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.
It’s still used as an input into illiquid 409a valuations.
It’s also frequently used to price stock options given to employees at publicly traded companies.
Black-Scholes assumes constant volatility and cannot compute option prices without a volatility input.
This volatility is backed out of nearby options prices, often using the formula for European options.
There isn’t any purely theoretical option price because an assumption depends on observed prices.
Great 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.
If 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 losses you might accumulate on the way to profit. The greater the volatility the more painful this process can be. If you can just sock away your investment and not look at it for a long time it will become more valuable. On a day-to-day time scale, as an actual human watching this risky bet you've made wobble back and forth, it can require a lot of fortitude to remain invested even as the value dips significantly.
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
No, this doesn't imply an "infinite amount of money", it's just a pricing model. You still need the parameters of the distribution (brownian motion / random walk), and these are unobservable. You can try to estimate them, but there is a lot of practical problems in doing so, primarily that volatility / variance isn't constant.
This is a pricing model, i.e. what is the value according to the assumptions the model does (which btw are known to be weak for BS) but as anything else the price is what you are going to pay in the market for whatever other reasons.
Imagine you have a model that establishes the price of used cars, it can be really really good but if you go to the market to buy one you will pay whatever is been asked for not what your model says.
EDIT: Although pricing models do not have direct affectation to market prices they do in an indirect manner. To manage risk are needed pricing models which somehow condition market participants and therefore prices indirectly. In the simile with cars, you can buy as many cars as you want at the price you want, but what you do when you have them and if you want to take wise decisions with them you have to know something about their value.
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).
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.
Considering they only choose top performers and inflation sounds like a safe guess :D
However the company stocks included in the S&P 500 aren't the same.
Safer bet would be to hold short term treasures.
I'd argue that's not a bet.
The competitive advantage is lessened because everyone knows it already. It’s “priced in” as they say
No. Just look at equations 6 and 7 in the link. The expected value of the move can be either positive or negative depending on the model parameters.
I think these models are self-defeating, since they stop working when enough people try to exploit them.
The opposite.
We have huge numbers of people 'rocking the boat' trying to create say.... a Gamma Squeeze.
The only reason everyone trusts a Gamma Squeeze can happen is because they trust the math in Black Scholes. The may not even understand the math, just trust that the YouTuber who told them about Gamma Squeezes had enough of an understanding
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Today's problem IMO, is now a bunch of malicious players who are willing to waste their money are trying to make 'interesting' things happen in the market, almost out of shear boredom. Rather than necessarily trying to find the right prices of various things.
Knowing that other groups follow say, Black Scholes, is taken as an opportunity to mess with market makers.
I used to think charting was bullshit for day and swing trading. Because on paper it sure seems to be, but in reality so many other players are also charting that it becomes useful and somewhat predictive. Largely because you’re all using the same signals. Sure it’s impossible difficult to time things perfectly, but perfect is the enemy of profit. You don’t need to catch the absolute bottom and you don’t need to catch the absolute top.
Specific to Black-Sholes the best option plays, when going long, are the ones which have incorrect assumptions about the volatility of the underlying. You can have far outta the money options, absolutely print, with a sufficient spike in the underlying. Even if the strike price will never be met (though you’ll also give that back if you ride them to expiration or let things settle down).
Why? the mean can be negative?
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.
This guy's other notes are also well thought through and written. Thanks for the link.
Here is a replacement for the Black-Scholes/Merton model: https://keithalewis.github.io/math/um1.html#black-scholesmer...
the 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
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