G14SFM-E1 The Universit\ of Nottingham SCHOOL OF MATHEMATICAL SCIENCES A LEVEL 4 MODULE, SPRING SEMESTER 2017-2018 672CHA67IC FI1A1CIAL 02DEL6 Time allowed THREE Hours Candidates ma\ complete the front coYer of their ansZer book and sign their desk card but must NOT Zrite an\thing else until the start of the e[amination period is announced. CUedLW ZLOO be gLYeQ fRU WKe beVW FOUR aQVZeUV. Onl\ silent, self-contained calculators Zith a Single-Line Displa\ or Dual-Line Displa\ are permitted in this e[amination. Dictionaries are not alloZed Zith one e[ception. Those Zhose first language is not English ma\ use a standard translation dictionar\ to translate betZeen that language and English proYided that neither language is the subject of this e[amination. Subject specific translation dictionaries are not permitted. No electronic deYices capable of storing and retrieYing te[t, including electronic dictionaries, ma\ be used. DO NOT WXUQ e[aPLQaWLRQ SaSeU RYeU XQWLO LQVWUXcWed WR dR VR G14SFM-E1 Turn oYer 1 G14SFM-E1 1. (a) Consider a one-period investment model with three risk\ assets having random returns৬ > )হ2- হ3- হ4*, where ৬ has mean ਆ > )3- 5- 4* and covariance matri[ ঽ > 3 2 12 6 11 1 4 / What is the investment strateg\ corresponding to the mean-variance efficient portfolio in this market with initial wealth
2000 and mean return 3? [10 marks] (b) Consider now a model with onl\ two risk\ assets; having returns ৬ > )হ2- হ3* with mean ਆ > )2- 2* and covariance matri[ ঽ > 2 2323 2 / Given that the mean and variance of the return on the two individual assets are identical, e[plain wh\ the distributions of returns হ)ಝ* and হ)ಝ༚* on two portfolios ಝ Ӎ ಝ༚ ѵ ϓ3 might be different. [10 marks] (c) Sketch the region in the )౨- ౡ*-plane of the possible standard deviations and means of portfolios in a market of purel\ risk\ assets and discuss its significance. Include the two named portfolios ಝৈ and ಝ in \our sketch and discussion. Describe in just a sentence or two the changes to this sketch and discussion if a riskless asset is included in the model market. [10 marks] G14SFM-E1 2 G14SFM-E1 2. Consider an investor with initial capital 1 ? 1. If at the beginning of \ear ( > 2- 3-Ϳ - ৎѿ2) she has capital ি then she will decide to consume some amount ৄ ѵ \1-ি^ during that \ear and invest the remainder, with the proceeds to be used as capital for the following \ear. In \ear ৎ she will consume all available capital. The investments that the investor will use are such that an investment of units at the start of \ear will \ield হ units at the start of \ear ,2, where হ1- হ2-Ϳ -হৎѿ2 are independent random variables and the value of হ is known onl\ at the end of \ear (that is, the start of \ear , 2). Assume that the investor aims to ma[imise her total e[pected consumption over all \ears1- 2-Ϳ - ৎ. (a) Find an e[pression for ি,2 in terms of ি, ৄ and হ; being careful to define which values of it is valid for. [2 marks] (b) Find an e[pression for the total reward ʊ from \ear onwards associated with the investment strateg\ ৄ- ৄ,2-Ϳ - ৄৎѿ2. [2 marks] (c) Let ঽ)* be the ma[imal total e[pected reward from \ear to \ear ৎ (inclusive), starting with ি > . Write a formula for ঽৎ)* and show carefull\ that, for > 1- 2-Ϳ - ৎ ѿ 2,ঽ)* > nbyৄѵ\1-^|ৄ , ব\ঽ,2)হ) ѿ ৄ**^~/ [10 marks] (d) Assuming that ব\হ^ > ? 2 for all > 1- 2-Ϳ - ৎ ѿ 2, find ঽৎѿ2)* and then show carefull\ that ঽ)* > ৎѿ)* ( > 1- 2-Ϳ - ৎ) and the optimal strateg\ is to consume nothing at times 1- 2-Ϳ - ৎ ѿ 2. [10 marks] (e) How will the optimal strateg\ and ঽ1 change if ব\হ^ > ? 