The returns per annum on two securities are denoted by R1 and R2 , and are modelled as follows:

Question 1 The returns per annum on two securities are denoted by R1 and R2 , and are modelled as follows:   

R1 ∼ N 0.1, 0.22 , R2 ∼ N 0.05, 0.042 , Corr(R1 , R2 ) = 0.5 where Corr(R1 , R2 ) denotes the correlation between R1 and R2 .

(i) Suppose that $1000 are invested in the minimum variance portfolio formed from the two securities. Calculate the composition of the minimum variance portfolio and state how much is invested in each security.

 Next, suppose that, at the beginning of the year, the balance on an investment account is $1000. The account is invested in securities 1 and 2 only, and short-selling is permitted. Suppose also that, at the end of the year, a random sum of $1000 × (1 + Q) is withdrawn from the investment account. The random variable Q is modelled as follows: Q ∼ N (0.08, 0.12 ), Corr(Q, R1 ) = 0.8, Corr(Q, R2 ) = 0.2 If x is the portfolio weight for security 1, then the end-of-year account balance is: W1 = 1000(1 + xR1 + (1 − x)R2 ) − 1000(1 + Q)

 (ii) Calculate the composition of the portfolio such that the end-of-year balance on the investment account has the least variance.

(iii) Let m(x) = E[W1 ] and v(x) = V ar[W1 ].

(a) Derive expressions for m(x) and v(x).

(b) Let x∗ be the weight of security 1 invested in the portfolio which maximises the probability of a positive end-of-year balance on the investment account.

Show that: m(x∗ )v 0 (x∗ ) = 2m0 (x∗ )v(x∗ )

Question 2 (i) (a) State the no-arbitrage principle.

(b) Explain why only a few arbitrageurs are required in a market to enforce the no-arbitrage condition.

(ii) The returns on securities in a market are modelled using a 2-factor model: Ri = ai + bi1 I1 + bi2 I2 + ci The following data, concerning three securities, are known: Security i Ri bi1 bi2 1 12% 0.5 1.2 2 8% 1.0 0.8 3 9% 0.7 -1.0

(a) Derive an equation for the arbitrage pricing plane in Ri − bi1 − bi2 space.

 (b) A fourth security exists with R4 = a, b41 = b, b42 = c Explain carefully why an opportunity for arbitrage arises.

 (c) Construct an arbitrage portfolio from securities 1, 2, 3 and 4 (stating relevant proportions) to exploit this arbitrage portfolio.

Group Group Group Group Group Group Group Group Group 20 21 22 23 24 25 26 27 28 a = 10%, b = 0.8, c = −0.2 a = 10%, b = 0.8, c = −0.2 a = 10%, b = 0.8, c = −0.2 a = 9%, b = 0.8, c = −0.2 a = 9%, b = 0.8, c = −0.2 a = 9%, b = 0.8, c = −0.2 a = 8%, b = 0.8, c = −0.2 a = 8%, b = 0.8, c = −0.2 a = 8%, b = 0.8, c = −0.2 Group Group Group Group Group Group Group Group Group Group 29 30 31 32 33 34 35 36 37 38 Table 1: Parameter Values. a = 10%, b = 0.9, c = −0.2 a = 10%, b = 0.9, c = −0.2 a = 10%, b = 0.9, c = −0.2 a = 12%, b = 0.7, c = −0.2 a = 12%, b = 0.7, c = −0.2 a = 12%, b = 0.7, c = −0.2 a = 11%, b = 0.8, c = −0.2 a = 11%, b = 0.8, c = −0.2 a = 11%, b = 0.8, c = −0.2 a = 10%, b = 0.8, c = −0.1

Question 3 (i) A model for the price Pt of a share is: Pt = P0 exp {at + bBt } where Bt denotes the standard Brownian motion and a, b are constants.

(a) Calculate the expected share price at time t given that its price is 115 pence at time 2.

(b) Calculate the probability that at time t the share price is greater than 118 pence, given that it is 115 pence at time 2.

