8 ms·

An Intuitive Explanation of Black–Scholes

212 points by alexmolas 2y ago

ComplexSystems 2y ago
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 2y ago
  I think these models are self-defeating, since they stop working when enough people try to exploit them.
  - deleted 2y ago
    [deleted]
  - dragontamer 2y ago
    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 2y ago
    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 2y ago
  The competitive advantage is lessened because everyone knows it already. It’s “priced in” as they say
- yold__ 2y ago
  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 2y ago
  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 2y ago
    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).
    - twoodfin 2y ago
      Sorry, which limits? How do those apply to the increasing economic value of turning the same amount of sand into faster and faster GPUs, for example?
      - sokoloff 2y ago
        There are still finite people willing to buy whatever the intermediate or end product of that fancy sand is. And finite energy and space. And only 5 billion years until the sun goes red giant. The limits may be very large, but they aren’t infinite.
        twoodfin 2y ago
        Yeah but a couple centuries? What’s the evidence for exhaustion of demand and supply for economic goods on that kind of time horizon.
      - SJC_Hacker 2y ago
        Human populations cannot expand indefinitely. Indeed many are predicting population to peak latter in century and then decline. Many first world countries are in a demographic decline if not outright collapse. See most of east Asia and Europe. Without expansion of population, consumption and production both stagnate. See what has happened in Japan in since the 90s.
- melenaboija 2y ago
  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.
  - listenallyall 2y ago
    Yes, but also no. Because you don't have to buy a mispriced asset (mispriced against you) and also, in many cases, you can construct what you want from pieces of other assets. One car dealer trying to sell a 2023 Honda Accord with 60,000 miles can't just decide, independently, to forget the high mileage and price the car based solely on it being 1 year old. Sure that's "whatever is being asked" but that car will never sell until he brings the price down in line with other 60k mile cars - and that is because the pricing models are essentially agreed upon by all market participants.
    - melenaboija 2y ago
      Yes, but also no. The value of things is only what the market wants to pay for it, and it does not matter if it is a 2023 Honda Accord or a financial product, currency... In one you might trust the engine reliability and on the other on the government behind the currency, whoever is writing the option, issuing the bond, ... But still, it is a matter of faith and bid/ask.
- stackghost 2y ago
  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 2y ago
    However the company stocks included in the S&P 500 aren't the same.
  - smabie 2y ago
    Safer bet would be to hold short term treasures.
    - stackghost 2y ago
      I'd argue that's not a bet.
      - sokoloff 2y ago
        It’s a bet on the continued existence, and willingness/ability to honor its obligations, of the US federal government. If that bets goes bad, the typical investor in Treasuries has perhaps bigger problems to worry about, but it’s still a bet IMO (and one which will inevitably eventually go bad).
    - deleted 2y ago
      [deleted]
  - gyudin 2y ago
    Considering they only choose top performers and inflation sounds like a safe guess :D
- nicolapede 2y ago
  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 2y ago
  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 2y ago
    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 2y ago
  Why? the mean can be negative?
  - bibouthegreat 2y ago
    2 parts: 1. Interest rates can be negative 2. Volatility reduces the average. Take an example of +10% then -10% (1+0.1)*(1-0.1) = 1 - 0.1² = 0.99 < 1. It's due to the "log normal returns"
- crystal_revenge 2y ago
  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.
- FabHK 2y ago
  Well, yes. If you buy a stock with positive drift and hold it, the model predicts "infinite growth" (in the sense that for any number N you give me I can give you a time t at which the E[S(t)] > N). But it might take quite some time, and it's still random, it might be much smaller or much bigger. You could be tempted to employ leverage. However, that introduces the chance of being wiped out. ETA: Real rates are normally positive. So you can achieve the same result by investing in long term bonds with less risk. Just have to wait even longer.
