This problem extends Example 1037 to a more general setting

This problem extends Example 10.3.7 to a more general setting. Again, suppose a very volatile stock rises 70% or drops 50% in price each day, with equal probabilities and with different days independent. Suppose a hedge fund manager always invests half of her current fortune into the stock each day. Let Y_n be her fortune after n days, starting from an initial fortune of Y_0 = 100. What happens to Y_n as n right arrow infinity

Solution

We know that she invests only half of her fortune.

Let the fortune on n^th day be Y_n

Then, it is equal to Y_n-1/2 + Y_n-1*1.7^x*0.5^n-1-x

Here x is the number of days the stock has risen. As n tends to infinity, x tends to infinity, as the likelihood of the rise and fall in price is the same.

Therefore,

Y_n can be obtained using the above expression, and is equal to 50 + 50*(1.7 - 0.5)= 110