Determine where P (a) is a martingale measure associated with a power option Ra. Check that your…

Determine  where P (α) is a martingale measure
associated with a power option Rα. Check that your result agrees with the
special cases

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Find the price and the hedge of the contract V with the
payoff  

You can apply Black–Scholes formula using the assets R
and X.

Consider a perpetual barrier option that pays off a unit
of Y when XY hits a level L > XY (0) in a geometric Brownian motion model.
Let   be
the first time that the price hits the barrier. Obviously, the price of the
perpetual barrier option is given by  

Determine  

(a) Use the Optional Sampling Theorem (Theorem A.1: E Y
[XY (τǫ)] = XY (0)) to determine P Y (XY (τǫ) = L).

(b) Compute  You may easily generalize the formula to
determine P Y t (τ

(c) Determine the hedging portfolio of this contract.
Show that your hedging portfolio delivers a unit of Y when the barrier is hit.

 

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