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This is the Exam of Statistical Science which includes Stochastic Differential Equation, Brownian Motion, Solution, Measurable Function, Markov Process, Starting, Bounded Functions, Local Martingale, First Time etc. Key important points are: Returns Matrix, Each Column Represents, State Contingent Return, Arbitrage, Opportunities, Future Inflows, Commit Zero Capital, Arbitrage Opportunities, Supporting Price, Vector
Typology: Exams
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Friday 31 May 2002 1.30 to 3.
Attempt THREE questions There are five questions in total
The questions carry equal weight
1 (a) In the returns matrix below each column represents the state-contingent return of an asset. Is this matrix arbitrage free for the three assets shown? If not, try to find arbitrage opportunities where you commit zero capital now, and arbitrage opportunities where your position yields no future inflows or outflows of capital.
(b) (i) The supporting price vector in a four state economy with a complete market is given by
p =
If a risk-free asset exists, what is its return? Use the risk-free return to derive the risk-neutral probability measure.
(ii) An agent has a von Neumann-Morgenstern utility function u defined over future state-contingent consumption ˜c u(˜c) := ln(˜c).
Use the supporting pricing vector in (i) to compute the optimal consumption plan for this agent when the states are equally likely and he has 100 currently available for investment.
(c) (i) Now suppose asset returns have the form
˜ri = ai + bi f˜ 1 + ci f˜ 2 ,
where f˜ 1 and f˜ 2 are independent random variables.
If the return on the risk-free asset is 1.05, the expected return on an asset i with bi = 1 and ci = 0 is 1.10 and the expected return on an asset j with bj = 0 and cj = 1 is 1.15, what is the expected return on an asset k with bk = β and ck = γ?
(ii) Suppose there is a risk-free asset with return R. In the absence of arbitrage, for any three risky assets (linearly independent) the following matrix must be singular:
a 1 − R a 2 − R a 3 − R b 1 b 2 b 3 c 1 c 2 c 3
Explain.
(iii) Use the fact that the matrix in (ii) is singular to derive the arbitrage-pricing theory for the special case where assets have no idiosyncratic risk.
4 Consider an infinite maturity put option written on a stock price that follows the risk adjusted process dSt = (r − δ)Stdt + σStdWt.
(a) For a given exercise trigger S price an infinite maturity put option with a strike price X written on the above security price (considering Merton’s version of the Black-Scholes PDE).
(b) Derive the optimal exercise price S∗^ for the put option. (c) Show how the trigger price is affected by changes in δ.
5 Write a short essay on credit derivatives. Your answer should include a description of the basic products and their current market, the uses of credit derivatives for both risk management and investment and a discussion of their pricing.