zyklu5 2y ago
This guy's other notes are also well thought through and written. Thanks for the link.
yieldcrv 2y ago
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 2y ago
  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 2y ago
    You’re judged by the company you keep
- mhh__ 2y ago
  If you mean LTCM then the story is far more dull (i.e. too much leverage, fund goes boom) Ed Thorpe did originally want to setup an options fund (he was the first to trade the model) that he later estimated would've blown up due to various market conditions at the time IIRC
javitury 2y ago
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 2y ago
  Thanks! I'll fix this.
ncclporterror 2y ago
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 2y ago
  It’s still used as an input into illiquid 409a valuations.
  - nknealk 2y ago
    It’s also frequently used to price stock options given to employees at publicly traded companies.
    - dumah 2y ago
      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.
- klysm 2y ago
  Kinda, but it’s not great because of the volatility smile
- wavemode 2y ago
  Sure, but isn't most of supply and demand in the market driven by large investors who use such formulas to derive the fair price of the option? That is, if the real price ever differred significantly from what Black-Scholes predicts, wouldn't algorithmic trading very quickly correct this deviation?
keithalewis 2y ago
Here is a replacement for the Black-Scholes/Merton model: https://keithalewis.github.io/math/um1.html#black-scholesmerton https://keithalewis.github.io/math/um1.html#black-scholesmer...
erehweb 2y ago
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 https://ieeexplore.ieee.org/document/261595 (don't see an ungated version)
- jesuslop 2y ago
  The "Loeb Measures in Practice" book also by Cutland has a survey chapter.
jesuslop 2y ago
I jotted a time ago a Sage snippet for options pricing in elementary calculus terms, pasted here https://pastebin.com/tTMp6fPk 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.
charlie0 2y ago
Brownian motion is what happens when people lose their life savings on meme stocks.
bryan0 2y ago
Another good explanation from Terence Tao’s blog: https://terrytao.wordpress.com/2008/07/01/the-black-scholes-equation/ https://terrytao.wordpress.com/2008/07/01/the-black-scholes-...
bee_rider 2y ago
Of course, Black-Scholes is a very famous and important mathematical model. However, it is Saturday night, so let’s be a little silly. I’ve always thought that one reason it became so well known is that it sounds kind of badass. A shoal is, of course, a shallow bit of water, general associated with running aground and that sort of thing. Black-Shoals sounds like an area where Blackbeard the pirate will hang out steal all your stuff if you get stuck. I’ve always thought quants secretly want to be pirates, but of course the era of going around pirating is over, so they learned how to do it on the market instead. In the time of piracy, they could probably have been navigators, that job was pretty mathy. The would have presumably gone around the Black-Shoals.
- putcher_willow 2y ago
  You're not alone in finding the name poetic: see https://www.blackshoals.net/ https://www.blackshoals.net/ "Black Shoals Stock Market Planetarium is an art project created by Joshua Portway and Lise Autogena. The project takes the form of a darkened room with a domed ceiling upon which a computer display is projected, like a planetarium. Audiences are immersed in a world of real-time stock market activity, represented as the night sky, full of stars that glow as trading takes place on particular stocks. In Black Shoals each traded company is represented by a star, flickering and glowing as shares are traded. The stars slowly drift in response to the complex currents of the market, while outlining shapes of different industries and the huge multinational conglomerates like the signs of the zodiac. The movement of the stocks is based on calculated correlations between the histories of each stock and those of its near neighbours. The stronger the correlation between the histories of the stock prices of any two companies, the more powerful the gravitational attraction between them. Although they start out randomly distributed in the planetarium, over time the stars clot together and drift into slowly changing constellations, nebulae and clusters. Through this technique different industries naturally start to emerge as galaxies. Any general disturbance in a section of the market will have a visible effect on the sky – the collapse of Enron, for instance, would have caused a sort of black hole - all the companies affected would glow very brightly due to the level of trading and would be pulled in to a single point in a very powerful vortex." It goes on...
