----------------------------------------------------- # # # Chapter 6: Price and Greeks of Plain Vanilla # # Options and the Black-Scholes Formula # # # ----------------------------------------------------- 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".

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