Thread by @LilyWrites4

Let's talk about pricing models. One of the most active and by definition important areas of quantitative finance is pricing derivatives. Derivatives are a gigantic market, being worth notionally much, much more than the entire GDP of the planet, so there's a lot of interest 1/x

in being able to properly price them. Most of us are familiar with the basic vanilla American option (call, put), but most of pricing theory is interested in the less well known exotics. A good recent example comes from @paradigm, which created a new class called the 2/x

perpetual option. On further look, most optionistas would be able to tell it was an existing type of option called a cliquet, which essentially is a perpetual option that resets strike price at pre-determined dates (similar to the Fed put lol). But unlike the simple vanilla 3/x

options, most exotics are not well behaved by something as 'simple' as the Black-Scholes. We can really separate pricing models into two types: closed-form (analytical) models and non-closed form (numerically derived). Analytical models are nice, because they're computational 4/

cheap. To step back, closed-form isn't a trick word; it simply means you can use a finite number of basic math operations (+, -, /, * for example) to evaluate a solution. A non-closed-form solution on the other hand can involve infinite number of math operators, and we can't 5/x

really solve it with math; we can for example, implement it, run it 1000 times, and take the average solution generated (a Monte Carlo approach). Alternatively we can use approximation techniques (Gaussian quadrature, for example) to get "close" enough quickly, or if our 6/x

solution space is well behaved (for instance, few parameters, goes towards a global maximum or minimum properly), we can solve it using regression or machine learning approaches. In general, these are all slow and consistently massive pain in the ass(es). Therefore, in practice 7

most will prefer an analytical pricing model to a numerical one, unless speed is really unimportant and the cost of minimal error is high. So now that we've wasted some time on this, let's move on to some basic pricing models! In essence, all pricing models start at the asset 8/

level; we want to know how the asset we're pricing the derivative for moves. This is important because the price of the derivative depends on the price of the asset at some point in time. That rad dude Ito and his lemma helps us here. In general, most assets (or processes, like 9

interest rate derivatives) will follow a stochastic process - some combination of a non-random process (the drift coefficient) and a random process (usually a Wiener process). The simplest version of this is the Ito process, shown below. It essentially says "we have a drift

component a(x,t) (some function a that depends on current stock price x and time t), and a random process that combines a Brownian motion (will talk about that another time) with a volatility process b(x, t). This is the basis for a ton of fairly complex models, including BSM. 11

However, some very smart folks realized that this equation isn't enough to price reality usually. There's a couple of reasons. One of that volatility is not constant, and autocorrelates! Or in English, in high volatility periods we expect to see continued high volatility, and low

volatility periods we see continued low volatility. This is a problem for most simple equations, because it means volatility itself is following a random process. Volatility changes randomly over time, but more important we need to create the observed volatility clustering 12/

to properly price asset derivatives! So around the late 90s or so, a guy named Steve Heston augmented our basic Ito process with some cool gadgets from the field of interest rates, namely the CIR model, to try and fit the observed properties of volatility better. 13/x

This was one of the first of a class of models we call stochastic volatility - models where not only the stock price itself is sorta random, but the volatility process used in the stock price is sorta random too. The other factor we have to talk about before putting it all 14/x

together is what we call the spot-vol correlation, or kinda tangentially vol beta. Spot-vol is the observation that in real world assets, volatility isn't constant as price moves; in equities, if assets sell off, vol tends to explode. Interestingly in some commodities like oil 15

the opposite pattern occurs (oil vol tends to go up as the price increases). This is important for pricing! Without understanding and modeling how volatility changes as price changes, you can't really properly hedge your options! So Heston included this in his famous Heston model

too, which ignoring the math jargon aims to represent a few things:
1) The Heston is an Ito process where we have a random stock process and a random volatility process
2) Our random volatility process has some correlation, rho, to our random stock price (spot-vol corr)

3) We expect while volatility may spike or fall randomly, over time we expect it to return to a long term historical average, theta
These are summed up in two funky looking equations below. The nifty thing about Heston's model is it is complicated as hell and has multiple inputs

but it also has an analytical solution, using what is called the Fourier transform! This makes it a lot quicker for helping us price options than an even more complicated model which might incorporate even more real world phenomena (jumps, JPow, trump tweets, etc).

We're almost done, trust me. There are two important steps when using a model to price options - choosing the proper model for your asset, and the much much worse step, called calibration. The truth of a model is you can make it do whatever you want; the value is in setting

your parameters properly. In Heston, we have a lot of parameters to pick from:
- Long term historical variance
- Vol of vol
- Expected rate of return
- Spot-vol correlation
Etc. You can and usually do fuck it up. Worse yet, when you calibrate on historical data, you're doing

exactly that - the parameters you determine (usually from fitting it to real world option prices) will be conditioned based on the data/period you chose to view! You have no way around this, but extra turbobrain folks use systems like Kalman filtering or phase models here to

track or modify the parameters provided over time. The less turbobrain folks just simply accept all models are wrong, and retrain their pricing model daily or thereabouts instead of trying to big brain their way into production losses anyway.
So here you have it! Fin.