# Understanding the asset pricing theory and numeraire

## Short summary of asset pricing theory from my book

1. We start our journey with a risky asset $$S_t=\mu S_tdt+\sigma S_t dW_t$$.
2. Define a risk-free stochastic process, $$M_t=r_t M_t dt$$ (as numeraire).
3. Form a relative price process or Quotient process $$f$$ (discounted value of risky price).
4. Use Ito’s product, or quotient lemma to find $$df$$.
5. State the results of Girsanov’s theorem, get $$d\widetilde{W}_t$$.
6. Form a new stochastic process under $$\mathbb{Q}$$, and show that it should be driftless.
7. Implement the Martingale Representation Theorem. State equation of $$f$$ in terms of $$\varphi$$ $$(f_t=f_0+\int_0^t\varphi_u d\widetilde{W}_u)$$.
8. 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.

• You can take the value of any asset as numeraire as long as it is always positive. The ratio $\frac{\mu_t-r_t}{\sigma_t}$ is called price of risk or Sharpe ratio (or reward-to-risk ratio if you like). It's a key quantity in finance and often used for changes of measures. The goal of applying Girsanov's theorem is that the drift of the stock price equals its funding cost (interest rate minus potential dividend yield). The discounted (and reinvested) stock price is then a martingale under $\mathbb{Q}$ which is used for derivatives pricing. Commented Nov 14, 2022 at 18:47
• Notice that by the fundamental theorem of asset pricing all discounted assets should be martingales under the corresponding constructed measure. Girsanov is only a tool to find the corresponding measure. One can also view the discounting to compare different investments, this shows that it only makes sense to use assets that are tradable as discounting assets, or in other words, as numeraire.
– rrnl
Commented Nov 14, 2022 at 19:08
• To put it together, you indeed can choose a different N as long as it is a tradable asset in your defined market. The drift in the risk neutral asset under the change of measure compensates for the martingale restriction.
– rrnl
Commented Nov 14, 2022 at 19:08
• "Girsanov is only a tool to find the corresponding measure." - find the corresponding measure using Radon-Nikodym derivative, right? @rrnl What make me still in question was that we are not constructing the new measure or at least not get from direct computation (according to my book). Commented Nov 14, 2022 at 19:15
• If you are specifically interested in understanding the Girsanov theorem, it may be interesting to start from the Cameron Martin theorem. In this video this is explained in more detail. However, for application I would not take this approach. There, I would use the approach based on the fundamental theorem of asset pricing (see for example the last part of this lecture).
– rrnl
Commented Nov 14, 2022 at 21:13