This algorithm can be accomplished with O ( log n ! ) comparisons. To output the distinct items of a given list a 1 , a 2 , .... , a n the following method is proposed. For each k, search the elements proceeding a k and if a k is not found, output a k .

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Chapter 8.3, Problem 23E

To determine

This algorithm can be accomplished with O(logn!) comparisons. To output the distinct items of a given list a1,a2,....,an the following method is proposed. For each k, search the elements proceeding ak and if ak is not found, output ak.

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In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)

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