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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