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#     Chapter 3: Real World and Risk Neutral      #
#                   Probabilities                 #
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Summary: In chapter 2 we did not specify any proba-
bility in the definition of the Binomial model. We 
did that since, from the view point of option pricing, 
the decisive property of that model is that, in going 
from one time step to the next, there are only two 
possible choices, an up move or a down move. As a 
consequence, we could prove in Theorem 2.1 that in 
this model every option payoff H = H(S_0,S_1,...,S_N) 
can be replicated exactly by a suitable trading strategy 
in the underlying S.

Despite the fact that we did not need any probabilities 
to calculate option prices in chapter 2, we can never-
theless introduce some probability p and write the 
dynamics of the price process S_k as
 
                    1 + ret_up      with prob p
    S_k = S_{k-1} *                                  (1)
                    1 + ret_down    with prob 1-p

Then, and this is the central result of chapter 3, 
there is some number p such that we can actually write 
the option price as an expectation value of the payoff 
function H = H(S_0,S_1,...,S_N) with respect to the 
stochastic process given by (1). 

Since this number p is actually not determined by some 
kind of time series analysis performed on the historical 
data of the underlying stock, but merely comes from the 
payoff replication logic detailed in chapter 2, this 
probability p is not a "real world probability" but 
it is refered to as the "risk neutral pricing probabi-
lity".

pdf-file: Chapter 3: Real World and Risk Neutral Probabilities