FixedIncomeRisk Forum

The name “risk-neutral measure” implies that in absence of arbitrage, valuation can be done as if investors have risk-neutral preferences. However, this is true only in the special case of continuous-time complete market models. The historical origins of the risk-neutral measure are based upon the continuous-time complete market models of Black and Scholes [1973], Merton [1973], and Cox and Ross [1976], and hence, the name of this measure is associated with doing valuation as if investors have risk-neutral preferences. However, this measure is no longer identified by doing valuation as if investors have "risk-neutral preferences."

The path breaking work of Harrison and Kreps [1979], significantly generalized the concept of risk-neutral measure to mean an equivalent martingale measure that uses the money market account as the appropriate numeraire asset for valuation. The so called risk-neutral measure may use information related to the risk-aversion behavior of investors in an incomplete market, in order to identify a specific preference-dependent measure, from an infinity of potential equivalent martingale measures. For example, the market price of volatility risk enters the Heston [1993] option pricing formula, and so information regarding risk aversion enters in the "risk-neutral measure" for obtaining a preference-dependent solution of contingent claims using this formula. So, despite its origins, the risk-neutral measure is now used in the finance literature to denote the more general concept of the Harrison and Kreps type “equivalent martingale measure,” in the context of incomplete markets, when the money market account is used as the appropriate numeraire asset for valuation. Hence, a more appropriate name for this measure should be MONEY MARKET MEASURE.

To see this more clearly, note that the most celebrated result in financial valuation (i.e., implied by the fundamental theorem of arbitrage-free pricing), which does not require any assumptions about return distributions or risk preferences is given as follows:

'An equivalent probability measure specific to asset Yt, MUST exist under which the ratio of two asset prices Xt/Yt is a martingale, simply by absence of arbitrage.' Even power law distributed returns of Nassim Taleb in his bestseller The Black Swan (which unfairly criticized many financial valuation models), MUST obey this result simply by absence of arbitrage.

When the asset Yt is a zero coupon bond or an annuity, this measure is called the forward measure (application are for pricing caps and swaptions, using term structure models). When asset Yt is say price of IBM, this measure can be called IBM measure (applications are for pricing options to exchange some asset for IBM asset). Using similar logic, when asset Yt is the value of the money market account, it should be called Money Market Measure (applications include pricing of all standard equity options under all kinds of models with stoch vol, jumps, etc.).

Why should we call the money market measure, a risk-neutral measure, when valuation is not risk-neutral, under models such as Heston (price of volatility risk enters the model), fundamental Vasicek model (market price of risk is needed under the fundamental Vasicek model, but not under the preference-free Vasicek+ model, as shown in a recent book, Dynamic Term Structure Modeling, by Nawalkha et al.), and under discrete-time binomial model with more general parameter choice as mentioned in the following paper:

http://ideas.repec.org/a/bla/finrev/v30 ... 05-15.html

This issue is relevant since any scientific field should use the names of its most important concepts with a lot of clarity and care. It is understandable how the name "risk-neutral measure" originated from the continuous-time complete market models. But why should this name be used under cases when arbitrage-free valuation can no longer be done as if investors are risk-neutral.

So in conclusion, the finance field should seriously consider renaming the risk-neutral measure as the Money Market Measure. The risk-neutral measure would then become a special case of the money market measure, for the case of continuous-time complete market models, such as Black and Scholes [1973], Merton [1973], and Cox and Ross [1976]. This will add much clarity and avoid needless confusion for students and practitioners of financial valuation models.