Brownian Bridge™

1. The implied volatility smile in equity options markets refers to the observation that:

Implied volatility is constant across all strikes and maturities, consistent with Black-ScholesOut-of-the-money puts trade at higher implied volatility than at-the-money options, while OTM calls trade at lower implied volatility — producing a downward-sloping skewOut-of-the-money calls and puts both trade at lower implied volatility than at-the-money options, producing a volatility frownImplied volatility increases monotonically with strike, reflecting greater risk for higher-priced options

2. Before the 1987 crash, equity option markets exhibited:

A pronounced negative skew, with OTM puts already trading at a significant premium to ATM optionsAn approximately flat implied volatility surface across strikes, broadly consistent with Black-Scholes lognormal dynamicsA positive skew, with OTM calls trading above ATM implied vol due to speculative demandAn inverted smile, with ATM options the most expensive relative to the wings

3. The leverage effect as a driver of the equity volatility skew refers to:

The use of derivatives leverage by hedge funds, which amplifies demand for OTM puts and drives their implied vol higherThe negative empirical correlation between equity spot returns and volatility: when the stock price falls, firm leverage rises, making equity riskier and pushing implied vol higher for downside strikesCentral bank leverage of monetary policy, which suppresses short-dated volatility and steepens the vol term structureThe positive relationship between option gamma and implied vol, which causes market makers to charge more for OTM puts

4. The 25-delta risk reversal for equity index options is typically:

Positive, because OTM call implied vols exceed OTM put implied vols due to speculative demand for upsideZero, because put-call parity requires calls and puts at the same delta to have equal implied volatilitiesNegative, because OTM put implied vols exceed OTM call implied vols — reflecting the leverage effect and excess institutional demand for downside protectionPositive and increasing with maturity, reflecting the growing cost of long-dated call protection

5. The Breeden-Litzenberger formula states that the risk-neutral density of the underlying at terminal value K equals:

The first derivative of the call price with respect to strike, divided by the discount factore raised to rT, multiplied by the second derivative of the call price with respect to strike KThe Black-Scholes delta of a digital call option at strike KThe implied volatility surface evaluated at strike K, scaled by the inverse of the option's vega

6. A fitted implied volatility surface has butterfly arbitrage when:

The ATM implied vol decreases with maturity, violating calendar spread no-arbitrageThe implied vol at any strike exceeds the implied vol at the forward, which violates convexity constraintsThe second derivative of call prices with respect to strike is negative at some strike, implying a negative risk-neutral density — equivalently, a butterfly spread has a negative market priceThe implied vol surface is not differentiable at quoted strikes due to piecewise interpolation

7. The calendar spread no-arbitrage condition on the implied volatility surface requires that:

The implied vol at each strike is non-decreasing in maturity T for all KThe total implied variance w(K,T) = sigma^2(K,T) * T is non-decreasing in T for every fixed KThe implied vol surface is differentiable in both K and T at all interior pointsThe ATM implied vol term structure is upward sloping, reflecting increasing uncertainty at longer horizons

8. The VIX model-free variance measure is typically higher than the at-the-money implied vol for the same maturity because:

The VIX uses realised volatility data from the past 30 days, which tends to exceed forward-looking ATM implied volThe VIX integrates the entire smile across all strikes, including OTM puts and calls that trade above ATM implied vol due to the convex shape of the smile — the excess is the butterfly premium of the variance swap over the ATM straddleThe VIX formula double-counts the contribution of deep OTM options, systematically biasing the estimate upwardATM implied vol is quoted as a one-standard-deviation move, while VIX is quoted as a two-standard-deviation move

Quiz: The Equity Volatility Smile

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