The 1-Period Binomial Model

In this lesson we focused on a 1-period binomial model.  This is similar to the binomial tree in my last post, but with only 1 time increment past t=0 (t=1).

W start with the assumptions that the initial price of the stock = S(0) is equal to 100.  It can increase by a factor of u=1.07 with probability of p or decrease by a factor of d = 1/u = .9346 with probability of 1-p.  We also assume that you can borrow or lend at a risk free rate, R.  We also assume that short-selling is allowed.  A binomial tree is shown below:

Now we want to ask two questions which will help us in pricing options.

1. How much is a call option worth that pays max{S(1) - 107, 0} at t=1?
2. How much is a call option worth that pays max{S(1) - 92, 0} at t=1?

A1:  At time t=1, the maximum price of the stock is 107, so max{S(1) - 107, 0} = 0 and the call price must be 0.
A2: At time t=1, the minimum price of the stock is 93.46, so max{S(1) - 92, 0} = S(1) - 92, so the call price must be S(0) - 92/R (since we must discount to present value).



We were then introduced to two different types of arbitrage.

Type A arbitrage: a security or portfolio that produces immediate positive reward at time t=0 and has non-negative value at t=1.  This is a security with a cost less than 0 (i.e. you are paid to hold this security) and it has a value at time t=1 that is greater than or equal to 0.  An example: finding $10 on the street

Type B arbitrage: a security or portfolio that has a non-positive initial cost, has a positive probability of yielding a positive payoff at t=1, and zero probability of producing a negative payoff at t=1.  This is a security with a negative or zero initial cost and will have a positive or zero payoff at time t=1. An example: someone comes up to you in the street and hands you a free lottery ticket.  This costs you nothing and you might gain some money.



We then returned to the binomial model and found conditions that would allow no arbitrate (a condition that we view as true, since in actual financial markets any arbitrage would be immediately dispelled by market forces).


Theorem: There is no arbitrage if and only if d < R < u

Proof: (i) suppose that R < d < u.  Then we can borrow S(0) and invest it in the stock

Cash flows under R < d < u
                        t = 0            t=1
borrow S(0)     -S(0)          -S(0)*R
invest S(0)       +S(0)          either u*S(0) or d*S(0)
net                       0            (u-R)*S(0) or (d-R)*S(0)
since R < d < u there is always a net gain in t = 1

Proof: (ii) suppose that d < u < R.  Then we can short sell the stock and invest it earning interest R.  This is simply the opposite of the argument above.

In both cases, we have a type B arbitrage.
In the future we will always assume d < R < u so there is no arbitrage.