Student Question #1: Dear Professor Kang, I was wondering if you could give me some guidance on
Problem 1 of the Term Project. On Task a), we should use Monte Carlo to price the chooser option. I
decided to use conditional Monte Carlo. Thus, the price of the chooser option can be computed as:
exp(-r*T1)*max(calls, puts)
Where r is the risk-free rate (0.0250), and calls and puts are the values of the call option and put option at
expiration by using Black-Scholes. I realized that, when using Black Scholes, the values of the call and
put options are independent of the drift (mu). Can the drift of each client at time T1<>
in this case?
Thanks in advance for your help.
My Responses to Student Question #1: It is good that you are working on the term project. Thanks for
the question. Question 1 is all about derivative pricing. When you price a derivative security, you should
use the risk-neutral measurer where the drift is r.
Subjective drift
of each client is ignored. Later when you
do risk analysis, you need to respect those subjective mu’s, though. I hope this helps.
Student Question #2: I am working on Task a). When I use the “smart lattice” version of binomial lattice,
should I branch another tree at time T1?
My Responses to Student Question #2: Thanks for the question. If you want, you can do so, but you do
not have to. At time T1, the conditional expectations of the payoffs at T2
is
known. As I highlighted
many times in our class, you should use the conditional expectations whenever possible. So, you can use
them at the end of the tree ending at T1. I hope this helps.
Student Question #3: In Task a), I am working on the finite difference method part. What should be the
upper and lower boundary conditions?
My Responses to Student Question #3: In principle, you need to tell me what should be the upper and lower boundary conditions.
Without violating the ground rules of the term project, I can still give you some guidance. Please, review
the lower boundary conditions of European put option. In addition, please, review the assignment
solutions of the FDM to figure out how we set the upper boundary conditions for exotic options. The
solution for Assignment #5 will be available on Dec 1, 2017, but I can give you some ideas now: If
S_Max is violated, it is assumed that the time T value of S is S_MAX. Then you can calculate the time T
value
of f. Then, you can discount it back to time t.
I hope this helps.
Student Question #4: Hi, Prof. Kang. To address Task b), I tried the antithetic variables technique, but it
does not work. Could you help me? Thanks in advance.
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My Responses to Student Question #4: I should not tell you what variance reduction technique you
should use. You should tell me what variance reduction technique should be used for the “chooser
options”
and
support
evidence for your claim.
We have learned three variance reduction techniques: 1) Antithetic variables technique; 2) control
variable technique; and
3) the conditional Monte-Carlo. First of all, notice that we are already implicitly
using the conditional Monte-Carlo. With that being said, let me reiterate several important features of the
antithetic variables technique and the control variate technique:
Antithetic Variables Technique: To use antithetic variables technique, one should first determine
if the payoff of
derivative security
is monotone or not (“Monotonicity Condition”). If the
monotonicity condition is not met, there is no guarantee that the antithetic variables technique
reduces variance or not. I cannot tell you whether the “chooser
option” satisfies the monotonicity
condition or not. I cannot tell you how to double-check if the “chooser
option” satisfies the
monotonicity condition or not. You should tell me these as necessary.
Control Variable Technique: You should find a control variable which is highlight correlated with
the discounted payoff of the “chooser
option”. In addition, the expectation of a control variable
should be known in advance. You cannot tell you which control variate satisfies these two
conditions. You should find one and report to me.
I hope these help.
Student Question #5: I used two methods to compute the discounted payoff. The one is considering the
chooser option as one call option with strike price K*exp(-r*(t2-t1)) and expiration time
t1 ,
and one put
option with strike K and expiration time t2. The other is discounted payoff = exp(-r*t1)*max(call with
strike K, put with strike K). I had absolutely different results by the two methods. Could you tell me
which one is correct? And the strike of call and the strike of put should be equal or not in the project?
My Responses to Student Question #5:
Thanks for the question. Your first method is actually the closed-form solution of the chooser option. As I
mentioned in the class, the closed-form solution is valid only when 1) the call strike and the put strike are
the same; and
2) the volatility between time 0 to T1 is the same as the volatility between time T1 and T2.
If one of these two conditions are not met, you cannot use the closed-form
solution,
and need to use
numerical methods such as Monte-Carlo simulation, lattice, and the FDM. When you use these numerical
methods, you should use something like your second method.
The answer to your second question depends on
task. In task a), X_c=X_p=50. In task c), you need to
propose strike prices for each client. So, it is up to you if you equate X_c=X_p=50 or not.
I hope these help.
