MATH3090/7039: Financial mathematics Assignment 2 Semester I 2023

Due Thursday April 27 5pm   Weight 10%

MATH3090 total marks   24 marks

MATH7039 total marks   30 marks

Submission:

• Submit onto Blackboard softcopy (i.e. scanned copy) of (i) your assignment solutions, as well as (ii) Matlab code for Problem 1 (c and d). 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 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.  (10 marks)  (Yield curve and swap pricing)

Assume that you observe the following yield curve for government’s coupon paying bonds.

• There are a total of 30 bonds.

• For the k-th bond, k = 1, . . . , 30, the maturity is k years.

• The face value is F = $100, 000 and the coupon rate for the k-th bond, k = 1, . . . , 30, is c = 5%. Let C = cF .

• The prices of the bonds (P (k), k = 1, 2, . . . , 30) are given in Table 1.

• 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)  Compute y0, 1 . y0, 1  is

y0, 1  =  log ( )

b.  (1 mark)  Show that

(                                           )

Table 1: Bond prices

Table 2: Table for Question 1 (c)

c.  (3 marks)  Implement in a Matlab program to compute spot zero-coupon bond yield curve y0,k  and the implied one-year forward rates yk−1,k .

Submit Table 2 filled with computed values.

d.  (4 marks)  Suppose you enter into a 30-year vanilla fixed-for-floating swap on a notional principal of $1,000,000 where you pay the fixed rate of 7.5% and the counter-party pays the yield curve plus 1%.

Code in Matlab a program to compute the swap value. Submit a table of results, similar to the table on L5.15.

e.  (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 neeed to develop a new code).

2.  (8 marks)  Consider

Bτ(y) = Ct(1 + y)τ −t ,     0 ≤ τ ≤ T,    y > 0.

This is the time-τ -value of the wealth if you hold a bond with cashflow (Ct;t = 1, 2, . . . , T) until time τ and sell it, assuming the YTM stays at y . Answer the following.

a.  (3 marks)  Derive a second-order Taylor approximation of Bτ(y+∆y)   Bτ(y) (polynomial in ∆y). You do not need to simplify the expression.

Use the Taylor approximation, we can get an approximation of Bτ(y + ∆y)     Bτ(y) is Bτ(y + ∆y)     Bτ(y) = B(y)∆y + B(y)

Where B(y) and B(y) are

B(y) =  Ct(τ     t)(1 + y)τ −t−1

B(y) =  Ct(τ — t)(τ — t — 1)(1 + y)τ −t−2

Hence the second-order Taylor approximation is

 Ct(τ — t)(1 + y)τ −t−1∆y +  Ct(τ — t)(τ — t — 1)(1 + y)τ −t−2

b.  (3 marks)  Suppose τ  =  |D| (the absolute value of the duration when the YTM is y). See L4.41. Show that the formula derived in (a) reduces to

B|D|(y + ∆y) — B|D|(y) = —  (∆y)2  t(|D| — t)Ct(1 + y)|D|−t−2 .

Now set |D| = τ ,we can get

B D| (y) = — |D|B|D|(y)/(1 + y) = — |D|  Ct(1 + y)|D|−t−1 B | (y) =  Ct(|D| — t)(|D| — t — 1)(1 + y)τ −t−2

We can get

c.  (2 marks)  Explain why it makes sense to set τ = |D| for immunisation.

3.  (6 marks)  (Bounds for put/call options)

a.  (2 marks)  Let P0  be the time-0 price of a K-strike and T-expiry European put option on a non-dividend-paying stock S .  Also, let Z0  be the time-0 value of a zero-coupon bond maturing at time T . Using the same argument as that in L6.25, show

(KZ0  — S0)+  ≤ P0  ≤ KZ0 .                                                 (1)

b.  (2 marks)  Show (1) using put-call parity and the bound on the call option price as in L6.25.

c.  (2 marks)  Let 0 < K1  < K2 . Let P0(K1) and P0(K2) respectively be the time-0 price of K1- and K2-strike European put option written on the same non-dividend-paying stock,

and with the same expiry. Plot (by hand) ST  '! PT(K2) — PT(K1) and show that 0 ≤ P0(K2) — P0(K1) ≤ (K2  — K1)Z0 .

Assignment questions - MATH7039 students only

4.  (6 marks)  Given a stock whose time-t price is St, consider a derivative that pays ST(2) at maturity T (the writer pays ST(2)  to the holder; the holder pays nothing to the writer). We assume that there is also a 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

S0(2)

C0  

Hint: consider a tangent line of x '! x2 .

c.  (1 marks)  Suppose ST  only takes values on [0, 1] (i.e., P(0 ≤ ST  ≤ 1) = 1).  Show that C0  ≤ min(S0, Z0 ).