Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

An Intuitive Explanation of Black–Scholes

gregorygundersen.com

Post reply on HN
Print view View original

31–40 of 94 posts

Re: An Intuitive Explanation of Black–Scholes

#31

Post by gyudin »

stackghost wrote: post #12
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.

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

Re: An Intuitive Explanation of Black–Scholes

#32

Post by nknealk »

mikeyouse wrote: post #25
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…

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.

Re: An Intuitive Explanation of Black–Scholes

#33

Post by dumah »

nknealk wrote: post #32
mikeyouse wrote: post #25

Earlier quoted context omitted.

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.

Re: An Intuitive Explanation of Black–Scholes

#35

Post by jesuslop »

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.

Re: An Intuitive Explanation of Black–Scholes

#36

Post by jesuslop »

erehweb wrote: post #34

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.

Re: An Intuitive Explanation of Black–Scholes

#38

Post by listenallyall »

melenaboija wrote: post #10
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.

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 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.

Re: An Intuitive Explanation of Black–Scholes

#39

Post by klysm »

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…

Kinda, but it’s not great because of the volatility smile
Post reply on HN