Equivalent Measures and the Risk-Neutral Pricing Expectation
Summary
The note examines a pricing argument for a replicable option payoff. If a trading portfolio and the derivative have equal terminal values almost surely, their discounted terminal values also agree, so their conditional expectations under a risk-neutral measure agree. The question asks whether equality of the pricing expectations can in turn prove almost-sure replication.
The response explains the change-of-measure identity using the Radon–Nikodym derivative: weighting expectations under the physical measure by the density process yields expectations under an equivalent risk-neutral measure. The risk-neutral measure is specified so discounted asset values behave as martingales, supporting conditional-expectation pricing. This identity explains the transformation of expectations but does not by itself make equality of expectations equivalent to pathwise equality. The excerpt offers no complete theorem establishing replication from price equality; its central lesson is to distinguish a terminal payoff condition from the expectation identity used to price it.
Key ideas
- Almost-sure equality of terminal portfolio and derivative values implies equality of their discounted conditional expectations.
- The Radon–Nikodym derivative expresses expectations under one measure as weighted expectations under another.
- Risk-neutral pricing uses a measure under which discounted asset values are martingales.
- An expectation identity alone does not establish pathwise equality of terminal payoffs.
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# Risk Neutral Pricing Necessary Condition
# Risk Neutral Pricing Necessary Condition
Suppose that I have an option on a single stock expiring at time $T$ and I replicate the payoff of this derivative by investing in the stock market and the money market. So this condition reads $$X(T) = V(T) \quad \text{almost surely}$$ where $X(T)$ is the value of my portfolio and $V(T)$ is the payoff of the derivative.
This condition holding under the actual probability measure is equivalent to it holding under the risk neutral measure, which I assume to exist and to be unique.
We then price the option by saying $$D(t)X(t) = \widetilde{E}[D(T)X(T)|F(t)] = \widetilde{E}[D(T)V(T)|F(t)]$$
I have a problem with the last equality. If the "almost surely" condition holds, then the last equality is implied. However, that equality does not necessarily imply the "almost surely" condition. Am I missing something here or is the fact that the price that comes of this method is unique and that it is a necessary condition for the almost surely condition to hold good enough for our purposes?
## Answer by emcor (score 2)
https://quant.stackexchange.com/a/14484
The theorem which justifies the equality of the expectations is Radon-Nikodym theorem. It says:
$$E^P(DX)=E^Q(X)$$
where $D=dQ/dP$ is a change of measure process with $E(D)=1$, $D>0$ and $Q\sim P$.
Note that $Q$ is further specified as the special riskneutral measure under which $X$ becomes a martingale.
You can see it easily by writing out the expectations:
$$E^P(DX)=\int DX \cdot dP=\int X \frac{dQ}{dP}\cdot dP=\int X dQ=E^Q(X)$$
## Answer by user7056 (score -1)
https://quant.stackexchange.com/a/14526
I guess that the unclarity comes from the fact that hedge pricing is an incomplete model, as it does not take the FVA into account. Not from flaws of the mathematics used in the model.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.