marxisttemp 2y ago
Like all economics, this uses massive oversimplifications that never apply in the real world to imply some incontrovertible nature to free markets that simply does not exist. Spherical cows indeed. There was an article posted here recently about “mathy” equations that this reminds me of. Anyways read Das Kapital if you want to actually understand economies.
- LudwigNagasena 2y ago
  > this uses massive oversimplifications that never apply in the real world If you've read Das Capital, you have noticed it also uses massive oversimplifications in its models. > imply some incontrovertible nature to free markets that simply does not exist. Spherical cows indeed. Das Kapital (as one can guess from its name) also studies the spherical cow of the free market. The implication of incontrovertible nature, that's something in people's heads though, not in the models. > There was an article posted here recently about “mathy” equations that this reminds me of. Any math model (including models described in Das Kapital) is either going to be oversimplified or "mathy". The only other choice is non-math models, which doesn't seem very useful if you want to talk about money, prices, profits and other numerical stuff.
1htfp 2y ago
One of the most fascinating things about working on a trading floor is that models such as BSM transcend their normative aspect and become mental models. Pricing an option? Basically only two things matter: where the underlying asset forward price is at maturity (this is related to the concept of drift) and what the volatility is. At any time, your job is choose “bumps” (which you add to market prices) in order to maximize your odds of making money on a trade subject to beating your competition on price. There are some people who make a living making these markets who likely have never heard of “Ito’s lemma” or diffusion equations.
- MuffinFlavored 2y ago
  > where the underlying asset forward price is at maturity What models do people use for SPY/SPX forward price?...
  - blitzar 2y ago
    Futures. Decomposes down to price + dividend + time value/cost of money
    - MuffinFlavored 2y ago
      How often is front-month /ES not right around 20-50 points ahead of whatever SPX is trading at?
      - blitzar 2y ago
        Whenever the maths says so - the range you suggest is due to dividends typically collectively paying slightly higher than the risk free rate. Were we to have higher rates and companies not paying dividends en mass then that would be a negative number. How often? I would guess often - especially over the ~100 year history - and not something you would want to have wrong when writing billions in options.
        MuffinFlavored 2y ago
FabHK 2y ago
A few points: 1) Very nice exposition. 2) Near eq. (4) it is claimed that one cannot compute the delta \frac{\del C}{\del S} without stochastic calculus, since S is stochastic. That doesn't strike me as correct: C is just a deterministic continuous function of S, C, K, T, t, r, sigma; and computing partial derivatives does not require stochastic calculus. 3) It captures the notion that when you hedge, you use risk-neutral probabilities. 4) Generally, in practice, BS is written as follows: C = df ( F N(d1) - K N(d2) ), where d1 = (ln(F/K) + 1/2 s^2)/s, d2 = d1 - s, s = sqrt(sigma^2 (T-t)), df is the discount factor, and F is the forward price of S. This abstracts away the whole discounting business. Note that sigma never occurs except in the expression sigma^2 (T-t), which is dimension less, thus sigma has physical dimension 1/sqrt(year), usually ("annualised vol"). C has the same dimension as F and K.
- gwgundersen 2y ago
  2) Thanks for pointing this out. I've fixed.
lowkey 2y ago
I’ve always found it strange that BSM is used for calculating implied volatility of American style options when it was specifically designed only for European style options. Can anyone comment if there are more suitable models for American style options?
- FabHK 2y ago
  Generally, you back out local vols (as a function of S, t) of the BS vols (as a function of K, T) by a process described first by Dupire, and then you price American options (and other products that are not sensitive to vol of vol) with that using a numerical PDE solver. https://en.wikipedia.org/wiki/Local_volatility https://en.wikipedia.org/wiki/Local_volatility
SJC_Hacker 2y ago
Like everything in finance it mostly worked (and sometimes still does) because everyone uses it.
vismit2000 2y ago
Veritasium video on the topic is also very good: The Trillion Dollar Equation - https://youtu.be/A5w-dEgIU1M https://youtu.be/A5w-dEgIU1M

