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