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The Black-Scholes/Merton equation [video]

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Re: The Black-Scholes/Merton equation [video]

#71

Post by PheonixPharts »

tel wrote: post #69
btdmaster wrote: post #13

How does this square with "past market returns are do not (entirely) determine future market returns"? Surely the same applies to the historical stddev?

The variance component, implied volatility, is more often than not treated as the _output_ of the equation. By looking at the prices of options you can determine what the market currently, implicitly, estimates the future variance of underlying to be. Lots of options trading involves taking a position on whether you think that implicit estimate is too high or too low. Generally, a long options position encodes belief…

Adding to this, it's very worthwhile exercise for any curious programmer to work out the IV of a stock based on options pricing and compare it to other measure of volatility (for example historic volatility, or even your own beliefs about volatility based on what you think future returns might be).

Black-Scholes/Merton makes a lot more sense once you work it all out yourself in code.

I'd actually suggest doing this through modeling the underlying geometric Brownian motion and ensuring that your simulated results match up to the analytic formula.

Re: The Black-Scholes/Merton equation [video]

#72

Post by PheonixPharts »

exclipy wrote: post #44
vikramkr wrote: post #38

Earlier quoted context omitted.

Yep. You can make money off of using options as a way of betting on what the volatility measure itself will be. If you think the historical standard deviation is lower than what it will be because of some new change to the company or the world environment, and your view is different from the market's view. It's why sometimes very out of the money call options will paradoxically go up in price after really bad news -…

Is that a bug in the equation that one could take advantage of?

The underlying assumption Black-Scholes makes, that stock price movements can be modeled by a log-normal distribution, is known to be false. However not since the 1980s has this lead to the ability to make money of the model itself being imperfect.

The true distribution of the market beliefs in future stock prices can be understood by empirically studying the volatility smile [0]. That is, because investors know Black-Scholes is not a perfect mathematical model of real world stock behavior, every strike price has a different implied volatility. By looking at these different IVs you can get a sense of what the market believes are the true probabilities of "long tail" events.

In theory, the opportunities you have to make money should be cases where you believe the market has mispriced risk. In my amateur experience, I have found that virtually every time you think the market has mispriced some extreme event, when you look at the volatility smile, you realize you are mistaken.

0. https://en.wikipedia.org/wiki/Volatility_smile

Re: The Black-Scholes/Merton equation [video]

#73

Post by dboreham »

RandomLensman wrote: post #56
RichardCA wrote: post #52

There was a PBS NOVA episode about this. https://vimeo.com/302855460 The question that seems obvious, but no one ever seems to talk about, is how the failure of LTCM paved the way for the subprime crisis of 2008. I mean, did the elite financial world fail to learn the lesson, or did it simply learn the wrong one?

How would you connect the two events?

The idea that you can somehow predict unpredictable things, if you wrap it in fancy math.

Re: The Black-Scholes/Merton equation [video]

#74

Post by skzv »

What I find fascinating about this equation is that is in the form of the diffusion (heat) equation. The fundamental solution to the diffusion equation is the Gaussian. So you can think of solving the equation as applying a Gaussian blur to the boundary conditions (given by the terminal price of the option, which turns into an initial condition after a change of variables).

The implication is that the further out the option, the blurrier the picture gets, because we're applying more Gaussian blur.

One thing that Veritasium glosses over is that BS assumes that the stock price is lognormal distributed, so it follows a geometric brownian process instead of just a regular one. That's why the drift term on a stock price is (r - sigma^2/2) instead of just r in terms of geometric returns. Volatility lowers compounding returns. This is called volatility drag [0].

[0] https://www.kitces.com/blog/volatility-drag-variance-drain-m...

Re: The Black-Scholes/Merton equation [video]

#75

Post by slt2021 »

tel wrote: post #69
btdmaster wrote: post #13

How does this square with "past market returns are do not (entirely) determine future market returns"? Surely the same applies to the historical stddev?

The variance component, implied volatility, is more often than not treated as the _output_ of the equation. By looking at the prices of options you can determine what the market currently, implicitly, estimates the future variance of underlying to be. Lots of options trading involves taking a position on whether you think that implicit estimate is too high or too low. Generally, a long options position encodes belief…

everybody has different model of pricing options, but because everyone knows BS model then the IV (implied vol) becomes as a quoting instrument.