Student Question #6: Is the “chooser
option” price computed by Monte-Carlo very different from the
price computed by the FDM?
My Responses to Student Question #6: If you correctly implement the Monte-Carlo and the FDM, the
two
method
should give you very close answers. If not, one of these two methods is incorrectly
implemented. You should keep your debugging until these two methods give similar results.
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Two additional advice may help:
1) If the FDM result falls inside the 95% confidence interval of Monte-Carlo method, it is a good
news. If it does not, try to change the random seed of the Monte-Carlo method.
To
my experience,
these two methods can be even closer than the confidence interval of the Monte-Carlo suggests.
2) If the call strike and the put strike are the same and the volatility does not change over time, you
can also use the closed-form solution to figure out whether the Monte-Carlo method and/or the
FDM is wrong.
I hope these help.
Student Question #7: Dear professor, I have a problem with part a2. Does the lattice start at T1 or T0?
My Responses to Student Question #7: Because the goal is to calculate the "chooser
option" price at
time 0, you should start your lattice at time 0.
Student Question #8: Hi, Prof, I have a question about profit-risk analysis. How do I decide the type of
the option at t1 when I do
profit-risk analysis? Because I don't know the cash flow at T2 if the type of the
option is not decided at t1.
My Responses to Student Question #8: Good question.
The decision should be made at T1 by a client. The client should choose a call with a predetermined strike
price or a put with a predetermined strike price.
What should be the criterion for making such a choice? You should tell me and justify your criterion.
There is no single right answer for the decision but there are at least two ways:
1) Comparing the two conditional expectations (put and call) under
risk neutral
measure at time T1.
2) Comparing the two conditional expectations (put and call) under
subjective physical measure
(mu2 and
sigma2) at time T1.
Again, a job of
quantitative analyst
is to make a modeling decision and justify the decision.
I hope these help.
Question 1 (20 Marks)
Part (a) (12 marks)
Basic Blinds Ltd issued a prospectus on 1 August 2015 to raise additional capital to expand their business. The prospectus invited applications for 1,500,000 ordinary shares at an issue price of $2.50 each.
On application for these shares, the applicant was to pay $2.00 per share.
A
second instalment of $0.50 was payable on 1 March 2016. According to the prospectus, any excess application money was to be refunded in full.
By the prospectus closing date of 1 September 2015, 1,600,000 applications for ordinary shares had been received. Only application money was sent in with all applications.
On 1 October 2015, the company’s directors decided that only the 1,500,000 shares would be allotted in proportion to their applications. No additional shares were issued.
By 1 March 2016, all the second instalment payments had been made, with the exception of one holder of 5,000 ordinary shares.
Required:
Prepare, in General Journal format, entries to record:
(i) the application for
and
issue of the shares, including any refund;
(ii) the payment of the second instalment
Question 2 (20 Marks)
Prepare, in General Journal format, entries to record the following situations.
Part (a) (6 Marks)
STRETCH Ltd intends to raise funds to launch their product in a new state. STRETCH Ltd decided to issue 8% Debentures. They issued 12,000 Debentures on 1 November 2015, each worth $100.
Question 3 (20 Marks)
Part (a) (14 Marks)
Small Ltd were approached by the owners of Medium Supplies to join forces and created a bigger company called Large Supplies Ltd. It was agreed that Small Ltd would acquire the assets and liabilities of Medium for $1,900,000 and create a new company on 1 July 2015. The acquisition would be funded through the issue of new shares in Small Ltd to the vendors. Prior to the acquisition and conversion, Medium Supplies’ trial balance was:
Question 5 (20 Marks)
Part (a) (8 marks)
For each of the following assets or liabilities:
• calculate the carrying value of the asset or liability
• calculate the tax base of the asset or liability
• calculate the temporary difference between the carrying amount and the tax base
• indicate whether the temporary difference will create a deferred tax asset or a deferred tax liability
(i) Interest income of $875 relating to next year had been received by 30 June. Interest income is taxed on receipt.
(ii) At the balance date, the Accounts Receivable general ledger account has a balance of $21,300 and the Allowance for Doubtful Debts account has a balance of $1,150.
(iii) The business had an opening balance in the Provision for long service leave account of $9,000. During the year, long service leave paid totalled $3,500 (this was paid from the provision account) and a further transfer to the provision of $6,000 was made on 30 June.
(iv) The business purchased factory equipment for $350,000 on 1 July last year. The accounting depreciation rate is 30%; but the tax depreciation rate is 25%.
Question 6 (4 Marks)
List 4 benefits or advantages of using a computerised accounting package rather than a manual recording system.