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MATH3090/7039: Financial mathematics

创建日期：2024-06-14 14:32:54

所属分类：数学mathematics

标签： MATH3090

Financial mathematics

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MATH3090/7039: Financial mathematics

Assignment
Submission:
Submit onto Blackboard softcopy (i.e. scanned copy) of (i) your assignment solutions, as well
as (ii) Matlab/Python code for Problem 2. Hardcopies are not required.
Include all your answers, numerical outputs, figures, tables and comments as required into
one single PDF file.
You also need to upload all Matlab/Python files onto Blackboard.
General coding instructions:
You are allowed to reuse any code provided/developed in lectures and tutorials.
Notation: “Lx.y” refers to [Lecture x, Slide y]
Assignment questions - all students
1. (3 marks) You have just invested in a 3-year coupon paying bond with 8% semi-annual coupons
and a face value F = $100, 000. Suppose the coupon-paying bond yield curve is flat at 9%.
a. (1 mark) Calculate the present value and the (absolute value of) duration |D| of the
bond.
b. (1 mark) Now calculate the value of the bond in |D| years time.
c. (1 mark) Suppose that, immediately after buying the bond, the yield curve shifted up
to be flat at 10%. Now calculate the value of the bond again in |D| years time under
the new yield curve (don’t calculate D again). Compare your answer with what you
obtained in (b).
You are welcome to reuse Excel sheets provided in class/tutorial. You are not required to
submit Excel files.
2. (7 marks) Assume that you observe the following yield curve for government’s coupon paying
bonds.
There are a total of 20 bonds.
For the k-th bond, k = 1, . . . , 20, the maturity is k years.
The face value is F = $100, 000 and the coupon rate for the k-th bond, k = 1, . . . , 20, is
c = 4%. Let C = cF .
MATH 3090/7039 – 1 – Kazutoshi Yamazaki
– Assignment 2 –
The prices of the bonds (P (k), k = 1, 2, . . . , 20) are given by
[P (1), P (2), . . . , P (20)]
= [99412, 97339, 94983, 94801, 94699, 94454, 93701, 93674, 93076, 92814,
91959, 91664, 87384, 87329, 86576, 84697, 82642, 82350, 82207, 81725].
Denote by y0,k the spot zero-coupon bond yield curve, and by yk−1,k the implied one-year
forward rates.
Assume that all the coupon payments are made annually. Use continuous compounding.
a. (1 mark) Show that
y0,k =
1
k
log
(
C + F
P (k)− C∑k−1j=1 e−y0,j×j
)
, 1 ≤ k ≤ 20.
b. (2 marks) Implement a Matlab/Python program to compute spot zero-coupon bond
yield curve y0,k and the implied one-year forward rates yk−1,k. Submit Table 1 filled
with computed values.
Table 1: Table for Question 2 (b)
period k spot y0,k forward yk−1,k
1 . . . . . .
2 . . . . . .
...
...
...
20 . . . . . .
c. (3 marks) Suppose you enter into a 20-year vanilla fixed-for-floating swap on a notional
principal of $1,000,000 where you pay the fixed rate of 6.5% and the counter-party pays
the yield curve plus 1%.
Code in Matlab/Python a program to compute the swap value. Submit a table of results,
similar to the table on L5.15.
d. (1 marks) Test with different fixed rates and provide a better approximation of the swap
rate so that the swap value is near zero (you do not need to develop a new code).
3. (6 marks) Assume annual time periods, T = 3, a binomial model of the yield curve, and
y0,1 = 2%. Suppose over the whole forward rate lattice that the next period’s forward rates
can either go up by a factor of u = 1.3 with a probability of p = 60% or down by a factor of
d = 0.9. (For example y(u) = y0,1× u = 0.02× 1.3 = 0.026 or 2.6%, y(uu) = y0,1× u× u and
so on.) Use discrete compounding.
a. (4 marks) Construct the forward rate lattice and the zero coupon bond yield curve y0,2
and y0,3.
b. (2 marks) Construct the 1-period forward rates y1,2 and y2,3, which are embedded in
this zero coupon bond yield curve (we already have y0,1).
4. (3 marks) Consider the payoff at maturity T in Figure 1. Show how to construct this payoff
using European calls with the same maturity only (you can use any combination of European
calls with any strike price). You must state long/short, strike prices as well as the number
of units. In addition, express the current value of the (replicating) portfolio in terms of the
current prices of strike-K European calls C0(K), K > 0.
MATH 3090/7039 – 2 – Kazutoshi Yamazaki
– Assignment 2 –
0 2 4 6 8 10 12 14 16
ST
0
2
4
6
8
10
12
Pa
yo
ff
Figure 1: Payoff diagram.
5. (5 marks) Given a stock whose time-t price is St, consider a derivative that pays e
ST at
maturity T (the writer pays eST to the holder; the holder pays nothing to the writer). We
assume that there is also a (risk-free) zero-coupon bond with maturity T and face value 1,
whose time-0 price is Z0. Let C0 be the arbitrage-free time-0 price of the derivative. Answer
the following.
a. (2 marks) Suppose ST can take any positive value with a strictly positive probability
(under the physical probability measure P), and hence P(ST > M) > 0 for any M > 0.
Show that the considered derivative cannot be super-replicated if only the stock and
bond are available in the market.
b. (3 marks) Show that
C0 ≥ e
S0
Z0Z0.
Assignment questions - MATH7039 students only
6. (3 marks) A binary call option with strike K > 0 and maturity T > 0 pays $1 if the terminal
stock price satisfies ST ≥ K and it pays nothing otherwise. Similarly a binary put option
with strike K > 0 and maturity T > 0 pays $1 if ST ≤ K and pays nothing otherwise.
Consider the payoff in Figure 2. Show how to construct this payoff using European calls,
European puts, binary calls and binary puts only (you can use any combination of these
options with any strike price with the same maturity T ). You must state long/short, strike
prices as well as the number of units.

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