Riddle

Dusty the gold dust trader floats down a remote river stopping at ten towns to buy or sell along the way. The last of these stops is a city where she must sell all the gold dust she has at $1 per unit. Travel and communication are difficult outside of the city. Prices are somewhat random. They tend to be less than $1 farther away from the city, but converge to $1 the nearer she gets to the end of the journey. The price she pays per unit also depends on the amount she is buying.

Starting with $1,000 and no dust, Dusty’s goal is to maximize expected profit. Devise and justify a trading strategy that specifies how many units of gold dust she should buy or sell at each stop as a function both of the price in that town and also of the holdings in cash and gold dust she has when she arrives there. What should the expected profit be?

One way to model this is by assuming p(t), the price per unit of gold dust at stop number t, is a random variable satisfying:

Log p(t)= (1-t/10) (log p(t-1) + d(t) + c[x(t)]), 2<=t<=10  

Where: X(t) represents the units of dust she buys (x(t)>0) or sells (x(t)<0) at stop t; the initial price is p(1)=0.95, and for each positive t, d(t) is normally distributed with mean zero and standard deviation 10%; and the constant c is a small nonnegative parameter, say c=1/10,000

Or try another specification that is consistent with Dusty’s scenario. Additional credit will be given for modifying the problem to deal with risk aversion, with borrowing, and with different numbers of stops or different values of other parameters.