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whitepaper_main - 1 Denitions Denition 1.1 European Call...

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1 Definitions Definition 1.1. European Call Option It gives holder the right but not the obligation to buy a stock at a fixed (strike) price K at time T . Hence the payoff is payoff = max[0 , X T - K ] (1.1) Definition 1.2. Barrier Option An option which comes into existence or becomes worthless if the underlying asset reaches some prescribed value before expiry. Down and out The option expires worthless if the barrier X t = b is reached from above before expiry. Thus, its domain is X t > b . Definition 1.3. Lookback Option An option whose payoff depends on the maximum or minimum realized asset price in some window [0 , T ] . payoff for minimum type call = max[0 , X T - Y T ] (1.2) where Y T = min 0 t T X t (1.3) 2 Pricing Minimum-Type Lookback Options Given the Black-Scholes partial differential equation L U = - ∂U ∂t + r U - r x ∂U ∂x - 1 2 σ 2 x 2 2 U ∂x 2 = 0 (2.1) and the minimum stock process is defined earlier in Equation 1.3, the price U ( x, y, t ) in ( x > y, t < T ) of a min-type lookback option satisfies L U = 0; U ( x, y, T ) = f ( x, y ); U ( y, y, t ) = 0 (2.2) Remark 2.1.

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whitepaper_main - 1 Denitions Denition 1.1 European Call...

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