CH6 , Equivalent martingale measure (EMM) => no arbitrage

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CH6 , Equivalent martingale measure (EMM) => no arbitrage

Postby StephQ » Tue May 06, 2008 11:02 am

I have some questions on CH6, and they are all related to the obj of this thread.

1)
on pag 138 you prove that EMM => no arbitrage. However, all the reasoning (pag 137-138, starting from no interest rates, adding them and finally considering equivalent measures) relates to the case of discrete time with only two possible times: 0 (now) and T (end).
The distribution of the assets (discrete or cont.) does not matter and it is a static setting.

Hence we can not apply this result to the case of the multi-step binomial tree, where one can use trading strategies: dynamic setting (the arb free unique price can be enforced with the arguments of CH3 anyway).
You actually make a point about this when at the beginning of CH6 when you write that we should impose the mart property to hold at all times, or only at zero but for all the self financing (SF) portfolios.

So my understanding is that we should prove that if the mart property holds (at all times) => every SF portfolio is not arbitrageable (i.e. the mart property holds for it at zero).

And the point is that we exclude non-SF portfolios (sort of restriction to endogenous variables), right ?

2)
When we move to the continuous setting we have the same problem.
We should prove that mart property in cont time => SF portfolio is not arbitrageable (i.e. the mart property holds for it at zero).

I have proved this in the B&S model with constant int rates. It should generalise easily to the deterministic case, but it is not clear to me how to generalise is to the stochastic case and so I have a problem with the change of numeraire argument at pag 157.

3)
If EMM => no arbitrage, and we trust that also the converse is true, then 6.69 holds and enforces the price of any option, even if the payoff depends on the whole path of S right?

We can enforce the price also via replication (completeness).
However, it seems to me that the Hedging section 6.10 (that actually gives the completeness of the B&S continuous settings) refers to an option C(S_t,t) that depends on (S_t,t) only. So only European options (and if it is the case then we have not proved that the mkt is complete).

Please correct me where I AM wrong :)

StephQ
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Postby mj » Sun May 11, 2008 10:19 pm

re 1, the point is that an integral against a martingale is a martingale so
self-financing trading strategies in martingales are martingales.

re 2, numeraire change is really a shift in viewpoint. A market is simply the right to change an asset for another. If we take an asset as numeraire then its value is always 1 in our new coordinates. This means that the same arguments work.

re 3. ill have to check what i actually wrote
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