Barrier Option Pricing with Binomial Trees || Theory & Implementation in Python
In this video we look at pricing Barrier Options using the Binomial Asset Pricing Model and show how you can implement the barrier tree model to price an up-and-out barrier option in Python. There are two types of barrier options which we look at in this tutorial, up-and-out barrier options and down-and-out barrier options. Here we go through the theory and implementation.
For those who just want to code, please skip ahead to the Python Implementation section. I will take you through two implementations of a simple barrier tree model in Python, one that will use ‘for loops’ to step through each node at each time step (a function I have defined as binomial tree slow), and the other (binomial tree fast) will vectorize these steps using numpy arrays, improving overall computation time as N time steps increase. Although not necessary for the example today, using numpy arrays and vectorizing our calculations will improve computations as we delve deeper into financial mathematics and implementation heading forward.
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.
For any observant viewers, I calculated the variable 'disc' incorrectly in this video. I used up factor 'u' in the numerator instead of the correct value with is the down factor 'd'. Also, in the slow tree only during backward recursion the stock price should be calculated as S = S0 * u**j * d**(i-j) instead of S = S0 * u**j * d**(N-j). Code is correct on my website (link above).
00:00 Intro
00:17 Theory || What are Barrier Options?
02:25 Theory || European vs Barrier Option Payoff
04:42 Theory || Multi-period Binomial Model with Barrier Value H
08:08 Python Implementation || Barrier Tree Slow
18:50 Python Implementation || Barrier Tree Fast
25:26 Python Implementation || Comparing the Slow vs Fast Implementation
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