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An Intuitive Explanation of Black–Scholes

gregorygundersen.com

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Re: An Intuitive Explanation of Black–Scholes

#51

Post by FabHK »

ndesaulniers wrote: post #49
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?

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.

Re: An Intuitive Explanation of Black–Scholes

#52

Post by FabHK »

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?

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.

Re: An Intuitive Explanation of Black–Scholes

#53

Post by FabHK »

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.

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.

Re: An Intuitive Explanation of Black–Scholes

#54

Post by lowkey »

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?

Re: An Intuitive Explanation of Black–Scholes

#55

Post by FabHK »

lowkey wrote: post #54

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

Re: An Intuitive Explanation of Black–Scholes

#56

Post by mhh__ »

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…

The real purpose of models is risk anyway e.g. implied vol is handy, delta is essential.

Re: An Intuitive Explanation of Black–Scholes

#57

Post by mhh__ »

yieldcrv wrote: post #16

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

Re: An Intuitive Explanation of Black–Scholes

#58

Post by lazide »

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).

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?

Re: An Intuitive Explanation of Black–Scholes

#59

Post by IAmGraydon »

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…

>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.

Re: An Intuitive Explanation of Black–Scholes

#60

Post by loveparade »

IAmGraydon wrote: post #59
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…

>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.

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.
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