Binomial Options Pricing

Introduction

In this post, I am going to explain how one can price options via binomial trees. As you perhaps know, a (european) call option \(C\) grants you the right to buy at a future time \(T\) an underlying \(S\) at a price \(E\), called strike price, which is known at time \(0\). I have already written something about options before. It is not necessary to read it, you just have to know what an option is. Also it would be good to know, that via the put-call-parity for european options, it is possible to price put options, assuming we know the value of the according call option. Well, that means that we can restrict ourselves in the following to the pricing of call options, since with the put-call-parity, the price of put options follows trivially.

Binomial trees

The Cox-Ross-Rubinstein model, or more simply, the binomial options pricing model assumes that we can model the price of the underlying \(S\) with a binomial tree. Normally the underlying would be something like a stock. Assume that the price of the underlying currently is \(S\). Then in a next step it can either go up by a factor of \(e^\sigma\), or it can go down by \(e^{-\sigma}\). In other explanations you will often see an increase by some constant \(\Delta\). This sucks, since then you have to take care that the price never drops below zero. For those of you in the know, I am using geometric Brownian motion instead of “regular” Brownian motion. In ASCII, that would be akin to the following:

  S*e^sigma
 /
S
 \
  S*e^(-sigma)

Now, why not simply take more branches in the tree? Let \(u=e^{\sigma\sqrt{\Delta t}}\), where \(\Delta t\) is a “sufficiently small” time difference and \(d=1/u\) to make the notation easier. Then we can have an even bigger tree.

  S*u^2
   /
  S*u
 /  \
S    S
 \  /
  S*d
   \
  S*d^2

and so on. I think you can imagine the principle behind the tree.

Options pricing

Now, what does this have to do with options pricing? Well, each of the paths in the tree correnspond to a possible development of the underlying. And we know that we can model the payoff at expiry time \(T\) of a call option as \(\max\{ 0, S(T)-E\}\), where \(S(T)\) is the price of the underlying at time \(T\) and \(E\) is the strike price. In the regular Cox-Ross-Rubinstein model you would now look at all paths and then go backwards. We are doing a more sloppy approach. We can simply observe enough possible developments of the underlying, via picking enough paths in the binomial tree. Now for each possible development of the underlying we have the according price of the option. That means the mean of the option at the possible outcomes converges to the “fair price” of the option. Note that the price of the option has to exist in the first place.

Why does this work? Because we simulate the expected payoff of the option. At least nearly. In a real-life scenario you would have to take care of the risk free interest rate, but for now we leave this out of the equation. Computing stuff by hand is difficult, so I have written a simulation in Python which computes the price of a european call option.

import numpy as np
import random
 
#Parameters
tbegin = 0
tend = 1
deltat = .0001
t = np.arange(tbegin, tend, deltat)

E=1.0

sigma = 0.3
S0 = 1

sqrtdt = np.sqrt(deltat)

N=np.zeros(10000) #do 10000 simulations
for i in xrange(0,N.size):
  y = S0
  for j in xrange(1, t.size):
      y = y * np.exp(random.choice([-1,1]) * sigma * sqrtdt)
  N[i]=y

N=np.maximum(0,N-E) #payoff for call option
print(N.mean())

If you want to see the answer to the option price, then you need to have python installed. And run my code. Learning experience is maximized if you hack around with it and try different parameters.

Hope you had fun.