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MATH 11158 : Optimization Methods in Finance Assignment 1

Due: 24 February, 2025, 4pm

Question 1 (Arbitrage (6 marks) ). Suppose we have m scenarios in the arbitrage detection problem with n risky assets S 1 , . . . , Sn and the risk-free asset S 0 . Choose any integer k ∈ {1, . . . , m} and consider the following linear program

State (with proof) a condition in terms of OP Tk that is both necessary and sufficient for type-B arbitrage to exist.

Question 2 (Sports betting (10 marks) ). For an upcoming Ashes test cricket match between England vs. Australia, the odds offered by bookmakers are: England win: 4/11, Aussie win 8/3,

draw: 5/2, tie: 41/3. Does this present an arbitrage opportunity?

Note: In a game of test cricket (which lasts up to 5 days!), either side can win (winner gets 3 points), or the match ends in a tie (if the runs scored are equal) and each team gets 2 points, or the game begins and ends in a draw and neither side wins and each side gets 1 point.

Note: Odds of a/b mean that if you were to bet amount b on an event, if the event happens you get a payoff of a (and additionally your stake b), if the event does not happen, you loose your stake, i.e. cost of investment is b. In case of win the payoff is a + b, in case of loss the payoff is 0.

The interest rate on borrowing cash is 1.5% and you can only place positive amounts as bets.

1. Show that there is no arbitrage of either type. (6 marks)

2. Suppose a new bet can be placed with payoff only when either team wins. Let the odds in this bet for an English win be 6/7 and for an Aussie win be δ/7. Compute the value of δ such that there is still no arbitrage in the problem. (4 marks)

Question 3 (Convexity (10 marks) ). Let X ⊂ R n be a convex set, R(x) be the return on portfolio x ∈ X, and δ be any positive scalar. Consider the following function over X.

Here E denotes the Expectation operation. Argue whether this function is convex or concave or neither.

Question 4 (Quadratic function (14 marks) ). Consider the function f : x ∈ ℜ5 → ℜ defined as

f(x) := −3x1x5 + 6x1x2 + x1x3 − 4x 2 1 + 11x2x3 − 7x2x4 + 2x2x5 + x 2 2 + 5x 2 3 − 8x3x4 + 9x 2 5

This is a homogenous quadratic since it has only degree 2 monomials in x and no linear terms.

1. Determine whether f is convex or concave or neither. (4 marks)

2. Solve the following quadratic program using the MATLAB function quadprog from its Opti mization toolbox (this does not require CVX), and state the optimal value. (4 marks)

3. Write the Lagrangian relaxation vD(λ) for the above quadratic program where the single linear constraint outside X is dualised with a multiplier λ, and solve it for many different values of λ ≥ 0. Plot vD(λ) versus λ and comment on whether you observe any convergence to vP . (6 marks)

Question 5 (Stochastic dominance (15 marks) ). We have three different risky assets S 1 , S2 , S3 . Let x b = (25%, 60%, 15%) be a benchmark portfolio on these three assets. The asset returns in the previous five weeks are

The previous five weeks are likely scenarios for next week with probabilities 0.3, 0.1, 0.2, 0.25, 0.15 respectively. Consider the portfolio x = (30%, 40%, 30%).

1. Find the cumulative distribution functions for returns on x and the benchmark. (6 marks)

2. Determine whether x has first order stochastic dominance over the benchmark. (3 marks)

3. Determine whether x has second order stochastic dominance over the benchmark. (6 marks)

Question 6 (Portfolio Computations (15 marks) ). Suppose you are trying to create an investment portfolio (without short selling) of stocks of the following 10 companies that are listed in the FTSE 100 index, Abrdn (ABDN), AstraZeneca (AZN), Diageo (DGE), HSBC (HSBA), Next (NXT), Prudential (PRU), Rolls-Royce (RR), Tesco (TSCO), Unilever (ULVR), Vodafone (VOD)

Note that these stocks present an asset class diversification since the companies are in different sectors – finance, pharma, beverages, banking, retail, insurance, aerospace, grocery, consumer goods, telecom, respectively.

Download historical data with weekly frequency from 1 Jan 2022 – 31 Dec 2022 for each stock from Yahoo Finance using the steps outlined here https://help.yahoo.com/kb/SLN2311.html.

Use the “Adjusted Closing Price” for calculating return rates.

We are interested in creating investment portfolios that satisfy the following criteria – (i) must not be invested more than 20% in the banking & finance sectors, (ii) investment in retail must be no more than consumer goods and the latter must be no more than the combined investment in beverages and grocery, and (iii) must have at least 10% in the aerospace sector (how else will we reach Mars one day!!)

1. Give the MATLAB code for the optimization model that minimises variance while achieving an expected return at least as much as the expected value of “iShares Core FTSE 100” (ISF.L), which is an ETF that tracks the FTSE 100. (4 marks)

2. Give the MATLAB code for the optimization model that maximises the expected return while beating a benchmark portfolio that is fully diversified between the following ETFs with symbols ISF.L, VUSA.L (Vanguard S&P 500 tracker), and SGLP.L (Invesco Physical Gold). Here, we consider beating in the sense of Second-Order Stochastic Dominance. (4 marks)

3. For question 1, draw the efficient frontier (with risk = standard deviation on x-axis) and the portfolio composition when you also add a constraint on target return, where the target varies in {−2%, , −1.8%, −1.6%, . . . , 8%}. (4 marks)

4. Repeat question 3 using the weekly data from 1 Jan 2023 – 31 Dec 2023 and plot the frontier and compositions. (3 marks)

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MATH 11158 : Optimization Methods in Finance Assignment 1

Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due

MATH 11158 : Optimization Methods in Finance Assignment 1

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