Stochastic Calculus for Quants | Risk-Neutral Pricing for Derivatives | Option Pricing Explained
In this tutorial we will learn the basics of risk-neutral options pricing and attempt to further our understanding of Geometric Brownian Motion (GBM) dynamics. First, we will learn why risk-neutral valuation is an important tool for quantitative financial analysts, and its advantages over other valuation methods. This leads us to discussing the one period binomial asset pricing model and the derivation of risk-neutral expectations under this simple model.
We will understand how this method links back to the fundamental theorem of asset pricing and discuss change of measures while introducing the radon-nikodym derivative. We then present this change of measure from real world to risk-neutral world through Girsanov’s Theorem. We then work through a practical example using Black-scholes model stock and bank account dynamics to show the usefulness of risk-neutral pricing.
We briefly discuss the generalisation of risk-neutral pricing to the risk neutral expectation pricing formula and establish its usefulness in valuing complex derivatives by using the Monte Carlo method of discounted risk-neutral path simulations.
00:00 Intro
01:30 Why risk-neutral pricing?
03:43 1-period Binomial Model
06:18 Fundamental Theorem of Asset Pricing
07:00 Radon-Nikodym derivative
10:20 Geometric Brownian Motion Dynamics
12:00 Change of Measures - Girsanov’s Theorem
14:00 Example of Girsanov’s Theorem on GBM
19:45 Risk-Neutral Expectation Pricing Formula
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