In modern finance the Black-Scholes formula is not used to "price" options in any meaningful sense. The price of options is given by supply and demand. Black-Scholes is used in the opposite way: traders deduce the implied volatility from the observed option prices. This volatility is a representation of the risk-neutral probability distribution that the markets puts on the underlying returns. From that distribution we can price other financial products for which prices are not directly observable.
An Intuitive Explanation of Black–Scholes
21–30 of 94 posts
Re: An Intuitive Explanation of Black–Scholes
#22Earlier quoted context omitted.
Indeed, hence the meme "stocks only go up". There's a grain of truth to the meme, though. The safest bet I can think of to make is that, on average, the S&P 500 will be higher in the future than today. Obviously there are temporary down trends but on a time horizon of years to decades I can't think of a safer bet.
Safer bet would be to hold short term treasures.
I'd argue that's not a bet.
Re: An Intuitive Explanation of Black–Scholes
#23Earlier quoted context omitted.
Indeed, hence the meme "stocks only go up". There's a grain of truth to the meme, though. The safest bet I can think of to make is that, on average, the S&P 500 will be higher in the future than today. Obviously there are temporary down trends but on a time horizon of years to decades I can't think of a safer bet.
Safer bet would be to hold short term treasures.
[deleted]
Re: An Intuitive Explanation of Black–Scholes
#24If you found a stock price that actually follows the geometric Brownian motion pattern this model is built on, wouldn't that basically just print you an infinite amount of money? The expected value of the price movement one time-unit later would be positive.
No. In fact, the fundamental principle of all quantitative finance is that your results in the ideal scenario are arbitrage-free meaning that nobody stands to make any money off any transaction. That's how you determine the ideal price given the ideal asset.
edit: To address your specific observation, that the price of the stock is expected to go up, it's assumed that if the stock goes up, so do all other assets. In mathematical finance you never keep you money as cash, so if you sell the stock you put that money in an account that expected to grow at the "risk-free" rate. The major difference between the "risk-free" account and the stock is the variance of these asset prices.
However, in your scenario, you wouldn't need Black-scholes for the price of the stock itself since that should be theoretically equal to it's expected (in the mathematical sense of "expectation") future value assuming the risk-free rate.
Black-Scholes is used to price the variance of the underlying asset over time for the use of pricing derivatives. But again, if the stock moved exactly as modeled then the model would give you the perfect price such that neither the buyer nor the seller of the derivative was at a disadvantage.
The way you would make use of such a perfectly priced stock would be to search for cases where either buyers or sellers had mispriced the derivative and then take the opposite end of the mispriced position.
However you don't need a perfect ideal stock to make use of Black-Scholes (this is a common misconception). Black-Scholes can also be used to price the implied volatility of a given derivative. Again, derivatives fundamentally derive their values from the volatility/variance of an asset, not it's expectation. By using Black-Scholes you can assess what the market beliefs are regarding the future volatility. Based on this, and presumably your own models, you can determine whether you believe the market has mispriced the future volatility and purchase accordingly.
One final misconception of Black-Scholes is that it's always incorrect because stock price volatility is "fat-tailed" and has more variance than assumed under Black-Scholes. This was the case in the mid-80s and people did exploit this to make money, but today this is well understood. The "fat-tailed" nature of assets prices is modeled in the "Volatility smile" where the implied volatility is different at different prices points (which would not be expected under pure geometric Brownian motion), but this volatility can still be determined using Black-Scholes for any given derivative.
tl;dr Buying stocks is about your estimate of the expected future value of a stock, but Black-Scholes is used to price derivatives of a stock where you actually care about the expected future variance of a stock. Even in an unideal world you can still use Black-Scholes to quantify what the market believes about future behavior and buy/sell where you think you have an advantage.
Re: An Intuitive Explanation of Black–Scholes
#25In modern finance the Black-Scholes formula is not used to "price" options in any meaningful sense. The price of options is given by supply and demand. Black-Scholes is used in the opposite way: traders deduce the implied volatility from the observed option prices. This volatility is a representation of the risk-neutral probability distribution that the markets puts on the underlying returns. From that distribution w…
It’s still used as an input into illiquid 409a valuations.
Re: An Intuitive Explanation of Black–Scholes
#26If you found a stock price that actually follows the geometric Brownian motion pattern this model is built on, wouldn't that basically just print you an infinite amount of money? The expected value of the price movement one time-unit later would be positive.
Generally these parameters are unknown and the drift parameter is often quite a bit smaller than the volatility. As a consequence, you cannot be sure your investment is secure and its value is likely to wobble significantly in the short term even if it ultimately produces value in the long term. If you actually knew that the drift on a certain investment was positive, you still have to be prepared to survive the loss…
How does this hold on assets that trend today wards the whole market if we assume that governments will not let markets crash too long before printing money?
What I mean is that if we can assume that the wiggle for VTI or SPY on the long term is positive because of outside factors, does that make options on those larger market assets become a game of who has a large enough reserve
Re: An Intuitive Explanation of Black–Scholes
#27Great article and very intuitive explanation. I also wanted to point out a (minor) typo. On equation 3, dZt is multiplied by sigma squared, but it should be multiplied just by sigma instead.
Thanks! I'll fix this.
Re: An Intuitive Explanation of Black–Scholes
#28If you found a stock price that actually follows the geometric Brownian motion pattern this model is built on, wouldn't that basically just print you an infinite amount of money? The expected value of the price movement one time-unit later would be positive.
Yes. That’s basically how the stock market works. If you buy and hold an S&P 500 index fund you can expect to make an infinite amount of money, in an infinite amount of time. But few have the patience for that.
We'll hit the limit in a few decades or at most a couple centuries due to ecological limits on growth though (unless a robust space economy develops).
Re: An Intuitive Explanation of Black–Scholes
#29Here is a replacement for the Black-Scholes/Merton model:
https://keithalewis.github.io/math/um1.html#black-scholesmer...
Re: An Intuitive Explanation of Black–Scholes
#30the creators of Black-Scholes destroyed their options selling fund based on their flawed belief that everyone else had mispriced options, or the black swan possibility should have been part of the formula also Black-Scholes doesnt factor in the liquidity of the underlying asset, in modern times I think this is relevant in determining the utility of an options contract there are other options pricing formulas
LTCM wasn't really an options selling fund though selling equity options did become a big trade for them Also they were more of advisors in the fund then anything else
You’re judged by the company you keep