Finite Differences Option Pricing for Quant Finance

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*Roman's Overview of Option Pricing Methodology*

When we go about pricing options the first step is to select a model framework and accept the associated assumptions: Black-Scholes, Heston, the list goes on... The goal in this capacity is to capture as much variation observed empirically while saying as little as possible about the true distribution observed of corresponding market prices. We aim to price consistently and efficiently - we can't subject our quotes to arbitrage and need to select a reasonable model capable of producing quotes consistent with a market implied volatility surface.

Once we select a model we can develop prices in several ways:

A.) Develop and Analytically Solve a Pricing Partial Differential Equation

B.) Develop and Numerically Solve a Pricing Partial Differential Equation (Finite Differences)

C.) Leverage the Law of Large Numbers (LLN) and Risk-Neutrality to Simulate Prices via Monte Carlo Simulation

The instrument you are pricing and the model framework you select will largely dictate which technique you will use to go about developing prices. In some cases we can produce prices slowly and consistently with models that perhaps better capture the dynamics observed in the market (i.e. the Rough Model Family) though there is literature and conjecture about the efficacy of rough volatility, neural networks and other machine learning techniques make it possible to provide quotes for such models in an effective manner - a video for another day!

I hope you enjoyed, this was a long one!

Roman
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📖 Chapters:
00:00 - Introduction
03:39 - Pricing Differential Equations
04:37 - Understanding Pricing Differential Equation
06:38 - Why Finite Differences is Necessary
07:50 - Understanding and Approximating Derivatives
12:00 - Why Finite Differences Works
14:21 - Visualizing Finite Differences (2D)
16:02 - Extrapolation and Interpolation with Finite Differences
17:32 - Finite Differences
18:48 - Example: Finite Differences, Ordinary Differential Equation
21:20 - Coding: Finite Differences, Ordinary Differential Equation
24:48 - Partial Differential Equations, 1-D Heat Equation
26:40 - Visualizing Finite Differences (3D)
27:27 - Example: Finite Differences, Partial Differential Equation
30:31 - Coding: Finite Differences, Partial Differential Equation
39:23 - Finite Differences Applied to the Black-Scholes Equation
41:28 - Closing Thoughts and Future Topics
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