212 points by alexmolas 2y ago

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 2y ago
I think these models are self-defeating, since they stop working when enough people try to exploit them.
- deleted 2y ago
  [deleted]
- dragontamer 2y ago
  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 2y ago
  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 2y ago
The competitive advantage is lessened because everyone knows it already. It’s “priced in” as they say
yold__ 2y ago
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 2y ago
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 2y ago
  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).
  - twoodfin 2y ago
    Sorry, which limits? How do those apply to the increasing economic value of turning the same amount of sand into faster and faster GPUs, for example?
    - sokoloff 2y ago
      There are still finite people willing to buy whatever the intermediate or end product of that fancy sand is. And finite energy and space. And only 5 billion years until the sun goes red giant. The limits may be very large, but they aren’t infinite.
      - twoodfin 2y ago
        Yeah but a couple centuries? What’s the evidence for exhaustion of demand and supply for economic goods on that kind of time horizon.
    - SJC_Hacker 2y ago
      Human populations cannot expand indefinitely. Indeed many are predicting population to peak latter in century and then decline. Many first world countries are in a demographic decline if not outright collapse. See most of east Asia and Europe. Without expansion of population, consumption and production both stagnate. See what has happened in Japan in since the 90s.
melenaboija 2y ago
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.
- listenallyall 2y ago
  Yes, but also no. Because you don't have to buy a mispriced asset (mispriced against you) and also, in many cases, you can construct what you want from pieces of other assets. One car dealer trying to sell a 2023 Honda Accord with 60,000 miles can't just decide, independently, to forget the high mileage and price the car based solely on it being 1 year old. Sure that's "whatever is being asked" but that car will never sell until he brings the price down in line with other 60k mile cars - and that is because the pricing models are essentially agreed upon by all market participants.
  - melenaboija 2y ago
    Yes, but also no. The value of things is only what the market wants to pay for it, and it does not matter if it is a 2023 Honda Accord or a financial product, currency... In one you might trust the engine reliability and on the other on the government behind the currency, whoever is writing the option, issuing the bond, ... But still, it is a matter of faith and bid/ask.
stackghost 2y ago
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 2y ago
  However the company stocks included in the S&P 500 aren't the same.
- smabie 2y ago
  Safer bet would be to hold short term treasures.
  - stackghost 2y ago
    I'd argue that's not a bet.
    - sokoloff 2y ago
      It’s a bet on the continued existence, and willingness/ability to honor its obligations, of the US federal government. If that bets goes bad, the typical investor in Treasuries has perhaps bigger problems to worry about, but it’s still a bet IMO (and one which will inevitably eventually go bad).
  - deleted 2y ago
    [deleted]
- gyudin 2y ago
  Considering they only choose top performers and inflation sounds like a safe guess :D
nicolapede 2y ago
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 2y ago
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 2y ago
  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 2y ago
Why? the mean can be negative?
- bibouthegreat 2y ago
  2 parts: 1. Interest rates can be negative 2. Volatility reduces the average. Take an example of +10% then -10% (1+0.1)*(1-0.1) = 1 - 0.1² = 0.99 < 1. It's due to the "log normal returns"
crystal_revenge 2y ago
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.
FabHK 2y ago
Well, yes. If you buy a stock with positive drift and hold it, the model predicts "infinite growth" (in the sense that for any number N you give me I can give you a time t at which the E[S(t)] > N). But it might take quite some time, and it's still random, it might be much smaller or much bigger. You could be tempted to employ leverage. However, that introduces the chance of being wiped out. ETA: Real rates are normally positive. So you can achieve the same result by investing in long term bonds with less risk. Just have to wait even longer.

zyklu5 2y ago

This guy's other notes are also well thought through and written. Thanks for the link.

