Many people, some of them otherwise quite clever, misunderstand risk neutrality. Examples? (You don't have to name names!)P

The forward price of X is an unbiased predictor of the expected future price of X.

Paul sorry for been a little bit sarcastic. It seems misunderstanding stems from the fact that if one studied first the measure change subject from a mathematical book and then looked the same subject at any financial handbook-benchmark. It was not be a problem and it would much better if there was only one financial vision of the measure change subject. And it always these mathematicians try to prevent or even correct the high soar of the financial ideas.

1. That the value of a vanilla option is independent of the future expected price of the underlying. (This runs completely counter to intuition, yet people are so convinced by their math, ignoring any doubts about the validity of their assumptions, that they are willing to ignore the most obvious fact that, other than hedgers, people trade options looking for a profit from how much the underlying changes in price more or less that the risk-free rate.)2. That (1) follows from put-call parity.

Last edited by Fermion on June 15th, 2008, 10:00 pm, edited 1 time in total.

I get the feeling this could lead to a discussion on Taleb/Haug ideas... but here goesyou dont need risk neutrality to price options

knowledge comes, wisdom lingers

There are two different things with risk neutrality.First, is it possible to present BSE solution using measure change techniques applied to linear SDE? The answer is not.Second, is it possible to write SDEs on risk neutral probability space? The answer is yes. But is it good way to set the problem? The answer is not. The correct way is to set the problem on the real world. This means to write all equations that one is going to use on risk neutral world on real world. Define risk free rate. Make measure change and be convince that you need such transformation. For example let one write a SDE (ZZZ) model of the risky rate on risk neutral world with prob_measure Q. What it does mean? To answer to the equation one needs exact image on real world: original probability space. As far as Q(dx) = g(x)P(dx) where g is Girsanov density never explicitly presented then the question is there any sense to use ZZZ model remains open deliberately or not.

- CrashedMint
**Posts:**2591**Joined:**

QuoteOriginally posted by: PaulMany people, some of them otherwise quite clever, misunderstand risk neutrality. Examples? (You don't have to name names!)PPaul, how would you explain risk neutrality in layman's terms? I remember that some day I was talking to my girlfriend (she's a biologist) about finance and tried to explain the risk neutrality concept. She's quite clever and I think I understood it myself, somehow, but I failed terribly in explaining the concept. So, i guess, the question is: How to explain risk neutrality in a few sentences without oversimplification in a way that makes sense to everybody.

- Clopinette
**Posts:**258**Joined:**

A trader is "neutral to risk" when he is hedged.With hedges in place, in theory, he should be indifferent to investing in cash or more volatile assets. So he values all his assets with the same expected return as the cash.Now you will tell me: why bother with complicated assets if all he want is the same return as the cash? Because as a bank you get bigger fees from customers when you sell structured notes than if you simply take their money and put them in a savings account.With that in mind, "risk neutral" valuation suits well exotics desks in investment banks for example.In the other hand, prop traders when they take directional or open positions may not use risk neutral prices. Basically "risk neutrality" has nothing to do - at least conceptualy- with measure change.

Last edited by Clopinette on June 17th, 2008, 10:00 pm, edited 1 time in total.

There are two approaches here. One is intuitive. We are talking what is close to common sense. In this case when one find an argument the distance to the common sense might increase remarkably. Other approach that might look for many people is a mathematical one. Mathematical setting. In this case mathematics converts real world into mathematical equations. Quantitative finance is an example. Thus for instance the above statements" in theory, he should be indifferent to investing in cash or more volatile assets."risk neutral" valuation suits well exotics desks ,he should be indifferent to investing in cash or more volatile assets"would be sound much stronger if it was expressed in formulas. Otherwise they sound as intuitive that might be close at certain degree to the real or not.

- Clopinette
**Posts:**258**Joined:**

List," in theory, he should be indifferent to investing in cash or more volatile assets"A practicle translation/application into a formula of this argument is the Feynman-Kac formula (used to derive PDEs)<=>In a single-currency market, If U(t) is the spot value of a security and r(t) is the money market rate then E[ dU(t) ] = r(t)U(t)If you have a stochastic model for U(t), all you have to do is to apply Ito to find dU(t) and then deduce E[ dU(t) ] .You normally end up with a differential equation, and then ...you call Cuchulainn or Paul wilmott.They will discretise it and actually solve the PDE for you!Hope this helps

Last edited by Clopinette on June 17th, 2008, 10:00 pm, edited 1 time in total.

