CRR: Option Pricing

This post dwells on CRR tree with the application of Option Pricing.

I admit the first method is the simpler one but only used for European Option Pricing. However if I wanted to generalize European and American Option Pricing, it is better explained in the second method.

Inputs:

• Share Price (S) in dollars
• Exercise Price (K) in dollars
• Dividend yield (q) in percent (continuously paying)
• Risk-free rate (r ) in percent and p.a.
• Volatility ( σ ) in percent
• Time of maturity (T) in years
• Number of periods/ steps (n)  

Output: European Call and Put Option

Method 1: Direct Approach (similar to Haug book and  Jackson and Staunton book)

First we get the necessary $u$, $d$ and $p$ by using the following formulas:

• $u = e^{\sigma \Delta{t}}$
• $d = e^{-\sigma \Delta{t}}$  or simply $d = \frac{1}{u}$
• $p = \frac{ e^{(r-q)\Delta{t}} - d }{u-d} $

where $\Delta{t} = \frac{T}{n}$.

European Options are only exercised at the end of the period. That means we are just looking at the end nodes of the tree. In this case, we solve for the probability at each of the end nodes, multiply it by each payoff, sum it up and then get its present value. The formula for call and put option are as follows:

\[ c= e^{-rT} \sum_{i=0}^{n}  {_n}C_{i}  p^i  (1-p)^{n-i} \text{max}[Su^i d^{n-i} - K, 0] \]
\[ p= e^{-rT} \sum_{i=0}^{n}  {_n}C_{i}  p^i  (1-p)^{n-i} \text{max}[K-Su^i d^{n-i}, 0] \]
where

• $e^{-rT}$ is the discounting factor
• ${_n}C_{i}  p^i  (1-p)^{n-i}$ is the probabilities at each final node $n$ at step $i$
• $Su^i d^{n-i}$ is the prices at each final node $n$ at step $i$
• $\text{max}[Su^i d^{n-i} - K, 0]$ is the payoff for call option.
• $\text{max}[K-Su^i d^{n-i}, 0]$ is the payoff for put option.

Notice that the only difference for put and call options are their payoffs.

Example: same as in Haug approach sheet in CRRforProj.xlsm 

Given the following values, we want to compute for the European Call and Put Option.

Solution:

Method 2: Using Trees (also similar to Haug book, Jackson and Staunton book and the presentation found in earlier post.)


First we get the necessary $u$, $d$ and $p$ by using the following formulas:

• $u = e^{\sigma \Delta{t}}$
• $d = e^{-\sigma \Delta{t}}$  or simply $d = \frac{1}{u}$
• $p = \frac{ e^{(r-q)\Delta{t}} - d }{u-d} $

where $\Delta{t} = \frac{T}{n}$.

Then we create the tree for price. The formula is  $Su^i d^{n-i}$ at each node $j$ at step $i$.

Then we create the tree for the option price. The end nodes, we compute for the payoff. It is the same whether it is European or American options.

• $\text{max}[Su^i d^{n-i} - K, 0]$ is the payoff for call option.
• $\text{max}[K-Su^i d^{n-i}, 0]$ is the payoff for put option.

At the remaining nodes, European and American Option Prices differ even if the process is the same. It is because European option only exercises the end of the period while American options can be exercised earlier. So the formula for the remaining nodes $j$ at step {i} are as follows:

• European Call and Put: $O_{i,j}= e^{-r \Delta{t}} (p O_{i+1,j+1}+(1-p)O_{i,j+1} ) $
• American Call: $OV_{i,j} = \text{max}[Su^i d^{n-i} - K, e^{-r \Delta{t}} (pO_{i+1,j+1}+(1-p)O_{i,j+1} )]$
• American Put: $OV_{i,j} = \text{max}[K-Su^i d^{n-i},e^{-r \Delta{t}} (pO_{i+1,j+1}+(1-p)O_{i,j+1} )]$

Example:

for European Option Example: please see earlier post.

For American Option Example: same as in American Tree sheet in CRRforProj.xlsm 

Given the following values, we want to get the American Put Option Price.

Solution:

Source: 

• http://r2-d2.bu.edu/AT__The_Complete_Guide_to_Option_Pricing_Formulas__2nd_ed_.pdf
• Advance Modelling in Finance using Excel and VBA by Jackson and Staunton