Review the Option Pricing Model

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

2. Replicating Portfolio Value of Call = Asset Value *  - Amount Borrowed () After each time period asset moves up to value Su and the call will be worth Cu OR asset 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 Change The 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 period u = 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 Example Various assumptions about the variables must be made as well a common one is that d = 1/u Now If “u” = 1.4 “d” = .71 the value of the asset = 50 and the option exercise price is we get the following binomial tree

5. The Binomial Tree The Tree Call Value 100 50 70 50 50 0 35 25 0 t0 t1 t2 The 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 Infinitum Similar calculations at T1 give a call value of $13.2 The call price today is $13.20 And So Like Merlin we grow backward picking up knowledge until we find our answer at the beginning.

8. Not Exact but Practical We calculated the option value without knowledge of its expected price We only need to know the Assets Current Value the Risk Free rate of return the 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 Price If the option price varies from value of the replicating portfolios, another position can be set up without cash without risk with profit Investors will drive the price up or down to the portfolio price