A company's cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.1 per month and a variance rate of 0.16 per month. The initial cash position is 2.0.

(a) What are the probability distributions of the cash position after one month, six months, and one year?

(b) What are the probabilities of a negative cash position at the end of six months and one year? (c) At what time in the future is the probability of a negative cash position greatest?

Suppose thatxis the yield on a perpetual government bond that pays interest at the rate of $1 per annum. Assume thatxis expressed with continuous compounding, that interest is paid continuously on the bond, and thatxfollows the process

dx=a(x0x)dt+sxdz

wherea,x0, andsare positive constants anddzis a Wiener process. What is the process followed by the bond price? What is the expected instantaneous return (including interest and capital gains) to the holder of the bond?

A financial institution plans to offer a security that pays off a dollar amount equal toST2at timeT.

(a) Use risk-neutral valuation to calculate the price of the security at timetin terms of the stock price,S, at timet. (Hint: The expected value ofST2can be calculated from the mean and variance ofSTgiven in section 15.1.)

(b) Confirm that your price satisfies the differential equation (15.16).