MTH319 Financial Engineering 1st SEMESTER 2020/21 FINAL EXAMINATION
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MTH319
Financial Mathematics
1st SEMESTER 2020/21 FINAL EXAMINATION
BSc FINANCIAL MATHEMATICS - Year 4
Financial Engineering
FINAL
Answer FOUR Questions
Q1 (Pricing Measure and Financial Calculus) (25 Marks)
1) Consider the securities model with one risky security and one risk-free security whose discounted values at t=0 and t=1 are given by:
( 1 4)
S* (0) = (1,3), S* (1, 业) = 1(1) 2(3))||| ,
Now there is a claims Y* with the payoffat time t=1:
( 5)
Y* (1, 业) = 3(4))|||
Work out the price of the claim Y* at t=0, e.g., Y*(0). (10 marks)
2) Suppose that the stock price and bank account follow two stochastic processes under a probability measure:
= rdt +adWt
dBt = rdt
t
where dWt presents a Brownian motion process. Now define a new random variable
of Yt = . Apply Ito’s lemma to derive the stochastic process for dYt. (15 marks)
Q2 (Option Prices and Finite-Difference Methods) (25 Marks)
1) Assume that in the continuous setup, St follows a Geometric Brownian process under the probability measure Q:
= rdt+QdWt
where dWt presents a Brownian motion process.
Show that the first two moments (e.g., mean and variance) of the stock price over the time interval of Δ are given by:
E[] = erV and Var[] = e2rV (eQ2V 一 1) (10 marks)
2) Suppose that the annual risk-free rate (r) is continuously compounded. Consider the
time-zero pricing formula for a European call with strike price K and maturity T:
c0 = e一rT EQ [(ST 一 K)+ ]
Work out prob(ST>K) and further apply the finite-difference methods to approximate prob(ST>K). Outline the key steps in the algorithm. (15 marks)
Q3 (Monte Carlo Simulation) (25 Marks)
Suppose that the stock price follows a Geometric Brownian process under the probability measure Q:
= rdt +QdWt , S(0) = S0
= rdt, B(0) = 1,
where dWt presents a Brownian motion process. Consider that a European option on this stock, with strike K and time to maturity T is traded. Its value at time 0 is given by:
C(0) = e rT EQ [(ST K)+ ]
Answer the following questions:
1) Work out the delta-hedge ratio of this option within the Black-Scholes model,
9C(t)
2) Suppose that an investor takes a short trading position in this option (e.g., nc = - 1). Develop a (STATIC) delta-neutral hedging strategy for the option trading position with the share of stock (e.g., nS) and the amount of cash (e.g., nB) together. Outline the algorithm for the Monte Carlo implementation of this strategy. (15 marks)
Q4 (Interest Rate Products and Modelling) (25 Marks)
1) Suppose that a trader has entered into a new interest rate swap with the 1.5-year maturity. Under the terms of this swap, the trader receives the 6-month LIBOR and pays a fixed rate (k) per annual on a notional principal of £10 million. At the moment, the LIBOR rates with continuous compounding for different maturities are attained as follows:
Maturity (Months) LIBOR Rate
6 10.25%
12 10.75%
18 11.25%
What is the swap rate k for this new swap with semiannual payments that is initiated now? Assume that the first payment will be made in six months. (10 marks)
2) Consider the CIR model for the spot rate r(t):
dr (t) = a (b 一 r(t))dt + Q dW (t), r (0) = r0 ,
Outline a Monte Carlo simulation (algorithm) for the discretized process of r(t), based on the control-variates sampling method. (15 marks)
Q5 (Portfolio Management) (25 Marks)
A global equity manager is assigned to select stocks from a universe of large stocks throughout the world. The manager will be evaluated by comparing her returns to the return on the MSCI World Market Portfolio, but she is free to hold stocks from various countries in whatever proportions she finds desirable. Results for a given month are contained in the following table:
Weight in MSCI
Index
Manager's
Weight
Manager's Return in Country
Return of Stock
Index for That
Country
U K |
0.15 |
0.3 |
20% |
12% |
Japan |
0.30 |
0.1 |
15 |
15 |
U.S. |
0.45 |
0.4 |
10 |
14 |
Germany |
0.10 |
0.2 |
5 |
12 |
Answer the following questions:
1) Calculate the total value added of all the manager’s decisions this period. (5 marks)
2) Calculate the value added (or subtracted) by her country allocation decisions. (10 marks)
3) Calculate the value added from her stock selection ability within countries. Confirm that the sum of the contributions to value added from her country allocation plus security selection decisions equals the total over- or underperformance. (10 marks)
2023-01-10