Risk Neutral Pricing = Martingale Pricing?

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In Shreve's book, we see the discounted stock price can be changed driftless in some equivalent measure. And then by the Martingale Representation theorem, we can discount the expectation of the derivative values at maturity to price the derivative.

However, there are other chances of numeraire. i.e. the stock. In this case, the derivative or the stock price does not follow a risk-neutral (drift=interest rate) process.

Thus I doubt if the risk neutral pricing is a necessary part of martingale pricing or just a coincidence?

Any advice would be appreciated.

 

[moretodo]

yes risk neutral pricing is an integral assumption in all pricing techniques since if it werent, the price would be a function of the model investor's risk averseness ( or lack of it) as measured by the first derivative of the utility function and therefore arbitrary.

 

[C S]

In Shreve's book, we see the discounted stock price can be changed driftless in some equivalent measure. And then by the Martingale Representation theorem, we can discount the expectation of the derivative values at maturity to price the derivative.

However, there are other chances of numeraire. i.e. the stock. In this case, the derivative or the stock price does not follow a risk-neutral (drift=interest rate) process.

Thus I doubt if the risk neutral pricing is a necessary part of martingale pricing or just a coincidence?

Any advice would be appreciated.

Let's clear up some confusion in terminology, just in case. Some people call any kind of numeraire pricing "risk neutral pricing" and call all kinds of things "risk neutral measure". Others are more strict and (justifiably) only use "risk neutral pricing" to refer to numeraire pricing with respect to the (domestic) money-market account as numeraire. In that case, you consider discounted price processes, as that's what you get when you divide price by money-market. This will become clearer when you read Chapter 9 of Shreve Vol 2, when he starts discussing numeraires in general. In Shreve's terminology, he mostly uses "risk neutral" measure to mean using domestic money-market as numeraire until Chapter 9, when he starts talking about domestic versus foreign money markets. Then from that point in the book, he uses "risk-neutral measure" to mean what other people might call the "numeraire measure".

Your comment about martingale representation indicates some kind of potential misunderstanding. Girsanov's theorem is what allows you to change the drift of your price process by changing measure. Once you change your discounted price process into a martingale, by *definition* of a martingale the future expectation is equal to the current value of the discounted process. No martingale representation theorem needed. What it's needed for is to show uniqueness of a risk-neutral measure, i.e. there is only one measure under which your asset prices discounted by the given numeraire are martingales.

 

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Let's clear up some confusion in terminology, just in case. Some people call any kind of numeraire pricing "risk neutral pricing" and call all kinds of things "risk neutral measure". Others are more strict and (justifiably) only use "risk neutral pricing" to refer to numeraire pricing with respect to the (domestic) money-market account as numeraire. In that case, you consider discounted price processes, as that's what you get when you divide price by money-market. This will become clearer when you read Chapter 9 of Shreve Vol 2, when he starts discussing numeraires in general. In Shreve's terminology, he mostly uses "risk neutral" measure to mean using domestic money-market as numeraire until Chapter 9, when he starts talking about domestic versus foreign money markets. Then from that point in the book, he uses "risk-neutral measure" to mean what other people might call the "numeraire measure".

Your comment about martingale representation indicates some kind of potential misunderstanding. Girsanov's theorem is what allows you to change the drift of your price process by changing measure. Once you change your discounted price process into a martingale, by *definition* of a martingale the future expectation is equal to the current value of the discounted process. No martingale representation theorem needed. What it's needed for is to show uniqueness of a risk-neutral measure, i.e. there is only one measure under which your asset prices discounted by the given numeraire are martingales.

Yes, that is what I mean - money market numeraire is not the only and essential option to martingale pricing. But I don't know why there is a need to show the uniqueness of a martingale measure of a given numeraire? Or like moretodo just said, there must be a riskless money market numeraire to price any option?

 

[C S]

Yes, that is what I mean - money market numeraire is not the only and essential option to martingale pricing. But I don't know why there is a need to show the uniqueness of a martingale measure of a given numeraire? Or like moretodo just said, there must be a riskless money market numeraire to price any option?

You don't need a riskless asset to price an option. Obviously you'll need some kind of assumptions on the assets you do have however. For example, Margrabe's formula lets you price an option where you have the right to exchange one risky asset for another. In the model assumptions for the formula, no riskless asset is assumed, although enough is assumed so that under numeraire pricing, one discounted asset behaves like a riskless asset.

Numeraire pricing turns on the hinge provided by Girsanov's theorem. If you look through the assumptions of that theorem, you'll see there is no assumption there is a money market. However, in order to make all your discounted price processes into martingales, you'll need to solve the analogous market price of risk equations. So it's necessary to make whatever assumptions are needed in your context so that you can solve those equations. Otherwise there is no risk-neutral measure you can switch to.

The neat thing about the standard BS assumptions (and usual variations) is that the market price of risk equations can be solved, even uniquely. Why is uniqueness important? From a certain standpoint, it's not important. Risk-neutrality will give you a price that can't be arbitraged. But if there are many risk-neutral measures, then you can have many prices that can't be arbitraged. Which is the "right" one? You can try and extract the risk-neutral measure given by the observed option prices, but in the real world, we don't have a continuum of strike prices, so this can only get you so far. On the other hand, some people aren't bothered by the lack of one no-arbitrage price. They say you should be using economic arguments anyway. But from a theory viewpoint, the lack of uniqueness can be bothersome.

 

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You don't need a riskless asset to price an option. Obviously you'll need some kind of assumptions on the assets you do have however. For example, Margrabe's formula lets you price an option where you have the right to exchange one risky asset for another. In the model assumptions for the formula, no riskless asset is assumed, although enough is assumed so that under numeraire pricing, one discounted asset behaves like a riskless asset.

Thanks for your answer. Would you be more specific on the Margrabe's formula? Any paper or textbook? I think I have found the same conclusion that two risky assets can price an option on their exchange rate.

 

[C S]

Thanks for your answer. Would you be more specific on the Margrabe's formula? Any paper or textbook? I think I have found the same conclusion that two risky assets can price an option on their exchange rate.

There's a straightforward explanation and derivation in Kerry Back's book on derivative securities. This kind of thing is also used a lot in commodities derivatives pricing.

 

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There's a straightforward explanation and derivation in Kerry Back's book on derivative securities. This kind of thing is also used a lot in commodities derivatives pricing.

2.9 Numeraires and Probabilities?

 

[C S]

2.9 Numeraires and Probabilities?

Just look for Margrabe's formula in the book. Try the index, table of contents...

 

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Just look for Margrabe's formula in the book. Try the index, table of contents...

Thanks for the recommendation. I will list Kerry Back's book as the next book to read after Shreve's.

 

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Thanks everyone.
Here is my answer - straight forward derivation without risk-neutral measure

http://ssrn.com/abstract=2397010