PROBLEM 7

Consider an investor who owns (t) shares of a money market account (price per share M(t)) plus (t) shares of a stock (price per share S(t)).

The differential equation for the evolution over time of M(t) and the starting value are given by:

dM = rM dt

with M(0) = 1

which of course leads to M(t) = e^rt

The total value of his/her portfolio at time t is

X(t) = (t)S(t) + (t)M(t)

[1] Assume the self-financing condition; show that this condition implies

S(t)d(t) + dS(t)d(t) + M(t)d(t) + dM(t)d(t) = 0.

[Hint: Note that in continuous-time stochastic calculus there are three terms in expressions of the form d(X(t)Y(t)) for processes X(t) and Y(t).]

Also show that

dX(t) = (t)dS(t) + r(X(t) (t)S(t))dt.

Next, we want to show why the above self-financing holds when there isnt any cash flow into nor out of the portfolio.

[2] We replace the continuous time variable t by the discrete time nt. Show that the portfolios wealth X_n is given at time n by

X_n = _nS_n + _nM_n

[3] Express the wealth X_n+1 at time n + 1 in terms of the number _n of shares of risky asset held in the time interval [n, n + 1). In other words, write an expression for the wealth at time n + 1 before rebalancing the portfolio.

[4] Write the wealth at time n + 1 after rebalancing the portfolio, using _n+1 as the number of shares held. Can the total wealth change at this point if we are in the self-financing situation? Why not?

[5] Equate the expressions derived in the last two questions to derive the self-financing condition in discrete time.

[6] Compare the previous self-financing condition to the self-financing condition in continuous time.