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An Intuitive Explanation of Black–Scholes

gregorygundersen.com

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Re: An Intuitive Explanation of Black–Scholes

#81

Post by blitzar »

MuffinFlavored wrote: post #78
1htfp wrote: post #46

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 ma…

> 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

Re: An Intuitive Explanation of Black–Scholes

#82

Post by [deleted] »

thaumasiotes wrote: post #75
onerandompotato wrote: post #48

Earlier quoted context omitted.

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 futu…

> 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 cut…

[deleted]

Re: An Intuitive Explanation of Black–Scholes

#83

Post by Renevith »

dahfizz wrote: post #65
FabHK wrote: post #51

Earlier quoted context omitted.

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)…

> 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?

The parent is assuming that you can always sell your option to someone else for its fair value. If that's the case, there would never be a time where it's optimal to exercise a call option, because the optionality will always make the option value higher that the value of owning the stock.

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.

Re: An Intuitive Explanation of Black–Scholes

#84

Post by MuffinFlavored »

blitzar wrote: post #81
MuffinFlavored wrote: post #78

Earlier quoted context omitted.

> 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

How often is front-month /ES not right around 20-50 points ahead of whatever SPX is trading at?

Re: An Intuitive Explanation of Black–Scholes

#85

Post by putcher_willow »

bee_rider wrote: post #41

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…

You're not alone in finding the name poetic: see 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...

Re: An Intuitive Explanation of Black–Scholes

#86

Post by FabHK »

sokoloff wrote: post #69
FabHK wrote: post #51

Earlier quoted context omitted.

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)…

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.

By put-call parity, C - P = S - df K, thus C = S - df K + P > S - K.

This is contingent only on the discount factor df being = 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).

Re: An Intuitive Explanation of Black–Scholes

#87

Post by blitzar »

MuffinFlavored wrote: post #84
blitzar wrote: post #81

Earlier quoted context omitted.

Futures. Decomposes down to price + dividend + time value/cost of money

How often is front-month /ES not right around 20-50 points ahead of whatever SPX is trading at?

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.

Re: An Intuitive Explanation of Black–Scholes

#89

Post by SJC_Hacker »

twoodfin wrote: post #47
tehjoker wrote: post #28

Earlier quoted context omitted.

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?

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.

Re: An Intuitive Explanation of Black–Scholes

#90

Post by MuffinFlavored »

blitzar wrote: post #87
MuffinFlavored wrote: post #84

Earlier quoted context omitted.

How often is front-month /ES not right around 20-50 points ahead of whatever SPX is trading at?

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.

Thank you. I was wrong and thought "/ES contract 24 months out will show what investors/market makers/whoever think the market will grow to in 24 months"

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?

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