(c) Evaluate:

[1] 3

 1 × E P2 |P1 P13

 (ii) Outline the advantages of the model in part (i) above compared to the following alternative model:

 Pt = P0 (1 + at + bBt )

 Group 20 a = 0.05, b = 0.2, t = 4 Group 29 a = 0.06, b = 0.2, t = 7 Group 21 a = 0.05, b = 0.1, t = 4 Group 30 a = 0.06, b = 0.1, t = 7 Group 22 a = 0.05, b = 0.15, t = 4 Group 31 a = 0.06, b = 0.15, t = 7 Group 23 a = 0.04, b = 0.2, t = 5 Group 32 a = 0.07, b = 0.2, t = 8 Group 24 a = 0.04, b = 0.1, t = 5 Group 33 a = 0.07, b = 0.1, t = 8 Group 25 a = 0.04, b = 0.15, t = 5 Group 34 a = 0.07, b = 0.15, t = 8 Group 26 a = 0.03, b = 0.2, t = 6 Group 35 a = 0.08, b = 0.2, t = 9 Group 27 a = 0.03, b = 0.1, t = 6 Group 36 a = 0.08, b = 0.1, t = 9 Group 28 a = 0.03, b = 0.15, t = 6 Group 37 a = 0.08, b = 0.15, t = 9 Group 38 a = 0.02, b = 0.2, t = 10

Table 2: Parameter Values.

Question 4 (i) Let {Bt , t ≥ 0} denote the standard Brownian motion and let Mt := max0≤s≤t Bt . Show that the random variables |Bt |, Mt and Mt − Bt have the same distribution.

(ii) Let {Bt , t ≥ 0} denote the standard Brownian motion.

(a) For any 0 ≤ s < t, show that the joint distribution of (Bs , Bt ) is a bivariate normal distribution and determine the mean vector µ and covariance matrix Σ of this bivariate normal distribution.

(b) Find a matrix ? A=? ass ast ats att ? ? so that [Z1 , Z2 ] defined as follows has a standard bivariate normal distribution. ? ? ? ? B Z ? 1? = A ? s? Bt Z2

(iii) If the Brownian motion {Bt , t ≥ 0} has natural filtration denoted by Ft then

(a) Compute E[Bt4 |Fs ] for t > s ≥ 0

(b) Show that {Mt , t ≥ 0} such that Mt = Bt4 −6tBt2 +3t2 is a martingale adapted to the filtration Ft .

Question 5 Suppose that the standard Brownian motion {Bt } is sampled at discrete points in time: t0 , t1 , t2 , . . . . The points are equi-distant, such that t1 − t0 = t2 − t1 = · · · = h, where h is a constant. Define: ej = Bt and ?B ej = B ej − B ej−1 B j e1 − B e2 .

 (i) State the distribution, mean and variance of B

(ii) Let Yn = Pn j=1 ej−1 ?B ej . Show that {Yn } is a martingale with respect to the B en . (Note: You do not need to show that E|Yn | < ∞.) filtration of B

(iii) Let Xn = Pn ej ?B ej . Derive an expression for E[Xn ], and hence state whether B en . {Xn} is a martingale with respect to the filtration of B j=1

Question 6 You are considering investing in a project in the food and beverages industry. This can involve any product of your choice. After some investigation, you decide that the market is liquid enough and that the risk of the project you want to invest in can be explained by the risk of a similar (homogeneous) product or a combination of such products that are already traded in the market. In order to determine the fair rate of return for your project, you decide to put together a two-factor model. As discussed in the lectures, in terms of explanatory variables you are free to use a wide range including:

i. market indices: for whole market or country-index or industry index.

 ii. macroeconomic factors: GDP-growth, unemployment factor, inflation, etc.

iii. fundamental company-specific factors: dividend yield, company size, earnings, etc. When answering this question, make sure that you:

(1) State the assumptions of your model and define clearly all the associated notation.

(2) Use empirical data over a sufficiently long period of time (e.g. five years) for the explanatory variables.

 (3) Use Excel or any other software of your choice to estimate the values for the parameters of the regression model as well as the equation of the regression model.

 (4) Give a brief intuitive interpretation of the regression plane.

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