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#   Chapter 6: Price and Greeks of Plain Vanilla    #
#       Options and the Black-Scholes Formula       #
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Summary: In the last chapter we saw that the Black-Scholes 
model can be obtained as the continuous time limit of a 
suitably defined binomial model. As a consequence, exact 
payoff replication is also possible in the Black-Scholes 
model, the trading strategy holding delta_t stocks at time 
t, where delta_t is the derivative of the option price with 
respect to the underlying price, replicates the option 
payoff. If the option is not path dependent, H=H(S_T) if 
S_T denotes the underlying price at option maturity T, 
we obtained a one-dimensional integral representation for 
the theoretical fair value, the price of the option.

In this chapter we use this one-dimensional integral 
representation to derive analytic closed form expressions 
for the price of call- and put-options. These options are 
also referred to as "plain vanilla" options because of the 
simplicity of their payoffs. Furthermore we define the 
"Greeks" which are simply derivatives of the option price 
with respect to certain parameters: the delta is the deri-
vative with respect to the underlying price, the vega is 
the derivative with respect to the Black-Scholes model 
volatility and the rho is the derivative with respect to 
interest rates. Analytic closed form expressions for these 
quantities are derived. 

The analytic closed form expression for the price of call- 
and put-options or "plain vanilla" options is referred to 
as "the Black-Scholes formula".

pdf-file: Chapter 6: Price and Greeks of Plain Vanilla Options and the Black-Scholes Formula