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Archive for October 2009

Einstein meets Markowitz: Relativity Theory of Risk-Return

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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}

to

\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 X_i, the linear combination

X = \sum_i \omega_i X_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

X_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

|X|^2 = g(X,X) = \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 X and X_0, we can also look at their difference

\bar{X} = X - X_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

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