yieldcrv 2y ago

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 2y ago
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 2y ago
  You’re judged by the company you keep
mhh__ 2y ago
If you mean LTCM then the story is far more dull (i.e. too much leverage, fund goes boom) Ed Thorpe did originally want to setup an options fund (he was the first to trade the model) that he later estimated would've blown up due to various market conditions at the time IIRC

javitury 2y ago

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 2y ago
Thanks! I'll fix this.

ncclporterror 2y ago

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 2y ago
It’s still used as an input into illiquid 409a valuations.
- nknealk 2y ago
  It’s also frequently used to price stock options given to employees at publicly traded companies.
  - dumah 2y ago
    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.
klysm 2y ago
Kinda, but it’s not great because of the volatility smile
wavemode 2y ago
Sure, but isn't most of supply and demand in the market driven by large investors who use such formulas to derive the fair price of the option? That is, if the real price ever differred significantly from what Black-Scholes predicts, wouldn't algorithmic trading very quickly correct this deviation?

keithalewis 2y ago

Here is a replacement for the Black-Scholes/Merton model: https://keithalewis.github.io/math/um1.html#black-scholesmerton https://keithalewis.github.io/math/um1.html#black-scholesmer...

erehweb 2y ago

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 https://ieeexplore.ieee.org/document/261595 (don't see an ungated version)

jesuslop 2y ago
The "Loeb Measures in Practice" book also by Cutland has a survey chapter.

jesuslop 2y ago

I jotted a time ago a Sage snippet for options pricing in elementary calculus terms, pasted here https://pastebin.com/tTMp6fPk 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.

charlie0 2y ago

Brownian motion is what happens when people lose their life savings on meme stocks.

bryan0 2y ago

Another good explanation from Terence Tao’s blog: https://terrytao.wordpress.com/2008/07/01/the-black-scholes-equation/ https://terrytao.wordpress.com/2008/07/01/the-black-scholes-...

bee_rider 2y ago

Of course, Black-Scholes is a very famous and important mathematical model. However, it is Saturday night, so let’s be a little silly. I’ve always thought that one reason it became so well known is that it sounds kind of badass. A shoal is, of course, a shallow bit of water, general associated with running aground and that sort of thing. Black-Shoals sounds like an area where Blackbeard the pirate will hang out steal all your stuff if you get stuck. I’ve always thought quants secretly want to be pirates, but of course the era of going around pirating is over, so they learned how to do it on the market instead. In the time of piracy, they could probably have been navigators, that job was pretty mathy. The would have presumably gone around the Black-Shoals.

putcher_willow 2y ago
You're not alone in finding the name poetic: see https://www.blackshoals.net/ https://www.blackshoals.net/ "Black Shoals Stock Market Planetarium is an art project created by Joshua Portway and Lise Autogena. The project takes the form of a darkened room with a domed ceiling upon which a computer display is projected, like a planetarium. Audiences are immersed in a world of real-time stock market activity, represented as the night sky, full of stars that glow as trading takes place on particular stocks. In Black Shoals each traded company is represented by a star, flickering and glowing as shares are traded. The stars slowly drift in response to the complex currents of the market, while outlining shapes of different industries and the huge multinational conglomerates like the signs of the zodiac. The movement of the stocks is based on calculated correlations between the histories of each stock and those of its near neighbours. The stronger the correlation between the histories of the stock prices of any two companies, the more powerful the gravitational attraction between them. Although they start out randomly distributed in the planetarium, over time the stars clot together and drift into slowly changing constellations, nebulae and clusters. Through this technique different industries naturally start to emerge as galaxies. Any general disturbance in a section of the market will have a visible effect on the sky – the collapse of Enron, for instance, would have caused a sort of black hole - all the companies affected would glow very brightly due to the level of trading and would be pulled in to a single point in a very powerful vortex." It goes on...

marxisttemp 2y ago

Like all economics, this uses massive oversimplifications that never apply in the real world to imply some incontrovertible nature to free markets that simply does not exist. Spherical cows indeed. There was an article posted here recently about “mathy” equations that this reminds me of. Anyways read Das Kapital if you want to actually understand economies.

