The Black-Scholes formula, explained (2019)
cantorsparadise.com
cantorsparadise.com
Thoughts:
- Since you have a right but not an obligation to buy/sell, that creates asymmetry. Since it's asymmetric, a wider range of outcomes, ie higher vol (imagine your gaussian curve on top of the hockey stick), makes the option worth more.
- Similarly having more time to expiry makes the option worth more, the range of outcomes is more spread out.
- There's a whole bunch of Greeks that the books will go through, but the intuition is the same for all of them. You can work out what's good or bad for you from thinking about how the distribution of outcomes is affected by a change in whatever.
- To trade the vol and not a mix of the vol and the direction, you flatten your delta by trading the underlying. If you do this at some point on the option price vs underlying price curve, you can get the graph to be flat, ie neutral to small price moves. But you can't make it flat everywhere with a hedge, because of course the graph is bendy.
- Near the strike where the bend is in the hockey stick is where it curves the most. On one extreme the option is worthless, on the other it's the same as having the underlying.
- As time passes it's got to get more curvey at the strike, less curvey on the sides.
- Curveyness on the price graph is called gamma. This is the gamma that ended up biting with the GME squeeze, by the sound of it. The problem is if you are short options, the graph looks like an upside down parabola, so if the underlying moves up a lot you will be short and getting shorter. If it moves down a lot, you'll be getting longer and longer. This is bad.
- It doesn't actually matter whether you are buying the right to buy or the right to sell. If you're buying, you have a positive gamma. But how? Well since owning a put and shorting a call of the same strike (or vice versa) should give you no curvature (looks like a straight line) they must have the same curvature. In the business people just call options with higher strikes than the current underlying price "calls" and options with lower strikes "puts" regardless of what they actually are. In-the-moneys just have some more premium attached to them, but act the same (in terms of everything other than delta) as their partner out-of-the-money option at that strike.
- Why do people get short gamma, knowing that movement is bad for them? Of course the option costs something to the guy who buys them. As long as the movement isn't too much it might be worthwhile to be short. In fact, most of the time the movement isn't enough to justify the price.
If the market is too high on volatility: sell the option and you'll end up paying less in delta-hedging costs.
If that market is too low on volatility: do the opposite.
These two observations become unified when one observes that the two variables sigma (vol) and (T-t) (time to expiry) only ever enter the Black-Scholes formula together, as sigma^2 (T-t) (which is the variance of the log return to expiry).
(That leads to the observation that the unit of vol is 1/sqrt(time), normally a^(-1/2) [a=annum=year]).
Next, you alluded to it, but let's make explicit the way to trade implied vol: Suppose (WLOG) that you think implied vol is too low. Then you buy it (buy cheap, sell expensive...) in the market (buy buying calls and/or puts). Now you're long gamma, but short theta, and long vol in two senses: a) if the stock doesn't move, but implied vol goes up, you can turn around and sell the option back for a profit. b) you can delta hedge the option. Every day, you lose money on theta, but gain money on the movements of the stock. If it doesn't move at all, you lose the theta. If the stock moves "as expected" (=as implied by the implied vol), you'll just break even. But if the stock moves more than expected (that is, realised vol is higher than the implied you bought it at), then you make money. If the market comes around to your estimate of the future vol, you can then sell the option at a profit, or you can hold it to the end and keep delta hedging, and that will be profitable if realised vol turns out to be higher than implied.
...and has caused a lot of catastrophic losses. The formula depends on a normal distribution and financial returns are random but not independent. They are not normal.
The formula works, mostly, but when it does not it is worse than useless. Financial gains and losses are in the tails, and the tails are no where near normal
This is sort of like throwing out physics because Newtonian mechanics don’t account for fluid dynamics.
Yes, the original theory assumed that away. And yes, the original theory is taught in undergrad. But the work has been developed far past its original, adjusting for or incorporating away those initial assumptions, and—to a large degree—having been shown, empirically, to work.
(This is not your fault. Popular writing on the topic is terrible, elevating drama over accuracy. Against the Gods is one of the better ones, and doesn’t require much math.)
What does this refer to? And, no. Disagreement on inputs doesn’t convey much about a model—it’s a negotiation. Any model will have procurer and vendor using different inputs when negotiating purchase and sale. That Boeing and steel mill don’t agree on tensile strength assumptions doesn’t mean aircraft designers are winging it.
