1. CRR and American Options Date: Oct.20.2014 Authors: LÓPEZ LÓPEZ, VÍCTOR LUTEMBEKA, SHEDRACK YUAN, BO

2. OVERVIEW  1. Problem statement  2. Method  3. Examples using Python: Price an American option using the CRR binomial model

3. Problem Statement Implement in Python the Binomial model (Cox- Ross-Rubenstein) and calculate the price as function of time to maturity and strike and show in a graph how the solution converge to the hockey-stick.

4. Method STEP 1: Create the binomial price tree At each step, it is assumed that the underlying instrument will move up or down by a specific factor (u or d) per step of the tree. So, if S is the current price, then in the next period the price will either be: The up and down factors are calculated using:

5. Method  STEP 2: Find Option value at each final node[edit]  At each final node of the tree — i.e. at expiration of the option — the option value is simply its intrinsic, or exercise value.  Where K is the strike price and S(n) is the spot price of the underlying asset.

6. Method  STEP 3: Find Option value at earlier nodes  The following formula to compute the expectation value is applied at each node:  “probability” p: an up move in the underlying  “probability” (1-p) denotes down move.

7. Examples using Python  Price an American option using the CRR binomial model  For an American option, since the option may either be held or exercised prior to expiry, the value at each node is: Max (Binomial Value, Exercise Value).  The input values is as follows:  n = 10 #number of steps  S = 100 #initial underlying asset price  r = 0.06 #risk-free interest rate  K = 105 #strike price  v = 0.4 #volatility  t = 1y #time to maturity

8. Examples using Python  Results:  Pegging the strike: a solution to get rid of oscilations.  (1)Pricing American call option in one year:

9. Examples using Python  (2) Pricing American put option in one year:

10.  Thank you for watching!