How to Choose Binomial Parameters - Binomial Option Pricing || Theory & Implementation in Python
In this video we look at pricing a European Call option using the Binomial Asset Pricing Model with four different methods to define the binomial parameters in Python.
Here we explore four methods to choose the binomial tree parameters up-factor u, down-factor d and risk-neutral probability q. All methods attempt to approximate the risk-neutral asset dynamics as accurately as possible; this is done by matching the mean and variance of the binomial asset pricing model to the dynamics of the underlying - Geometric Brownian Motion (GBM). The Cox, Ross and Rubinstein (CRR) method assumes that the up and down factors have the same jump size while the Jarrow and Rudd (JR) method assumes the same risk-neutral probabilities.
If the underlying binomial tree is represented in terms of the natural logarithm of asset prices (where x=ln(S)) then the risk-neutral dynamics of the black-scholes model can be used for accurate and robust approximation. Using the GBM, the log-normal distribution has constant mean and variance and hence the mean and variance is easy to match with that of the binomial tree model. The Equal Probability method (EQP) proposed exactly that, using equal probabilities with the underlying tree represented by the natural logarithm of asset prices. The Trigeorgis method (TRG) is again represented by equal jump sizes, however implemented on a binomial tree represented by the natural logarithm of asset prices.
In this tutorial series we will be breaking down the theory described and published in Steven Shreve’s book’s Stochastic Calculus for Finance I & II. As a guide for implementing these concepts in Python, we will refer to the numerical methods and practices outlined in Les Clewlow & Chris Strickland’s book Implementing Derivatives Models.
00:00 Intro
02:12 Theory || Choosing u, d, q
03:42 Theory || Jarrow Rudd (JR) & Cox, Ross & Rubinstein (CRR) methods
04:24 Theory || Binomial Tree in terms of the natural logarithm of asset prices
05:35 Theory || Trigeorgis (TRG) & Equal Probabilities (EQP) methods
06:10 Python Implementation || European Call Option Pricing
07:00 Python Implementation || Cox, Ross & Rubinstein (CRR) method
08:44 Python Implementation || Jarrow Rudd (JR)
10:12 Python Implementation || Equal Probabilities (EQP) method
14:10 Python Implementation || Trigeorgis (TRG) method
15:46 Python Implementation || Comparison of Methods
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