Thread by @LilyWrites4

Okay, today's lesson is on implied volatility. Implied volatility is a weird concept. In fact, all the option greeks are. Essentially for an option you are paying for the intrinsic value, which is easy to calc (current - spot), but more so for *extrinsic* value, or what we call

optionality. Optionality is a weird concept related at a deep level to the idea of expectancy - the expected value of the difference between the current price and the price at expiry. This is slightly different for American options, but also who cares because those are annoying.

The price of an option contract is controlled at a deep level by the random process that drives the stock price. The stock price as you may remember is controlled by the drift (not random) and the volatility (random). We can relate the price of the option contract to the stock

by this weird quirk of mathematics called Ito's lemma. It is kind of related to the chain rule of calculus, but I won't go into too much detail since it's not important here. Essentially there are many ways to frame the pricing of an option, but at their core, they all fall back

to what we call the no-arbitrage principle, which isn't named to trick you. In finance, we price things according to the idea of risk versus return, and arbitrage (riskless profit) violates that. If your risk is 0, your return should be 0. If arbitrage exists, we expect given

enough time/market liquidity, it will be closed by another party. This is a key distinction - arbitrage *may* exist for short or even noticeable periods of time, it just happens in the infinite limit it should not exist. But going back to options. You can inductively show that

the price of an option has to be the point of zero profit rationally for both buyer and seller over the risk free rate. This is intuitive. If there is an expected profit on one side (buyer or seller), another party can undercut and drop the price. Basic game theory in a

competitive market. Of course in practice this isn't exactly accurate and you have quirks like the variance risk premium and implicit collusion, but it's an important realization when we talk about implied volatility. Implied volatility means a lot of things. The most logical

definition is a plug to the Black-Scholes equation which lets us gel realized option pricing with the model. It's in many senses a deficiency between reality and mathematics. But why does it arise? Black-Scholes is a model, and as @EmanuelDerman once noted, the use of models

especially popular ones adds extra variables to the actual market related to the deficiency of said models. IV is of course no exception here. A more intuitive explanation of IV goes to the idea of supply and demand. A high IV option simply is an option where both the buyer and

seller have agreed it is worth more than what the BSM should predict. In a competitive market, this isn't gouging you; this is simply because it may be in more demand (more seller power), there may be unmodelable risks (jump/gap risk, regulatory, etc), and so on. It is not

'worse' in expectancy for either the buyer or the seller in the infinite limit than a low IV option! This of course gets more complicated in the real world because we don't have infinite time and games to play out things. There are very real variant effects between high and low

IV options, but you can't blanket apply a rule like "Low IV is better for me as a buyer". It isn't the case. The reason an option has IV is simply that the market, the aggregation of all buyers and sellers, has agreed it should be priced higher - for whatever reason. Vol traders

model the IV surface and know all sorts of empirical facts, because the supply/demand curves of IV are often well more predictable than price itself, given real-world effects like volatility clustering, convexity of vol, etc.
Fin.

Also if you want to delve more into the ideas of optionality and expectancy, s/o to @volmagorov for teaching me about the Feynman-Kac method for option pricing which essentially relies on this