First of all, Black-Scholes is not a model of price variations of stocks. It is a model for putting values on options contracts, contracts that give their owner the right to buy or sell a stock at a particular price at a particular time in the future.
You are saying Black-Scholes is a model of stock prices probably because you recall correctly that Black-Scholes includes a model for stock price variations. To find the Black-Scholes options prices, one assumes (or models) stock price variations so that the logarithm of the variations is a normally distributed random variable. The variance, or width, of the normal distribution is simply determined by the volatility of the stock, which is determined from plotting the histogram of price variations of the stock.
The histogram of the logarithm of the variations fits a normal distribution quite well out to one or two sigma. But beyond 2 sigma or so, there are many more high variation events than the normal distribution predicts.
So if you just multiplied the variance by 7 or whatever to account for the prevalence of 20-sigma events, yes, you would now correctly predict the prevalence of 20-sigma events, but you would severely underestimate the prevalence of 1, 2, 3, 4… 19 sigma events. So this is not at all a good fix. You are attempting to fit something that is NOT bell-curved shape by covering it with a wider bell-curve.
Meanwhile, how good is black-scholes, and is there something better for options? For short dated options (less than a year to expiration), its pretty good. The calculation is more influenced by the −2sigma to 2sigma part of the curve which is accurately predicted as a log-normal, than it is by the outliers. But for longer dated options, the outliers become more and more important.
Are there better models than black-scholes for predicting options values? Yes, if you don’t mind dealing with GIGO. There are models that effectively let you specify the probability distribution function curve completely, and then calculate various options prices from that. The GIGO, garbage in garbage out, arises because the “correct” probability distribution, in detail, is not known a priori. If it turns out the stock is going up, there will be more excess probability on the positive variations side then the negative variations side. If it turns out the stock is going down, it will be the other way. If you knew ahead of time whether the stock was going up or going down, you wouldn’t need a calculation as complex as black-scholes to know how to get rich from that knowledge.
First of all, Black-Scholes is not a model of price variations of stocks. It is a model for putting values on options contracts, contracts that give their owner the right to buy or sell a stock at a particular price at a particular time in the future.
You are saying Black-Scholes is a model of stock prices probably because you recall correctly that Black-Scholes includes a model for stock price variations. To find the Black-Scholes options prices, one assumes (or models) stock price variations so that the logarithm of the variations is a normally distributed random variable. The variance, or width, of the normal distribution is simply determined by the volatility of the stock, which is determined from plotting the histogram of price variations of the stock.
The histogram of the logarithm of the variations fits a normal distribution quite well out to one or two sigma. But beyond 2 sigma or so, there are many more high variation events than the normal distribution predicts.
So if you just multiplied the variance by 7 or whatever to account for the prevalence of 20-sigma events, yes, you would now correctly predict the prevalence of 20-sigma events, but you would severely underestimate the prevalence of 1, 2, 3, 4… 19 sigma events. So this is not at all a good fix. You are attempting to fit something that is NOT bell-curved shape by covering it with a wider bell-curve.
Meanwhile, how good is black-scholes, and is there something better for options? For short dated options (less than a year to expiration), its pretty good. The calculation is more influenced by the −2sigma to 2sigma part of the curve which is accurately predicted as a log-normal, than it is by the outliers. But for longer dated options, the outliers become more and more important.
Are there better models than black-scholes for predicting options values? Yes, if you don’t mind dealing with GIGO. There are models that effectively let you specify the probability distribution function curve completely, and then calculate various options prices from that. The GIGO, garbage in garbage out, arises because the “correct” probability distribution, in detail, is not known a priori. If it turns out the stock is going up, there will be more excess probability on the positive variations side then the negative variations side. If it turns out the stock is going down, it will be the other way. If you knew ahead of time whether the stock was going up or going down, you wouldn’t need a calculation as complex as black-scholes to know how to get rich from that knowledge.
Yes, I should be more careful when using terms. As you said, I used B-S as informal term for the log-normal variation assumptions.