2 for all > 1- 2-Ϳ - ৎ ѿ 3 but ব\হৎѿ2^ ѵ )1- 2*? Justif\ \our answer b\ describing which parts of \our answer to (d) will change and which will sta\ the same; giving details of the parts which will change. [6 marks] G14SFM-E1 Turn OYer 3 G14SFM-E1 3. (a) i) What do \ou understand b\ the statement that probabilit\ measures ষ and স are equiYalent? Give a formal definition; an equivalent condition when the underl\ing sample space is discrete; and an informal interpretation. ii) Let ষ, স and া be three probabilit\ measures on the sample space ʊ > |2- 3- 4~ given b\ ষ )|~* > 24- স)|~* > 7- া )|~* > ѿ 24 ) > 2- 3- 4*/ Classif\ each pair of these probabilit\ measures (ষ and স, ষ and া, স and া) as equivalent or not equivalent. [6 marks] (b) Consider a one-period arbitrage-free Binomial model for a stock and a riskless asset. Show that an\ contingent claim (প, sa\) to be paid at time 1 ma\ be hedged; and show how its price ma\ be e[pressed in terms of the risk-netural probabilit\ for the Binomial model. [12 marks] (c) A utilit\-ma[imising investor in the one-period arbitrage-free Binomial model has initial capital ৗ1 ? 1 and utilit\ function ে)* > 3 ѿ 3. Determine the portfolio that will ma[imise the e[pected utilit\ of the investor¶s final wealth. (You ma\ e[press this portfolio in terms of the possible time-1 wealth values ৗ2 andৗ3, which \ou should have e[plicit e[pressions for.) [12 marks] 4. Consider a stock over two time periods. Assume that the stock prices at times 0,1 and 2 are1 > 2, 2 > হ21 and 3 > হ32; where হ2 and হ3 are independent random variables with ষ )হ > 3* > 34 > 2 ѿ ষ )হ > 34* ( > 2- 3). (a) Determine the values of the risk-free interest rate ౦ for which there is no arbitrage opportunit\ in this model; and find the risk-neutral probabilit\ measure in the case౦ > 23 . [6 marks] (b) Determine the arbitrage-free price of the contingent claim প > )3*3 at time 0 and find the hedging portfolio for প between times 0 and 1. [15 marks] (c) Given that 2 > 3, find the quantit\ of stocks in the hedging portfolio for প between times 1 and 2. [5 marks] (d) Suppose now that the distribution ofহ3 is changed so that ষ )হ3 > 43* > 23 > ষ )হ3 > 23*, with the distribution of হ2 and ౦ > 23 unchanged. Determine whether or not there is an arbitrage opportunit\ in this new two-period model. [4 marks] G14SFM-E1 4 G14SFM-E1 5. (a) i) What does it mean to sa\ that |- Ӓ 1~ is a Brownian motion with drift ౡ and variance ౨3? ii) Define a geometric Brownian motion |- Ӓ 1~ in terms of a Brownian motion. [6 marks] (b) What assumptions on (i) the stock price |- Ӓ 1~, (ii) the interest rate ౦, and (iii) the contingent claim, do we need to make for the Black-Scholes formula to be valid? [4 marks] For the remainder of this question assume that the necessar\ assumptions from (b) hold true. (c) Using the Black-Scholes formula, find the arbitrage-free time-0 price of the contingent claim that pa\s an amount mph ༡ at time ༚, where is the stock price at time . [6 marks] (d) Using a suitable multi-period Binomial model appro[imation, derive the Black-Scholes formula for the time-0 price of a European call option on the stock with strike price ৄ e[piring at time ༚. Hint: You ma\ wish to use in \our answer the e[pression)2 , ȡ౦*ѿৎবস\) ȡ༡ ѿ ৄ*,^ for the time-0 price of a European call option under a multi-period Binomial model; where \ou should define all the terms in this e[pression. [14 marks] G14SFM-E1 EQG
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