Einstein meets Markowitz: Relativity Theory of Risk-Return

When working with Gaussian processes, the observation that you can interpret covariance geometrically comes in very handy (see Visualizing Market Risk: A Physicist’s Perspective). Of course, when you’re given a new toy, you’ll want to take it apart. In this post, we’ll extend the analogy

$\text{Risk}\leftrightarrow\text{Geometry of Space}$

$\text{Risk-Return}\leftrightarrow\text{Geometry of Space-Time}.$

The basic idea is to recall that Gaussian processes form a vector space. For instance, given Gaussian processes $dX_i$ , the linear combination

$dX = \sum_i \omega_i dX_i$

is also a Gaussian process. Gaussian distributions are particularly nice because everything you can know about them is encoded in the two parameters $\mu$ (mean of the process) and $\sigma$ (standard deviation of the process).

Each of the differentials may be expressed as

$dX_i = \mu_i dt + \sigma_i dW_i$

where $dW_i$ is a standard Brownian motion with $\mu = 0$ and $\sigma = 1$ .

We can interpret the $dW_i$ as spanning a cotangent space of some risk manifold.

If you have $N$ such processes, the dimension of the space they span depends on the rank of the covariance matrix

$\Sigma_{i,j} = \text{cov}(dW_i,dW_j).$

The matrix $\Sigma$ is symmetric and positive semi-definite. However, if the rank of $\Sigma$ is $n$ , we can find $n$ uncorrelated differentials $de_i$ and construct a covariance matrix

$g_{i,j} = \text{cov}(de_i,de_j) = \delta_{i,j}.$

This matrix is symmetric and positive definite. Therefore, on the space spanned by the $de_i$ , we can think of the covariance matrix $g$ as a metric tensor.

Now, we can re-express any Gaussian process via

$dX_i = \mu_i dt + \sum_j \sigma_{i,j} de_j.$

Comparing this with the previous expression we see that

$\sigma_i dW_i = \sum_j \sigma_{i,j} de_j.$

Furthermore,

$\sigma^2_i = \sum_j \sigma^2_{i,j}$

which is just the familiar expression for the variance of the sum of uncorrelated processes.

The neat thing comes when you bring $dt$ back into the picture.

Since we can interpret the covariance matrix $g$ as a metric tensor on a space, we can extend this to a Lorentzian metric on spacetime by specifying

$g_{t,i} = g(dt,de_i) = 0$

and

$g_{t,t} = g(dt,dt) = -\frac{1}{c^2}.$

With this extension of the metric, we have

$|dX|^2 = g(dX,dX) = \sigma^2 - \frac{\mu^2.}{c^2}$

Therefore at each point of our manifold, we have a risk-return cone in analogy to the light cone of special relativity with a velocity-like value given by

$c = \frac{\mu}{\sigma}.$
Risk Return Cone
Note that $\sigma$ is the radius of the risk-return cone at the particular security.

If you have two processes $dX$ and $dX_0$ , we can also look at their difference

$\bar{dX} = dX - dX_0.$

Then the relative process

$d\bar{X} = (\mu-\mu_0) dt + \bar{\sigma} d\bar{W}$

gives rise to a relative risk-return cone

Relative Risk Return Cone II

Note that $\bar{\sigma}$ is the radius of the relative risk-return cone.

The relative risk-return cone has a velocity-like value given by

$\bar{c}= \frac{\mu-\mu_0}{\bar{\sigma}}.$

Now, this applies to finance by letting $X_i$ be the log of the price of some security and letting $X_0$ be the log of the price of some benchmark. With this financial interpretation, the process $dX_i$ is the return of the security and $d\bar{X}_i$ is the excess return.

The “velocity” of the security $X_i$

$c_i = \frac{\mu_i}{\sigma_i}.$

is known as the Sharpe ratio and the “relative velocity” of the relative security $\bar{X}_i$

$\bar{c}_i= \frac{\mu_i-\mu_0}{\bar{\sigma}_i}.$

is known as the information ratio. The radius $\bar{\sigma}_i$ is the tracking error.

To further the analogy, you could define an absolute velocity for which all other light cones are compared. This absolute light cone is closely related to an investor’s risk aversion.

Written by Eric

October 11, 2009 at 10:27 am

Posted in Market Risk, Mathematical Finance

5 Responses

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something came up today : differential geometry meet finance http://arxiv.org/abs/0910.1671 (disclaimer i haven’t read the paper in detail yet, but that seems related to stuff you’re interested about)

benji

October 12, 2009 at 12:41 am

Reply
But…but…markets are not Gaussian!

You need to include jump processes, non-ergodicity, and you don’t want to penalize upward variance, so you need to use a quasimetric.

quantum probability

November 13, 2010 at 8:54 am

Reply
- Hi QP,
  
  You’re absolutely correct about markets being non-Gaussian. That is a very important and under appreciated point. That is why I use stable distributions in practice. For example, see
  
  https://phorgyphynance.wordpress.com/2009/08/08/daily-sp-500-value-at-risk-estimates/
  
  and
  
  https://phorgyphynance.wordpress.com/2009/08/06/80-years-of-daily-sp-500-value-at-risk-estimates/
  
  Most everything I presented here can be generalized in ways similar to what you are suggesting.
  
  phorgyphynance
  
  November 13, 2010 at 9:05 am
  
  Reply
  - I don’t see how you can easily generalize to dependent processes, especially multiply dependent. How would you form a vector space then?
    
    Moreover the prices add up to Gaussian over most short time periods but not over long time periods.
    
    quantum probability
    
    November 18, 2010 at 4:55 am
  - What do you mean by ‘stable distribution’ ?
    
    Those links don’t clear it up for me.
    
    quantum probability
    
    November 18, 2010 at 4:56 am

Phorgy Phynance