Numeraires, Risk-Neutral Measures, and Asset Price Martingales
Summary
The post outlines a textbook route from a risky asset’s stochastic differential equation to risk-neutral pricing. It describes choosing a numeraire, forming the risky asset’s value relative to it, applying stochastic calculus, changing probability measure with a Radon–Nikodym derivative and Girsanov’s theorem, and representing the resulting process as a martingale. The author asks whether any positive asset can serve as numeraire and how the market price of risk relates to the measure change.
A further question concerns why the asset’s drift becomes the risk-free rate under the risk-neutral measure if the aim is to obtain a martingale. The post does not provide answers or resolve its displayed formulas. It is a set of conceptual questions rather than a derivation, so readers should consult a consistent model treatment for assumptions on the numeraire, integrability, and existence of an equivalent martingale measure.
Key ideas
- A numeraire expresses asset values in units of a chosen positive reference asset.
- The post traces how a measure change can transform relative asset prices into martingales.
- It asks how the market price of risk, Radon–Nikodym derivative, and Girsanov theorem fit together.
- The risk-neutral asset may have risk-free drift while its discounted value is a martingale.
- The document poses conceptual questions but does not supply a derivation or answers.
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Full text
# Understanding the asset pricing theory and numeraire
# Understanding the asset pricing theory and numeraire
While reading about asset pricing theory and numeraire, I had faced some confusion.
### Short summary of asset pricing theory from my book
- We start our journey with a risky asset $S_t=\mu S_tdt+\sigma S_t dW_t$.
- Define a risk-free stochastic process, $M_t=r_t M_t dt$ (as numeraire).
- Form a relative price process or Quotient process $f$ (discounted value of risky price).
- Use Ito’s product, or quotient lemma to find $df$.
- State the results of Girsanov’s theorem, get $d\widetilde{W}_t$.
- Form a new stochastic process under $\mathbb{Q}$, and show that it should be driftless.
- Implement the Martingale Representation Theorem. State equation of $f$ in terms of $\varphi$ $(f_t=f_0+\int_0^t\varphi_u d\widetilde{W}_u)$.
- Substitute $d\widetilde{W}_t$ back in to the original risky stochastic differential equation $dS_t$. Rearrange, cancel stuff, factor, and get final $\mathbb{Q}$-martingale.
### Breakdown the issues
- In step $(2)$, Can we choose any other asset with price process $N(t)$ such that $N(t)>0,$ for all times $t$?
- In step $(5)$ we can get a probability measure $\mathbb{Q}$ which is called an equivalent martingale measure by Radon-Nikodym derivative $\left(Z=\frac{d\mathbb Q}{d\mathbb P}\right)$, $$Z := \exp\left( -\int_0^s \lambda du - \frac{1}{2}\int_0^s \lambda^2 dW_s \right)$$ Where The value $\lambda$ is called the market price of risk and it will have a different formula for different market models. In our book the numeraire was taken as risk-free asset and directly defined the $\lambda:=\frac{\mu_t-r_t}{\sigma_t}$ saying that $\lambda$ is the coefficient of $dt$ that occurs when you do the re-arranging once $dW_t$ is isolated by factoring $\left(df=\sigma_t f \left(\frac{\mu_t-r_t}{\sigma_t}dt+dW_t\right)\right)$. Is it always work like that? I didn't get any intuition for that. Like then what's the role of Radon-Nikodym derivative and Girsanov’s theorem ( Guarantee the existence of such measure? ) at all?
- In step $(8)$, we get $dS_t=r_t S_t dt +\sigma_t S_t d\widetilde{W}_t$ where the drift of the risky asset $\mu_t$ become the risk free interest rate $r_t$. Isn't our goal was to make it martingale (driftless SDE) from the very beginning? To extract some useful information from $\mathbb E_t^\mathbb{Q}[-|\mathcal F_t]$.
Sorry to put that so many question in a single thread but as all of them are inter-related I was expecting it won't go against the community rules.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.