For a model that is supposed to measure the objective value of an asset, that’s clear failure. BS is at this point just a vague market consensus which stinks more and more, the farther you stray away from vanilla European options.
Let me know when we make a plane or rocket that doesn’t need to recalibrate it’s flight model every millisecond.
As I said, market neutral market makers will use different assumptions from directional buy side shops. To say nothing of their vastly different funding costs and structures. Using input heterogeneity or calibration frequency as an estimator of model quality is...odd.
Sure. But we do update all manner of atmospheric, gravitometric and similar factors in our flight and orbital models. Once again, calibration frequency is a poor predictor of model quality. There are useless models in every domain involving immutable constants. And there are very good numerical methods that have no constants per se.
Wilmott covered model robustness and calibration frequency. Any time you have to adjust an unobserved term in your SDE (like market price of risk, for example) you are changing the model. So if your model has to be changed every x minutes else it gives wrong results, you're not "adjusting your inputs", you are just using T/x shitty models.
Comparing financial markets to physics is pretty much what made it all explode last two times.
It is getting ready to explode again.
The asset market has been sedately rising for a decade. Which makes models like really good.
\sarcasm{ON} This time is different \sarcasm{OFF}
No it is not.
>having been shown, empirically, to work.
What exactly do you mean by this? That they are correct most of the time? Or that the person that uses them won't go bust?
This parallel between physical theories and assumptions about how the market works is bogus. In trading you can have strategies that are correct most of the time yet when they fail the impact of the loss can take you out.
Yup. Options market makers, the critical mass of dynamic hedgers, don’t blow up any more [1]. I left the business ten years ago, and was probably among the last well-paid people to do it. There isn’t much risk anymore which means there isn’t room for ingenuity—it’s execution, mechanical.
Between market circuit breakers limiting instantaneous price moves; the tremendous amount of liquidity in option-covered symbols; tail-risk estimating options models; and fully electronic options, equities and money markets, there simply isn’t empirical evidence for hidden risks in the model. Cash equities execution was once super complicated. It’s now commoditised. Same for options.
It sells books to claim otherwise. But you largely need to re-tell stories from the 90s, where LTCM bet on short-term Russian debt, or recount crisis-era structured products on illiquid mortgages to fill the pages.
[1] American OCC-cleared options market makers who aren’t making directional bets but manufacturing options and hedging their books
In the same way a rocket flight model is forecasting the arrangement of air molecules it’s about to run into. They’re instantaneous forecasts that are dynamically updated. No long-term forecasting involved.
At the end of the day, options market makers haven’t blown up since the early noughties. (LTCM got sunk by non-options bets.) They are low-margin, low-risk businesses. It’s fun to talk about them like they’re black boxes. And traders trying to defend their compensation will keep pitching them that way to senior management. But options pricing is a boring, largely solved—if still interesting—problem.
Since the model to convert parameters to prices and quantities would be known in advance to all traders, the next step could be providing a feed of current parameters which traders could use to build their own view of the public book. They'd still need plenty of quotes published piecemeal as now (as not all quotes would be model-generated) but the messaging could be much more efficient, and that could lead to more options trading.
No market works like this. The smaller-still message of a price is sent and disseminated. When you buy a flight, you want the price of the ticket—not the airline’s fuel and tariff costs.
Btw when interest rates went negative for the first time, many trading shops were caught pants down, because of course their models did not have a provision for negative interest rates.
> There isn’t much risk anymore which means there isn’t room for ingenuity—it’s execution, mechanical
What do you mean by ingenuity here? Like coming up with your own model that was better than other people's models, or new strategies, etc?
Also, what the heck is an "aerospace investment banker"? Someone in IB who only works on aerospace stuff?
Creativity. What you said. No more 10x improvement opportunities. Just marginal adjustments. Maintenance. Running the same model a bit more efficiently, carving off minuscule edge cases here and there.
> what the heck is an "aerospace investment banker"
A made-up moniker. I raised money—and did deals, e.g. IP licensing, M&A, PPP, et cetera—for rocket, satellite, drone and adjacent start-ups before that was a thing. I had enough technical knowledge to know we were and are on a precipice. Computer-aided design, singularly, as well as new fluid dynamics numerical methods being unsung and recent game changers; falling launch costs our transcontinental railroad. But not enough to do the work myself. So I sold and structured, things I am good at, while reading Banks and Nivens and K.S. Robinson on the weekends.
