The Black-Scholes formula, explained (2019)
Former options trader here. This all checks out correctly, but there&#x27;s maybe some intuition that enlightens it. BTW option traders are often called volatility traders, because when you look at the formula there&#x27;s this one free variable (all the rest are somehow given by the market). So when you&#x27;re trading options, you&#x27;re trading vol and the actual price is just a sort of formality.<p>Thoughts:<p>- Since you have a right but not an obligation to buy&#x2F;sell, that creates asymmetry. Since it&#x27;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.<p>- Similarly having more time to expiry makes the option worth more, the range of outcomes is more spread out.<p>- There&#x27;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&#x27;s good or bad for you from thinking about how the distribution of outcomes is affected by a change in whatever.<p>- 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&#x27;t make it flat everywhere with a hedge, because of course the graph is bendy.<p>- 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&#x27;s the same as having the underlying.<p>- As time passes it&#x27;s got to get more curvey at the strike, less curvey on the sides.<p>- 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&#x27;ll be getting longer and longer. This is bad.<p>- It doesn&#x27;t actually matter whether you are buying the right to buy or the right to sell. If you&#x27;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 &quot;calls&quot; and options with lower strikes &quot;puts&quot; 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.<p>- 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&#x27;t too much it might be worthwhile to be short. In fact, most of the time the movement isn&#x27;t enough to justify the price.