The Black-Scholes formula, explained (2019)

> That they are correct most of the time?

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

Former options trader here. This all checks out correctly, but there's maybe some intuition that enlightens it. BTW option traders are often called volatility traders, because when you look at the formula there's this one free variable (all the rest are somehow given by the market). So when you're trading options, you're trading vol and the actual price is just a sort of formality.

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.

All of the inputs of the BS model are forecasts. All of them can be wrong, and they have been wrong countless times. It’s like saying that your linear extrapolation for the stock market mostly works, except for the times it doesn’t.

This comment is really interesting.

> 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?

I don’t know why you’re getting downvoted. While what you said isn’t exactly correct, it’s pretty close.

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.