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.
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.
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.
I have seen the insides of an options market maker, and can say this is not really true (at least for some regions of the market). Black-Scholes is used to derive theoretical prices for options. Good option traders will have an opinion on volatility and won't just take whatever the market says.
However, one of the interesting aspects of serious option trading is that Black-Scholes is merely your bread and butter. There is a lot of information that goes into option pricing, including supply/demand signals. The mix of signals also depends on the time scale on which you are trading.
What rings true to me with this comment is the correlation between products. Option traders are often concerned with many relationships between product pricing: between underlying and option, across expiries, across strikes, between products in indices, between products in sectors, etc .
>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.
I'm not sure what your point is. Yes, actual market prices are determined by...the market. The Black-Scholes formula is widely used in modern finance to MODEL the price of an option given different sets of inputs in theoretical situations.
The way the article is written, it appears that the formula is used as: 1. Observe market parameters (volatility of the underlying and risk free rate) 2. Plug into formula 3. Deduce a price for the option.
My point is that it is used in the opposite way: observe prices to deduce market parameters. You claim my point is obvious, but I'm not sure it would be obvious to a reader unfamiliar with modern finance reading this article, which is the target audience.
> 1. Observe market parameters (volatility of the underlying and risk free rate) 2. Plug into formula 3. Deduce a price for the option.
In the FX market (interbank), the quoted and "traded" number is Implied Vol - the price of the option then follows from there (via the Black–Scholes model).
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?
If there was a way to directly formulate every parameter of the black Scholes formula you would be correct. The problem that you run into is how to calculate volatility itself? Without the volatility value, your algorithm cannot trade on it.
Using history of volatility is insufficient, because volatility is a forward looking measure. Just because the stock was volatile in the past does not mean it will be in the future, and vice versa. There are even more nuances with this, as volatility is a smile (or a surface), not a singular number https://en.wikipedia.org/wiki/Volatility_smile.
TLDR Trading in volatility is a very complicated topic. However, volatility is a useful parameter, and black Scholes is typically used to deduce the forward looking volatility from option price, in addition to volatility -> option price.
That article says that implied volatility is inconsistent, with options at strike prices that are very far from the current market price having costs that imply a different level of volatility than options at strike prices that are close to the current price. The cute question here is "should an option be priced according to the actual level of volatility in the price of the underlying asset, or should it be priced according to the level of volatility that exercising the option would require?"
In a sense, BS and the option market enable trading in volatility itself.
Specifically, you trade in the estimate of the stock's volatility over the time from now to expiry of the option.
If you don't want to trade options directly to do that (it is cumbersome, as it involves "continuous" delta hedging), you can trade in VIX futures for the same purpose. Or variance swaps.
Black Scholes is not used for any otc option pricing, except perhaps to provide an instantaneous estimate to get in the ballpark, but no one would use it for the final price.
European and American calls cost the same on non-dividend paying stocks (on dividend paying stocks, it might make sense to exercise an American just before the ex-date).
Either way, as was pointed out, in reality BS is used as a deterministic one-to-one mapping between option prices and BS vols. Then, from market quotes (either as prices or as BS vols) a vol-surface is fitted (as a function of strike and expiry time), from which a stochastic process is fitted that correctly re-prices all these points (using a model such as "local vol" or "stochastic vol" or a combination of those two, or others), and then everything is priced of that.
American style options are inherently more valuable. Imagine you had options on a stock that experienced a sharp but possibly temporary move. As a holder of an American style option, you could benefit from that temporary move, making it more valuable.
> European and American calls cost the same on non-dividend paying stocks
All else being equal, I would prefer to buy an option contract I can exercise at any time vs one I can only exercise on a certain date. It doesn’t make intuitive sense they would be priced the same, can you please elaborate?
> traders deduce the implied volatility from the observed option prices.
I've only ever seen one thing:
Black-Scholes models say IV should be less but your broker/brokerage/the market are overpaying for it.
I always figured it was closer to a Vegas juice/vig.
I never understood the benefit really.
Complicated math to tell you lots of people want to play roulette on NVDA earnings and whatever you are going to pay for it is going to be "overpriced/overvalued" in at least one way.
I've never seen the opposite where it helps you find an edge and something was undervalued.
For valuing financial products with no directly observable price, BS or its descendants matter quite a lot. For actually pricing a transaction on those, it becomes more complex but model value is typically an important input.
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.
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, 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.
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.
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.
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.
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.
What you’re describing is just a variant of Malthusianism [https://www.intelligenteconomist.com/malthusian-theory/], which may not be wrong - but has not proven right either (in modern times) with advances in technology.
Especially improvements in energy generation, fertilizer production, and efficient usage of both (often through information technology).
Given any stable state of technology/energy/space, a society will generally reach a high point, then go through cycles of growth/retraction.
But improvements in technology and energy generation means it won’t be at a stable state, eh?
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).
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.
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).
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"
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.
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)
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.
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?
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.
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
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
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.
> 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.
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.
> where the underlying asset forward price is at maturity
What models do people use for SPY/SPX forward price?...
Futures. Decomposes down to price + dividend + time value/cost of money
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.
Another good explanation from Terence Tao’s blog: https://terrytao.wordpress.com/2008/07/01/the-black-scholes-...
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.
I have seen the insides of an options market maker, and can say this is not really true (at least for some regions of the market). Black-Scholes is used to derive theoretical prices for options. Good option traders will have an opinion on volatility and won't just take whatever the market says.
However, one of the interesting aspects of serious option trading is that Black-Scholes is merely your bread and butter. There is a lot of information that goes into option pricing, including supply/demand signals. The mix of signals also depends on the time scale on which you are trading.
