@quantfinancetogo: ➡️ Quantitative Finance from Scratch, part 38: the volatility smile Read the implied volatility out of the option price at every strike and every expiry, and the Black-Scholes model says you should get one flat surface. Real option markets do not give a flat surface. Implied volatility often rises at both ends, so the curve across strikes looks like a smile, hence the name. For options on a stock index the left end, at the low strikes, is much higher, which is called the skew. Low strikes belong to puts far below today's price, which work like insurance against a crash, and since the crash of October 1987 this insurance on index options has been priced with a higher volatility than the rest. The code (volsmile.py, 16 lines of plain Python) builds its own market: a stock at $100 that swings with 15 % volatility and, with a chance of 1 in 10, also crashes by about a quarter in one year. It simulates 100,000 years, scales them so the stock grows like money in the bank at 5 %, prices a call at nine strikes from 60 to 140 and turns each price into an implied volatility with the bisection search from part 31. Result: 19.9 % at strike 60, 17.3 % at 100 and 15.9 % at 140, a skew, because the crashes make big falls more likely than one bell curve allows. Without the crashes the same run gives a flat line near 15 %. Sources: L. Benzoni, P. Collin-Dufresne, R. S. Goldstein, "Explaining asset pricing puzzles associated with the 1987 market crash", Journal of Financial Economics, 2011 (index option smiles relatively flat before the crash, a permanent steepening after it); D. S. Bates, "Post-'87 crash fears in the S&P 500 futures option market", Journal of Econometrics, 2000; J. C. Jackwerth and M. Rubinstein, "Recovering probability distributions from option prices", Journal of Finance, 1996; D. S. Bates, "Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options", Review of Financial Studies, 1996 (smile in currency options). Honest notes: the 1987 statement is about options on a stock index; single stocks show a weaker skew (Bakshi, Kapadia and Madan, Review of Financial Studies, 2003). All prices in the video come from our own simulated crash model, not from a market, with one fixed crash size and at most one crash per year. #quantfinance #options #impliedvolatility #volatilitysmile #trading

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Sunday 01 November 2026 15:06:14 GMT