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European Option is a contract that can be exercised on the expiration date only. A *call option* gives the holder the right to buy the underlying asset at a certain prespecified price, called *strike*. A *put option* give the holder the right to sell the underlying asset at the strike. If
is the strike and
is the underlying price at expiration, then the terminal payoffs can be written as

Over the last 40 years, substantial body of methods has been developed to price options, with the Black-Scholes formula, volatility interpolation schemes and stochastic volatility models being the centerpiece. At this point the field is well-researched but there are still some interesting questions remaining. For example, much profit depends on how we choose

The price of a typical call option is illustrated in the graph below.

Hull, J. (2011). Options, Futures, and Other Derivatives (8th ed). Pearson / Prentice Hall.

Duffie, D. (2001). Dynamic Asset Pricing Theory (3rd ed). Princeton University Press.

Bjork, T. (2009). Arbitrage Theory in Continuous Time (3rd ed). Oxford University Press.

Lipton, A. (2001). Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach. World Scientific.

Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley Finance, New York.

Passarelli, D. (2008). Trading Option Greeks: How Time, Volatility, and Other Pricing Factors Drive Profit. Bloomberg Press, New York.

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