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.

Interpretation and Limitations

Local volatility gives an exact calibration to any arbitrage-free surface. However:

1. Smile dynamics are wrong. Under local vol, the implied vol smile flattens as the spot moves forward (the "Derman smile dynamics" problem). Empirically, the smile roughly translates with the spot rather than flattening. This makes local vol inadequate for barrier options, cliquets, and other path-dependent products where dynamic smile behaviour matters.

2. Surface needs differentiating. Computing $\partial C/\partial T$ and $\partial^2 C/\partial K^2$ from noisy market data amplifies errors. In practice, the surface must first be smoothly fitted (e.g., via SVI), then differentiated.

3. No forward smile. Local vol implies a deterministic future smile (given current smile). Stochastic vol models produce uncertain future smiles, which better match cliquets and forward-starting options.

SABR Model and Smile Dynamics

The SABR (Stochastic Alpha Beta Rho) model of Hagan, Kumar, Lesniewski, and Woodward (2002) adds a stochastic vol driver to a CEV-type underlying:

$dF_t = \sigma_t F_t^\beta \, dW_t^{(1)}, \qquad F_0 > 0,$

$d\sigma_t = \alpha \sigma_t \, dW_t^{(2)}, \qquad \sigma_0 = \alpha_0 > 0,$

$d\langle W^{(1)}, W^{(2)} \rangle_t = \rho \, dt.$

Parameters:

• $\beta \in [0, 1]$: CEV exponent. $\beta = 1$ → log-normal backbone; $\beta = 0$ → normal model; $\beta = 1/2$ → square-root diffusion. Choice of $\beta$ controls the backbone but not the smile shape independently.
• $\alpha > 0$: initial volatility.
• $\nu \geq 0$: vol of vol. $\nu = 0$ reduces to CEV.
• $\rho \in (-1, 1)$: correlation between spot and vol. Negative $\rho$ produces left skew (typical in equity markets).

SABR Implied Volatility Formula

Hagan et al. derived an asymptotic expansion for the SABR implied vol $\hat{\sigma}_{\mathrm{SABR}}(F, K, T)$ valid for small expiry $T$ and near the money:

$\hat{\sigma}_{\mathrm{SABR}} = \frac{\alpha}{(FK)^{(1-\beta)/2}} \cdot \frac{z}{\chi(z)} \cdot \left[1 + \left(\frac{(1-\beta)^2}{24}\frac{\alpha^2}{(FK)^{1-\beta}} + \frac{\rho\beta\nu\alpha}{4(FK)^{(1-\beta)/2}} + \frac{2-3\rho^2}{24}\nu^2\right) T + \cdots\right],$

where

$z = \frac{\nu}{\alpha}(FK)^{(1-\beta)/2}\ln(F/K), \qquad \chi(z) = \ln\!\left(\frac{\sqrt{1 - 2\rho z + z^2} + z - \rho}{1 - \rho}\right).$

For at-the-money ($F = K$), this simplifies to:

$\hat{\sigma}_{\mathrm{ATM}} = \frac{\alpha}{F^{1-\beta}}\left[1 + \left(\frac{(1-\beta)^2}{24}\frac{\alpha^2}{F^{2(1-\beta)}} + \frac{\rho\beta\nu\alpha}{4 F^{1-\beta}} + \frac{2 - 3\rho^2}{24}\nu^2\right)T\right].$

Smile Dynamics Under SABR

The key advantage over local vol is the treatment of smile dynamics. In SABR:

• As the forward moves, the smile moves with it (sticky-delta behaviour), consistent with empirical observation.
• The skew is controlled primarily by $\rho$: more negative $\rho$ → steeper left skew.
• The curvature (smile convexity) is controlled by $\nu$: higher $\nu$ → more convex smile.

However, SABR has known pathologies: for $\beta < 1$, the absorbing boundary at zero can generate negative implied vols at very low strikes. Various extensions (shifted SABR, free boundary SABR) address this.

Heston Smile at Long Maturities

For completeness: in the Heston model (treated in full in the next module), the implied vol smile at long maturities approaches a symmetric smile centred at the money, with wings that widen at rate proportional to $\xi\sqrt{\theta T}$. At short maturities, the skew is dominated by the correlation $\rho$. Understanding these limiting behaviours is important for choosing between SABR (better at short maturities, single expiry) and Heston (better at capturing term structure and long-dated smiles).

