[944] HW6 - Q3 cand Q2

[David Itkin]

Hi Lucas,

Q3c) -- when deriving the PDE in the previous parts you are showing that
the price of the security must satisfy the PDE + BC's + FC. In part c) you
are essentially asked to show the converse -- that is if you have a
solution to the PDE then it must be the AFP of the option. This is
analogous to what is done in theorem 8.2 for the European Call.

Q2) -- When you look at the final answer you get you should notice that [image:
X(T) \ne 0] -- in fact you can determine a sign for [image: X(T)] depending
on the relationship of [image: \sigma_2] and [image: \sigma_1]. Furthermore
you should notice that [image: X(T)] is not stochastic (i.e. it's
deterministic/risk-free) so that you always get this value for your
portfolio at maturity. Of course if [image: \sigma_1 = \sigma_2]  then you
get back [image: X(T) = 0] which makes sense since then the call was
appropriately priced to begin with.

What this shows is that in the case when [image: \sigma_1 \ne \sigma_2]
following the delta hedging rule will always yield a positive (or negative,
depending on relationship of [image: \sigma_2] and [image: \sigma_1])
portfolio value and since we started off with 0 (i.e. we bought the call,
shorted [image: \Delta] shares and put/borrowed the remainder in the money
market account) -- this is precisely an arbitrage opportunity. This doesn't
contradict anything since [image: \sigma_1 \ne \sigma_2] implies that the
call was mispriced and so we actually expect there to be arbitrage.

Indeed in this case the portfolio value will depend on the stock price path
taken, so it will not just depend on [image: S(T)] and so wouldn't be
Markovian.

Hope this helps,

David

On Tue, Dec 5, 2017 at 11:50 AM, Lucas Duarte Bahia <lduarteb at andrew.cmu.edu
> wrote:

> Hi,
>
> Q3 -c) Is it enough to say that I derived the PDE assuming dX(t) = dc, and
> by uniqueness of semi-martigale we get that  X=c?
>
> Q2)
> I saw an email earlier and the question itself saying that there is an
> arbitrage and that the option was priced incorrectly.
>
> I don't see how you have an arbitrage if at the time you are creating the
> hedge you don't know the value of the future volatility, and you have a
> non-zero probability of loosing money. I also don't see how the price of
> the call is incorrect, for me it is correct but the future was different
> than what was expected so the position generated a profit or loss.
>
> Also I noticed that we have the final of the portfolio as an integral
> involving among other things the the stock price and gamma. Does that mean
> that our wealth value (long call short replicating portfolio) will not be a
> markov process?
>
> Best Regards,
> *Lucas Duarte Bahia*
> MS. Computational Finance Student
> Carnegie Mellon University
> Telephone: (412) 378-1892
>
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