ECO00040M Theory of Finance MSc Degree Examinations 2018-9
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ECO00040M
MSc Degree Examinations 2018-9
Economics
Theory of Finance
SECTI0N A. Answer at least 0NE question
1. (a) Explain how it is possible in principle to price an asset using contingent claims analysis. [4 marks]
(b) What is the signiÆcance of having an asset that has the same payoV in every state? [2 marks]
(c) If the number of possible states is S and the total number of assets is N , what is the signiÆcance of the 3 possible situations: S = N,S < N and S > N? [4 marks]
(d) Consider this representation of a contingent claims market: ┌''' ┐''' = ┌''' 1(1) 1(1) 1(1)0(0) ┐''''(┌) '(┐) ,
where the left hand column represents the prices of four traded securities; the right hand table their payoVs in the three states and the right hand column the associated claims prices. There is an arbitrage opportunity here. Set up a portfolio that would proÆt from this. [2 marks]
(e) Now consider this representation:
┌ ┐ ┌ ' 100 ' = ' 10 |
0 110 10 |
0(0) ┐ ┌ Q(Q)2(1) ┐ 110 ' ' Q3 ' . |
Solve for:
(i) the claims prices Q1 , Q2 and Q3 ;[8 marks]
(ii) the risk neutral probability of each state. [2 marks]
(f) Now consider this representation of a coupon bond market:
┌ ┐ ┌ ┐ ┌ D(D)2(1) ┐
' 100 ' = ' 10 10 110 ' ' D3 ' ,
where the left hand column represents the prices of three non-defaultable se- curities; the right hand table their payoVs in the next three periods and the right hand column the associated discount factors. What are the prices of zero coupon bonds paying one unit of currency in each future period? [2 marks]
2. Suppose that the price of interest rate risk is constant and that the spot rate rt follows the Ærst order autoregressive process on an annual basis:
rt+1 = (1 - f)µ + frt + xt+1
(where xt is an i.i.d. normal error term) so that the logarithm (dm |t ) of a discount bond price (Dm |t ) is an aYne function of this variable:
-dm |t = am + bm rt ; m = 1, ...,M. (1)
where am and bm are deÆned in the Aide demoire.
(a) How do am and bm behave as maturity becomes very large? [4 marks] (b) Derive a formula for the m-period discount yield ym . [l0 marks] (c) How does ym behave as maturity becomes very large? [5 marks]
(d) Suppose that f = 0.2. Calculate the eVect on the 1, 5, 10 and 20 year bond yields of a one point change in the spot rate. Comment on your results. [6 marks]
SECTI0N B. Answer at least 0NE. (Weight: 25% of Ænal mark)
3. Suppose that an investor can invest in a risk-free asset yielding y and N ri¥ky assets with random returns ri , means E(ri ) = µi , variances V (ri ) = sii , (i = 1, ...,N) and covariances Cov(ri ,rí ) = sií (i,j = 1, ...,N; i j). Let αi be the share held in the ith asset. DeÆne the N x 1 vectors r = (r1 , . . . ,rN )\ , µ = (µ1 , ...,µN )\ , α = (α1 , . . . ,αN )\ and {\ = (1, ..., 1). This investor maximises a utility function of the form:
F[E(rp ),V (rp )] = E(rp ) - gt V (rp ). (1)
(a) Interpret the parameter gt and explain how it inØuences the investorís attitude towards risk. [3 marks]
(b) Find the optimal value of the vector of risky asset holdings α . [l2 marks]
.
