K_2$-strike call ($K_2 > K_1$) for a net credit, and still have non-negative payoff — a static long call spread that is free. Equivalent condition on the surface: the implied vol smile cannot rise fast enough in $K$ to reverse the price monotonicity.

Butterfly Positivity (No Negative Density)

For fixed $T$, the second derivative of the call price in $K$ must be non-negative:

$\frac{\partial^2 C}{\partial K^2}(K, T) \geq 0.$

By Breeden-Litzenberger, this second derivative equals $e^{-rT} p^{\mathbb{Q}}_{S_T}(K)$ — the risk-neutral density. Negativity of the density is unacceptable: it would allow a long butterfly spread (long $K-\epsilon$, short $2K$, long $K+\epsilon$ calls) to have positive expected payoff while being initially net zero cost.

Calendar Spread (No Arbitrage Across Maturities)

For fixed $K$, the call price must be non-decreasing in $T$:

$\frac{\partial C}{\partial T}(K, T) \geq 0, \qquad r = 0.$

A longer-dated call can be exercised or held; a shorter-dated call cannot be. Violation allows a calendar spread to provide a guaranteed profit. In terms of total variance: the condition becomes

$\frac{\partial w}{\partial T}(k, T) \geq 0,$

i.e., total implied variance must be non-decreasing in maturity.

Dupire's Local Volatility

Motivation

The volatility smile shows that Black-Scholes is miscalibrated. One question is: does there exist a diffusion model — i.e., a model of the form $dS = \mu S\, dt + \sigma(t,S) S\, dW$ — that is consistent with the entire observed implied vol surface? Dupire (1994) and Derman-Kani (1994) showed the answer is yes, and gave an explicit formula for the local volatility function $\sigma_L(t, S)$.

Setup

Assume the risk-neutral dynamics:

$dS_t = r S_t \, dt + \sigma_L(t, S_t) S_t \, dW_t.$

Given a complete, arbitrage-free call price surface $C(K, T)$ (with $K > 0$, $T > 0$), the local volatility is uniquely determined by Dupire's equation:

$\sigma_L^2(K, T) = \frac{\dfrac{\partial C}{\partial T} + (r - q) K \dfrac{\partial C}{\partial K} + q\, C}{\dfrac{1}{2} K^2 \dfrac{\partial^2 C}{\partial K^2}}.$

Derivation Sketch

The key tool is the Fokker-Planck equation (forward Kolmogorov equation) for the transition density $p(t, S; T, K)$ of the diffusion. One derives the PDE satisfied by $C(K, T)$ as a function of the strike and maturity (not time and spot), using the fact that:

$C(K, T) = e^{-rT} \int_K^\infty (s - K) p(0, S_0; T, s) \, ds.$

Differentiating with respect to $T$ and twice with respect to $K$, substituting the Fokker-Planck equation, and using Breeden-Litzenberger yields Dupire's formula.