The Risk Neutral Measure

[NYConsultant]

I was wondering how different distributions (such as the normal distribution) are affected when calculated in a risk neutral measure (I'm new to this)

 

[Sanket Patel]

In brief, switching to a risk-neutral measure simply shifts the mean of the distribution such that the expected return becomes the risk free rate. The variance, however, remains unchanged.

 

[NYConsultant]

How is this shift of the expected return achieved? Is there an analogy to a transformation of basis (in the llinear algebra sense)?

Also, I'm probably missing something fundamental here, but how can something be priced by the assumption of risk neutrality? I'm sure investors in the real world want better compensation for more volitile instruments.

 

[Eugene Krel]

it's not a price, it's a value.

 

[Stefan Zota]

How is this shift of the expected return achieved? Is there an analogy to a transformation of basis (in the llinear algebra sense)?

Also, I'm probably missing something fundamental here, but how can something be priced by the assumption of risk neutrality? I'm sure investors in the real world want better compensation for more volitile instruments.

There is quite a bit of theory that is required to understand risk-neutral pricing. You can start even from the wiki page and then pick-up a stochastic calculus book.
http://en.wikipedia.org/wiki/Risk-neutral_measure

The core of risk-neutral pricing is the concept of martingale (stochastic process with 3 properties, most important being that given all information to the present, expected future value is the current value). Once the underlying asset (e.g. stock) can be "changed" into this martingale measure, then you can talk about hedging/replication.

Basically any derivative contract would need to be replicated/hedged with a portfolio (another stochastic process). Anything that can be hedged can be valued. In the end, the goal is to find the current value of the portfolio (stochastic process) with all data till current moment available (filtration) and with a precisely defined function in the future (at maturity or anywhere on the path)

In this explanation I skipped almost all details and it's not 100% precise. It just gives an idea of the complexity ...

 

[Sanket Patel]

Also, I'm probably missing something fundamental here, but how can something be priced by the assumption of risk neutrality? I'm sure investors in the real world want better compensation for more volitile instruments.

How else would you price it? Pricing an asset using real-world / physical measure would be impossible because the measure wouldn't be unique. Each investor would have his or her own risk preferences resulting in countless distributions.

For an excellent, yet simple discussion, read chapters 2, 14, and 15 from Salih Neftci's An Introduction to the Mathematics of Financial Derivatives.

 

[tobias elbert]

Pricing the an asset in using real-world / physical measure would be impossible because the measure wouldn't be unique. Each investor would have his or her own risk preferences resulting in countless distributions.

Platen suggests an alternative in his Benchmark Approach to Quantitative Finance. He develops a framework for pricing under the real world measure whereby your numeraire is the best performing (he calls it growth optimal portfolio) market portfolio. Since that benchmark portfolio must be unique by its definition, the (pricing) measure associated with it will be also. The risk-neutral valuation approach is just a special case in his framework.

 

[NYConsultant]

Platen suggests an alternative in his Benchmark Approach to Quantitative Finance. He develops a framework for pricing under the real world measure whereby your numeraire is the best performing (he calls it growth optimal portfolio) market portfolio. Since that benchmark portfolio must be unique by its definition, the (pricing) measure associated with it will be also. The risk-neutral valuation approach is just a special case in his framework.

This sounds interesting; I'd like to take a look once I have a bit more background.

Follow up question: what's the relation between a measure and a numeraire?

 

[quotes]

You have post a serious but often neglected question. Not all distributions can be easily shifted to a risk neutral measure to price an option via martingale approach. It has to be transformed through Radon-Nykodym derivative.

The book Brownian Motion Calculus may answer your question on Brownian Motion case.