The first project consists of the following 3 questions.

1. Consider the contract paying 100*ln(S(t)) at time t = 0.25: Suppose the current price of the stock S(0) is 125: 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 100*ln(S(t)): The continuously compounded interest rate for the three month maturity is currently 5.6%: Describe the arbitrage strategy to be conducted if the upper bound was violated.

2. The data for this question is in the sheet called FinancialCalls.xls 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) Use the call option prices at successive strikes to determine the price of a contract paying a dollar if the stock at maturity is above the
lower of the two successive strikes.

(b) Next determine the price of a contract paying a dollar if the stock at maturity is between two successive strikes.

(c) Let s_i denote the mid points of your successive strikes and let q_i determined in part b, be the price that the stock at maturity is s_i.
Graph the prices q_i against the stock prices s_i for all four underliers and their two maturities.

(d) For a continuously compounded interest rate of 3.5% per annum for the three month maturity, determine the spot price of the contract that pays in three months 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 one of the four underliers. The returns are defined by
R_i = (s_i - 100) / 100

and the contract pays

10,000,000 * (R_i²(.25) - R_i²(.0833)).

3. The data for this question is in the file SP XOptionData20160311.xls. For the four pairs of strikes of (1950, 2100); (1980, 2100); (1950, 2150) and (1980, 2150) consider a position of x options in the lower strike financed by a position in the upper strike. If you sell the lower strike then you buy the upper strike and the other way around if you buy the lower strike.

For a range of stock prices you are given their probabilities. You are also given the initial level of the index S: For an expected utility maximizing investor with exponential utility

u(C) = 1 - exp ((- 0.75 / S).C),

determine graphically the expected utility maximizing levels for x in the four option trades. You may evaluate the expected utility for each of level of x from say -10 to 10 in steps of .05 to then find the maximum value.

Attachment:- Call Price.rar

Attachment:- Data for SPX.rar