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I am reading chapter 2 in Neftci's book, where there is a statement regarding the probability measure that says that every asset becomes a martingale when discounted under the appropriate probability measure. If we can convert any asset into a martingale by altering the probability measure, then why do we bother about numeraires? E.g. In the two binomial stock process, where s(0) = 100, S(up)=110 and S(down) = 90, r=10%, why can we not set p(up)=0.5 and p(down)=0.5 so that 's' is a martingale? However Neftci states that the stock price is not a martingale, rather the discounted stock price is a martingale. I am a bit confused- why does 'r' enter the picture at all if once could change the probability measures to make the process drift-free? Is there an underlying economic rationale that needs to be satisfied by the probability measures? Finally how does the dynamic hedging argument fit into the scheme of things?Thanks!

Changing the numeraire and changing the probability measure are basically two terms defining the same - to change measure by Girsanov you change the drift of the process, or change the numeraire, i.e. you kind of represent one asset in terms of the other's path. r is riskless rate, so when you use dynamic hedging to make your position riskless, this position should earn riskless rate of retrun... that's rough intuition... when you change the measure, you want to make all discounted prices a martingale... including that of a bond... you can change the drift of a stock, as stock has a stochastic term, but you cannot change the drift of a bond process as it's a deterministic process. If you make the drift of a stock 0, then under technical conditions it will be martingale without discounting, but your bond will not be... so it not convenient as you may see. Hope it helps, DROPlet

Read the article of F. Delbaen and W. Schachermayer What is a Free lunch?, it will solve your questions (I hope).A.

see the "why does risk neutral pricing work" thread in the faqsthe riskless bond is deterministic and so if there are non-zero rates, it will not be a martingale in any measure. You therefore have to use a numeraire to get it's discounted value to be a martingale.

Thanks all...I've done what everyone of you has suggested (and again and again!!) and have the following questions:1. Does risk neutrality go hand in hand with 'change of measure'. Is this why we always try to make the discounted asset price (discounted by risk-free rate) a martingale? 2. I still dont have an intuitive feel for what happens when we change the probability measure in order to make an asset price a martingale. Can someone explain to me with a simple example (no equations please!) what the 'physical' interpretation of a change in measure is?3. How does the no-arbitrage concept fit into the whole scheme of things? Would really appreciate some help!Thanks

Anyone please!

QuoteOriginally posted by: longvegaThanks all...I've done what everyone of you has suggested (and again and again!!) and have the following questions:1. Does risk neutrality go hand in hand with 'change of measure'. Is this why we always try to make the discounted asset price (discounted by risk-free rate) a martingale? 2. I still dont have an intuitive feel for what happens when we change the probability measure in order to make an asset price a martingale. Can someone explain to me with a simple example (no equations please!) what the 'physical' interpretation of a change in measure is?3. How does the no-arbitrage concept fit into the whole scheme of things? Would really appreciate some help!ThanksLots of pple can probably answer this better than me, but here's an attempt1. Prices of securities are usually obtained by taking expected value under the risk-neutral measure. The change of measure is used to go from the real-world probabilities to the risk-neutral probabilities (its like defining how to adjust the probabilities for different events from the old measure to the new measure so you can actually calculate the expectation). I think the main point is just to find a martingale, doesn't have to be discounted price.2. For example, in the case of the geometric brownian motion model of stock price, the actual stock price has a certain drift. But we want to do a change of measure (i.e. adjust probabilities of events) such that under the new set of probabilities, the stock price as drift equal to the risk-free rate, so you get the nice property that the discounted stock price is a martingale.3. If you can find a risk-neutral measure, then there is no arbitrage. The proof is somewhat technical but if you just search for a proof of the fundamental theorem of asset pricing, there should be plenty.

selcon gave you good answers, here are my attempts:QuoteOriginally posted by: longvega1. Does risk neutrality go hand in hand with 'change of measure'. Is this why we always try to make the discounted asset price (discounted by risk-free rate) a martingale?The most common reason for changing measures is to get to risk neutrality. So they go hand-in-hand like hole and shovel. But you can make a hole without a shovel and use a shovel for jobs other than digging holes.Changing the probability measure is like changing coordinates. Some problems are easier to solve using polar coordinates than Cartesian, or by using log or log-log coordinates. These are just tricks to make the solution easier, they do not fundamentally change the problem.Quote2. I still dont have an intuitive feel for what happens when we change the probability measure in order to make an asset price a martingale. Can someone explain to me with a simple example (no equations please!) what the 'physical' interpretation of a change in measure is?Let's start with risk-neutral pricing. Under certain assumptions, a derivative can be replicated by transacting in the underlying and a risk-free asset. If that's true, the value of the derivative does not depend on the risk preferences of an investor. If eggs can be bought or sold freely for $0.10 each, a dozen eggs is worth $1.20, regardless of how many eggs you want for yourself.Since the value of a derivative does not depend on the risk preferences of an investor, we can assume any risk preference, price the derivative, and know that price is applicable to all investors. We generally choose the risk neutral investor, because that's the simplest for computation. Everything is priced at its expected value. It is not important that a risk-neutral investor actually exist.But to be consistent, we must estimate the probabilities that make current asset prices look correct to the hypothetical risk-neutral investor. If a stock sells for $100 and can only go either to $120 or $90, we must assume the probability of the up move is 1/3, otherwise the risk-neutral investor wouldn't value the stock at $100. We know the real probability of an up move need not be 1/3. We have changed the measure, going from real probabilities to risk-neutral ones. In the new measure, all securities are priced at their expected values, so they are all martingales. I can easily price any derivative of the stock price; I just compute the derivative's value at stock price $120 and $90, multiply the first value by 1/3 and the second by 2/3 and sum them. A call option at $105, for example, would be worth $5.A real investor might think the odds are 1/2 of each move, but demand a 5% risk premium to hold the stock. This investor thinks the expected value of the call option at $105 is $7.50. However, she will demand a 50% premium to hold the option, making its price $5. How do I know she will demand a 50% premium? If she doesn't, I can trade with her and make unlimited amounts of money.Quote3. How does the no-arbitrage concept fit into the whole scheme of things?See the last sentence of the answer to (2). If arbitrage is allowed then investors can have inconsistent preferences and the whole pricing argument falls apart.

QuoteOriginally posted by: longvegaI am reading chapter 2 in Neftci's book, where there is a statement regarding the probability measure that says that every asset becomes a martingale when discounted under the appropriate probability measure. If we can convert any asset into a martingale by altering the probability measure, then why do we bother about numeraires? Our professor tried to drum the fact into us that we shall forget "discounted" asset prices. They are asset prices under the money market account as the numeraire.X * exp(-rT) = X / exp(rT)hope this helps