American Option Pricing with Binomial Trees || Theory & Implementation in Python

In this video we look at pricing American Options using the Binomial Asset Pricing Model and show how you can implement the binomial tree model to price an American option in Python. We also show why it is not optimal to early exercise an American Call option with no dividend (or positive interest rates).
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 American 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.
00:00 Intro
00:35 Theory || What are American Options?
01:05 Theory || Early exercise is not optimal for American Call
04:44 Theory || American Put Options
06:52 Theory || Some other considerations
08:30 Python Implementation || American Tree Slow
18:10 Python Implementation || American Tree Fast
21:55 Python Implementation || Comparing the Slow vs Fast Implementation
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[@lornemalvoo]

This channel is certainly a gold mine. Please sir keep it up.

[@thetrickingcorner439]

This video is amazing. Thank you so much :)

[@daves1413]

Awesome tutorial, keep it up

[@schnohr3292]

Great video!
Have you done a video on pricing in trinomial trees or do you plan on doing one? More specifically on pricing American call options using a two-factor Hull-White model?

[@QuantPy]

Yes will get to trinomial trees and can include hull-white model 👍

[@bhdkid1]

Hi @QuantyPy , great video. when proving the idea that you should never discount earlier for American call option, why do we use e (assume continuous compounding) for the interest earned for our bank account given that none pay that? I always wondered that when seeing e in many formula.
Thank

[@wajihchtiba34]

i really like your channel !

[@siddharthsidselvamohan2051]

Hey, Thanks for the video. I am getting different values when using different # of steps. that is, for N = [3,10,1000,5000], I think we should define u and d inside the function and compute them with varying N, if using constant u of 1.1, then the option price is varying a lot

[@AlgoFrenchTrader]

Right, u and d are fonctions of the time step value. So : u = exp[sigma * sqrt(T / N)] where N is the time steps number and T the maturity given in years. The expression for d is straightforward.

[@joshuasmilde5198]

Legend

[@tejusrawal3871]

❤

[@tim2138]

I dont think you can compare valuation of the same portfolio for different time

[@lebagnard6018]

could we get the trinomial tree ? please ? :D

[@QuantPy]

Yes 👍 I’ll make some videos soon