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The Black-Scholes/Merton equation [video]
- mhh__ 3y agoBlack-scholes is a hedging argument, the eqn isn't the essence of it
- JumpCrisscross 3y agoEh, 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 https://en.m.wikipedia.org/wiki/Put–call_parity
- sixers2329 3y agoWhat 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?
- lordnacho 3y agoFormer 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.
- MuffinFlavored 3y agoWhy do any of the other variables matter if the market is collectively fighting between "overpriced and underpriced" on premium/implied volatility?
- kflansburg 3y agoYou 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.
- sheepscreek 3y agoThe 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.
- lordnacho 3y agoThey 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.
- eggdaft 3y agoWas BS actually of any practical use? It just seems to be a fantasy like most maths in economics.
- lordnacho 3y agoYes, 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.
- scherlock 3y agoImplied 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
- kflansburg 3y agoIt'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 https://en.wikipedia.org/wiki/Volatility_smile
- dist-epoch 3y agoThis 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", ...
- JumpCrisscross 3y ago> 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.
- KMag 3y agoOne 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.
- drexlspivey 3y agoThe 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.
- RandomLensman 3y agoIt's no really circular, just think of IV as the price/what is traded.
- dxbydt 3y agoI 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.
- MuffinFlavored 3y ago> 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.
- JumpCrisscross 3y ago> 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.
- Fade_Dance 3y agoWasn'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.
- JumpCrisscross 3y 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.
- mhh__ 3y agoTaleb and Derman have argued that put call parity implies BS but other disagree quite strongly.
- lupire 3y agoUnder reasonable assumptions, put call parity is true so it's not an important statement to say that it implies BS.
- mhh__ 3y agoYou still need additional assumptions about the delta hedged PnL, no?
- JumpCrisscross 3y ago> 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.)
- RandomLensman 3y agoId say the underlying hedging argument is continuous delta hedging.
- glutamate 3y ago[flagged]
- zeroonetwothree 3y agoYou win some you lose some
- lordnacho 3y agoI thought they mostly lost money on interest rates, not options?
- sgjohnson 3y agoDerivatives are a zero-sum game. If someone lost a trillion dollars trading them, someone else made a trillion dollars trading them. It will always result in a net 0.
- glutamate 3y agoThat assumes that both counterparties are solvent. If one party isn't and is too big to fail, the tax payer will very much not see this as a sum game.
- runarberg 3y agoI agree that they are a zero sum game, but I’d argue that the sum is a net positive for traders over a long period of time, and the people who are systematically loosing are the workers who’s true earnings were siphoned away from them and onto the stock market, where they are sold as derivatives. This manifests in e.g. how much cheaper it is for people to borrow money who hold a large stock portofolio, how interests in a normal savings account are usually much less than people who keep stocks, etc. If you trade in derivatives perhaps every individual trade is a zero sum game, but taken together a whole year of trading yields benefits regardless, and that is also a zero sum game, but this time an unfair one, where you are guaranteed earnings at the cost of the workers.
- sgjohnson 3y ago
- Mistletoe 3y ago[flagged]
- btdmaster 3y agoHow does this square with "past market returns are do not (entirely) determine future market returns"? Surely the same applies to the historical stddev?
- fancyfredbot 3y agoIn 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!
- dr1ver 3y agoVolatility is a bit more predictable than price. And there are more complex formulae that also model volatility rather than treat it as a constant.
- lupire 3y agoIf volatility is predictable, it would quickly be traded until it became unpredictable and unprofitable.
- richrichie 3y agoIt is a well known empirical fact that volatility is mean reverting.
- slt2021 3y agothe 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
- throwawayFinX 3y agodr1ver 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.
- cubefox 3y agoThis got me thinking: Can the backpropagation algorithm be expressed as an equation? I assume this should always be possible using functional programming.
- eternauta3k 3y agoAre you talking about backpropagation in neural networks? Do you mean as a differential equation? It already is a normal algebraic equation.
- cubefox 3y agoYes 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.
- lamontcg 3y agoBackpropagation 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.
- mhh__ 3y agoSomething 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.
- curiousgal 3y agoPure 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.
- MuffinFlavored 3y agoI 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.
- rvcdbn 3y agoI'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????
- ripjaygn 3y ago[flagged]
- TacticalCoder 3y agoThis LLM generated garbage is pure rubbish. BTW I'm not saying all LLMs outputs are garbage: but that one is the usual confident fart. > discusses the work of Louis Bachelier, who in 1900, derived a formula to price options. This formula, known as the Black-Scholes-Merton formula Fuck no: BSM is called BSM because Black, Scholes and Merton build up on the work of Bachelier. Otherwise it'd have been called the "Bachelier formula" or something like that. Downvoted and flagged for having generated and posted vomit.
- JackFr 3y agoThere 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.
- wslh 3y agoIgnorant question: is it just bare speculation priced there as a component?
- anonymouse008 3y agoThat’s the market clearing volatility
- imtringued 3y agoFisher black has written some interesting books. For anyone who is interested, check out his Wikipedia page.
- NovemberWhiskey 3y agoJust FYI: Fischer ... not Fisher.
- SirLJ 3y agoIf 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"
- AlexCoventry 3y agoThe video refers to him.
- SirLJ 3y agoYou 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 https://en.wikipedia.org/wiki/When_Genius_Failed
- lvl102 3y agoEd Thorp worked with Claude Shannon. That time in Cambridge, MA must’ve been amazing.
- SirLJ 3y agoEspecially for Shannon :-)
- RichardCA 3y agoThere was a PBS NOVA episode about this. https://vimeo.com/302855460 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?
- RandomLensman 3y agoHow would you connect the two events?
- RichardCA 3y agoI 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.
- andruby 3y agoHousing 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.
- RandomLensman 3y agoI'd say the maths wasn't that central compared to beliefs about the housing market or regulations focused on ratings.
- seanhunter 3y ago
- lvl102 3y agoBefore recent events where derivatives became the underlying. Different games being played.
- skzv 3y agoWhat 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-mean-arithmetic-vs-geometric-average-investment-returns/ https://www.kitces.com/blog/volatility-drag-variance-drain-m...
- Hayatabad 3y agowasn't this debunked by NNT?