This document describes the binomial tree method for pricing options. It explains how to construct a binomial tree based on the asset price, risk-free rate, and volatility. The value of the option is then solved for by working backwards through the tree. An example is provided where the asset price can move up 40% or down 29% at each time step. Working back from the expiration date, the number of asset units to purchase and amount to borrow is calculated to replicate the option payoff. Arbitrage ensures the option price equals this replicating portfolio value.

1. Getting Hit on the Head by an Apple While Lying Under the Binomial Tree

2. Replicating PortfolioValue of Call = Asset Value *  - Amount Borrowed ()After each time period asset moves up to value Su and the call will be worth CuORasset moves down to value Sd and the call will be worth Cd value = the units of the asset purchased (Cu-Cd)/Su-Sd)

3. How Much Does the Price ChangeThe amount the asset increases or decreases each time period can be shown to be an exponential function of “” , the risk measure or variance of the price movements of the asset, “r” the risk free interest rate, the life (“T”) of the option and the pricing periods (“m”) until exercise “u” is the % increase each period “d” is the % decrease each periodu = EXP[(r- 2/2)*(T/m) + (2(T/m))1/2]d = EXP[(r- 2/2)*(T/m) - (2(T/m))1/2]

4. An ExampleVarious assumptions about the variables must be made as well a common one is that d = 1/uNowIf “u” = 1.4 “d” = .71 the value of the asset = 50and the option exercise price is we get the following binomial tree

5. The Binomial TreeThe Tree Call Value100 507050 50 035 25 0t0 t1 t2The Problem for a given exercise price $50 and risk free rate =11%Solve backwards from time t2 the value of  = the units invested in the asset and“” the amount borrowed

6. Solve for the Two Cases at T2(100*)-(1.11*) = 50( 50*)-(1.11*) = 0= 1, = 45 (Buy 1 share borrow $45)Now Calculate the call value at T1 (70 *-)=(70-45)=25 (50*)-(1.11*) = 0(25*)-(1.11*) = 0=0,  =0 (Buy “0” and borrow $0) Now Calculate the call value at T1 (70 *0-0)=0

7. And So On Ad InfinitumSimilar calculations at T1 give a call value of $13.2The call price today is $13.20And So Like Merlin we grow backward picking up knowledge until we find our answer at the beginning.

8. Not Exact but PracticalWe calculated the option value without knowledge of its expected priceWe only need to knowthe Assets Current Valuethe Risk Free rate of returnthe Riskiness of the underling Asset “”We calculated the value in discrete time increments the way many real world problems are posed.We made No assumption about the underlying distribution.

9. Arbitrage Guarantees One PriceIf the option price varies from value of the replicating portfolios, another position can be set upwithout cashwithout risk with profitInvestors will drive the price up or down to the portfolio price