WITH SKETCH ANSWERS. Postgraduate Certificate in Finance Postgraduate Certificate in Economics and Finance

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1 WITH SKETCH ANSWERS BIRKBECK COLLEGE (University of London) BIRKBECK COLLEGE (University of London) Postgraduate Certificate in Finance Postgraduate Certificate in Economics and Finance SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS INTRODUCTION TO MATHEMATICAL FINANCE EMEC03P Monday 1 June 006, (with 15 minutes reading time) There are FIFTEEN questions. SECTION A contains TEN questions and you should answer ALL of them. SECTION B contains FIVE questions and you should answer THREE of them. Section A is worth 40% of the overall mark. Sections B is worth 60% of the overall mark. Non-programmable calculators may be used. Credit will be given for clear, compact and purposeful use of mathematics. Write clearly and leave space between and around your answers Turn over 1

2 SECTION A Answer ALL questions. Say whether each of the following statements is TRUE or FALSE.Give a BRIEF explanation for your answer. 1. In the absence of arbitrage opportunities a forward contract, F (t) on the underlying asset price S(t), to deliver in period T, when the price is fixed at period t, will satisfy the condition F (t)e r(t t) = S(t) where r is the risk-free rate, with continously compounded interest. True. Assume the the expected return on asset i satisfies the condition E(R i )=E(R z )+β i [E (R m ) E (R z )] where R m is the market return, β i = cov(r i,r m )/var(r m ) and R z is the return on any portfolio for which β z =0. If this condition holds for all assets, all portfolios combining asset i and the market portfolio offer the same tradeoff between risk and return. False. Only at the margin where the weight on asset i is zero. 3. The return, R, on a portfolio of two assets with returns R 1 and R will have variance var(r) =a var(r 1 )+(1 a). var(r ) where a is the weight on asset 1. False: A gift to weaker students. Neglects the covariance term.a.(1 a).cov(r 1,R ) 4. If x is an Ito Process of the form dx = a 1 xdt + a xdz where z is a Wiener Process and a 1 and a are constants, then ln x is a generalised Wiener Process of the form where b 1 and b are constants True:byIto slemma d ln x = b 1 dt + b dz

3 5. Assume there are two equally probable states of nature and two pure securities. Pure security s, with price p s pays off 1 in state s, and zero otherwise, s =1,.If p 1 =0.45, p =0.4theexpectedreturnonpure security is lower than that on pure security 1. False: 1+E(R 1 )=.5/.45 = 10/9; 1 + E(R )=.5/.4 =10/8 > 1+E(R 1 ) 6. The probability that a stochastic variable x will be greater than a given value bx is given by prob(x >bx) = f(x)dx where f(x) is the density of x. True: 7. If asset A second-order stochastically dominates asset B at current prices, no risk-averse investor will buy asset A. false: other way around: A dominates B 8. If there are N states of nature, and N assets, itisalwayspossibletoderive the implicit prices of N "pure securities", P i,i=1..n, where pure security i has a payoff of unity in state i, and zero in all other states. false: the payoffs must be linearly independent. ideally show using brief linear algebra (matrix A of payoffs must be invertible) as in lectures 9. If x y = α + βx z = γ + δx where x is stochastic and α,β γand δ are constants, then True cov(y, z) =βδvar(x) 10. If dividends at time t are given by D 0 (1 + g) t, and there is a constant cost of capital, r, at all points in the future, the ex-dividend value of a firm at time zero, S 0,willbegivenby: S 0 = D 0 r g False: replace D 0 with D 1 3

4 SECTION B Answer THREE questions (a) Assume that investors have a utility function U(R) defined in terms of R, thereturnontheirwealth,withu 0 > 0, U 00 < 0.Assuming R N(μ, σ) show that investors will have mean-variance expected utility. Answer:Define E(U(R)) = U(R)f(R)dR whichwecanrewritebyusingthefactthat dr = σdz and, from the definition of the normal density, f(r) = 1 σ f(z).thus we can substitute for R and write E(U(R)) = = = = U(R)f(R)dR U(μ + σz) f(z) σ σdz = V (μ, σ) U(μ + σz)f(z)dz (b) Show that, on the same assumptions, if all agents can lend or borrow at the risk-free rate, and R m is the return on the market (the portfolio of all assets, weighted by by their market value), then there is a constant price of risk (the market tradeoff between volatility and expected return) given by E(R p ) σ p = E(R m) R f σ m where R p is a portfolio of the risk-free asset and the market portfolio, and σ p and σ m are the standard deviations of the return on the portfolio and the market respectively. 4

5 answer: Form portfolio with a in market and (1 a) in risk-free asset, implies E(R p ) = R f + a [E(R m ) R f ] E(R p ) a = E(R m ) R f var(r p ) = a σ m σ p = aσ m hence E(R p ) = E(R p)/ a = E(R m) R f σ p σ p / a σ m (c) Assume that in equilibrium all assets are fairly priced and in the market portfolio. Use a no arbitrage argument to derive the CAPM security market line for asset i, E(R i )=E(R z )+β i [E (R m ) E (R z )] where β i = cov(r i,r m )/var (R m ) (a) A consumer has the intertemporal consumption and portfolio choice problem max u(c 0 )+ 1 c 0,X 1...X N 1+δ E(u(c 1)) subject to W = c 0 + c 1 X j (1 + R j )X j where δ istherateofpuretimepreference,andu() is the per period utility function, defined in terms of consumption, c; c 0 is (certain) consumption at time zero, and c 1 is (uncertain) consumption in the next period, when asset returns are realised. X j is the quantity of asset j bought in period 0, yielding return R j in period 1. Show that the first order conditions for this problem yield for every asset j, an optimality condition of the form u 0 (c 0 )= 1 1+δ E ((1 + R j)u 0 (c 1 )) answer: substute from the nd constraint into the maximand, and 5

