1. Realised vs implied volatility

  • Realised / historical vol: Measured from past returns of the underlying.

  • Implied volatility: The σ\sigma you must plug into a model (Black / Black–Scholes) so that the model price matches the market price of the option.

If Cmkt(K,T)C^{\text{mkt}}(K,T) is the observed option price, implied vol σimp(K,T)\sigma_{\text{imp}}(K,T) solves:

Cmodel(K,T;σimp)=Cmkt(K,T) C^{\text{model}}(K,T;\sigma_{\text{imp}}) = C^{\text{mkt}}(K,T)

Practitioner view:

“Price is what you pay; implied vol is how expensive it is in volatility units. Traders quote and hedge in implied vol, not in raw prices.”

2. Volatility smile / skew

If Black(-76) were literally correct:

  • One σ\sigma per maturity TT would fit all strikes.
  • Implied vol would be flat in strike.

Markets don’t look like that. Plot σimp(K)\sigma_{\text{imp}}(K) for fixed TT:

  • Equities: “smirk” / downside skew.
    • Low strikes → higher implied vol (crash risk).
  • FX, many commodities: more symmetric smiles.
    • Both deep ITM and deep OTM have higher vols than ATM.

Reasons (intuitively):

  • Returns are not lognormal:
    • Fat tails, jumps, asymmetry.
  • Supply/demand for protection:
    • Heavy demand for downside puts lifts their implied vol.
  • Structural features:
    • Constraints, leverage, physical flows (especially in power/gas).

Mental picture:

For each maturity TT, you have a curve K↦σimp(K,T)K \mapsto \sigma_{\text{imp}}(K,T); that curve is the smile (or skew).

3. Vol surface and term structure

Across strikes and maturities you get a surface:

(K,T)↦σimp(K,T) (K,T) \mapsto \sigma_{\text{imp}}(K,T)

Two important cross-sections:

  • Smile/skew: fix TT, vary KK.
  • Term structure: fix moneyness (e.g. ATM), vary TT.

Typical shapes:

  • Short-dated vols higher if market is stressed; longer-dated vols smoother, often lower.
  • Commodities may have seasonal vol term structures:
    • e.g. winter gas months more volatile than summer.

Use in practice:

  • Traders quote implied vols per strike/maturity.
  • Quant builds a smooth surface through these quotes to:
    • price off-the-run strikes,
    • compute Greeks consistently,
    • run risk on structured products.

4. Interpolation sanity: no-arbitrage constraints

If you interpolate implied vols badly, you can create arbitrage in your own model.

Key no-arb constraints on the price surface (for each TT):

  1. Monotonicity in strike:
    • Call prices decrease with strike.
    • Put prices increase with strike.
  2. Convexity in strike (butterfly no-arb):
    • Call price as a function of strike is convex.
    • If not, you can build a butterfly with negative cost and non-negative payoff.
    In practice:
    • Call spreads with wider widths must not get cheaper per unit width.
  3. Calendar no-arb:
    • For fixed strike, total variance w=σimp2Tw = \sigma_{\text{imp}}^2 T should be non-decreasing in TT.
    • Otherwise, a longer-dated option could be cheaper in a way that violates time value.

Because implied vol is a non-linear transformation of price, you must interpolate carefully (e.g. in total variance, or using arbitrage-free parametrisations) to keep these properties.

Practitioner stance:

“I don’t interpolate vols directly; I interpolate objects that keep prices monotone and convex. After fitting, I back out implied vols again.”

5. Simple modelling levels (from desk point of view)

  1. Flat vol per expiry (old-school BS/Black-76):
    • One number σ(T)\sigma(T) per maturity.
    • Good for teaching, rarely enough in production.
  2. Smile per expiry:
    • Parametric smile (e.g. quadratic in log-moneyness).
    • Or SABR-like parametrisation per maturity.
    • You still treat expiries independently.
  3. Global surface model:
    • A model over (K,T)(K,T) (e.g. SVI, SABR surface, stochastic vol).
    • Calibrated jointly to all market quotes.
    • Designed to be arbitrage-free by construction.

For many commodities desks:

  • You start with Black-76 implied vols from broker screens,
  • Fit a simple smile per expiry,
  • Then refine towards an arbitrage-free global surface if you care about exotic structures.

6. How to talk about it in an interview

Short sentences you can use:

  • On implied vol:

    “Implied vol is the volatility that, plugged into Black-76 or Black–Scholes, reproduces the market option price. It’s how the market quotes option richness, because prices themselves mix spot, forwards, and rates.”

  • On smiles:

    “In a pure lognormal world the smile is flat. In reality we see skew and smiles: downside puts are bid in equities, wings are expensive in FX/commodities. So implied vol becomes a function of strike and maturity, giving us a surface.”

  • On interpolation:

    “When we build a surface, we respect no-arbitrage: call prices should be decreasing and convex in strike, and total variance should increase with maturity. That’s why we interpolate in stable quantities like total variance or use parametrisations like SVI, rather than just spline implied vols naively.”