Binomial Option Pricing Model

Q. What is the fair price (=No-Arbitrage) of a European call option at time ?

  1. Find the purchase of stock and borrow of money that would have the equivalent payoff as the call option (=reproducing portfolio) Use that to price the stock
  2. Then, according to the No-Arbitrage that must be the same as the current price of the call option
  3. Do this for periods for a binomial tree

Call Option Details

Payoff (=price of call option at time )

  • is the uptick factor and the downtick factor

Reproducing Portfolio Details

  • : number of stocks you purchase
    • will increase due to dividend payouts:
    • is the dividend rate
  • : amount you borrow at time
  • ⇒ Value of portfolio

Calculating Fair Call Price

⇒ Equate and solve for to obtain the reproducing portfolio.

  • Calculate :
  • ⇒ Thus and simplifying:

Generalizing into terms

  • With the No-Arbitrage we have .
  • Then define the risk-neutral probability of an uptick such that

thm. Risk-Neutral Binomial Call Pricing Formula


  • is the risk-free rate and dividend rate

By taking we see

  1. becomes a Stochastic Process
  2. Binomial sum can be approximated by a Normal Distribution