An Intuitive Explanation of Black–Scholes
gregorygundersen.com
gregorygundersen.com
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.
"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...
What models do people use for SPY/SPX forward price?...
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.
I didn't realize it was exclusively (as you said, dividends + ~risk free rate).
Is there anything that you are aware of options chain wise (for example, a call 12 months out or 24 months out) that holds any statistical accuracy/merit in "oh, the market collectively thinks S&P will end up around this price range by this time period?
That's for boring dollar cost averaging.
I was saying, is there anything we can look at "proof" on the options chain or futures contracts on what people/"the big boys" are "hedging for/against"?
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.
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.
That is, if the real price ever differred significantly from what Black-Scholes predicts, wouldn't algorithmic trading very quickly correct this deviation?
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?"
Volatility is just a quantity.
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.
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.
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?
This is shown in the article: the curved lines representing the option value are always above the straight lines of the final option payoff (the value if exercised).
This is not necessarily true for put options or for call options if the stock pays dividends. In those cases the option value can be below the payoff line and early exercise would be better than selling the option.
This is contingent only on the discount factor df being <= 1, and P >= 0, which is basically always the case. Thus, the value of the call exceeds the exercise value, making exercise never optimal.
Exercise for the reader: Understand why the same argument doesn't work for puts (or calls on dividend paying stocks).
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.
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.
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).
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 .
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.
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.
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.
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).
The limits may be very large, but they aren’t infinite.
Without expansion of population, consumption and production both stagnate. See what has happened in Japan in since the 90s.
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?
It isn't crazy to think that there are physical limits to things.
Malthus projected that populations would grow exponentially, but agricultural yields would grow linearly. He was wrong in that ag yields did keep up and population growth slowed down. One thing to keep in mind is we used fossil fuels and fertile lands to do that, but we are hitting the limits for fossil fuels and we are burning through arable land.
However, there are other physical limits, and some of these are a bit harder to work around. Infinite growth isn't necessary, but a decent life and an equitable distribution of wealth is.
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.
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.
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).
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.
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
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"
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.
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.
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.
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.
Can anyone comment if there are more suitable models for American style options?
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
Also they were more of advisors in the fund then anything else
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
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.
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.