[HN Gopher] An Intuitive Explanation of Black-Scholes
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An Intuitive Explanation of Black-Scholes
Author : alexmolas
Score : 80 points
Date : 2024-10-03 13:28 UTC (2 days ago)
(HTM) web link (gregorygundersen.com)
(TXT) w3m dump (gregorygundersen.com)
| ComplexSystems wrote:
| 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.
| pfdietz wrote:
| I think these models are self-defeating, since they stop
| working when enough people try to exploit them.
| dragontamer wrote:
| 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
|
| ------------
|
| 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.
| rubyn00bie wrote:
| 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).
| pram wrote:
| The competitive advantage is lessened because everyone knows it
| already. It's "priced in" as they say
| yold__ wrote:
| 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.
| bjornsing wrote:
| 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.
| tehjoker wrote:
| 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).
| melenaboija wrote:
| 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.
| stackghost wrote:
| 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.
| pigeons wrote:
| However the company stocks included in the S&P 500 aren't the
| same.
| smabie wrote:
| Safer bet would be to hold short term treasures.
| stackghost wrote:
| I'd argue that's not a bet.
| gyudin wrote:
| Considering they only choose top performers and inflation
| sounds like a safe guess :D
| nicolapede wrote:
| 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.
| tel wrote:
| 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.
| eclark wrote:
| 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
| smabie wrote:
| Why? the mean can be negative?
| crystal_revenge wrote:
| 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.
| zyklu5 wrote:
| This guy's other notes are also well thought through and written.
| Thanks for the link.
| yieldcrv wrote:
| 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
| smabie wrote:
| 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
| yieldcrv wrote:
| You're judged by the company you keep
| javitury wrote:
| 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.
| gwgundersen wrote:
| Thanks! I'll fix this.
| ncclporterror wrote:
| 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.
| mikeyouse wrote:
| It's still used as an input into illiquid 409a valuations.
| nknealk wrote:
| It's also frequently used to price stock options given to
| employees at publicly traded companies.
| dumah wrote:
| 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.
| keithalewis wrote:
| Here is a replacement for the Black-Scholes/Merton model:
| https://keithalewis.github.io/math/um1.html#black-scholesmer...
| erehweb wrote:
| You can also use nonstandard analysis to derive Black-Scholes,
| replacing stochastic calculus by a random walk with infinitesimal
| steps. https://ieeexplore.ieee.org/document/261595 (don't see an
| ungated version)
| jesuslop wrote:
| I jotted a time ago a Sage snippet for options pricing in
| elementary calculus terms, pasted here
| https://pastebin.com/tTMp6fPk.
|
| The idea is that the clean picture is done in terms of log-prices
| (not prices). Probability of log-prices follows a diffusion with
| an initial Dirac delta at-the-money. At expiration the profit
| function is deterministic (0 out of the money, a ramp if in the
| money) and the probability is certain gaussian. The expectancy of
| the value of a function applied to a random var of given density
| is like a weighted sum of the values, weighted by the
| frequency/density, as in a dot product (an integral here). Add to
| that the "time value of money" (see Investopedia) that works as
| linear drift, and you are done.
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