2.  How to price a European call option
     --a simplified model (basic idea of the binomial method)

Q-What's a "European call option"??

Let's go to the overhead/handout, and check out a recent Wall Street Journal...


        --time out--




We first present a simplified model, then add more and more structure.

(1) Say the asset price is at $50 per share today, March 29, 2001; and depending on the upcoming report on management restructuring, the asset price will either go

to...  

(How such numbers can be predicted will be discussed below; they are part of the stock's "volatility".  Notice that we are not assuming we know anything about the likelihood of the stock going up or down, just the size of the fluctuation.)

(2) Next, say the option that we want to price allows the option holder to purchase stock at $51 on March 30.  That means that the value of the "call" option becomes...

       
  
  
Summary so far:
  
           today           tomorrow
  
  STOCK:    $50            
                  
  
  OPTION     ??  
  
(3) Now we want to know what a good price for that option is.  We can look in the Wall Street Journal to see what the option is trading at, and we want to know: should we buy that option today, or not?

(4) The analysis begins:

The idea is to construct an equivalent portfolio of stocks and cash, one whose possible values tomorrow are identical to what the call option's values are, and see what this equivalent portfolio is trading at today.  Then that should be the same as what the option should be trading at today, i.e. we will have deduced a fair price for the call option.

We let  ∏ denote the equivalent, or replicating, portfolio:  ∏ = Δ * S + C

Δ denotes how many shares of stock we will have in the portfolio (the notation is traditional from finance, not from mathematics nor physics, so Δ does not denote the more traditional "change in").  C denotes how much cash we will have in this portfolio.

Then depending on what the report indicates, the portfolio will be worth



We want the value of the portfolio to match that of the option; so
we simply solve the system
    $9 = Δ * $60 + C
    $0 = Δ * $44 + C
    
We need no fancy solving function here; simply subtracting the equations yields
    $9 = Δ * $16
so  Δ = 9/16, i.e slightly more than half a share, and similarly we solve for C to
get C =  - $24.75

Here's where you have to pay attention:

So our portfolio consists of borrowing or owing $-24.75, and owning 9/16 of a share of stock.  This portfolio is worth (9/16)*50+$-24.75, or $3.375.  So THAT'S the price of the call option: a portfolio on which the market places a worth of $3.375 will produce the same return as the option.  So the market should place a worth of $3.375 on the option.


Here's where you have to pay even more attention:

So here's one way to make NO money:
    Sell one option at $3.375.
    Also borrow $24.75, and with the total, you can afford to
    ...purchase exactly 9/16 shares of stock.  

Then tomorrow, sell the stock and see what happens:
    --If the stock price goes up, use the revenue (9/16*$60) to repay the loan AND come up with the $9 you owe the person who owns the option.  
    --If the stock price goes down, use the revenue to repay the loan.
    
In either case, NO money is made.  You start with $0, then sell - borrow - buy, and by the
end of the next day, you've got $0 regardless of what the market does.

This simple model (using nothing more than intermediate algebra) neglects things like interest rates, but it does illustrate quite well what is the basis for the so-called binomial method (simulated in greater detail below).