This is much nicer to see. Now you have a statement E[ dU(t) ] = r(t)U(t) dt I use a hint to add 'dt' on the right. It could be assumption as you use it or it could be consistent with the historical observations. In latter case we need to apply randomization of the historical data which in turn should be correspond to statistical setting such as observation are taking in equal environment. Right?

- Clopinette
**Posts:**258**Joined:**

List,I am not sure I understand what you meant by "such as observation are taking in equal environment".But still, applying historical data to calibrate your model will be a bad idea: Because if you do so you will have a model that implies a price for your hedge that is different than the prevailing market one.Basicaly, model parameters need to be set so that the price of you hedge when calculated using you model is the same as the market price. It seems to me that statistical models are done under the "real measure" as opposed to the "risk neutral" one. So they should include estimation of the price of risk.You need to read Harisson & Kreps about all this.

When you use math statistics observation or a sample should be a representative one equally represent population = the original probability space. Your statement that "historical data to calibrate your model will be a bad idea" is rather questionable. If a model has visible divergent with historical data it is better do not use it for the predictions at least there is no evidence that in future it would be better match than in the past.Many people consider possibility of hedge as an argument in favor of smthng. But first is the pricing itself and hedging is an application of pricing. In theory we always assume that all model parameters are known. In practice data is used to present estimates of the model. Of course the implied distribution is also a parameter of a model. Unfortunately the assumption regarding distribution never check in finance. Years ago I checked myself log normal hypothesis for the first stock of DJI over a period might be a year. The hypothesis that diffusion is driven by Wiener process could be accepted say about 50%. Obviously there is no sense to use the model for calculation or to by such software."It seems to me that statistical models are done under the "real measure" as opposed to the "risk neutral" one. So they should include estimation of the price of risk." The question is How to present your basic equation E[ dU(t) ] = r(t)U(t) in risk neutral world? or wheter we need it? or when do you think one need to consider this equation on "risk neutral" world

- Clopinette
**Posts:**258**Joined:**

"Your statement that "historical data to calibrate your model will be a bad idea" is rather questionable. "==> Not only it is not questionable but it is the most basic of things on exotic desks: if your model can't even reprice the most liquid instruments what do you think you can do with it?Again prop desk business is a different story. They take a position against the market so they don't care about not repricing the market."Years ago I checked myself log normal hypothesis for the first stock of DJI over a period might be a year. The hypothesis that diffusion is driven by Wiener process could be accepted say about 50%."==> Nothing prevents you from having a good model AND price under risk neutral asumption....The choice of model has nothing to do with risk neutral valuation."How to present your basic equation E[ dU(t) ] = r(t)U(t)dt in risk neutral world?"==> This WAS risk neutral world.Realy you may want to read a bit about finance theory.

Clopinette and list,you should both read:1. "A Perfect Calibration! Now What?" Wim Schoutens, Erwin Simons and Jurgen Tistaert, Wilmott Mar 2004 6678 (Some of the dangers of calibration.)2. "Stochastic volatility and mean-variance analysis." Hyungsok Ahn and Paul Wilmott, Wilmott Nov 2003 8490 (How to use real probabilities and yet still calibrate!)3. "Which free lunch would you like today, Sir? Delta hedging, volatility arbitrage and optimal portfolios." Riaz Ahmad and Paul Wilmott, Wilmott Nov 2005 6479 (Prop trading for quants!)4. "The Market Price of Interest-rate Risk: Measuring and Modelling Fear and Greed in the Fixed-income Markets." Riaz Ahmad and Paul Wilmott, Wilmott Jan 2007 6470 (Why calibration is unstable, it's because people are! Nice plot of the market price of risk.)These papers question the idea of risk neutrality and calibration. Clopinette, you can't say things have to be accepted without questioning them!P

GZIP: On