Talk:Risk-neutral measure
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Cite: "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)
I think the example is wrong. I think the physical dynamics, instead of
Should read
- dSt = μStdt + σStdWt
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
- dSt = rStdt + σStdWt
where r is the short term risk-free rate of interest.
[edit] the aricle 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).
