The Black-Scholes/Merton equation [video] | Hacker News Reader

The Black-Scholes/Merton equation [video]

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126 pointssurprisetalk86 comments

There is some valuable intuition in that the value of an option can be broken into three components, the intrinsic value - that is the difference between the asset price and strike price on the option, the time value which is dependent on the time to expiry and the risk free rate, and the “insurance” value which is dependent on the volatility and the time to expiry.

In the swaptions market, for instance quotes are typically for an “at-the-money forward” rate, which makes the first two components zero and all the value is tied to the vol.

Ignorant question: is it just bare speculation priced there as a component?

That’s the market clearing volatility

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...

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?

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 really existed until after the assumptions failed and the collapse was inevitable.

My point is you can't plead 20-20 hindsight.

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.

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

LTCM wasn't caused by anything like mortgage-backed securites.

LTCM was doing ultra-leveraged short-term trades of various kinds. When they started these were arbitrages of various kinds so they had low risk and a solid edge but as more capital flowed into their fund, the capacity of those trades was exhausted and they put money into riskier other trades. Some of their counterparties were big banks who parked overnight funds in LTCM and then when they got spooked by some losses and yanked those funds, LTCM lost a staggering amount of money very quickly as a result and went bust. If you want an excellent book about LTCM, "When Genius Failed" is one of the best books ever written about the history of financial markets.

The origins of the subprime mortgage crisis were a lot more complicated than most people give credit for and in particular I really wouldn't take "the big short" as any kind of reliable guide. For a good critical view of the crisis written by someone who actually knows what they are talking about I would recommend "Fools Gold" by Gillian Tett.

Financial crises and crashes have happened since the dawn of human history and will probably continue to happen. Suffice to say that the 2008 crisis had nothing to do with thing things that caused LTCM to fail, and neither of them have anything to do with the insight behind the Black/Scholes/Merton model other than the fact that Scholes and Merton were I think on the board of LTCM.[1]

[1] Fun fact, the other one of the three, Fischer Black was a quant at Goldman Sachs. So there's your 2008 crisis connection[2]

[2] Or not. Fischer Black died in 1995.

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

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

In practice the standard deviation used is implied from option prices, making the whole thing completely circular. To match the market you need to use different standard deviations (volatilities) for different options!

Volatility is a bit more predictable than price. And there are more complex formulae that also model volatility rather than treat it as a constant.

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

dr1ver is correct here and you are not.

Actual volatility (not implied!) is much easier to predict than price.

It’s also much more difficult to trade than price changes. So your intuition about this is correct though.

It is not super difficult to predict tomorrows volatility sign (up/down compared to today) with +60% success. Even textbook GARCH models do well here.

If you could do that with the price, you’d quickly become filthy rich.

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

People have been trading options and futures on the volatility of the S&P500 for years:

https://www.cboe.com/tradable_products/vix/vix_options/

This is also (slightly) incorrect.

When trading the VIX, you are trading the implied volatility not the actual (realized) volatility.

VIX represents the implied vol of options on S&P500 expiring 30 days into the future.

Trading the realized volatility is not easy :)

It’s about using the best information you have to quantify and systematize risk.

This is why options traders (prototypical ones at least, I suspect most trading is still directional) trade on the implied volatility, as a projection of future volatility.

You take a view on volatility by buying or selling an option, if you are right then you will make money proportional to the options gamma (i.e. the convexity of the option is where the money comes from)

Need to separate two different situations here:

1) where there are pretty complete markets for implied volatility, looking at the past matters less to little, because there is a market for the "future volatility" you can hedge and interact with

2) when there isn't a good volatility market and hedging future volatility exposure is difficult, looking towards the past for some guidance increases in importance

Both things can get complicated at times and in both cases it isn't strictly speaking the stddev you care about, but the quadratic variation (which can be the same under some assumptions).

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 - you're so far away from the price of the share that the increase in volatility from the price drop increases the option's value even though it's moved even more out of the money

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

In a way, yes. A lot of money follows these standard formulas for pricing which do not necessarily reflect accurate probabilities of the underlier price movement. After an idiosyncratic price shock (disappointing earnings, geopolitical news etc), people blindly following a trailing 1 month volatility or something will misprice the option as volatility reverts back to the mean. This probably has been arbed away to a large extent by trading algorithms.

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

Not really. The equation is just saying "based off these assumptions here is the best price" and you would make money if your assumptions differ from market assumptions in a favorable direction. Arbitrage is the closest to exploiting "bugs" in finance to get risk free returns but in a liquid enough market all these obvious opportunities quickly close (if there's free money on the ground, someone will pick it up, and then there's no more free money ond the ground. Some hedge funds build ultra fast private internet networks just to be able to pick up that free money nanoseconds faster than someone else). It's more that the equation is telling you if you think you have a better estimate for some of these values, what you should be willing to pay.

You can make money just indiscriminately selling option premium, many do. You just have to do it in a way that you're sure you won't blow yourself up. You make money because you are paid for taking on that volatility risk. Just like an insurance company.

