The Price of a stock starts with the value $S.
After N time periods (which may be days or years or minutes) in which there are m stock increases and N-m decreases, it changes to U^mD^N-mS where U is the UP gain-factor and D is the DOWN gain-factor.

According to this Binomial stuff on Stock Evolution (a la Binomial distributions), the Expected value of the stock price is:

Expected Stock Price = Σ pm(1-p)N-m UmDN-mS

where the typical term in this sum is for m Ups (and N-m Downs).

>And all this has something to do with options?

We assume that the option expires after N time steps.
If the stock price after N steps (involving m Ups and N-m Downs) is U^mD^N-mS
then C_m, the value of a call option, is:
      C_m = Max[U^mD^N-mS - K,0]

>But there are lots of possible stock prices hence lots of possible option prices, at expiry, right?
Yes, and each has an associated probability.

However, the "Expected Value" of the option price is a sum of things which look like:
      (Probability of achieving a particular option price)x(the option price)
namely:
      N_Np^NC_N + N_N-1p^N-1(1-p)C_N-1 + N_N-2p^N-2(1-p)²C_N-2 + ... + N₀(1-p)^NC₀
where we now know the associated probabilities (from the Binomial stuff).

We'll rewrite this formula using a more familiar notation for the binomial coefficients:

Expected Option Premium (at expiry) = Σ pm(1-p)N-m Max[UmDN-mS - K,0]

>But the option price could be 0 if there are lots of Downs.
Yes, it could ... if the stock price falls, at expiry, below the strike price. That means that, although the sum is for all m
from m = 0 to m = N, if there are too few Ups ... meaning m is too small ... then that term has the value 0.

So, we'll let m = L be the Lowest value of m for which the option will expire "in the money".
In other words, if there are at least L Ups, then the associated option price will be greater than 0.
In other words, C_m = Max[U^mD^N-m - K,0] = U^mD^N-m - K     provided m > L.
In other words ...
>How many other words do you have?
Okay, we'll just cut to the chase and write the sum like so:

Expected Option Premium (at expiry) = Σ pm(1-p)N-m {UmDN-mS - K}

We assume it's worth $C today and $E at expiry. If we invested our $C at the risk-free rate (meaning a gain factor of R per time step) it would grow (after N time steps) to C R^N which, if we want to assume no risk, should equal the expected option premium of E. Hence C R^N = E so C = E R^-N.
In other words, the current value of the option should be the expected value at expiry divided by R^N.
In other words, we're discounting the expiry value to the current value by discounting at the risk-free rate.
In other words ...
>That's enough other words.

Okay, we'll write as and divide by R^N which we'll split up as R^mR^N-m and rearrange/combine stuff to get:

Current Option Price = S Σ {pU/R}m{(1-p)D/R}N-m - (K/RN) Σ pm(1-p)N-m

Do you recognize this expression?
>Are you kidding?
I mean, its form. The way it looks. Is it familiar?
>Are you kidding?

Notice that there are two curious factors in the first summation, namely P = {pU/R} and {(1-p)D/R}.
The first of these can be rewritten as P = {(R-D)U}/{R(U-D)}, since p = (R-D)/(U-D) ... see figure 2.
Further, 1 - P = 1-{(R-D)U}/{R(U-D)} = {(U-R)D}/{U-D)R} = {(1-p)D/R} which is the second curious factor.

Hence, we can rewrite the formula above as:

Current Option Price = S Σ {P}m{1-P}N-m - (K/RN) Σ pm(1-p)N-m where p = (R-D) / (U-D) and P = pU/R

The binomial distribution, defined by B(N,m,p) = p^m(1-p)^N-m, is your basic, all-purpose probability distribution.

>Yes. I read the Binomial stuff. Please proceed.

We'll let the time to expiry be fixed at, say, t years and let the number of time-steps be N so each step has length t/N. We're going to let N become infinite, implying a continuous (rather than discrete) evolution of prices. The Binomial distribution will magically turn into the Normal distribution.

As we saw in Part I, the discounting at the risk-free rate, as evidenced by the factor K/R^N in the second summation, becomes Ke^-rt where "r" now represents the annual risk-free rate.

>What, exactly, are we looking for?
We're looking to reproduce the Black-Scholes formula which looks like this:
      Current Option Price C = S N(d₁) - Ke^-rt N(d₂)

and we know (from the Binomial stuff) that the Binomial distribution looks like the Normal distribution when N gets large
... like so:

for Part III