Hello, I just started studying stochastic calculus and mathematical finance in general, and I need some help with a topic.Don't laugh, but I'm having a difficult time understanding the concept of risk neutral probablility and a complete market.I mean, I know what their formal mathematical definitions are, but I'm having difficulty concretizing these definitions in my mind. (i.e., explaining it to myself in layman's terms).Also, how does risk neutral probability differ from a real world probability? I mean, they're kind of equivalent right?I do understand the concept of martingales, stopping times, and filtrations and what they mean.Thank you for your help!

Risk neutral probabilities are really only a mathematical concept, i.e. they are not probabilities in the sence of that something will happen with a given probability, or that we can view the world as having a new "risk-neutral" dimension, or anything of that sort. Risk neutral probabilities are only probabilities in the sence that the fulfill the requirement that each element should be in the range 0 to 1 and that the sum to 1.Take a look at the first 3 pages of Baxter's book "Financial Calculus : An Introduction to Derivative Pricing" (you can read these pages free of charge on Amazon using the Search Inside function) for a good intro to Risk Neutral probabilities in Horse Racing. I thought it was a real eye-opener the first time I read it. Basically, what these pages tell you is that there are many different ways you can calculate the odds of game. The two most obvious are (1) trying to estimate the probability of something actually happening in the future (such as a specific hourse winning a race, or a stockprice being $100 on a given future date) and the other (2) being that you use the information on the market to derive the odds (in the case of a house race you use the amount of money waged on each horse to find suitable odds that guarantee that you always make some small(?) profit independant of which hourse actaully wins the race, in the case of a stockprice, you create a risk-neutral portfolio). In each of the two examples (hourse-race and stockmarket) you can derive a measure which we can call "risk-neutral probabilities".The equivalence of risk-neutral probabilities and real-world probabilities are due to properties of the assumed process of the stockprice and in general there is no guarantee that these two probabilities are related (at least as far as I understand/have read about it).Hope this helps!

I've been wrestling with risk-neutral valuation for several years and still don't fully get it. However, in all this time, I have come to some simple ideas that help to understand it.First of all, risk-neutral valuation is only relative valuation. It only works when you price something in terms of something else to which you have a known relationship. (This is where the notion of complete markets comes in.) Thus for instance you can price a derivative in terms of its underlying if you know all the variables that govern its relationship. The reason for this is the hedging connection and the absence of arbitrage opportunities. Consider futures and spot. Risk-neutral valuation tells us they are related by risk-neutral discounting because otherwise you could arbitrage one against the other. But unless you know the true value of one, you cannot obtain the true value of the other.In the case of options, the relationship is given by the real probability distribution (assuming you know it). You'll often hear it said that options cannot depend on the real future expected value of the underlying. This is actually false. (They can depend on it through the volatility. Only if the random fluctuations in spot are independent of the real future expected value (e.g. GBM) can you know that the option is independent of it too.) Nevertheless, you can always obtain a spot-relative price using risk-neutral valuation -- even if that value depends on the real future expected value of the underlying.But it is critical to realise that risk-neutral pricing cannot tell you anything about the true value of a derivative, unless you know the true value of the underlying and its true future pdf.

see the why does risk neutral valuation work thread in the faqs

hey, thanks for the replies guys.I think I'm starting to get it. Risk neutral probability exists because information about the whole market is known (risk neutral world?) and that everyone views the value of an asset in exactly the same way. Because of this, the price of an asset is equal to the discounted value of its expected future pay-offs, the price of which everyone agrees with. Is this understanding correct?Also, is it correct to say that the discounted value of the expected future payoffs of an asset is a martinagale? (or is it the asset only?) I read this somewhere.

Isalandboy,The risk neutral probability exists because there exists a riskless portfolio which yields the same payoffs as the option.I agree with Grabben: take a look at Baxter's book; very nice and intuitive.HTH

QuoteOriginally posted by: HiboumalinIsalandboy,The risk neutral probability exists because there exists a riskless portfolio which yields the same payoffs as the option.I agree with Grabben: take a look at Baxter's book; very nice and intuitive.HTHAh! can I get me some of that riskless portfolio!

