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An Intuitive Explanation of Black–Scholes

gregorygundersen.com

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Re: An Intuitive Explanation of Black–Scholes

#41

Post by bee_rider »

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.

Re: An Intuitive Explanation of Black–Scholes

#42

Post by marxisttemp »

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.

Re: An Intuitive Explanation of Black–Scholes

#43

Post by bibouthegreat »

smabie wrote: post #18
ComplexSystems wrote: post #2

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.

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"

Re: An Intuitive Explanation of Black–Scholes

#44

Post by melenaboija »

listenallyall wrote: post #38
melenaboija wrote: post #10

Earlier quoted context omitted.

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 you…

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 neve…

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.

Re: An Intuitive Explanation of Black–Scholes

#45

Post by wavemode »

ncclporterror wrote: post #21

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 w…

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?

Re: An Intuitive Explanation of Black–Scholes

#46

Post by 1htfp »

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.

Re: An Intuitive Explanation of Black–Scholes

#47

Post by twoodfin »

tehjoker wrote: post #28
bjornsing wrote: post #8

Earlier quoted context omitted.

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?

Re: An Intuitive Explanation of Black–Scholes

#48

Post by onerandompotato »

wavemode wrote: post #45
ncclporterror wrote: post #21

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 w…

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.

Re: An Intuitive Explanation of Black–Scholes

#49

Post by ndesaulniers »

ncclporterror wrote: post #21

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 w…

Also, isn't it only used for European style options, not American?

Re: An Intuitive Explanation of Black–Scholes

#50

Post by FabHK »

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.

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