STOCK MARKET. Today is Monday. An investor wants to buy before next Thursday a package of 100 shares of a listed company. To this aim, the investor contacts a broker that allows the investor to decide either to buy the shares at the closing price by the end of the day or wait until the next day. Thus, the decision process is as follows: on Monday, the investor must decide whether to buy the shares at the closing price or wait until Tuesday with the risk of having three possible prices; on Tuesday, after observing the price of Tuesday, the investor can buy the shares at the closing price or wait until Wednesday with the risk of also having three different prices. On Wednesday, she will buy the shares at the corresponding closing price. On Monday, the closing price is 10 per share. On Tuesday we estimate the price of shares could be either 9 (10% lower), 10 (unchanged) or 11 (10% larger), with probability 0.3, 0.3 and 0.4, respectively. On Wednesday, we expect the price of the share could be 10 % larger than the one of Tuesday, it could remain unchanged or 10 % lower than the price of Tuesday with the following probabilities:

	10% Larger	Unchanged	10% Lower
9	0.1	0.3	0.6
10	0.6	0.2	0.2
11	0.7	0.2	0.1

Note: for instance, if the closing price of Tuesday has been 9, the probability that on Wednesday the price of the share increases 10% is 0.1.

a) Build the decision tree that allows to determine the optimal buying strategy: that is, you must specify whether it is better to buy the shares on Monday or wait until Tuesday; in case of waiting until Tuesday, you must specify for each of the three possible prices of Tuesday, whether it is better to buy the shares on Tuesday or wait until Wednesday. The aim of the investor is to minimize the expected purchasing cost.

b) What is the expected monetary cost of its decision? Suppose now the investor has not bought the shares on Monday and that now it is Tuesday just after knowing that the closing price of Tuesday is 9. That is, let us suppose that now the investor can only buy the shares at the price of 9 or wait until Wednesday with the risk of having three different prices with the probabilities indicated in the first row of the table above.

c) What amount is willing to pay the investor for having perfect information. Suppose the investor has now the possibility of reading for free a daily report about the evolution of the share price. The issue of Tuesday includes a prediction for the closing price of Wednesday. For simplifying, the prediction of the report can be an price increase by 10%, an invariance of the price, or a price decrease by 10%. However, the report has the following probabilities of error:

	10% Larger	Unchanged	10% Lower
Increase 10%	0.9	0.1	0.1
Invariance	0.05	0.8	0.1
Decrease 10%	0.05	0.1	0.8

Remark: To correctly read the table, it says for instance that the probability that the report predicts a price increase by 10% conditioned to the case where, afterwards, the price increases by 10% is 0.9.

d) Having the possibility of taking this report into account, build the decision tree that allows to first decide whether to use or not the report, and then, to buy the shares on Tuesday at the price of 9 or wait until Wednesday. Again, the aim of the investor is to minimize the expected purchasing cost