Hi Islandboy!No, it's not correct that "risk neutral probabilities exist because ... everyone views the value of an asset in exactly the same way".I would recomend that the concept of "probability" be scraped in the concept of risk-neutral probabiliteis. A better word would be "pricing-vector (or pricing-distribution) as a result of existance of replicating portfolio" or something along those lines. The only reason why we call it a probability is that the sum of the vector (or the integral of the distribution) equals 1. The problem with calling it a probability is that it leads people to think of it as a probability and get deeply interested in what a risk-neutral world would look like and it's implications. Idéas about this "world" are important, however it is not just important, but rather crucial, that people understand what this world really is and why it exists.The only reason why this vector (or measure) exists is that we can create a replicating portfolio, hence we can create a risk-neutral position by taking one option long and one recplicating portfolio short, and when you put all these pricing-equations togeather you notice that you can find the price of the option in a slightly shorter fashion, using the risk-neutral "pricing-vector" (or pricing-distribution). No replicating portfolio, no "risk-neutral probabilities". It does not have any connection to peoples views of the future or how an asset should be priced.Just as Fermion says, this is a method using relative valuation as you derive the risk-neutral probabilities from the market prices of other instruments, however he touches a number of other issues, such as if real distribution and risk-neutral distribution and their relationship and these discussions become more difficult as they depende on the acutal process of the underlying.

dear Islandboylook through the papers http://papers.ssrn.com/sol3/papers.cfm? ... d=844484it may clarify somthing.

This means : if I want to calculate the fare price for all market participant then I should go with Risk-Neutral measure. however in case if do the same for my client who has a particular view on future movement of underlying then I should use Physical measure.Is my understanding correct ?

my personal view is that the term 'risk-neutral' was poorly chosen and creates a lot of confusionI much prefer the concept of 'forward-neutral' measure which is closely connectede.

There two primary confusions in derivative studies. The first one is that the benchmark states that set of stocks with arbitrary expected return could be delivered for the same BS price if the volatility of the set is the same. We do not need to look for real market examples what we need is to check the formal logical basis of such result. We need to assume that two stocks have equal price at a date t and equal volatility. Actually equal price at t is not a restriction. Indeed if prices are different we can chose a constant q and consider one stock against the share that contains q other stocks. The BS pricing states that one can get junk stock and AAA stock at maturity T with equal premium at t using the option type of the contract settlement. After such ridiculous pricing that contradicts the common sense the only question whether or not we have correct definition of the price?The second problem is the risk-neutral world. This problem is the pure stochastic calculus problem. We have the parabolic equation with the coefficient r at the partial derivatives of the first order. There is the SDE that has drift µ. Someone probably heard that in stochastic calculus there exist techniques called measure change. Using this technique the SDE with drift coefficient µ can be transformed to the SDE with other coefficient r. This idea inspired to consider this transformed probability space named in finance risk neutral world as original for derivatives pricing. Here we have two aspects of the problem. The first is it possible to present the original Cauchy problem for parabolic (BS) equation as a functional risk neutral process on risk neutral world. The answer is no. It is the explicit mathematical mistake. And why the people could not see that mistake is only they lost Girsanov exponent density that ties original probability space with risk neutral one. It looks about 95% people who are dealing with derivatives follow this way. Other 5% having much better mathematical background do not make such mistake. Recently I look through a paper in which they first determined equity market having N stocks. Then it is possible of course to define risk neutral world. They did not highlighted explicit formula for Girsanov density that should be a stochastic matrix exponent. Next they started to consider canonical models of the spot interest rates. The critical point of view is that the canonical models on original (physical) probability space would be converted to the equations with very uncommon types. That is from their setting it follows that we are observing very strange objects. Besides when one needs to calculate correctly expected value of a functional. This functional can be a derivatives payoff on risk neutral world we have to be sure that original expected return should be come up in the valuation formulas. Otherwise calculation is incorrect.In mathematics it is common to set a problem in original probability space (physical world ). Then we can move into transformed world (risk neutral ) and no problems exist in this transition.

I jsut read the bookies example in "Financial Calculus : An Introduction to Derivative Pricing" and I do not understand how they come up with 1667 net profit for second horse to win... can anyone explain?I feel as dumb as I look

I jsut read the bookies example in "Financial Calculus : An Introduction to Derivative Pricing" and I do not understand how they come up with 1667 net profit for second horse to win... can anyone explain?I feel as dumb as I look