I am trying to understand why we use risk-neutral pricing for derivatives. We are just computing the expected present value of cash flows but why do we do this in the risk-neutral setting ?In risk-neutral pricing, we compute :Price = exp(-rT) Integral ( Cash flows * dQ ) where dQ is the risk-neutral probability and r the risk-free rateBut why can't we compute :Price = exp(-rT) Integral ( Cash flows * dP ) where dP is the physical probability and r the discount rate (corresponding to the risk of the asset)????

go and buy Neftci's "Introduction to the Mathematics of Financial Derivatives" - chapter 14 of the second edition, dare I say it, should even be understood by you

It should even be understood by me ? Thank you for telling me I'm stupid !

Level 1 knowledge: It works because of hedging. If you can hedge then suddenly you don't care whether stocks rise or fall. So you may as well pretend that you are in a world with drift being the interest rate.Level 2: It doesn't work because you can't hedge! Volatility is unknown, stocks jump, you can't hedge discretely. Risk neutrality doesn't work!Level 3: Actually Black-Scholes is very robust. You only need to delta hedge once every few days to get rid of a lot of risk. Throw in some static hedging and all is fine. Back to risk neutrality.Level 4: You don't really care whether it works or not, you are pricing lots of options. Some make you money some lose you money. On average it works out.Level 5: Did you get a big bonus? If so, stop worrying.You've got a long, but interesting, journey ahead of you!P

no, what i was trying to say was that before ask such an open-ended question you should at least try to find out more yourself - once you show that you've done your homework (had a search in this forum where similar questions have been asked before and bought the odd good book), people such as me will be happy to help you furtherand Paul's such a softie for giving of his time to answer your question

spursfan, you could have at least directed him to PWOQF2 where 3 of my 5 levels are discussed!P

Paul,It seems to me that in the credit market, people basically skipped steps 1-3 and went straight for 4. Of course this was very quickly followed by 5 and then by 6 which we would do better not to bring up.In any case, do you agree with my statement that they skipped 1-3 ?

MrMartingale,A very good question! I would agree. I suspect that people are so used to 'risk-neutral pricing' that they use its results and methods (e.g. Monte Carlo and pricing via expectations) without asking whether they are relevant to their markets. (You also see this in some papers where people just arbitrarily make drift the same as the risk-free rate without any comment!)I'd go further and say they missed Step 4! Step 4 relies on diversification to work, i.e. lots of different contracts. In credit the contracts are all very similar. And, of course, we know what happened eventually!P

see the faqs

Sorry, I did not realise that it was such a common question !Well after reading throuh the FAQ, I understand that risk-neutral valuation is useful because of its link with hedging. However, there is still something I don't grasp :Would it be technically impossible to valuate options using real (physical) probabilities ? Or would it be just too difficult to find the appropriate discount rate in that case? The big question is : Would using physical probabilities just be too annoying (but possible) or is there a theoretical reason why it should not be done ?

There isn't such a thing as a physical probability distribution. Probability theory is epistemology, not ontology. Probability theory is like theology. If you believe in the God of the Christians, then you can use that to figure out useful things. When should you go on holiday? What should you eat? Other Christians will also agree, and so you can predict their behaviour, such as where you could find them on a Sunday. This doesn't mean there is any real reason for me to be a Christian, or that Christianity will tell me anything about the real world, because there are other equally compelling religions.The only time that you could use probability theory to calculate an observable quantity, is when there are enough constraints - symmetries of some kind or another - to fully determine the distribution. This would be the case in a casino for example. We can assume that a roulette wheel really does have an equal chance of hitting each number, because any other distribution implies that we have some degree of knowledge about which number will come up next. If you could find a way of obtaining that knowledge, then you would be banned from the casino. The simple fact that you are allowed in the casino means that you are a sucker, and suckers should use the discrete unform distribution.If you did choose some arbitrary probability distribution for a stock (power laws seem to be quite fashionable, apparently this God is the smightiest God because he brings forth credit crunches upon the withered blasphemers and their Gaussian whores) then you could calculate the price of an option. That price might even mean something to you, but it wouldn't mean much to me unless we agree about the distribution, and why would we do that?

Harrison & Pliska (1981) is a good read.

QuoteOriginally posted by: Redox33Would it be technically impossible to valuate options using real (physical) probabilities ? Or would it be just too difficult to find the appropriate discount rate in that case? This is exactly what the risk neutral distribution is. The RN distribution is the real distribution reweighted by a stochastic discount factor.

QuoteOriginally posted by: MrMartingaleQuoteOriginally posted by: Redox33Would it be technically impossible to valuate options using real (physical) probabilities ? Or would it be just too difficult to find the appropriate discount rate in that case? This is exactly what the risk neutral distribution is. The RN distribution is the real distribution reweighted by a stochastic discount factor.The 'risk-neutral' distribution is (almost) any distribution reweighted by a stochastic discount factor. In the same wayE' = MC^2 + Xwhere X is the number of angels that can dance on the head of a pin, and E' is the 'holy energy' defined as E' = E + X.

Thank you for all these replies. I clearly have to do further reading.