Put-Call Skew Analysis

WHAT SKEW IS. Black-Scholes assumes one volatility for all strikes. Traded option prices do not obey that: implied volatility varies by strike, and in equity-index options it is usually HIGHER for out-of-the-money puts than for comparable out-of-the-money calls. Crypto skew is less stable and can REVERSE, with calls richer than puts when upside demand dominates - so do not carry the equity intuition across without checking. Plot IV against strike and you get a smile or, more commonly, a downward-sloping smirk. Twenty-five-delta skew measures it directly: the implied volatility of the 25-delta put minus that of the 25-delta call.

WHY 'DELTA' AND NOT A STRIKE. Delta is the option's sensitivity to the underlying, and quoting by delta rather than by strike keeps the measure comparable as spot moves and across assets and tenors. It is the market convention for exactly that reason.

THE CONFUSION WORTH CLEARING UP. Delta is often described as the probability of finishing in the money. It is not. In the zero-dividend Black-Scholes model a call's delta is N(d1) while the RISK-NEUTRAL probability of finishing in the money is N(d2), with d2 = d1 - sigma x sqrt(T). (With a continuous dividend yield q, spot delta is exp(-qT) x N(d1).) The two are close for short-dated near-the-money options, and their gap can become material as total volatility sigma x sqrt(T) grows - though it is not monotonic in maturity, since moneyness, rates and carry move both arguments. A 25-delta option is a convention for 'roughly this far out of the money', not a 25% chance.

WHY THE SKEW EXISTS. Three overlapping contributors - not a clean causal decomposition, since the literature does not settle their relative weight. The LEVERAGE EFFECT (Black, 1976): as equity value falls, leverage rises and volatility with it, so lower prices genuinely carry higher volatility. DEMAND: portfolio hedgers systematically buy downside protection and are less systematically sellers of it, so the price of puts embeds that imbalance. And the shape steepened durably in index options after October 1987 - crash risk was repriced and never fully unpriced, which is why some of the literature calls it crash-o-phobia.

THE MOST IMPORTANT CAVEAT. These are RISK-NEUTRAL prices, not forecasts. An option's implied volatility is what the market CHARGES, and it embeds a risk premium - the tendency, found in many equity-option samples, for option-implied variance to exceed expected or subsequently realised variance. So elevated put skew means downside protection is expensive, which is a statement about the price of insurance and about demand for it. It is not a direct estimate of the PHYSICAL probability of a fall. Option prices do imply a risk-neutral distribution, and that distribution embeds risk premia and state prices - so converting it into real-world probabilities requires assumptions about those premia. Reading skew straight off as 'the market thinks there is an X% chance' is the standard error.

TENOR MATTERS. Seven-day and 30-day skew answer different questions - the short tenor is generally more sensitive to imminent events, while the longer places relatively more weight on persistent volatility and hedging demand - so they can disagree without either being wrong.