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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