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.
Student Question #9: Hi professor, we can value Euro call with strike price Kexp(-r(T2-T1)) and
expiration T1 and Euro put option with strike K and expiration T2, and then chooser option
is equate
to
sum
of both. However, we have sigma1 from 0 to T1 and sigma2 from T1 to T2. when we value put
option with expiration T2, we do not have sigma for the two
period.
In addition, I have a question about a binomial tree. If there are two sigmas for
different period, the
binomial will not recombine after T1, because u and
d is
related
with
sigma, generally, we build binomial
tree by using constant u and d. Therefore, if we use sigmas
varing
with time, we have to calibrate
nonrecombine
node. It is not easy.
My Responses to Student Question #9: Thanks for your questions. Let me try to help you.
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There are two sigmas for two periods (time 0 to time T1; and time T1 to Time T2). In other words, sigma
is time-varying as you can see in the table
in
page 3 of the term project description. I guess you
understand this.
As I highlighted in the class, you can use the closed-form solution (i.e., Euro call with strike price Kexp(-
r(T2-T1)) and expiration T1 plus Euro put option with strike K and expiration T2) only when sigma is
constant over time and the call strike is the same as the put strike. In task a), sigmas are time-varying for
Clients #1 to #4. (In the case of Client 5, sigma is constant over time.) Therefore, you cannot use the
closed form solution. In task b), you should recommend strike prices for your clients. So, you can even
recommend different strikes for the call and the put. That is an additional reason that you cannot simply
use the closed-form solution. Because you cannot simply use the closed-form solution, you need to use
numerical methods such as Monte-Carlo simulation, lattice, and the finite difference method. This kind of
situation occurs a lot in practice, and that is why those numerical methods are valuable.
For your question about binomial tree, please, carefully read Questions #1 and #2 of the “student question”
document
that I have uploaded to Blackboard. Remember this: whenever you can use conditional
expectation, you should use the conditional expectation. I think I highlighted this several times in our
classes. You can branch your binomial tree from time 0 to time T1 using sigma1. Because the tree is no
longer recombining after time T1, you can "re-branch" your tree from each terminal node at time T1 using
sigma2, if you want. However, you do not have to do so, because at each terminal node at time T1, you
know the value of the chooser option, which
is max
[(European call price at time T1 with underlying
price S(T1), strike K_C, volatility sigma2, and time to expiration (T2-T1)), (European put price at time
T1 with underlying price S(T1), strike K_P, volatility sigma2, and time to expiration (T2-T1))].
I hope these help you