An index's 25-delta put trades at 28% implied volatility, the 25-delta call at 20% and at-the-money at 23%. Price a zero-cost risk reversal, explain what the skew is paying for, and say who is on the other side.

Sell the 18,377 put and buy the 20,986 call for zero premium. The 25-delta put is worth about 449 points at 28%, and a 25-delta call only 284 at 20%, so the put pays for a call about 34-delta. The skew is the price of crash insurance, bought by hedgers of long equity books. The client takes the other side of that insurance and loses heavily in a crash.

Read the three quotes as prices of insurance. A put five percent out of the money protects against a fall; a call five percent out of the money pays on a rally. The market charges 28% volatility for the downside and 20% for the upside, so protection against a fall costs far more than a ticket to a rise of similar size. A risk reversalBuying an out-of-the-money call and selling an out-of-the-money put, or the reverse. Quoted as the call volatility minus the put volatility at the same delta. quoted at 25 delta is 20 minus 28, minus 8 vols: a steep downside skew.

Price the legs you know. The 25-delta put at 28% is worth about 449 index points. The 25-delta call at 20% is worth about 284. Selling the put and buying the same-delta call leaves a credit of about 165 points, so the client can afford a call closer to the money. Walk the call strike down the smile until its premium equals the put's: that happens at about 20,986, where the interpolated volatility is 20.9% and the call's delta is about 0.34.

Compare the put with what it would cost at the at-the-money volatility: about 296 points instead of 449. The extra 153 points are the premium for crash insurance, and they exist for two reasons: index volatility rises when prices fall, and institutions holding equities keep buying puts to protect their books. A household buys fire insurance for more than the expected cost of fires because it cannot afford the fire; pension funds and insurers pay over the odds for index puts for the same reason. Covered-call writers sell upside calls at the same time, which pushes call volatility down.

The client is selling that insurance: short the put the hedgers want, long the call the overwriters are selling. Most of the time this earns the skew premium, which is how the upside comes free. The cost arrives all at once: a 15% fall to 17,000 loses about 1,377 index points on the sold put, with no premium cushion because none was received. Hemadri's desk, having sold the call and bought the put, is on the hedgers' side and should say plainly that the zero cost is paid for in crash risk.

Close with the limit of the model. Black-Scholes with a quadratic smile is a convention for converting quotes into prices, not a forecast of the crash probability. The fair question for the client is whether the skew overpays for crash risk over the coming three months, and that is a judgement about the market, not something the model can answer.