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 that volatility is cheap and visa versa. Options are also a very specific kind of instrument and can be used to craft very specific bets on volatility. For instance, you might feel that the options at a $200 strike are pricing too high of an implied volatility compared to those at the $195 and $205 strikes.

Traders build an intuition around the model instead of treating it as in and of itself predictive. They instead try to price or take bets on certain derived quantities from it (the "greeks").

The saying goes that implied volatility is "the wrong quantity put into the wrong model in order to make the right decision".

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.

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).

If someone is interested in all this, I would strongly recommend looking into Ed Thorp, he discovered pretty much the same thing earlier, but instead of publishing, he made money with the knowledge...

Great book about all this, 2017 Autobiography: "A Man for All Markets: From Las Vegas to Wall Street, How I Beat the Dealer and the Market"

The video refers to him.

You made me watch this :-) still another book worth reading and not mentioned there:

"When Genius Failed: The Rise and Fall of Long-Term Capital Management"

https://en.wikipedia.org/wiki/When_Genius_Failed

Ed Thorp worked with Claude Shannon. That time in Cambridge, MA must’ve been amazing.

Especially for Shannon :-)

Pure Black Scholes is not really that useful nowadays because of its key limitations, some can be easily fixed (dividends, no risk free rate, etc.) other cannot (constant volatility) which makes it only useful in areas like vol targeting where you want the volatility to be constant.

I feel like none of these really matter. It's "what is a 0DTE trading for on Robinhood" which is largely driven by supply/demand which does nothing but jack up the IV variable of the equation.

Black-scholes is a hedging argument, the eqn isn't the essence of it

Eh, put-call parity is the hedging argument [1]. Black-Scholes-(Merton) was a breakthrough because it lets one understand why the hedge works, and thereby hedge and price more precisely.

[1] https://en.m.wikipedia.org/wiki/Put–call_parity

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?

Former option trader here. The free parameter is actually the "thing" that you're actually trading when you trade an option. All the other parameters are just environmental, you look them up.

The short story is that the implied vol is a sort of balancing price between how much the option loses in value over time vs how much you can make performing the hedge.

Why do any of the other variables matter if the market is collectively fighting between "overpriced and underpriced" on premium/implied volatility?

You can find an IV that makes sense for a single option with invalid other parameters, but things will break down when you go to price other expirations / strikes.

When trading, you don't want to wait to see an "updated" IV, you would want to respond directly to changes in important and well understood parameters like underlying price.

The Black-Scholes equation describes the unconscious biases that influence the price of options. It is able to beautifully separate the one quantity that every trader prices differently from the rest. That’s volatility.

All the other factors, time including, are the same for everyone.

They don't really matter. They are just things you look up in order to a get a number out for what the option costs in dollars.

Was BS actually of any practical use? It just seems to be a fantasy like most maths in economics.

Yes, it has some use. First of all, if you don't have a common model, it becomes impossible to talk about vol. So even if everyone uses their own model, they convert back into BS vol to talk about vol. Second, all models are wrong, but some models are useful. BS is a good starting point because it captures something that is relevant in option pricing, namely that uncertainty matters.

Implied volatility is really the standard deviation of the price over time. You can calculate it by look at prices in the market. Then interpolate values. Where banks get funky is that the market for options go out about 3 years, but a banks will write options going out much much further. For those options, they are really just guessing, no matter how much fancy math they do, it's all to dress up a guess. And the traders don't care since they won't be around when the option expires

> Implied volatility is really the standard deviation of the price over time.

Maybe you mean that “implied volatility is really the implied standard deviation of the price over time”.

Sorry, yes, that is the better definition.

It's not really implied volatility, it's the actual volatility between now and expiration. Everyone can only estimate at what that will be.

IV is essentially using prevailing prices to understand what everyone else has estimated that forward volatility to be.

Beyond that, you will also find that IV differs across strikes [1]. Still, being able to fit a vol smile from incomplete market data (and some other adjustments if you are very sophisticated) and then price an arbitrary option is pretty useful.

[1] https://en.wikipedia.org/wiki/Volatility_smile

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.

One thing that seems not widely understood is that the assumption of normal or log-normal distributions in stochastic calculus is like the assumption of linearity in most engineering fields. It's known to be incorrect, but you can do a heck of a lot with piecewise-linear model. Similarly, using a normal or log-normal assumption within a range of parameters (such as the volatility smile) is really useful.

Part of the reason a Gaussian distribution is used so much is that you need a stable distribution if you want to be able to perform algebra on your random variables. The variance of the Cauchy distribution is undefined and the variance of the Levy distribution is infinite, so Gaussian is really the go-to distribution.

The parameter (sigma) is the historical volatility (stdev of annualized returns). Implied volatility is what you get if you run the formula backwards and input the observed price to solve for volatility. In practice though many people use implied volatility as the input making the whole thing circular.

