### What is change of numeraire and why is it useful in finance?

Posted:

**March 16th, 2003, 8:34 pm**from reza.P

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Posted: **March 16th, 2003, 8:34 pm**

from reza.P

Posted: **March 17th, 2003, 1:38 am**

as usual I let the rigorous definitions to our brilliant frinds Omar, Pat & Martingale to me the intuitive idea behind a numeraire is (as Shreve said) an asset with strictly positive price. we then can denominate all other assets in *units* of this assetclassiacl example in Quant Finance, havingdS/S=r dt + sigma dWusing beta = exp(r t) the cash money-market-account numeraire, posing S*=S/betadS*=S* sigma dWso S* is a Martingale ...

Posted: **March 17th, 2003, 1:50 am**

Reza...if I am not mistaken, the numeraire need not be an asset (eg. temperature). But, true..the usefulness of this method appears only when the numeraire is an asset. As Reza points out, asset pricing is done only "in terms" of the price of another asset which can be chosen as anything.

Posted: **March 17th, 2003, 1:54 am**

you're right, I'm using the word asset in a very general sense

Posted: **March 17th, 2003, 2:46 pm**

A change of numeraire allows you to convert a difficult problem into a slightly easier problem. Classic case: Black Scholes with stochastic interest rates. The standard B-S arguments are still valid but the evaluation of the expectation is much more difficult due to the presence of 2 stochastic terms (the stock and the interest rate) in the discounted payoff function. Converting to a new numeraire (a risk free zero coupon bond), allows you to remove one stochastic element and thus evaluate the expectation relatively easily.Of course this requires a very liquid (almost complete?) market of zeros, but hey, it's just a modelSam

Posted: **March 17th, 2003, 3:51 pm**

QuoteOriginally posted by: supernaut20Reza...if I am not mistaken, the numeraire need not be an asset (eg. temperature). But, true..the usefulness of this method appears only when the numeraire is an asset. As Reza points out, asset pricing is done only "in terms" of the price of another asset which can be chosen as anything.the numeraire has to be an asset of positive value. Definition of asset: something you can assign a dollar value to.MJ

Posted: **March 19th, 2003, 4:17 am**

On a fundamental level numeraires are necessary to reduce general price processes to local martingales.Cash prices will not in general be a local martingale in any probablity because of inflationary drift upward(the money illusion).Cash prices divided by the numeraire can become local martingales in a probablity associated with the numeraireif suitable conditions are satisfied (loosely speaking: absence of arbitrage). The idea is that the numeraire capturesthe general drift and division by the numeraire eliminates this drift.Local martingales are much simpler than general stochastic processes.There is a large theory. For example a continuous local martingale which is adaptedto the filtration generated by a Brownian motion is an integral with respect to this Brownian motion.This is already much more concrete.On a practical level we might see the effect of a numeraire as follows: suppose you have to take into account short term interest rates (the short rate). It might be stochastic. This could seriously interfere with your pricing PDEs.At the very least it will complicate the PDE and necessitate a short rate model.On the other hand you could use the risk free bond as a numeraire and express prices in units of the risk free bond.In this new numeraire the short rate is zero. In order to go back to cash prices all you need is to multiply with the risk freebond, that is you need some way to retrieve the risk free bond from market data not necessarily a short rate model andthe associated complications with the pricing PDEs.

Posted: **March 19th, 2003, 9:32 am**

If N is a numeraire with martingale measure Q^NFor any asset f: f/N is a martingaleD(f/N) = sigma f (f/N)d z ^NIf we consider a traded asset MUnder measure Q^N the process M/N is a Martingale with volatility lambdad (M/N) = lambda (M/N) dz^NThe value of derivative f must be the same if calculate with different numeraires N,MLet’s describe change of numeraire practical use in finance. Under risk-neutral measure Q^B the stock price follows the process dS = rSdt + sigma S dzThink about a self-financing portfolio, it’s identified numeraire if security prices calculated in units of this portfolio, accepting an equivalent martingale measure.I think change of numeraire technique is useful in foreign currency derivatives markets: an American call option To buy 1 AUD with dollar price process S, for K dollars is equivalent to an American put option to sell K dollars, with AUD price process K/S. Dollar price of the call must equal product of the current exchange rate, So, and the AUD price of the put.Same alteration of numeraire could be used to get interest parity theorem which states the time zero dollar forward price, Go(T) for time T distribution of one AUD as the spot exchange rate times the ratio of 2 discount bond prices: Go(T) = So B1(T) / Bo(T)Bo(T) is the time zero dollar price of a discount bond paying $1 at time T and B1(T) is the time zero AUD price at T