Limitations

Surface fitting vs. dynamic hedging. Local vol and SABR fit the current smile but differ radically in their predictions of how the smile will move. Products sensitive to future smile dynamics (cliquets, Napoleon options) require models calibrated to dynamical properties, not just static fit.

No-arbitrage of interpolated surfaces. When the surface is parametrised and the butterfly and calendar conditions are not explicitly imposed, calibrated surfaces can violate no-arbitrage locally. Negative densities produce mispriced digitals and instability in PDE solvers. Always validate the implied density after fitting.

High-strike extrapolation. Implied vol at extreme strikes is unobservable. For risk management (tail scenarios, barrier options at very high strikes), the tail behaviour of the assumed distribution matters enormously. The Roger Lee moment formula bounds the wing slope:

$\limsup_{k \to +\infty} \frac{\hat{\sigma}^2(k,T)}{|k|/T} \leq 2,$

where $k = \ln(K/F)$. Parametrisations that violate this produce moment explosions.

SABR not suitable for long tenors. The SABR lognormal density can become negative at very low strikes for $\beta < 1$. For swaption surfaces with tenors beyond 10 years, more robust parametrisations (SSVI, Heston) are preferred.

Interview Angle

L1. What is implied volatility? Why is it not constant across strikes? Sketch the typical equity skew and explain qualitatively why it exists.

Implied vol is the unique value of $\sigma$ that, plugged into the Black-Scholes formula, reproduces the observed market price. It is not constant because Black-Scholes assumes constant vol (one of its core assumptions), while markets price in the possibility of jumps, leverage effects, and stochastic volatility. The equity skew (higher implied vol for lower strikes) reflects: (a) demand for downside protection (puts at low strikes are expensive), (b) leverage effect (as stock falls, equity volatility rises), (c) jump-at-default risk.

L2. State the three static no-arbitrage conditions on a call price surface. Explain what a violation of each implies in terms of a trading strategy. State Dupire's formula and explain the role of the Fokker-Planck equation in its derivation.

Call spread violation: $\partial C/\partial K > 0$ at some $K$. Sell $K_1$-call, buy $K_2$-call ($K_2 > K_1$) for a net credit, zero payoff at maturity — free money.

Butterfly violation: $\partial^2 C/\partial K^2 < 0$ at some $K$. Long a butterfly spread (long $K - \epsilon$, short $2K$, long $K + \epsilon$ calls) at zero cost with non-negative payoff — free money.

Calendar violation: $\partial C/\partial T < 0$ at some $(K, T)$ (with $r = 0$). Buy the $T_2 > T_1$ maturity call, sell the $T_1$ call. At $T_1$, the long call is still alive; the short call expires. If the stock is below $K$, both are worthless; if above $K$, you exercise the short at a loss, but the long still has value — but the initial premium was negative, so this is a net profit. More precisely: the call price is the expectation of the intrinsic value, and longer maturities allow more time for the option to expire in the money.

L3. Derive Dupire's equation from first principles using the Fokker-Planck equation. Explain why local vol produces incorrect smile dynamics for forward-starting products. Compare SABR and Heston smile dynamics at short and long maturities.

Fokker-Planck derivation. Under risk-neutral dynamics $dS = rS\,dt + \sigma_L(t,S)S\,dW$, the transition density $p(T,K)$ satisfies the forward equation:

$\frac{\partial p}{\partial T} = -\frac{\partial}{\partial K}[rK \cdot p] + \frac{1}{2}\frac{\partial^2}{\partial K^2}[\sigma_L^2(T,K) K^2 \cdot p].$

Since $C(K,T) = e^{-rT}\int_K^\infty (s-K)p(T,s)\,ds$, differentiating once in $T$ and twice in $K$ and applying the Fokker-Planck equation yields Dupire's formula after integration by parts. The derivation is exact (no approximation) given a smooth, arbitrage-free surface.

Smile dynamics. Local vol is deterministic: given the current state $(t, S_t)$, the future smile is uniquely determined. Under the local vol model, the smile at future time $t > 0$ is a function of the new spot level, and because $\sigma_L(t,S)$ is calibrated to today's smile, the future smile (at the current forward) is flatter than today's smile. This is the opposite of what is empirically observed (the smile tends to translate rather than flatten). For cliquets and other products that are long forward-skew, local vol systematically underestimates the value because it underprices the future implied skew.