6 rewrite the problem as a Lagrangian max u(c 0 )+ 1 c 0, X 1...X N 1+δ E u (1 + R jt )X j λ c 0 + X j W where everything is now chosen in the first period. The first order conditions are u 0 (c t ) λ = δ E [u0 (c 1 )(1 + R j )] λ = 0 for all j c 0 + X j W = 0 (b) Show how to re-express this optimality condition in the form 1=E ((1 + R j ) θ) for all j and explain how in equilibrium, if all investors can optimise as in section (a), and have the same expectations, there must be a unique market stochastic discount factor, θ. (c) Show that E(θ) = 1 1+R where R is the risk-free rate, and use this and the general expression in (b) to derive and briefly interpret an expression for E(R i ) R, the risk premium on asset i, in the light of the optimality condition derived in part (a) 13. Assume that the log of the stock price is a generalised Wiener Process d ln S = µμ σ dt + σdz where μ is the expected return; σ is the variance rate, and z is a Wiener process. 1. (a) Show that ln S T is normally distributed, with the form ln S T N(ln S 0 + ηt, σ T ) and give the definition of η. answer: summary version of Hull (4th edition) pp1-4, and p38. η = μ σ 6

7 (b) Hence show that E(S T )=S 0 e μt answer: line) given assumption of log normality (no need to prove first (c) Using the result that: where E ln(st )+var(ln ST )/ E(S T ) = e = e ln S0+ηT+ σ T = e ln S 0+(μ σ )T + σ T = e ln S 0+μT = S 0 e μt E[max(S T X, 0)] = E(S T )N(d 1 ) XN(d ) d 1 = d = ³ ln(s 0 /X)+ σt 1 ³ ln(s 0 /X)+ σt 1 μ + σ μ σ and N() is the cumulative density of a standard normal variate, show how the formula needs to be adapted under risk-neutral valuation, given a constant riskless rate r. Carefully explain and interpret the impact of risk-neutral valuation on the probability N(d ), and hence on the term XN(d ). answer: μ needs to be replaced with r.this implies a fall in N(d ), which is the probability that S T >X (d) Hence derive the Black-Scholes formula for the price, c, ofaeuropean call option at time zero,with exercise price X and maturity date T. answer: T T c = e rt [ E(S b T )N(d 1 ) XN(d )] = e rt [S 0 e rt N(d 1 ) XN(d )] where d and d 1 are redefined replacing μ with r. Cancelling out of terms in the first expression gives us the Black-Scholes formula: c = S 0 N(d 1 ) Xe rt N(d )] 14. Ito s Lemma states that for any Ito Process, x: dx = a(x, t)dt + b(x, t)dz 7

8 where z is a Wiener process, then a function G(x, t) is also an Ito Process, with the form: µ G dg = x a + G t + 1 G x b dt + G x bdz 1. (a) Assuming that the stock price is a geometric brownian motion process, ds = μsdt + σsdz derive the process for any derivative security, f(s, t), by application of Ito s Lemma. (b) By assuming that derivatives and the underlying securities can be traded so that riskless portfolios can be constructed, use the resulting expression in (a) to.derive the Black-Scholes differential equation for the price of a derivative security: f f + rs t S + 1 f S σ S = rf answer toaandb: simply substitute into Ito s Lemma µ f f df = μs + S t + 1 f S σ S dt + f S σdz form portfolio Π = f + f S S change in portfolio is dπ = df + f S ds µ f f = μs + S t + 1 f S σ S dt + f S σdz + f [μsdt + σsdz] S terms in f f SμSdt and Sσdz cancel. dπ does not depend on dz and hence is riskless, so must yield the risk-free return, hence after cancelling terms we have µ dπ = f/ t 1 f S σ S dt = rπdt Substituting for Π on the right-hand side, we have µ f/ t 1 µ f S σ S dt = r f + f S S dt which after cancellation of terms in dt and rearrangement gives f f + rs t S + 1 f S σ S = rf which is the B-S DE 8

9 15. (c) Briefly explain why the expected return on the stock, μ,does not appear in the Black-Scholes differential equation. answer: The B-S DE is derived on the assumption that the underlying portfolio is made riskless (you can point to the fact that there are two offsetting terms in μs) (d) Assume a derivative is traded with the form f = αse qt derive the implied process for f, and show that it only satisfies the Black-Scholes differential equation if q =0. Interpret this result. 1. (a) State the key assumptions underlying the Arbitrage Pricing Theory and carefully show how these assumptions can be used to derive an expression for expected returns on assets. (b) Show how a special case of the Arbitrage Pricing Theory can be related to the Capital Asset Pricing Model, carefully explaining the differences in assumptions underlying the two models. 9