I am currently studying finance independently to take a standardized
exam, and I am getting frustrated with one particular problem that I
cannot solve and whose solution is not given.  I would be most
grateful if anyone could solve this for me:

Consider a two-period binomial model in which u = 13/10, d= 9/10, r=0,
and the initial price of the stock is 100.  Suppose that you have to
replicate the payoff of a European call option with strike 100 and
expiry 2; this will require a certain holding of the risky asset
during the first time period, then the holding of some different
amount during the second period, depending on the outcome of the first
period.  Suppose that the change of the risky asset at time 1 is
costly; buying a unit of the risky asset costs S(1+k) if the share
price is S, and selling a unit raises S/(1+k).  Taking k=1/5, show
that the time-0 cost of replicating the call is 575/36.  Compare this
with a policy which holds 4/5 units of the stock and -64.8 in the bank
through both time periods; show that the latter strategy dominates the
first.

[Note from poster: hence, if the current stock price is S, it goes up
to Su with probability p and falls to Sd with probability (1-p). In
one example in my book, p is given by (1+r-d)/(u-d), but I do not know
if this is true in general]

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Clarification of Question by dubois-ga on 03 Mar 2003 07:36 PST

Based upon other problems with which this one is grouped, I believe
that a solution should not be overly difficult for someone versed in
finance (however, please correct me if I am wrong).  If someone could
please provide an answer to this problem which is still confusing me,
preferably by tomorrow night (March 4), I will gladly leave an
additional tip for any extra time it may take.  Thanks a lot.

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Request for Question Clarification by livioflores-ga on 03 Mar 2003 14:48 PST

I am working hard on this exercise, but I need more info.
Can you clarify the following paragraph?

"Compare this with a policy which holds 4/5 units of the stock and
-64.8 in the bank through both time periods; show that the latter
strategy dominates the
first."

What -64,8 means?

livioflores-ga

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Clarification of Question by dubois-ga on 03 Mar 2003 16:04 PST

Hi Livioflores,

   To be honest, I am not entirely sure how to interpret the "-"; I
typed up the problem word for word as written (and just verified to
make sure that the paragraph in question was correct), but my main
concern is the calculation of the time-0 cost of replicating the call.
 Perhaps my book has a typo; regardless, if you can derive the 575/36
result and the next part of the problem still does not make sense,
then ignore it, and I will be quite willing to accept a solution to
the first part of the problem as an answer.

Thanks a lot.

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Request for Question Clarification by livioflores-ga on 04 Mar 2003 20:45 PST

Hi dubois!!

I am really sorry to say you that, after trying all the ways that I
know and after research for a guideline, my background is not good
enough to solve this problem.
I have worked on this very hard, but the only thing that I can give to
you are some links to pages and files wich have information that may
be useful to you and, I hope this, help you to find a solution.

You must have MS WORD, Powerpoint and Acrobat Reader.

"Securities Transaction Taxes and Financial Markets" by Karl
Habermeier and Andrei A. Kirilenko. Take a look to the paragraph 6.2
-A Simple Example with Transaction Taxes-, I feel that this is very
close to your goal:
http://www.imf.org/external/pubs/ft/staffp/2002/00-00/pdf/haberm.pdf

“Replicating Options and Portfolio Insurance”
http://faculty.cox.smu.edu/~achen/fina6220/OPTION-%20Note%205.doc

"MULTIPERIOD MODELS AND TREES"
http://galton.uchicago.edu/~lalley/Courses/390/Lecture2.pdf

"The binomial model and dynamic hedging strategies"
http://www.cob.ohio-state.edu/fin/829/stulz/book/ch11.pdf

"Numerical Methods for Option Pricing: Binomial and Finite-dierence
Approximations"
http://www.financialengineering.org/thesis/optionpricing.pdf

"An Introduction to Financial Mathematics"
http://www.wis.kuleuven.ac.be/stat/GM31.pdf

"An Introduction to Binomial Trees"
http://www93.homepage.villanova.edu/david.shaffer/Teaching/Derivatives/PowerPoint%20Slides/dm10.ppt

i hope this helps you a at least little.

Regards.
livioflores-ga


PS: This is not an answer, it is only a comment that I post in order
to put in your hands some things that I found in my research for this
question that I cannot solved. You will not be charged for this post.

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Clarification of Question by dubois-ga on 05 Mar 2003 05:25 PST

Dear Livioflores,

    It is exceedingly generous of you to give so much time to this
problem; I will thoroughly go through all the links you gave.  The
fact that you are incapable of solving the problem suggests that it is
more difficult than I initially believed, and hence unlikely to be
representative of the type of question I will see on my exam. 
However, due to the time I have devoted to this problem, if any of the
other researchers wish to try and solve it, I would still be quite
curious to see a solution.
    Thank you again, Livioflores, for all your help.

- dubois