# How to identify between Analytical, Numerical and ML Model based option pricing? [closed]

I am new to Quantitiative Finance. Coming from Computer Science domain, I wanted to clear the key distinguishing factor between analytical, numerical and ML based models for option pricing.

As far as I have learned, following are the differences between these three.

1- Given the required parameters, we can say that the price calculated for a call or put option using Black-Scholes Formula is an analytical approach as it gives a closed form answer while the underlying asset is operating with Black-Scholes asssumption.

2- If we use Monte-Carlo method to get the price for an option, it will be considered as a numerical approach because to get an answer for a given data point, the algorithm will have to extrapolate n paths for the under lying asset using geometric Brownian Motion formula hence making it numerical in nature.

3- If we use any Machine Learning or Deep Learning model given the input parameters from the data, it will be considered as a machine learning based approach as it will use machine learning or deep learning parameteric approach to learn the underlying pattern of the option price and hence predict the value.

Please let me know if I am missing something or am mistaken in any of the above concepts.

Leaving ML aside for now. Neither Monte Carlo simulation nor numerical solutions are a model generally speaking. They are methods that can be applied on a model to find solutions.

For example, Black Scholes has

In practice, I have not seen ML used (seriously) so far to price options. I think partially because

• listed options have prices and you use the information these prices convey to get implied vol surfaces which can be used to price more complex structures
• if OTC trading is the dominant way of pricing (e.g. FX options), you have direct IVOL quotes
• you only need more complex models like Local Vol (LV), Stochastic Vol (SV), Stochastic Local Vol (SLV) or Shifted LMM (if you look at interest rate options) to price complex derivatives: However, to use ML, you would need a lot of market data - which you usually (unless you are the market maker) don't have access to

Numerical methods or MC will usually only be used if there is no closed form solution. Generally, a finite-difference solver of the PDE or MC simulation of the SDE should result in the same value. However, for more exotic path-dependent structures MC is usually preferred despite being more computationally intensive. The reason is intuitively obvious if you require to know expected values of underlying(s) over time, simulating their path directly will be useful.

In my opinion, the real question should be what you need to price. That will determine what to use to capture the underlying dynamics of the product you're pricing.