Equivalent martingale measure vs. risk neutral measure

This forum is to discuss the book "the concepts and practice of mathematical finance" by Mark Joshi.

Equivalent martingale measure vs. risk neutral measure

Postby Anti » Wed Sep 19, 2012 12:03 am

Maybe this is a simple question, but any help would be appreciated!

I've seen both in this book and other places that the risk neutral measure (with respect to a numeraire) is defined as an equivalent measure (to the real world one) such that ALL tradeable assets (discounted by the numeraire) are martingales. Typically then one finds the measure which makes for example the stock/bond (or the bond/stock in the stock numeraire) a martingale and then says that the option/bond (option/stock) is also a martingale and prices this way.

Why does finding an equivalent martingale measure for only one process imply ALL other tradeable discounted processes are martingales with respect to this measure?!

If we have abstractly the existence of a risk neutral measure in the above sense (first fundamental theorem of asset pricing?), i.e. one that makes all discounted tradeables martingales, and if we're in a complete market, then by uniqueness finding the equivalent martingale measure for one process (like stock/bond) is sufficient to know that also the option/bond process is a martingale under this measure.

However, it seems like the same tactic is used in the incomplete models too. For example in the jump diffusion chapter, the stochastic process for the stock is written down (with the Poisson jumps), Girsanov's theorem and the martingale classification theorem are invoked to find an equivalent martingale measure for the stock/bond process, but then the option/bond process is also assumed to be a martingale in this measure and priced via discounted expectation! I understand that risk neutral measures are not unique in this setting (i.e. incomplete), why does finding an equivalent martingale measure for the single stock/bond process give us an equivalent martingale measure for the processes of other tradeables? i.e. Why does the equivalent martingale measure for only the stock/bond process work jointly for the stock/bond and option/bond processes?

Thanks for any help and sorry if I'm misunderstanding something fundamental!
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Re: Equivalent martingale measure vs. risk neutral measure

Postby Anti » Wed Sep 19, 2012 1:05 pm

Hmm, upon further thinking I realize that I don't understand the first fundamental theorem of asset pricing precisely and that zeroing out the drift of a fixed drift stock/bond process via Girsanov leads to a *unique* measure in which the stock/bond process is a martingale. I believe that the reasoning (say in jump diffusion case) is as follows:

Assume no arbitrage, first fundamental theorem implies that there exists a risk neutral measure, i.e. an equivalent measure in which all tradeables are martingales. Study drift of stock/bond process, for a fixed jump intensity and jump distribution (i.e. fixed drift), there exists a unique change of measure such that stock/bond process is a martingale. As there exists a risk neutral measure, it must be the one in which the stock/bond process is a martingale. Having identified a risk neutral measure, we are able to get an arbitrage free price for other tradeables via discounted expectation with respect to this measure.

I'm still confused as to the time dependence of the above paragraph, but I guess I'm less confused than 12 hours ago!
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Re: Equivalent martingale measure vs. risk neutral measure

Postby mj » Fri Sep 21, 2012 3:36 am

well the way I think of it is that we have 3 levels of instrument
1) things we model evolution of
2) things we calibrate to
3) things we price

you haven't mentioned 2 above so let's ignore them.

When we pick the martingale measure we make everything/numeraire in level 1 a martingale.
We then DEFINE price processes for everything in level 3 that makes them martingales as well.
So the process of everything is a martingale.

With incomplete markets the difference is that there are many choices that make the level 1's martingales.
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Re: Equivalent martingale measure vs. risk neutral measure

Postby Anti » Fri Sep 21, 2012 2:21 pm

Thank you for the response, and while you're here, thank you for all of the great resources/advice you provide to learn quantitative finance!

I guess that the word DEFINE (which is also what you used in the book) is what I don't like and what made me realize that I still had some confusion. That is, I don't like 3, and that made me realize that I don't understand 1. We have (at least) two processes, the stock/bond and the option/bond processes. A priori, why should changing the measure to ensure that one process is a martingale force the other to be a martingale? It seems perfectly reasonable that we swap measures for one process, it now has zero drift, however the other process simply does not. It's not very satisfying (in my opinion!) to flat out define the option/bond process to have the martingale property in the stock/bond measure, when in fact it is some process that exists in nature and at face value may very well not be a martingale in the stock/bond zero drift equivalent measure.

The martingale property guarantees no arbitrage, but is the converse true? I guess that's the content of the first fundamental theorem, and without knowing that simply defining the option/bond process to have the martingale property seems to cut a logical corner. Probably I'm making things harder than they are, but it seems like what I mentioned in the second post makes more sense in my (beginner's) mind:

Assume no arbitrage, apply first fundamental theorem to get the existence of an equivalent risk neutral measure in which all tradable assets are martingales. Notice that there is a unique change of measure for the stock/bond process to be a martingale. Uniqueness forces the risk neutral measure to coincide with the stock/bond equivalent martingale measure, hence the option/bond process is a martingale in the stock/bond equivalent martingale measure.

I know that the first fundamental theorem holds in a discrete market, but is there also a continuous analogue?
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Re: Equivalent martingale measure vs. risk neutral measure

Postby mj » Fri Sep 21, 2012 9:16 pm

there is a continuous time version.

We are trying to FIND the price of an option which is compatible with the stock price dynamics.
In the complete case, there is a unique such price and it has a unique associated process. So
writing down such a process for the option is sufficient since it must agree with any existing arbitrage-free price.

In the incomplete case, you have the issue that there are many compatible prices and associated
processes. So there is no reason to think that a given one will agree with the one observed in the market place.

Reading chapter 15 may illuminate these points since I do discuss them further there.
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Re: Equivalent martingale measure vs. risk neutral measure

Postby Anti » Fri Sep 21, 2012 10:09 pm

Okay will do. Writing things down here for myself and reading your comments has been very helpful. Thanks for the discussion!
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Re: Equivalent martingale measure vs. risk neutral measure

Postby coool » Wed Nov 21, 2012 6:58 am

The strike reset call is similar to the vanilla call, except an additional feature that you can reset the strike equal to the spot within [t, T1] as many times as you like. In other words, the effective strike will be the minimum of spot within [t, T1]. I think there is a closed form solution as this is similar to the Asian option with floating strike. My question is whether we can do a static replication in Chapter 10 for it? Thanks.
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Re: Equivalent martingale measure vs. risk neutral measure

Postby mj » Wed Nov 21, 2012 11:30 pm

i am not sure what this has got to do with this thread.

Essentially you are talking about a look-back option. The method used for feeble static replication of Asian options
could be applied to that case.
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