FE Problem Set #03
Team Latte Nov 10, 2005
Problem #1
A structurer is trying to value a contingent premium option. The underlying spot is trading at 100 and he wants to price a 6 month ATM (at the money) contingent premium call option such that the contingent condition is satisfied, i.e. zero premium is paid up front. The risk free rate is 5% and the dividend yield on the stock is 3%. The annualized volatility that the trader is using is 15%.
(a) Given the above data what should the price of the 6 month ATM contingent call option on the stock such that the contingent condition is satisfied, i.e. the buyer of the option (your client) pays no premium for the option upfront but if the option finishes in the money then he would pay an agreed price (the contingent premium).
(b) Assume that the spot is trading at 100 and you sold a contingent premium call to a client at $9.00. The market price of an ATM vanilla call is $5.00. How would you hedge this position? Explain.
Problem #2
Consider the following function:
What is the partial (mathematical) derivative of the function with respect to at the location ? At what point in space will the partial derivative of with respect to and the partial derivative of with respect to be equal (i.e. the slopes will intersect)?
Problem #3
Fed funds futures contract is designed as a hedge to a $5 million 30day deposit in fed funds. The final settlement price of a fed funds futures contract in a particular month is set to 100 minus 100 times the average of the effective fed funds rate over that month. Further, the marktomarket payment of the contract (per basis point) is estimated by calculating the interest payment on a $5 million 30day loan (on 30/360 day basis).
(a) If in November the average rate was 2.055% then the contract will settle at what price?
(b) If a trader gained 10 basis points on his fed funds contract then in Dollar terms what was his gain?
Problem #4
In a mean reversion model the expectation of a short term interest rate as a function of the horizon gradually rises from its current value (t = 0 value of ) and approaches a limiting value (which is usually denoted by ). This limiting value is the goal or the long term goal. It can be shown that the distance between the current value of the short rate and the goal decays exponentially at the mean reversion rate. A useful quantity, which is seldom mentioned in the nonacademic circles, is the half life of the interest rate, defined as the time it takes for the short rate to progress half the distance toward its goal.
If currently the short term rate is at 5% and the long term goal is 9%, then with a mean reversion speed of 0.025 what would be the half life of the short rate in years?
Problem #5
Consider a transition matrix for four rating categories Aaa, Aa, B and D, where D implies default by a company on its bond. The rating transition matrix P is given by:
Whereby, 0.04 in first row and second column in the above matrix signifies that there is a 4% probability that a bond in the rating category Aaa will move to category Aa (lower rating) in the next period and the number 0.01 in the first row and fourth column signifies that there is a 1% probability that a bond that is currently rated Aaa will be in category D (i.e. it will default) in the next period. Assume that this is an annual transition matrix. Further, from the prices of the bonds that trade in the market we deduce the following implied default probabilities:
Estimate the modified transition matrix by adjusting the matrix P such that the implied default probabilities are matched.
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