1. Consider the contract paying 100  ln(S(t)) at time t: Suppose the current price of the stock is S(0): Observe that the tangent to the log contract constructed at the point S(0) lies above the log contract. Use this observation to develop an upper bound for the price of the contract paying 100ln(S(t)): Describe the arbitrage strategy to be conducted if the upper bound was violated.

2. You are asked to prepare a quote in US dollars on the forward price you would charge for the delivery in 9 months of a British bond with a face value of $1000 and semiannual coupons of 10% per annum. The UK and US yield curves are quoted in the following schedule of simple and continuously compounded interest rates. The spot exchange rate is $2:4 per $:

Maturity in years UK            Simple interest rate US   Continuously compounded rate

              .5                                          5.0%                                       3.2%
             .75                                         5.25%                                     3.3%
             1.0                                         5.5%                                       3.45%

(a) Determine the price in pounds of the British bond.

(b) Determine the amount borrowed in US dollars to implement the replication strategy of buying the British bond in the spot market and selling intermediate incomes.

(c) Determine the US dollar forward price quote.

(d) Determine the forward price for the same British bond in the UK in pounds.

(e) Determine the forward exchange rate for 9 months.

(f) Determine the dollar cost of buying the bond forward in the UK using the 9 month forward exchange market.

(g) Compare your answers to c and f and describe the arbitrage if the answer to c was below that in f:

3. The Data file for this question is for SPX options on 20090415 in bmgt444S15Project1.xls. It contains the following information in the specified columns.

Column 1 2 3 4 5 6 7
Variable S(0) K r q t is call Option Price
where ìis callîis one for calls and zero for puts.

(a) Show that the current market price for the forward stock delivered at maturity T is S(0)e qT :

(b) Revise the put call parity relation for the presence of dividend yields and show that now one must have

c(K; T) p(K; T) = S(0)e qT Ke rT :

(c) Using market traded strikes design a position in bonds and options such that a total investment of one million dollars is fully invested
in stocks with a largest proportion of the portfolio value being held in the bond market. Specifically list the complete portfolio and the precise position in each asset.

4. The data for this question is in the sheet called FinancialCalls and it contains call prices for options on BAC, GS, JPM, and MS on September 10 2008 as calibrated using an arbitrage free model with spot prices set to 100; zero rates and dividend yields, with strikes ranging in two dollar intervals from 50 to 150 for the two maturities of one and three months respectively.

(a) Graph the risk neutral densities for the four stocks at both maturities.

(b) For a continuously compounded interest rate of 3:5% per annum, determine the spot price of the contract that pays the di§erence between the squared return at three months and the squared return at one month times a notional of 10 million dollars on each of the four underliers.

(c) Determine the optimal position in a bond, the stock and out of the money options on the stock. This is done by regressing the cash flow desired on the cash flows to a bond, stock and options on the stock. Use strikes ranging from 60 to 140 in steps of 10 dollars. The position is to be designed for the investment of a 1000 dollars in GS for a risk averse investor with exponential utility and absolute risk aversion coefficient 0:5: The investor has log normal beliefs with a mean rate of return of 7% and a volatility of 15%: The utility function in this case is

u(c) = 1 exp( c=2):