In the next sequence of modules we're going to discuss equity derivatives in

practice, but before we get on to discussing equity derivatives in

practice, we're going to spend some time in this first module discussing and

reviewing the binomial model. So we'll call again our pricing of a

European call option in the binomial model We're going to assume an

exploration of t equals to 3, a strike of $100 and a gross risk free rate of r

equals 1.01. So the pay off of the option is given to

us by the maximum of 0 and st minus k which is 100.

So we see here the pay off of the option. Its 22.5, 7, 0, 0 and recall how we

priced this option. We computed our risk neutral

probabilities, which are given to us down here, q u and q d and then we work

backwards in the binomial ladder. So for example the value 15.48 is the

value of the core option at this node and is given to us by one over the risk free

interest rate, 1.01 Times the expected value of the payoff of the option one

period ahead. So we can work backwards in the lattice

and price the option that way. Note also that when we price using this

mechanism here, we're guaranteed to have no arbitrage by construction.

And that's as long as D is less than or less than U.

Remember that this is our new Arbitras condition in the binomial model.

And it ensures that QU and QD are both strictly great in zero as we have down

here. And of course what that means therefore

is that in a price like this It's impossible to have an arbitrage because

it would be impossible to get a payoff here and here, which is strictly

positive, and have a value that's strictly negative here, for example.

And that's because qu and qd are strictly positive.

So this is how we price securities or derivative securities in the binomial

model. We ensure that this condition is

satisfied to ensure no arbitrage, and then we work backwards in the lattice as

usual. And so we can continue on in this

fashion. Compute the price of the option at every

node, working backwards until we find an initial option price of 6.57 dollars.

Now, we can also write the option price, or compute it in one shot.

When one calculation, using this expression here.

So, this just reflects the fact that the option price is the expected value under

Q of the discounted payoff of the option. The discount factors won over are acute.

And these are the probabilities, so for example, 3Q squared times 1 minus Q,

while this is equal to 3, reduced to. Times Q squared times one minus Q to the

power of three minus one. So this is a binomial probability, it

counts the number of ways in which the stock price can go up into periods and

fall in the third period. And in fact, this is equal to three here,

and we know that there's three ways to get up to this point, so down one and up

two, up one, down one, and up one, or up two and down one.

So we can also basically combine all the one period probabilities into three

period probabilities given to us by here, and compute the value of the option in

this manner. We also discussed, trading strategies in

the binomial model. So let's quickly review again what we did

there. St is going to denote the stock price at

time t. Vt denotes the value of the cash account

at time t, without any loss of generality we assume that b0 equals one dollar, so

that bt equals r to the power of t. So now we're explicitly viewing the cash

account as security. We let xt denote the number of shares.

Held between times t minus one and t. We also let yt denote the number of units

that the casher account have between times t minus one and t.

Then theta t equals x t y t as the portfolio held immediately after trading

at time t minus one and therefore is known at time t minus one.

And immediately before trading at time t. So, basically, if this is t minus 1, and

this is time t, then we know theta t at this point, and this represents the

portfolio that's held immediately after trading at time t minus 1, until trading

at time t. So theta t is a trading strategy.