Yanzhong(Eric) Huang

MQF of Rutgers Business School
3 years of FOF experience
Passionate about quant and data science.

Implied volatility (IV) is the volatility input that, when used in the Black–Scholes–Merton (BSM) formula, reproduces a market option price.
Across strikes and maturities, markets quote a surface of IVs rather than a single number. Understanding, cleaning, and modeling this volatility surface is central to pricing, risk, and hedging.

1) From Prices to Implied Volatility

Given a mid market price $C^{mkt}(K,T)$ (or $P^{mkt}$), implied volatility $\sigma_{\text{iv}}(K,T)$ solves \(C^{BSM}\!\big(S_0,K,T;r,q,\sigma_{\text{iv}}\big)=C^{mkt}(K,T).\) Inversion is typically done by Newton–Raphson with the derivative (Vega): \(\sigma_{n+1}=\sigma_n-\frac{C^{BSM}(\sigma_n)-C^{mkt}}{\text{Vega}(\sigma_n)},\qquad \text{Vega}=S_0 e^{-qT}\phi(d_1)\sqrt{T}.\) Good seeds: Brenner–Subrahmanyam for near-ATM, or bracket searches for deep OTM/ITM.

Quoting conventions. Equity & rates often use strike $K$ or log-moneyness $k=\ln(K/F)$ with $F=S_0e^{(r-q)T}$; FX often quotes by delta (25-risk-reversal, 25-butterfly, ATM).

2) Smile, Skew, and Term Structure (Intuition)

  • Equities: Typically downward skew (OTM puts rich) due to leverage effect, crash risk, and demand for downside protection.
  • FX: More smile-like; asymmetry reflects macro skew (e.g., risk-reversals).
  • Commodities: Smiles vary; inventory/risk premia can produce positive skew.
  • Term structure: Short maturities exhibit stronger skew/kurtosis; long maturities tend to be smoother with lower level and curvature.

3) No-Arbitrage Anatomy of a Surface

For undiscounted call price $C(K,T)$:

  • Butterfly (static) arbitrage free in strike $\Rightarrow$ convexity: \(\frac{\partial^2 C}{\partial K^2}(K,T)\ \ge\ 0\quad\Longleftrightarrow\quad\text{non-negative density.}\)
  • Calendar arbitrage free in maturity: \(\frac{\partial C}{\partial T}(K,T)\ \ge\ 0\quad\text{(holding $K$ fixed)}.\)
  • Vertical spread bounds: $0 \le \frac{\partial C}{\partial K} \le e^{-rT}$.
  • Lee’s moment bounds (wings): the slope of total variance $w(k,T)=\sigma_{\text{iv}}^2(k,T)\,T$ must obey constraints as $ k \to\infty$; helpful for robust extrapolation.

Practical check (discrete quotes). Ensure convexity across strikes (finite-difference second derivative $\ge 0$) and monotonicity in $T$ for each strike bucket.

4) Building a Production-Quality Surface (Workflow)

  1. Data cleaning
    • Use mid quotes; drop stale/outliers; enforce put–call parity to harmonize calls & puts.
    • Map to a consistent forward $F(T)$ and discount factor $D(T)=e^{-rT}$ using your curves.
  2. Transform to stable coordinates
    • Work in log-moneyness $k=\ln(K/F)$ or delta for FX.
    • Interpolate total variance $w(k,T)=\sigma_{\text{iv}}^2(k,T)\,T$ rather than $\sigma$—it behaves more linearly in $T$.
  3. Parametrize cross-sections (smiles)
    • SVI (Stochastic Volatility Inspired) per maturity: \(w(k)=a + b\Big(\rho (k-m)+\sqrt{(k-m)^2+\sigma^2}\Big),\) with constraints on $(a,b,\rho,m,\sigma)$ to avoid static arbitrage.
    • Alternatives: quadratic in $k$ near ATM with wing constraints; Spline-in-$k$ on $w$ with convexity enforcement.
  4. Interpolate in maturity
    • Interpolate forward variance $w(k,T_2)-w(k,T_1)$ linearly in $T$ (or use monotone splines) to preserve calendar monotonicity.
  5. Wing extrapolation
    • Anchor tails using Lee’s bounds or product-specific heuristics (e.g., FX risk-reversal/Butterfly extrapolation).
  6. Sanity & arbitrage checks
    • Numerically verify $\partial_{KK} C!\ge!0$ and $\partial_T C!\ge!0$ on a dense $(k,T)$ grid.

