# Ratio of real world to risk-neutral density

Suppose I have a risk-neutral pdf and a real-world pdf of an asset. Both functions are related by a scaling factor or the sdf which would transform the risk-neutral into the real-world density, is this correct? Would I be able to obtain this scaling factor by simply computing the ratio of the RW/RN density across X? If not, why?

The ratio of any two PDFs (if it can be defined: see below) is just the Radon-Nikodym derivative. The conditions for the Radon-Nikodym derivative to exist are given by the Radon-Nikodym theorem.

Whilst on a first read basis, the rigorous phrasing of the theorem might sound a bit abstract / difficult, the intuition is easy. The theorem says that given two probability measures $$\mathbb{P}_1$$ and $$\mathbb{P}_2$$, the Radon-Nikodym derivative $$\frac{\partial\mathbb{P}_2}{\partial\mathbb{P}_1}$$ exists if the two measures agree on "what is possible", which means they agree on where probabilities are zero and where they are non-zero (see footnote*).

Example 1:

Consider two log-normal random variables $$X$$ and $$Y$$, where:

$$\ln{(X)}\sim N\left(\mu -0.5 \sigma^2, \sigma\right), \quad \ln{(Y)}\sim N\left(r -0.5 \sigma^2, \sigma\right)$$

Let the PDF of $$X$$ be $$f_X$$ and the PDF of $$Y$$ be $$f_Y$$ (and the associated probability measures induced by $$X$$ and $$Y$$ are $$\mathbb{P}_X$$ and $$\mathbb{P}_Y$$). Note that the support of $$X$$ and $$Y$$ are the same: specifically the two random variables have positive densities on the real-line for $$x>0$$, whilst their densities are zero for $$x\leq0$$: therefore, these two random variables "agree on what is possible" (any $$x>0$$): they just assign slightly different probabilities to these outcomes. So the conditions for the Radon-Nikodym derivative to exist are satisfied and we can write:

$$\frac{\partial\mathbb{P}_Y}{\partial\mathbb{P}_X}=\frac{f_Y}{f_X}$$

Example 2:

Consider now a log-normal random variable $$X$$ and a normal random variable $$Z$$, where:

$$\ln{(X)}\sim N\left(\mu -0.5 \sigma^2, \sigma\right), \quad Z\sim N\left(\mu -0.5 \sigma^2, \sigma\right)$$

The problem here is that $$Z$$ has positive densities across the entire real line $$\mathbb{R}$$ whilst $$X$$ only has positive densities for $$x>0$$. So for the case where we want to go from $$\mathbb{P}_X$$ to $$\mathbb{P}_Z$$, the conditions for the Radon-Nikodym theorem are not satisfied and the Radon-Nikodym derivative does not exist. Indeed, if we tried to compute the ratio of the two PDFs, we would run into the following issues:

$$\frac{\partial\mathbb{P}_Z}{\partial\mathbb{P}_X}=\frac{f_Z}{f_X}=??$$

The derivative is clearly undefined $$\forall x \leq 0$$ due to division by zero.

*Footnote: strictly speaking, for $$\frac{d\mathbb{P}_2}{d\mathbb{P}_1}$$ to exsist, the theorem states that $$\mathbb{P}_2$$ must be absolutely continuous w.r.t. $$\mathbb{P}_1$$, which means that whenever $$\mathbb{P}_2(A)=0$$ for any $$A \in \mathcal{F}$$, it implies that $$\mathbb{P}_1(A)=0$$ (but the converse needs not to be true).

• Thanks for the reply. Assuming the Radon-Nikodym derivative exists, how would I go about computing it? Is it a matter of computing the ratio of probabilities for each point on the pdf? Commented Aug 7 at 10:30
• @OttoWinata: if you have the two PDFs expressed as functions, you can just take the ratio of the two functions. Say the risk-neutral PDF (denoted $f^{\mathbb{Q}}_X$ ) and the real-world PDF (denoted $f^{\mathbb{P}}_X$) are given as: $$f^{\mathbb{Q}}_X(h)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(h-r)^2}{2\sigma^2}} \quad f^{\mathbb{P}}_X(h)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(h-\mu)^2}{2\sigma^2}}$$ Then their ratio is $$e^{\frac{2h(\mu-r)+r^2-\mu^2}{2\sigma^2}}$$ If you only have the PDF at specific points, then indeed you just compute the ratio of the values at each point. Commented Aug 7 at 13:41
• Thanks again. So say I have a list of ratios across the range of X. How would these values relate to Girsanov's theorem where you go from risk-neutral to real-world densities through a change of measure? Would it be a reasonable approach to estimate the change of measure by fitting my data onto the equation laid out by the theorem? Commented Aug 7 at 19:19
• @OttoWinata: do you assume any specific dynamics for your asset? Standard Girsanov assumes log-normal dynamics of the asset (and the therorem is then applied to the Brownian within the GBM). Commented Aug 7 at 19:23
• The idea is that I have the real world density of returns of an asset which is estimated by fitting a kde on return data over a couple years. Then I'd multiply the kde by the radon-nikodym derivative to obtain the risk-neutral density. Does this approach make any sense? Commented Aug 8 at 0:11