tm V. =P(1+) Q. 3 (Arbitrage-free value of a bank account, 20 pts). The purpose of this exercise is to relate our discussion on compounding methods with the principle of no-arbitrage. Consider a bank account with an initial deposit P. The bank is free to use any com- pounding method as long as all these methods have the same effective rate. If discrete compounding is employed with m periods in a year and annual interest rate r > 0, then the value of this account is given by 1 2 for every teo, m' m In practice, the value of the account remains constant during a compounding period. This is because you would lose the interest earned in the last period if you were to withdraw your money before the completion of that period. However, this practice of valuation does not really comply with the principle of no-arbitrage. Show that the arbitrage-free value of the account is given by Vi = P (1+)*"for every t 2 0. For convenience, you may restrict yourselves to the values of t in [0, 1]. Hint: A starting point would be to re-think about our motivating discussion on the effective rates of compounding methods. tm