Dupire Local Volatility

From the (arbitrage-free) surface, the local volatility $\sigma_{\text{loc}}(t,S)$ satisfies \(\sigma_{\text{loc}}^2(t,K)= \frac{\partial_T C(K,T)+q\,C(K,T)-r\,K\,\partial_K C(K,T)} {\tfrac12\,K^2\,\partial_{KK} C(K,T)}\Bigg|_{T=t}.\) This yields a diffusion $dS_t=(r-q)S_t\,dt+\sigma_{\text{loc}}(t,S_t)S_t\,dW_t$ reproducing all vanilla prices by construction.

SABR (log-normal β=1 summary)

SABR dynamics (for forward $F_t$): \(\begin{aligned} dF_t &= \alpha_t F_t\, dW_t^{(1)},\\ d\alpha_t &= \nu \alpha_t\, dW_t^{(2)},\qquad d\langle W^{(1)},W^{(2)}\rangle_t=\rho\,dt. \end{aligned}\) Hagan’s approximation provides a closed-form implied vol $\sigma_{\text{iv}}(K,T)$ matching smiles for rates/FX; parameters $(\alpha_0,\nu,\rho)$ control level, skew, and convexity.

When to use what?
Local vol exactly matches today’s surface but can mis-represent dynamics (too “sticky-strike”).
Stochastic vol (Heston/SABR) captures smile dynamics and term structures more realistically but won’t fit surfaces exactly without extra freedom.

6) Surface Dynamics for Hedging

  • Sticky-strike: keep $\sigma_{\text{iv}}(K,T)$ fixed as $S$ moves.
  • Sticky-delta: keep $\sigma_{\text{iv}}(\Delta,T)$ fixed (common in FX).
  • Sticky-moneyness/log-moneyness: partial adjustment with $S$.

Your choice impacts P&L attribution and hedge performance (delta/vega rebalancing, vanna/vomma effects). Backtest under the desk’s house convention.

7) Implementation Notes (desk-ready)

  • Numerics: Use robust IV inversion (bounded Newton + vega floor).
  • Grids: Work on $(k,T)$ lattices; store $w=\sigma^2T$ for stability.
  • Calibration: Fit each maturity slice (SVI), then smooth through time.
  • Risk: Compute surface Greeks ($\partial \sigma/\partial K$, $\partial \sigma/\partial T$) for vanna/vomma management and slippage analysis.
  • Auditability: Persist raw quotes, cleaning flags, and fitted parameters for replay.

8) Quick Example (SVI slice)

Suppose for $T=1$ year you fit an SVI smile with parameters
$a=0.02,\ b=0.15,\ \rho=-0.4,\ m=0.0,\ \sigma=0.25$.
At $k=\ln(K/F)=-0.1$, \(w(k)=0.02+0.15\Big(-0.4(-0.1)+\sqrt{0.1^2+0.25^2}\Big)=0.02+0.15(0.04+0.269)=0.02+0.046=0.066.\) So $\sigma_{\text{iv}}(K,T)=\sqrt{w/T}\approx \sqrt{0.066}=25.7\%$.

9) Takeaways

  • The volatility surface encodes the market’s risk-neutral beliefs and risk premia.
  • Building a clean, arbitrage-free surface requires careful coordinates (log-moneyness), total variance interpolation, and SVI/other parametric fits.
  • Local vol reproduces the surface; stochastic vol reproduces smile dynamics; practitioners often blend both (e.g., local-stochastic vol).

Next up: Exotic Options

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