```-----------------------------------------------------
#                                                   #
#   Chapter 9: The Risk Neutral Pricing Measure     #
#            for the Black-Scholes Model            #
#                                                   #
-----------------------------------------------------

Summary: This is the continuous time version of the
chapter 3 on real world and risk neutral probabilities
in the binomial model. Let us quickly recall the pri-
cing logic of chapters 2 and 3 on the binomial model:
Arbitrary option payoffs could be replicated by a
suitable trading strategy in the underlying. The money
which is necessary to set up such a replicating stra-
tegy is the option price. In chapter 3 we saw then
that we can actually write this price as an expectation
value with respect to some stochastics, where this
stochastics is actually not given as a result of some
statistical analysis of the underlying dynamics (that
is, no "real world" pricing), but that stochastics
merely came from the requirement that in our basic
replication equation

V_N = V_0 + sum_{k=1}^N delta_{k-1}*(S_k - S_{k-1})     (1)

the sum on the right hand side of (1) should simply
vanish under this expectation value.

In continuous time, (1) is substituted by the Ito-
integral

V_T = V_0 + int_0^T delta_t * dS_t = H(S_T)           (2)

with H(S_T) being the option payoff (which also can be
path-dependent) and again we are able to arrive at a
compact pricing equation

V_0 = E[ H(S_T) ]                         (3)

if we define the stochastics with respect to which
the expectation on the right hand side of (3) is taken,
in a suitable way. It is a quite fundamental result
of mathematical finance that if the real world dynamics
of the underlying S_t is supposed to be a Black-Scholes
dynamics with some drift mu,

dS_t/S_t = mu*dt + sigma*dx_t                  (4)

with dx_t a Brownian motion, then the stochastics which
has to be used in (3) is given by a Black-Scholes dyna-
mics with exactly the same volatility sigma, but with
mu completely dropped out and substituted by the risk
free interest rate r,

dS_t/S_t =  r*dt + sigma*dx_t                  (5)

As a consequence, Black-Scholes option prices do not de-
pend on the real world drift parameter mu, but only de-
pend on the risk free rate r and the volatility sigma.

```
pdf-file: Chapter 9: The Risk Neutral Pricing Measure for the Black-Scholes Model

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