LudwigNagasena 2y ago
> this uses massive oversimplifications that never apply in the real world If you've read Das Capital, you have noticed it also uses massive oversimplifications in its models. > imply some incontrovertible nature to free markets that simply does not exist. Spherical cows indeed. Das Kapital (as one can guess from its name) also studies the spherical cow of the free market. The implication of incontrovertible nature, that's something in people's heads though, not in the models. > There was an article posted here recently about “mathy” equations that this reminds me of. Any math model (including models described in Das Kapital) is either going to be oversimplified or "mathy". The only other choice is non-math models, which doesn't seem very useful if you want to talk about money, prices, profits and other numerical stuff.

1htfp 2y ago

One of the most fascinating things about working on a trading floor is that models such as BSM transcend their normative aspect and become mental models. Pricing an option? Basically only two things matter: where the underlying asset forward price is at maturity (this is related to the concept of drift) and what the volatility is. At any time, your job is choose “bumps” (which you add to market prices) in order to maximize your odds of making money on a trade subject to beating your competition on price. There are some people who make a living making these markets who likely have never heard of “Ito’s lemma” or diffusion equations.

MuffinFlavored 2y ago
> where the underlying asset forward price is at maturity What models do people use for SPY/SPX forward price?...
- blitzar 2y ago
  Futures. Decomposes down to price + dividend + time value/cost of money
  - MuffinFlavored 2y ago
    How often is front-month /ES not right around 20-50 points ahead of whatever SPX is trading at?
    - blitzar 2y ago
      Whenever the maths says so - the range you suggest is due to dividends typically collectively paying slightly higher than the risk free rate. Were we to have higher rates and companies not paying dividends en mass then that would be a negative number. How often? I would guess often - especially over the ~100 year history - and not something you would want to have wrong when writing billions in options.
      - MuffinFlavored 2y ago

FabHK 2y ago

A few points: 1) Very nice exposition. 2) Near eq. (4) it is claimed that one cannot compute the delta \frac{\del C}{\del S} without stochastic calculus, since S is stochastic. That doesn't strike me as correct: C is just a deterministic continuous function of S, C, K, T, t, r, sigma; and computing partial derivatives does not require stochastic calculus. 3) It captures the notion that when you hedge, you use risk-neutral probabilities. 4) Generally, in practice, BS is written as follows: C = df ( F N(d1) - K N(d2) ), where d1 = (ln(F/K) + 1/2 s^2)/s, d2 = d1 - s, s = sqrt(sigma^2 (T-t)), df is the discount factor, and F is the forward price of S. This abstracts away the whole discounting business. Note that sigma never occurs except in the expression sigma^2 (T-t), which is dimension less, thus sigma has physical dimension 1/sqrt(year), usually ("annualised vol"). C has the same dimension as F and K.

gwgundersen 2y ago
2) Thanks for pointing this out. I've fixed.

lowkey 2y ago

I’ve always found it strange that BSM is used for calculating implied volatility of American style options when it was specifically designed only for European style options. Can anyone comment if there are more suitable models for American style options?

FabHK 2y ago
Generally, you back out local vols (as a function of S, t) of the BS vols (as a function of K, T) by a process described first by Dupire, and then you price American options (and other products that are not sensitive to vol of vol) with that using a numerical PDE solver. https://en.wikipedia.org/wiki/Local_volatility https://en.wikipedia.org/wiki/Local_volatility

SJC_Hacker 2y ago

Like everything in finance it mostly worked (and sometimes still does) because everyone uses it.

vismit2000 2y ago

Veritasium video on the topic is also very good: The Trillion Dollar Equation - https://youtu.be/A5w-dEgIU1M https://youtu.be/A5w-dEgIU1M