Option Traders can quote each other in IV without disclosing their asset pricing models and assumptions (trade secret tech).

Re: The Black-Scholes/Merton equation [video]

#76

Post by slt2021 »

richrichie wrote: post #41
lupire wrote: post #31

Earlier quoted context omitted.

If volatility is predictable, it would quickly be traded until it became unpredictable and unprofitable.

It is a well known empirical fact that volatility is mean reverting.

the hardest thing obviously is to time the moment when it will start mean reverting.

there is possibility you can take trading position with expectation of vol reverting to the mean, and vol will keep increasing (what happened to tesla and gamestop short sellers)

and vice versa

Re: The Black-Scholes/Merton equation [video]

#77

Post by andruby »

RichardCA wrote: post #59
RandomLensman wrote: post #56

Earlier quoted context omitted.

How would you connect the two events?

I don't know what the mathematical underpinnings of those mortgage-backed derivatives were, but I know someone somewhere looked at all that math and decided the risk burden was acceptable. No one said hey, wait, the model is only as good as the assumptions you feed into it. The movie The Big Short got at this, but it was just a movie with Steve Carell standing in for that shocked person who doesn't seem to have reall…

Housing prices hadn’t really gone down in the 20 or so years before 2008. So the “risk” of decreasing prices was just under-appreciated.

When the firms started making tons of money, it’s probably easy to ignore the risk department.

Margin call is another great movie. Doesn’t really explain the details of what went wrong, but I think it shows how a financial institution unravels when they accept that the risk is real and it is going to come down.

Re: The Black-Scholes/Merton equation [video]

#78

Post by RandomLensman »

RichardCA wrote: post #59
RandomLensman wrote: post #56

Earlier quoted context omitted.

How would you connect the two events?

I don't know what the mathematical underpinnings of those mortgage-backed derivatives were, but I know someone somewhere looked at all that math and decided the risk burden was acceptable. No one said hey, wait, the model is only as good as the assumptions you feed into it. The movie The Big Short got at this, but it was just a movie with Steve Carell standing in for that shocked person who doesn't seem to have reall…

I'd say the maths wasn't that central compared to beliefs about the housing market or regulations focused on ratings.

Re: The Black-Scholes/Merton equation [video]

#79

Post by JumpCrisscross »

MuffinFlavored wrote: post #58
dxbydt wrote: post #47

Earlier quoted context omitted.

I don’t believe any of the comments below address the meat of your question - what value are you getting from the equation ? I would answer- not much. You can think of BS as a curried function. Since all the other params are fixed, you can curry and get a reduced equation that only depends on IV and underlying. If you do that, then its just - you give me iv and underlying, i give you spot. So, for a given strike(fixe…

> If you do that, then its just - you give me iv and underlying, i give you spot. But... what really happens (in my opinion) is... options makers or writers or whatever might set a price based on what they feel is fair/good for them/whatever Then a bunch of people on Robinhood make memes over it, hammer the bid, IV goes to 160%, voila... Why does "spot" price matter in that equation? Robinhood buyers + supply/demand…

> options makers or writers or whatever might set a price based on what they feel is fair/good for them/whatever

Former options market maker. We basically made money because of (a) people setting prices based on gut feel and (b) retail investors buying options for leverage and then forgetting to exercise barely in-the-money contracts. The first has largely left the market; fortunately, the second came in with gale force.

> Robinhood buyers + supply/demand are what drives IV in reality

Of course. Supply and demand drive price. Volatility is a measure on price. Options are principally an instrument for trading volatility.

Re: The Black-Scholes/Merton equation [video]

#80

Post by JumpCrisscross »

dist-epoch wrote: post #26
sixers2329 wrote: post #6

Earlier quoted context omitted.

What I never fully understood is there’s a free parameter in the equation (Implied Volatility)- which has no solid definition besides “the number that makes the rest of the equation work”. At that point… how much value are you really getting from the rest of the equation?

This equation is an idealized option, a spherical cow. If one would actually use it to price options one would lose money. There are many empirical option pricing features that this equation can't explain - the "smile", the "skew", ...

> If one would actually use it to price options one would lose money

If you try to fly a rocket across the solar system using only Newton's equations, it will crash. That doesn't make Newtonian mechanics useless. Almost every option-pricing engine in the market starts with Black-Scholes-Merton. Smiles and skews are all dealt with on the vol surface--it's an expandable variable.

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