关键词 > MATH4090/7049
MATH4090/7049: Computation in Financial Mathematics Assignment 1
发布时间：2022-08-16
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH4090/7049: Computation in Financial Mathematics Assignment 1
Semester II
Assignment questions
1. (6 marks) Consider a three-period binomial tree with tn = n∆t, n = 0, 1, 2, 3. Let the interest rate r be such that exp(r∆t) = 5/4 and the stock price corresponding to node (n,j) be Sn(j) = 8 × 2j . Thus, the probability of up-movement under the risk-neutral measure is, for all (n,j),
pn(j) = 
= 
.
(a) (3 marks) Consider an‘up-and-out’barrier put option with strike K = 20, expiry date T = 3∆t, barrier H = 15, and barrier monitoring dates T = {t1 ,t2 }. Fill in the table below:
|
node (n,j) |
stock price Sn(j) |
knock-out I(Sn(j) ≥ H,tn ∈ T) |
option value (if no prior knock-out) Cn(j) |
|
(3,3) |
|
|
|
|
(3,1) |
|
|
|
|
(3,-1) |
|
|
|
|
(3,-3) |
|
|
|
|
(2,2) |
|
|
|
|
(2,0) |
|
|
|
|
(2,-2) |
|
|
|
|
(1,1) |
|
|
|
|
(1,-1) |
|
|
|
|
(0,0) |
|
|
|
(b) (3 marks) Consider an American put option with strike K = 20 and expiry date T =
3∆t. Fill in the table below and verify that it is optimal to exercise right away.
|
node (n,j) |
stock price Sn(j) |
exercise value |
hold value |
(unexercised) option value Cn(j) |
|
(3,3) |
|
|
N/A |
|
|
(3,1) |
|
|
N/A |
|
|
(3,-1) |
|
|
N/A |
|
|
(3,-3) |
|
|
N/A |
|
|
(2,2) |
|
|
|
|
|
(2,0) |
|
|
|
|
|
(2,-2) |
|
|
|
|
|
(1,1) |
|
|
|
|
|
(1,-1) |
|
|
|
|
|
(0,0) |
|
|
|
|
|
Programming instructions: This is a non-programming question. You are not required to submit a Matlab script. |
2. (6 marks) Combinations of European options are often used by traders to design strategies that better describe the positions they want to take.
(a) (3 marks) Draw using Matlab the payoff function (as a function of ST ranging from 0 to 200) of a bull spread which involves buying a call option with strike price Kl = 100 and selling another call option with a higher strike price Ku = 120.
(b) (3 marks) Draw using Matlab the payoff function (as a function of ST ranging from 0 to 200) of a strangle which involves buying a call option of higher strike Ku = 120 and a put option of lower strike Kl = 100.
|
Programming instructions: This question is a programming question. Write a single Matlab script file payoff .m that produces a single figure showing both functions for (a) and (b). Save the figure and name it payoff plot .jpg. Submit both payoff .m and payoff plot .jpg. |
3. (8 marks) Let r,σ,T > 0 and X0 ∈ R be fixed. Consider an N-period binary tree model with
N∆t = T. Define
N
XT = X0 +工 θk
k=1
where θ 1 ,θ2 , . . . are i.i.d. with
θ = { 
σ(^)
w(w)i(i)t(t)h(h) p(p)r(r)o(o)b(b).(.) 1(p) − p
where
p = 
(1 + r −12σσ 2 ^∆t).
Define also
N
T = X0 +工 ϕk
k=1
with
ϕk = (r − 
σ 2 )∆t + σ ^∆tYk, k = 1, 2, . . .
where Y1,Y2 , . . . are i.i.d. with
Y = { 
1,
with prob. 
,
with prob. 
.
(a) (2 marks) Show that EXT = E
T .
(b) (2 marks) Show that Var(XT) = Var(
T ) + O(∆t).
(c) (4 marks) Show using the central limit theorem that the distribution of 
T converges to Normal(X0 + (r − σ 2 /2)T,σ2T) as ∆t → 0.
|
Programming instructions: This is a non-programming question. |
4. (5 marks) In this question, you are asked to carry out experiments with a European-style strangle using the no-arbitrage binomial tree method discussed in L1.
The payoff of the option is given in Problem 2(b) with T = 1, σ = 0.3, S0 = 100, r = 0.02, Kl = 100 and Ku = 120.
In the numerical experiments, for the number of time periods N in the binomial tree, use N = Nk = 50 × 2k , k = 0, 1, . . . 8. We further denote by C0(0),k ≡ C0(0)(∆t)k , k = 0, 1, . . . 8, the numerical approximations to the exact (no-arbitrage) strangle value C(S0(0),t0 ), noting S0(0) = S0
and t0 = 0, using the tree method with the timestep size (∆t)k = T/Nk .
Fill in the following table.
|
k |
∆t |
value |
changes |
|
0 1 2
8 |
(∆t)0 = T/N0 (∆t)1 = T/N1 (∆t)2 = T/N2
... |
C0(0) ,0 C0(0) ,1 C0(0) ,2
... |
... |
|
C(S0,t0 ): |
|||