(c) Specialise this problem for the case of one risky asset and solve for the holding of the risk-free asset. [2 marks]
(d) Show that the shares zt = αt /{\ αt of the risky asset portfolio {\ αt held in each risky asset are independent of gt . [4 marks]
(e) Outline the implications of this result for the mutual fund industry. [4 marks]
4. (a) Under what circumstances is it realistic to assume that there is a safe asset in a mean-variance portfolio optimisation problem? Give three examples of situations in which this is unrealistic. [6 marks]
(b) Suppose that there are N ri¥ky assets with random returns ri , ex- pected returns E(ri ) = µi , variances V (ri ) = sii , (i = 1, ...,N) and covariances Cov(ri ,rí ) = sií (i,j = 1, ...,N; i j). Let αi be the share held in the ith asset. There is no ¥afe a¥¥et. Assume that all investors are mean variance portfolio optimisers with the same where g is the coeYcient of relative risk aversion.
(i) Derive the shadow safe rate of interest in this problem; [6 marks]
(ii) Derive the mean variance optimal portfolio shares; [6 marks]
(iii) Now suppose that the variances and covariances are constant but the expected returns µi are themselves Gaussian random variables. How is the
shadow safe rate of interest distributed? Is there a model that can be used for
pricing bonds in this situation? [7 marks]
SECTI0N C. Answer at least 0NE. (Weight: 25% of Ænal mark)
5. Consider the discrete-time observations of a homoscedastic stochastic process wt = {α1 + α2 + ... + αt }, where the increments are independent with:
E[αi ] = w〇 = 0; and V [αi ] = 1. (1)
(a) What are the means, variances and standard deviations of:
(i) w2 ; [l mark]
(ii) w1〇; [l mark]
(iii) wt ; [l mark]
(iv) s xwt (where s is a constant). [l mark]
(b) Consider the position wt of this process at time t and the eVect of taking smaller and smaller sub-intervals of time. Call the length of the sub-interval Dt. Assume homoscedasticity as in (a)
Find expressions in terms of Dt for:
(i) the number (N) of sub-intervals in the sample period running from 0 to t; [l mark]
(ii) the mean, variance and standard deviation of the change in the process during any sub-interval of length Dt.; [3 marks]
(iii) the mean, variance and standard deviation of wt (Hint: aggregate the changes over these N sub-intervals) ; [6 marks]
(iv) the mean, variance and standard deviation of the limit of the change dwt in wt as the time interval Dt becomes inÆnitesimally small.
[2 marks]
(c) Noting that wt /N is the sample mean of the changes over the N sub- intervals, what does the Central Limit Theorem tell you about the distribution of wt as Dt the time interval becomes inÆnitesimally small.? [7 marks]
(d) When is the homoscedasticity assumption not likely to be satisÆed? [2 marks]
6. (a) State Itoís Lemma. [2 Marks] (b) Prove Itoís Lemma. [l5 Marks] Hint : use the Ito rules:
E[dw2 ] = dt
V [dw2 ] = 0
E[dw .dt] = 0.
(c) Suppose that an asset price is described by the Geometric Brownian
Motion:
dP(t)/P(t) = µdt + sdw(t).
Use Itoís Lemma to derive the SDE for p(t) = ln[P(t)] and hence the distribution of p(t) and P(t) conditional on the initial values p(t) = 0,P(t) = 1.
[8 Marks]
7 (a) DeÆne the statistical term: martingale. [3 Marks]
(b) Suppose that a security price P follows a Geometric Brownian Motion under the physical probability measure:
dP(t)
P(t)
and also follows a Geometric Brownian Motion under the adapted probability
measure Q:
= ndt + sdwN .
S(t) =
= where : |
e一yt P(t) P(t) Py (t) Py (t) = eyt |
where y is the risk-free rate.
(i) Use (1) and (2) to show the relationship dwN between and dw(t). [4 Marks]
(ii) Use Itoís lemma to derive the SDE followed by S(t) under measure Q. [4 Marks]
(iii) Find the value n of that reduces the drift in this SDE to zero and thus makes S(t) a martingale under measure Q. [4 Marks]
(iv) Explain the signiÆcance of this result. [2 Marks]
(c) Show how this relationship can be used to
(i) Ænd the expected value of P(T),T > t conditional upon P(t). [4 Marks] (ii) value P(t) given the expected value of P(T),T > t. [4 Marks]
2023-05-23