Posted: **March 24th, 2003, 3:16 pm**

A numeraire is a traded asset that can be used as relative measure of value.In physics, quantities can be measured in units which are constant and well defined, meters or seconds for example. Unfortunately in finance the measurement of value is a little trickier. For example a cheque for $100,000 is worth maybe £61,000 today but could be worth £60,000 pounds next week because of exchange rate fluctuations. It might buy me a car now, but it won't buy me a car in fifty years time because of inflation. The problem is that the dollar is itself a traded asset. The solution is to model the behaviour of a traded asset and to use this as a relative unit of value.As an example consider the traditional numeraire, the zero coupon bond which guarantees to pay (say) $100 in one-years time.I know that today the bond can be bought for $50. My cheque is worth 2000*B, or two-thousand zero coupon bonds. In one years time it will only be worth 1000*B, its relative value is less even though its apparent value hasn't changed (money now is better than money later). Here expressing everything in units of zero-coupon bonds allows values assets at different times to be compared.Of course I can use any traded asset as a relative measure of value, but this is just a change in the way I make measurements so it shouldn't affect the real value of things. This is the principle of numeraire invariance and it is a very fundamental symmetry of finance which should be observed by any mathematical models. Being able to change units in a calculation is sometimes a very effective way of simplifying the calculation, however it is very important to remember that the pricing measure is the measure which makes all the assets written in units of the numeraire assets martingales. This means that the measure will be different if we decide to use a different numeraire. One simple example is the pricing of an option to exchange one share for another share at a future date. Measured in pounds the payoff of this option is (£)max(S_1(T)-S_2(T),0) where the assets satisfy the equations (usual Black-Scholes terminology),S_i(t)=S_i(0)exp[(mu_i-1/2 sigma_i^2)t+sigma_i w_i] (Log-Normal) and [dw_1, dw_2]=rho dtCalculating the option price requires the calculation of the expected value over both correllated brownian motions, which is a bit tricky. However, measured in units of the second stock price (at the appropriate time) the payoff is max(S_1(T)/S_2(T)-1,0)Now Y=S_1/S_2 is easy because of the Log-Normal distributions and shows that Y is also Log-Normal with a volatility sqrt(sigma_1^2+sigma_2^2-2rho sigma_1 sigma_2). The option price formula is (in units of S_2) E[max(Y-1,0)] or in pounds (S_2)(0)*E[max(Y-1,0)], where the expected value is taken under the measure which makes Y a martingale, which is just S_2*BS(S=S_1/S_2, K=1, r=0,sigma=sqrt(sigma_1^2+sigma_2^2-2rho sigma_1 sigma_2), T=T).This is known as the Margrabe formula.Im sure lots of people will provide lots of other examples in the rest of the thread.

Posted: **January 27th, 2004, 10:10 am**

I heard that the valuation of a put option derived from a chance of numeraire of a call option is another application of the change of numeraire. Can somebody explain this to me? Thanks!

Posted: **January 28th, 2004, 10:16 am**

Hi Rotscher!Sehr interessante und intelligente Frage!Sehr spannend!Also, das mit dem Numeraire Change funktioniert folgendermassen:Schritt 1u musst zuerst schauen, ob die Matrix des Hedge Portfolios transponierbar und invertierbar ist. Sollte dies gegeben sein, folgt Schritt 2.Schritt2: Extrahiere den Marktpreis des Riskos aus verschiedenen SWAP Saetzen mit vercshiedenen Laufzeiten (hier ist vor allem das Modell von Hoo/Lee der Zinsstruktur von Nutzen.Schritt 3: Transformiere die verschiedenen Swapsaetze mit einer stetigen Radon Nikodym Ableitung in verschiedene Numeraires. Anschliessens ist es sehr einfach, denn Du brauchst nur noch in die Black Scholes Formel einzusetzen. Viele Gruesse,Black Scholes

Posted: **March 1st, 2004, 9:55 am**

Hi Blacki,Could you translate your previous reply (which looks interesting) in english please, for those who don't understand german ?

Posted: **November 16th, 2005, 2:31 pm**

Hi Rotscher!Very interesting and intelligent question !. Very exciting !With the a Change in the Numeraire it works the following too:Step1: You must first check wether the Matrix of the hedge portfolio is transposable and invertible. If this is true, then you follow the Step 2.Step2: Extract the market value of the risk out of differents SWAPS rates with different expirations (here it is useful the Hoo-Lee Model for the interest rate structure) . What follow afterwards is simple: you need only to plug the previous result into the Black Scholes expression, Regards,Black Scholes

Posted: **May 13th, 2012, 1:18 am**

An example from the new (2012) book, Red-Blooded Risk, by Aaron Brown. It's about the statistics on scratches on bombers during WWII. The underside and leading edges of the plane took more than other parts. Air Force sent the data to a statistician, Abraham Wald, asking to point out areas where armor should be added. His answer was to put armor on the places for which no damage been recorded, for the obvious reason that bombers hit in those places did not came back.It's the case of numéraire inversion and it shows the importance of the whole idea.I guess, normally, people might not realise the presence of numéraire, the fact that we value financial instruments w.r.t. each other and operate with relative value -- that prices should be consistent across instruments or arbitrage opportunities arise. Regardless what you trade apples or oranges, there should a unique rate of exchange between two, usually expressed in currency. Currency itself is a spot numéraire, where distribution of value over time is defined by a risk-free rate of return/a collection of riskless ZCBs. So, for each point in time, there is a unique value equivalent for a money unit ($1).To continue with apples and oranges, if we trade oranges but have to operate with apples too, its natural to use 'an orange equivalent price' for an apple--a unique measure that corresponds to the numéraire. An apple's price will be a martingale in the orange-trading universe -- it will depend on equivalent measure, and the impact of the rules of apple-trading universe will be transformed by the measure.Using the same logic, if we use options to trade volatility rather than delta then, for market makers and dealers who manage a book of calls and puts, its only natural to quote option prices in volatility units rather than money units.