Of course you right Paul. There many interesting and important papers and this is only way to follow time. Generally speaking I have understanding how the risk neutral world RNW should be set. What I really never seen is the correct calculation a particular parameter in the risk neutral setting. More precisely To calculate expectation in the RNW in continuous time one should make calculation of the correspondent function or functional then multiply it by the Girsanov density that is also the stochastic functional. Because any formula in the RNW if it makes sense and written correctly could be transform in the corespondent formula in the real world it is not clear an advantage of the use RNW.

"The forward price of X is an unbiased predictor of the expected future price of X. "Is this statement an assumption or it can be justified or proved?

"The forward price of X is an unbiased predictor of the expected future price of X. "Is this statement an assumption or it can be justified or proved?

Neither -- it's an answer to the thread title question.

It seems that if "unbiased predictor" is close to the unbiased estimate in math statistics then it would be that expected value of the price in a future moment is about equal to theforward price. Risk nuetrality if it is interpreted with the help of Girsanov theorem relates to the possibility to change probability measure such that the drift of of the original SDE on transformed probability space would be a risk free constant or function in time.It might be not 100% that these two things are very close to each other.

It might be more accurately to say that neutral market and neutral world are different notions.

"No matter the stochatic process, just assume the underlying asset´s expected return is the risk-free rate, and the derivative´s price will be covered". That´s a common mistakePeople forget that if you can´t make a perfect hedge (markets are imcomplete, in economist´s dialect), there are infinite martingles measures. Picking one of them will not guarantee the perfect hedge.Perfect hedge (complete markets) implies a unique martingale measure. Existence of some martingale measure does not imply perfect hedge. Unfortunately, weather derivatives, energy derivatives, and some credit derivatives don´t take that into account

To me, the term "risk-neutral" is somewhat misleading. I rather see it as "growth-neutral", and here's my explanation:The underlying asset is expected to grow at its growth rate mu. However, today it has the price it has today, i.e., if we were to price the underlying asset based on real expectations (i.e., growing at mu), it is mispriced itself. Therefore, we also have to price derivatives by not considering their final payoff, but by the payoff as if it were today, i.e., without any growth priced in. As a seconds step, we have to discount this expectation to today using the risk-free rate because the payoff will not happen until expiration.Therefore, in my opinion, risk-neutral valuation should not be explained by "replace the growth rate by the risk-free rate", but rather "dump the growth rate altogether and discount the expected payoff using the risk-free rate". In fact, this is the same thing, but the latter just makes more sense to me intuitively.Any comments on this way of seeing it?

QuoteOriginally posted by: snooper77To me, the term "risk-neutral" is somewhat misleading. I rather see it as "growth-neutral", and here's my explanation:The underlying asset is expected to grow at its growth rate mu. However, today it has the price it has today, i.e., if we were to price the underlying asset based on real expectations (i.e., growing at mu), it is mispriced itself. Therefore, we also have to price derivatives by not considering their final payoff, but by the payoff as if it were today, i.e., without any growth priced in. As a seconds step, we have to discount this expectation to today using the risk-free rate because the payoff will not happen until expiration.I think that is an accurate description except that, instead of "i.e., without any growth priced in", I would write "because the present value of the expected risk compensation is already priced into the underlying (and its future contract)"QuoteTherefore, in my opinion, risk-neutral valuation should not be explained by "replace the growth rate by the risk-free rate", but rather "dump the growth rate altogether and discount the expected payoff using the risk-free rate". In fact, this is the same thing, but the latter just makes more sense to me intuitively.Any comments on this way of seeing it?I don't agree with this. Instead of "dump the growth rate altogether" I would rather say, "replace the expected value by the risk-free expected value" or "dump the present value of the expected risk compensation altogether". The real growth rate (and future risk compensation) may well still affect the shape of the risk-neutral distribution (i.e. variance and higher moments).

Last edited by Fermion on October 22nd, 2008, 10:00 pm, edited 1 time in total.

Just to make it clear, I think one of the most common and stupidest things people say about risk neutrality is that it is a sufficient condition to ensure your hedge. I can see it as a necessary condition, although.

Hi all, I still stuck with risk neutral concept but recently I see some lights which I put together as note as below link. Fermion thanks for your help earlier. http://cid-36b727b3731ca86f.skydrive.li ... ept.pdfIts a short note so pls take a look. In short, Risk Neutral describes only NOW to lock-in all time-space structure as of NOW(t=0) to make all components arbitrage free. While real expected return is the only device to make meaningful prediction of future market price level. In the end two has totally different objective and do not contradict each other. Look for any comment / criticism. Thanks !

Sorry forgot to make URL as link. PositioiningOfRiskNeutralConcept

2 preliminary remarks:1. It looks like that concepts of CAPM and risk neutral world are non related.2. In your page 3 you introduced Fig.1 Black Scholes Framework. It follows Hull Book. There it was defined a portfolio and in the next line d . If one takes differential in t from it is easy to see that in the expression d one term is missed as far as partial derivative g with respect to S is a function of t. You could check yourself when you replace function g by its value which is claimed to be solution of the BS equation. Put the explicit BS solution in and take differential. Note that Hulls needs both and d in the presented forms. Otherwise the derivation of the BS equation could not be completed. I put this remark in my first notice : http://ssrn.com/abstract=500303

Thanks list for your kind attention and remarks. Apologies for my errors. My math is poor. I think this issue is very much philosophical rather than mathimatical/financial. Logic can be build on t=0 discrete relationship/models. But in time-space we are living in now, there is huge GAP in between PRESENT and FUTURE. PRESENT is somewhat under control of human being and can be put in discrete model. But once we step in future, we tend to lose control. Best thing can be done is dynamic adjustment by moment to next moment. To have more comprehenseive model, I presume we need one more dimention in formulation that stretch out to real future time direction on top of existing dimension used. My math is too poor to handle this but hope somebody will make it in future. Many may feel my comment is crazy. I also welcome such criticism. Thanks in advance.

My remarks relate not to your derivation and in too many people teach or study the same derivation so you do not have to sorry about. Every time I think what is the easiest way to present RN idea. It seems that some time ago was the best. First where it comes from? As far as I know RN come from the attempt to present BSE solution in a probabilistic form. As real stock on the real world (= original probability space) has return mu and underlying of BS option price we have expected return r then there is an attempt to explain paradox. The paradox is that the option is actually takes its values not from the underlying stock but from heuristic imaginary security with expected return r and the same volatility. Actually ignoring financial applications we could construct RN world with a particular probability measure and consider SDE with expected return mu. This "real "process on real world will have expected return r. This is correct approach though it is very bad methodologically as far as real stock equation exists on real world but not on RN.If one starts with original probability space then making measure change would not present expected RN world.The gap between spot and future as you discussed does not relate to RN world. Intuitively you probably talk about neutral trading in which buyer and seller should correct their positions to eliminate arbitrage or in simpler words one side advantage. If you meant that then it is different risk neutral trading and risk neutral world.

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