What rings true to me with this comment is the correlation between products. Option traders are often concerned with many relationships between product pricing: between underlying and option, across expiries, across strikes, between products in indices, between products in sectors, etc .
>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.
I'm not sure what your point is. Yes, actual market prices are determined by...the market. The Black-Scholes formula is widely used in modern finance to MODEL the price of an option given different sets of inputs in theoretical situations.
The way the article is written, it appears that the formula is used as: 1. Observe market parameters (volatility of the underlying and risk free rate) 2. Plug into formula 3. Deduce a price for the option.
My point is that it is used in the opposite way: observe prices to deduce market parameters. You claim my point is obvious, but I'm not sure it would be obvious to a reader unfamiliar with modern finance reading this article, which is the target audience.
> 1. Observe market parameters (volatility of the underlying and risk free rate) 2. Plug into formula 3. Deduce a price for the option.
In the FX market (interbank), the quoted and "traded" number is Implied Vol - the price of the option then follows from there (via the Black–Scholes model).
And it's a cycle. Supply and demand are partially driven by pricing models used by hedge funds, and variants of Black Scholes is one of those.
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.
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?
If there was a way to directly formulate every parameter of the black Scholes formula you would be correct. The problem that you run into is how to calculate volatility itself? Without the volatility value, your algorithm cannot trade on it.
Using history of volatility is insufficient, because volatility is a forward looking measure. Just because the stock was volatile in the past does not mean it will be in the future, and vice versa. There are even more nuances with this, as volatility is a smile (or a surface), not a singular number https://en.wikipedia.org/wiki/Volatility_smile.
TLDR Trading in volatility is a very complicated topic. However, volatility is a useful parameter, and black Scholes is typically used to deduce the forward looking volatility from option price, in addition to volatility -> option price.
> There are even more nuances with this, as volatility is a smile (or a surface), not a singular number https://en.wikipedia.org/wiki/Volatility_smile.
That article says that implied volatility is inconsistent, with options at strike prices that are very far from the current market price having costs that imply a different level of volatility than options at strike prices that are close to the current price. The cute question here is "should an option be priced according to the actual level of volatility in the price of the underlying asset, or should it be priced according to the level of volatility that exercising the option would require?"
Volatility is just a quantity.
In a sense, BS and the option market enable trading in volatility itself.
Specifically, you trade in the estimate of the stock's volatility over the time from now to expiry of the option.
If you don't want to trade options directly to do that (it is cumbersome, as it involves "continuous" delta hedging), you can trade in VIX futures for the same purpose. Or variance swaps.
I’ve seen it used for OTC option pricing - there’s no liquid market, so you are more of a market maker than a market taker.
Black Scholes is not used for any otc option pricing, except perhaps to provide an instantaneous estimate to get in the ballpark, but no one would use it for the final price.
The real purpose of models is risk anyway e.g. implied vol is handy, delta is essential.
Also, isn't it only used for European style options, not American?
European and American calls cost the same on non-dividend paying stocks (on dividend paying stocks, it might make sense to exercise an American just before the ex-date).
Either way, as was pointed out, in reality BS is used as a deterministic one-to-one mapping between option prices and BS vols. Then, from market quotes (either as prices or as BS vols) a vol-surface is fitted (as a function of strike and expiry time), from which a stochastic process is fitted that correctly re-prices all these points (using a model such as "local vol" or "stochastic vol" or a combination of those two, or others), and then everything is priced of that.
American style options are inherently more valuable. Imagine you had options on a stock that experienced a sharp but possibly temporary move. As a holder of an American style option, you could benefit from that temporary move, making it more valuable.
The way the market is typically modeled, temporary moves are not a thing.
The way the market actually exists, temporary moves are definitely a thing.
> European and American calls cost the same on non-dividend paying stocks
All else being equal, I would prefer to buy an option contract I can exercise at any time vs one I can only exercise on a certain date. It doesn’t make intuitive sense they would be priced the same, can you please elaborate?
> traders deduce the implied volatility from the observed option prices.
I've only ever seen one thing:
Black-Scholes models say IV should be less but your broker/brokerage/the market are overpaying for it.
I always figured it was closer to a Vegas juice/vig.
I never understood the benefit really.
Complicated math to tell you lots of people want to play roulette on NVDA earnings and whatever you are going to pay for it is going to be "overpriced/overvalued" in at least one way.
I've never seen the opposite where it helps you find an edge and something was undervalued.
For valuing financial products with no directly observable price, BS or its descendants matter quite a lot. For actually pricing a transaction on those, it becomes more complex but model value is typically an important input.
Kinda, but it’s not great because of the volatility smile
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.
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.
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, 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.
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.
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.
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).
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?
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.
Yeah but a couple centuries? What’s the evidence for exhaustion of demand and supply for economic goods on that kind of time horizon.
Or just poor people hitting the zero bound.
What you’re describing is just a variant of Malthusianism [https://www.intelligenteconomist.com/malthusian-theory/], which may not be wrong - but has not proven right either (in modern times) with advances in technology.
Especially improvements in energy generation, fertilizer production, and efficient usage of both (often through information technology).
Given any stable state of technology/energy/space, a society will generally reach a high point, then go through cycles of growth/retraction.
But improvements in technology and energy generation means it won’t be at a stable state, eh?
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).
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.
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).
Why? the mean can be negative?
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"
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.
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)
The "Loeb Measures in Practice" book also by Cutland has a survey chapter.
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:
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.
2) Thanks for pointing this out. I've fixed.
This guy's other notes are also well thought through and written. Thanks for the link.
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?
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
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
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
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
Brownian motion is what happens when people lose their life savings on meme stocks.
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.
> 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.