Really rewarding work. Didn’t pay that much, unfortunately, though the resulting equity changed my life.
No creativity makes it sound like the market has been figured out. I know that isn't true, so does that mean that the risk/reward of strategies has flattened off (I.e. same risk for less reward) because there are less opportunities to exploit?
> No creativity makes it sound like the market has been figured out
That was my bet. It has, so far, been a good one.
> does that mean that the risk/reward of strategies has flattened off (I.e. same risk for less reward) because there are less opportunities to exploit?
Your instinct was on point. You don’t need someone with a feel for volatility to make money (or not lose it) in options market making anymore. The work consists of, and will for some time, re-implementing existing ideas. There are still mis-pricings. But they aren’t inherent to the pricing model. The commodity component can be isolated and run with an eye towards costs and economies of scale.
(This has been a fun conversation, by the way. Thank you.)
I wonder if that's true in insanely volatile stocks like GME? People were buying way OTM calls on that stock. Then the stock would move 50% in one day. A market maker would have to be very good at dynamic hedging to keep up with that.
Of course options market makers have one incredible thing going for them. While they have market risk for every individual option they write, their net exposure can potentially be very small. That only works for a market maker, not for someone making a directional bet because YOLO.
Edit: The few times I tried to study what was going on with GME options, I saw a lot of "no bid" on many OTM strikes. So it looks like the market makers were simply stepping away. Which totally changes the market dynamics. If there's no liquidity in an option, a punter's only choice is to hold to expiration? That's financially very risky and also counter to everything we've come to believe about an "efficient" market.
Bingo. Self help [1] and circuit breakers [2] negate the unsolvable edge case: large, instantaneous price movements.
[1] https://www.reuters.com/article/usa-options-cboe-idUSL2N1H40...
[2] https://www.npr.org/2020/03/09/813682567/how-stock-market-ci...
One reason for black scholes today is that it is a decent interpolation function. It is significantly easier to create an implied volatility function to interpolate with than it is to create a price function to interpolate with directly. Another is that regardless of the smile, the real delta of an option is pretty damn close to black-scholes delta. So, you can maintain prices in real time as a function of the underlying price pretty accurately. A third (and this is important) is that trading systems have it built in as a way to interface with them. People know black-scholes and it isn’t proprietary. So you can do all sorts of research on the dynamics of a volatility smile, and it can be orthoganol to someone doing research on expected dividends or what the actual value of the underlying is. And you can bring all those pieces back together via the black scholes equation. A fourth reason tied into machine learning: implied volatilities behave just much better than raw interpolated prices when running them through predictive algorithms.
I mean, I agree with you. But to me, the whole complexity is just moved to vol modeling. BS with its economic assumptions is just an empty shell now, so to speak.
I am deeply cynical
It is not "priced in"
It goes up up up... crash.
Every generation has to learn this. I just hope we put the bankers in jail where they belong next time rather than bail them out.
But nobody prices options while assuming all the good old innocent assumptions underlying the original derivation of the formula. That, indeed, can be seen from the fact that different vols will be quoted for different strikes at the same expiry.
All of the models (and techniques) I mentioned were well known and in use by 2000.
LTCM went bust because they thought they new better than the market and they were very very greedy. Very.
There is no formula for the market. The EMH in its weak form is correct. Has not been proved, but it is like P!=NP. True.
I am dismayed but unsurprised that financail models get so much support here.
You get money by working. Investments are savings. Just because there is some fool driving a Ferrari does not make that untrue, you cannot see the rest of the finance geeks flipping burgers.
Greed. Hubris. Bankruptcy.
> Since its introduction in 1973 and refinement in the 1970s and 80s, the model has become the de-facto standard for estimating the price of stock options
The only contemporary use for BS by professionals is as a convention for quoting volatility. As a pricing model it does not account for key effects such as the permanent "volatility smile" appearing in the aftermath of the 1987 crash (significantly increased price of downside options), and well understood behaviours like jumps and volatility clustering.
I feel like Medium is the new expertsexchange. I remember how much I hated the site always when I ended there and I seem to have very similar feelings towards Medium.
Edit: docked for bad sense of humor (mine or of downvoters - of that I am not sure)
Their fault for violating GDPR by trying to put cookies without my permission! Ha!