Talk:Risk-neutral measure

Example looks wrong

I think the example is wrong. I think the physical dynamics, instead of

$dS_{t}=\mu +{\frac {1}{2}}\sigma ^{2}S_{t}dt+\sigma S_{t}dW_{t}$

should read

$dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t}$

Most people in the finance world reserve "the risk neutral measure" to mean the measure under which the money market account is treated as numeraire security. Under this measure the risk neutral dynamics would be

$dS_{t}=rS_{t}dt+\sigma S_{t}dW_{t}$

where r is the short term risk-free rate of interest.

the article did not mention one very important definition of RN measure

I think the article missed one very important definition of Risk-neutral measure. It just described how Risk-neutral probability is used in asset pricing theory and an example in Black-scholes world.

One should ask what kind of information is offered from Risk-neutral probability and where can we find this measure in the real world.

The first question leads to an equivalent definition of risk neutral probability. A risk neutral probability is the probability of an future event or state that both trading parties in the market agree upon. (This definition is also related to the concept of state price.)

A simple example. For a future event,(eg, whether it rains tomorrow), two parties enter into a contract, in which party A pays party B $1 if it happens(or rains) and $0 if it doens't. For such an agreement, there is a price for party B to pay party A. Obviously, the price is in the range from 0 to 1, exclusive of the end points. If the two parties agrees that party B pays $0.4 to party A, that means the two party agree that the probability of the event that happens(eg, rains) is 40%, otherwise, they won't reach that agreement and sign the contract. So this price reflects the common beliefs of both parties towards the probabilty that the event happens. 40% is the risk neutral probability of the event that happens. It is not any historical statistic or prediction of any kind. It is not the true probability, either. Simply put, it is just a belief that shared between the two trading parties in the market.

For the simple example mentioned above, once the price is established, the risk-neutral measure is also determined. Whenever you have a pricing problem in which the event is measurable under this measure, you have to use this measure to avoid arbitrage. If you don't, it's like you are simply giving out another price for the same event at the same time, which is an obvious arbitrage opportunity.

A more complicated example is the Black-Scholes world, in which we assume the stock follows brownie motion. In this setting, the stock price itself is enough to reveal the common belief between the trading parties towards the stock return distribution.( The arguement is similar to the first example.) And as a result, we have the famous Black-Scholes formula for eroupean options. In the real world, the stock dynamics is not brownie motion, so the price given by Black-scholes formula is just a reference price. A more accurate information source for risk-neutral probability is the market price of the stock options. In practice, people use options price to get the risk-neutral measure and further price more complicated contigent claims(eg.exotic option). —Preceding unsigned comment added by Qiwen Chen (talk • contribs) 16 October 2005

A contrasting example to the above risk-neutral one might be where two parties enter into an insurance transaction. Typically the buyer of insurance will pay more than the risk-neutral premium, and is doing so to pass risk to the insurer. The insurer may have low risk over all for reasons including being a party to a large number of insurance contracts with low correlation. Elroch (talk) 19:53, 1 June 2009 (UTC)

"Brownie motion" sounds delicious!!! Is that the non-random process of a brownie going from the kitchen into my belly? —Preceding unsigned comment added by 159.53.46.147 (talk) 17:33, 7 April 2010 (UTC)

What is Eq

What is Eq in the article? Not explained in first Backround part. —Preceding unsigned comment added by 84.230.151.191 (talk) 17:30, August 25, 2007 (UTC)

Incorrect statement

The article says:

"This is in contrast to the physical measure - i.e. the actual probability distribution of prices where (almost universally 2) more risky assets (those assets with a higher price volatility) have a greater expected rate of return than less risky assets."

This statement - even taking the footnote into account - is wrong. As shown by equilibrium arguments only systematic risk can have a risk premium. Some very risky contracts e.g. a purchased insurance contract, a pure gamble or gold will typically have zero or negative risk premiums, since they have no systematic risk or are hedging systematic risk. --Olejasz 07:22, 19 November 2005 (UTC)

Bad English in terminology

This article uses the odd term "physical probability" as a synonym for "real-world probability". I would suggest this term be removed unless it is considered to be the normal one to be used for this concept despite being bad English - physical means related to matter, while many instruments (such as bonds) are not in any sense physical, beyond the tenuous fact that they are represented in the physical world by patterns of electrons (or pieces of paper). Elroch (talk) 19:41, 1 June 2009 (UTC)

Not sure I follow. The "P" in P-measure stands for "Physical", and "Physical probability" is a fairly standard term. "Objective probability" is also used but nobody ever says "O-measure". Btyner (talk) 14:11, 15 August 2009 (UTC)

Hm, I'm not sure if I buy that reason for the term P-measure. I would presume that the P is for probability. Usually there is only one probability measure in most mathematical discussions and you would denote the probability measure by "P". Then when you talk about a different measure you would then use the next letter, "Q". --DudeOnTheStreet (talk) 12:46, 22 April 2011 (UTC)

I don't know if P-measure stands for physical measure, but in finance literature (e.g. http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf page 16) the term "physical measure" or "physical probability" are used to describe the probability measure in the 'real-world'. Zfeinst (talk) 19:32, 22 April 2011 (UTC)

Binomial model

What is the market price of risk in the binomial model? If someone knows, please add it to the article. Thanks, Btyner (talk) 02:05, 27 May 2010 (UTC)

"S is a martinagle"

In my opinion, this statement in the last sentence of the article is wrong. It's the discounted(!) stock-price process that's a martingale under the risk-neutral measure. If you agree, please change it. — Preceding unsigned comment added by 82.220.78.140 (talk) 21:16, 7 June 2011 (UTC)

I believe you are correct, I have corrected the material in the page.Zfeinst (talk) 21:52, 7 June 2011 (UTC)

A prototypical case?

What does this (from the first sentence of the article) mean?

a risk-neutral measure, is a prototypical case of an equivalent martingale measure

Elsewhere, the article claims that a "risk-neutral measure" and an "equivalent martingale measure" are actually different names for the same thing. Are these the same thing, or is one a sub-case of the other? If the latter, what makes it "prototypical"?

(I also note that the first paragraph of this article, while full of buzzwords, does a spectacularly bad job of actually providing any information about what a risk-neutral measure actually is...) -- Foogod (talk) 21:26, 8 February 2012 (UTC)

I've always seen that a risk-neutral measure is the same as an equivalent martingale measure. Zfeinst (talk) 19:45, 1 March 2012 (UTC)

Flag for citations

The flag was restored for this article needing citations. Which sections need sources? Statoman71 (talk) 19:30, 1 March 2012 (UTC)

The motivation and origin sections certainly could use more sources. Also a citation for the 2nd example wouldn't be bad (that should be quick to find). Zfeinst (talk) 19:43, 1 March 2012 (UTC)

Ok, I will look into it. Statoman71 (talk) 01:36, 3 March 2012 (UTC)

Conflicting defns

In usage we have $E_{q}(H)=E_{p}({\frac {dq}{dp}}H)$

whereas the page Numéraire claims

$E_{q}(H)=E_{p}({\frac {dq}{dp}}H)/E_{p}({\frac {dq}{dp}})$

Darcourse (talk) 23:10, 10 March 2023 (UTC) Darcourse (talk) 23:09, 10 March 2023 (UTC)