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We consider a stock with current price of 8. After one year, the stock price will increase by 20% (with probability 40% ) or decrease by 20% (with probability 60% ). The same applies the following year. The process representing the evolution of the stock price in time is denoted by (Sn)n=0,1,2. We conclude that S1 will take the values S1u=S0u or S1d=S0d, while S2 will take the values S2uu=S0u2,S2ud=S0ud and S2dd=S0d2, where u is the growth factor and d the decreasing factor. a) Is (Sn)n=0,1,2 a martingale with respect to the above probability and with respect to the filtration generated by (Sn)n=0,1,2 ? b) We now define the stochastic processes (S~n)n0:S~0=S0,S~1u=S1uk,S~1d= S1d+k and S~2=S2. Is (S~n)n0 a martingale with respect to the probability and the filtration of the previous item for a suitable k>0 ? c) Is there a probability measure Q such that the probability that the stock price increases (respectively, decreases) in the first year is equal to the probability that the stock price increases (respectively, decreases) in the second year, and such that (Sn)n=0,1,2 is a martingale with respect to Q ? We consider a stock with current price of 8. After one year, the stock price will increase by 20% (with probability 40% ) or decrease by 20% (with probability 60% ). The same applies the following year. The process representing the evolution of the stock price in time is denoted by (Sn)n=0,1,2. We conclude that S1 will take the values S1u=S0u or S1d=S0d, while S2 will take the values S2uu=S0u2,S2ud=S0ud and S2dd=S0d2, where u is the growth factor and d the decreasing factor. a) Is (Sn)n=0,1,2 a martingale with respect to the above probability and with respect to the filtration generated by (Sn)n=0,1,2 ? b) We now define the stochastic processes (S~n)n0:S~0=S0,S~1u=S1uk,S~1d= S1d+k and S~2=S2. Is (S~n)n0 a martingale with respect to the probability and the filtration of the previous item for a suitable k>0 ? c) Is there a probability measure Q such that the probability that the stock price increases (respectively, decreases) in the first year is equal to the probability that the stock price increases (respectively, decreases) in the second year, and such that (Sn)n=0,1,2 is a martingale with respect to

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