It's no really circular, just think of IV as the price/what is traded.

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(fixed), with the prevailing time left(fixed theta) under current interest rate(fixed), given the underlying, the historical vol gives you the wrong spot. You fudge it until you get the right spot. Call the fudged quantity the IV. Now plot that fudged quantity for a few other strikes and you get a smile. Then you can mess with that smile, plot the vol surface etc but end of the day, does the BS equation matter if the price of spot is going to be off and you have to fudge it with IV ? Its a good question. From an operational standpoint, the equation doesn’t matter. You can use bopm and get a more intuitive price anyways. Traders can trade the iv without knowing what effect BS has on the system.

When I was in 5th grade, they took us to the top of a tall building. We dropped a ball and measured the time it took to hit the ground. So if you square that time and multiply by 5, that’s how tall that building is. At that age I thought wow this is such magic! Then I grew up and reached 8th grade and worked out equations of motion with some basic differential calc, and derived the canonical equation s equals ut plus half at square. So since u is zero and a on planet earth happens to be g which is 9.8, half of which is about 5, that’s why 5t^2.

ok but does this equation matter ? I could have gone my whole life measuring height of buildings without knowing what is gravity.

> 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 are what drives IV in reality I feel.

> 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.

Wasn't the main breakthrough it's utility and accessibility, not precision?

It's still quite generalized in that it assumes a flat volatility surface, which even traders in the pits intuitively knew was wrong (thus the emergent volatility smile after '87). What it did allow was for a single number (implied volatility) to function as the single knob to be dialed to move quotes up and down for convex instruments. Therefore, instead of calling a trade desk and working out direct price quotes, you could have an immediate frame of reference ("this is trading at 31 vol") and move the offer to, say, "30 vol", leaving the calculation to the computer because both parties shared a language.

As for the vol smile and lack of volatility surface uniformity in real markets, it wasn't an issue, because pit traders were fine pricing different strikes at different vols, and players deeper in the volatility space had their own more accurate models geared for each market/instrument.

Knowing an underlying's iVol gives a good general overview of the pricing landscape with just a single number, and then if you need more precision, you can pull up the list of strikes and ivols for each strike and see the shape of the vol surface. That just takes a few more seconds. It's very quick and very useful, from the pit trader crews with the proto-handheld computer to the sell side and buy side deals working the phones. Utility!

To expand a bit, it is also a great feature that the second level of granularity (breaking away from the theoretical flat vol surface by applying different iVol values to different strikes) isn't crammed into another overarching generalized model. It breaks the model and lets traders go, say after the '87 crash, "tail risk is trading much higher what it has been historically, and this stuff is staying permanently bid, looks like a regime shift. We don't have a generalized model for this yet but in the meantime, traders in the pit are working with this new pricing, we can all see it and speak the same language, and we'll work out the new generalized models at a later date." That's why these simple options pricing models are still useful today, even though there are far more known kinks in volatility surfaces than there were decades ago.

> Wasn't the main breakthrough its utility and accessibility, not precision?

Utility yes, accessibility no. It quantified the previously artistic. You may enjoy Peter Bernstein’s Against the Gods.

> pit traders were fine pricing different strikes at different vols

Former algorithmic options trader. We ate the former pit traders for breakfast. Modelling the volatility surface is an entire field, and to the extent problems in finance can be solved this is sort of one of them.

Taleb and Derman have argued that put call parity implies BS but other disagree quite strongly.

Under reasonable assumptions, put call parity is true so it's not an important statement to say that it implies BS.

You still need additional assumptions about the delta hedged PnL, no?

> Taleb and Derman have argued that put call parity implies BS but other disagree quite strongly

I mean sure. They're related. Same way the Michelson-Morley experiment implied special relativity. That doesn't detract from Einstein specifying just how. (Derman is legit but Taleb is a hack.)

Id say the underlying hedging argument is continuous delta hedging.

Fisher black has written some interesting books. For anyone who is interested, check out his Wikipedia page.

Just FYI: Fischer ... not Fisher.

Before recent events where derivatives became the underlying. Different games being played.

Something the intro of this video missed a little bit is that most derivatives are linear — no optionality.

Options are a relatively tiny market, but are nonetheles key to reasoning about markets.

This got me thinking: Can the backpropagation algorithm be expressed as an equation?

I assume this should always be possible using functional programming.

Are you talking about backpropagation in neural networks? Do you mean as a differential equation? It already is a normal algebraic equation.

Yes in neural networks. Backpropagation is probably worth more than the equation in the video. But it's an algorithm, which aren't usually expressed as equations.

Backpropagation is just the chain rule from basic calc. Done on a massive scale in something like an LLM. You can absolutely express it as equations.

wasn't this debunked by NNT?

I've been trading for 4 years on the side mostly on intuition. I like to think one can emotionally program the powerful computer that is the unconscious mind to serve the distributed monster of global finance for profit. Sometimes I even do tarot. Seems to kinda work LoL what could possibly go wrong????

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