Phorgy Phynance

This post is part of a series

• Network Theory and Discrete Calculus

As stated in the Introduction to this series, one of my goals is to follow along with John Baez’ series and reformulate things in the language of discrete calculus. Along the way, I’m coming across operations that I haven’t used in any of my prior applications of discrete calculus to mathematical finance and field theories. For instance, in the The Discrete Master Equation, I introduced a boundary operator

Although, I hope the reason I call this a boundary operator is obvious, it would be more precise to call this something like graph divergence. To see why, consider the boundary of an arbitrary discrete 1-form

A hint of sloppy notation has already crept in here, but we can see that the boundary of a discrete 1-form at a node is the sum of coefficients flowing into node minus the sum of coefficients flowing out of node . This is what you would expect of a divergence operator, but divergence depends on a metric. This operator does not, hence it is topological in nature. It is tempting to call this a topological divergence, but I think graph divergence is a better choice for reasons to be seen later.

One reason the above notation is a bit sloppy is because in the summations, we should really keep track of what directed edges are actually present in the directed graph. Until now, simply setting

if there is no directed edge from node to node was sufficient. Not anymore.

Also, for applications I’ve used discrete calculus so far, there has always only been a single directed edge connecting any two nodes. When applying discrete calculus to electrical circuits, as John has started doing in his series, we obviously would like to consider elements that are in parallel.

I tend to get hung up on notation and have thought about the best way to deal with this. My solution is not perfect and I’m open to suggestions, but what I settled on is to introduce a summation not only over nodes, but also over directed edges connected those nodes. Here it is for an arbitrary discrete 1-form

where is the set of all directed edges from node to node . I’m not 100% enamored, but is handy for performing calculations and doesn’t make me think too much.

For example, with this new notation, the boundary operator is much clearer

As before, this says the graph divergence of at the node is the sum of all coefficients flowing into node minus the sum of all coefficients flowing out of node . Moreover, for any node there can be one or more (or zero) directed edges from into .

Doyne Farmer is awesome. I first ran into him back in 2001 (or maybe 2002) at the University of Chicago where he was giving a talk on order book dynamics with some awesome videos from the order book of the London Stock Exchange. He has another recent paper that also looks very interesting:

Leverage Causes Fat Tails and Clustered Volatility
Stefan Thurner, J. Doyne Farmer, John Geanakoplos
(Submitted on 11 Aug 2009 (v1), last revised 10 Jan 2010 (this version, v2))

We build a simple model of leveraged asset purchases with margin calls. Investment funds use what is perhaps the most basic financial strategy, called “value investing”, i.e. systematically attempting to buy underpriced assets. When funds do not borrow, the price fluctuations of the asset are normally distributed and uncorrelated across time. All this changes when the funds are allowed to leverage, i.e. borrow from a bank, to purchase more assets than their wealth would otherwise permit. During good times competition drives investors to funds that use more leverage, because they have higher profits. As leverage increases price fluctuations become heavy tailed and display clustered volatility, similar to what is observed in real markets. Previous explanations of fat tails and clustered volatility depended on “irrational behavior”, such as trend following. Here instead this comes from the fact that leverage limits cause funds to sell into a falling market: A prudent bank makes itself locally safer by putting a limit to leverage, so when a fund exceeds its leverage limit, it must partially repay its loan by selling the asset. Unfortunately this sometimes happens to all the funds simultaneously when the price is already falling. The resulting nonlinear feedback amplifies large downward price movements. At the extreme this causes crashes, but the effect is seen at every time scale, producing a power law of price disturbances. A standard (supposedly more sophisticated) risk control policy in which individual banks base leverage limits on volatility causes leverage to rise during periods of low volatility, and to contract more quickly when volatility gets high, making these extreme fluctuations even worse.

I completely agree with this idea. In fact, I discussed this concept with Francis Longstaff at the last advisory board meeting of UCLA’s financial engineering program. Back in December, I spent the majority of a flight back to Hong Kong from Europe doodling a bunch of math trying to express the idea in formulas, but didn’t come up with anything worth writing home about. But it seems like they make some good progress in this paper.

Basically, financial firms of all stripes have performance targets. In a period of decreasing volatility (as we were in preceding the crisis), asset returns tend to decrease as well. To compensate, firms tend to move out further along the risk spectrum and/or increase leverage to maintain a given return level. The dynamics here is that leverage tends to increase as volatility decreases. However, the increased leverage increases the chance of a tail event occurring as we experienced.

On first glance, this paper captures a lot of the dynamics I’ve been wanting to see written down somewhere. Hopefully this gets some attention.

I hope to begin some research into currencies. Before I come out with any result though, I thought I’d ask an open question and hope someone comes by with a response.

First of all, as a former scientist, thinking about currencies is very fun. See, for example, my previous article

• Currency exchange rates, directed graphs, and quivers

This morning, I happened across a recent article that appeared on the arxiv:

• Topology of the correlation networks among major currencies using hierarchical structure methods

The second paragraph really stood out:

One of the problems in foreign exchange research is that currencies are priced against each other so no independent numeraire exists. Any currency chosen as a numeraire will be excluded from the results, yet its intrinsic patterns can indirectly affect overall patterns. There is no standard solution to this issue or a standard numeraire candidate. Gold was considered, but rejected due to its high volatility. This is an important problem as different numeraires will give different results if strong multidimensional cross-correlations are present. Different bases can also generate different tree structures. The inclusion or exclusion of currencies from the sample can also give different results.

This is interesting because financial modeling is often about prices of securities or changes in prices. Currencies are about the relationship between prices. In graph theoretic (or category theoretic) terms, it is tempting to say that currency models should be about directed edges (or morphisms).

Is the best way to model currencies to choose some numeraire as is done in this paper? Or is there a way to study the relationships (morphisms) directly?

This post is in response to a question from Tim van Beek over at the Azimuth Project blog hosted by my friend John Baez regarding my paper

• Financial Modelling Using Discrete Stochastic Calculus

The basic fact needed to address the question is that we have a set 0-dimensional objects and 1-dimensional objects that obey the following geometrically-motivated multiplication rules:

To see the geometrical meaning of these multiplications, it might help to consider discrete 0-forms

and

Multiplication 1.) above implies

which is completely reminiscent of multiplication of functions where we think of as the value of the function at the “node”

Multiplications 2.) and 3.) introduce new concepts, but whose geometrical interpretation is quite intuitive. To see this consider

and

Multiplying the function on the left of the “directed edge” projects out the value of the function at the beginning of the edge and multiplying on the right projects out the value of the function at the end. Hence, the product of functions and edges do not commute.

In the paper, it was shown that the exterior derivative of a discrete 0-form is given by

Here, we show that this may be expressed as the commutator with the “graph operator”

The result is quite simple and follows directly from and the multiplication rules, i.e.

and

so that

or

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

to

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

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

Each of the differentials may be expressed as

where is a standard Brownian motion with and .

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

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

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

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

Now, we can re-express any Gaussian process via

Comparing this with the previous expression we see that

Furthermore,

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

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

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

and

With this extension of the metric, we have

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

Note that is the radius of the risk-return cone at the particular security.

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

Then the relative process

gives rise to a relative risk-return cone

Note that is the radius of the relative risk-return cone.

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

Now, this applies to finance by letting be the log of the price of some security and letting be the log of the price of some benchmark. With this financial interpretation, the process is the return of the security and is the excess return.

The “velocity” of the security

is known as the Sharpe ratio and the “relative velocity” of the relative security

is known as the information ratio. The radius 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.

In a recent article, Cheng and Madhaven from Barclays Global Investors published a good article on leveraged ETFs

The Dynamics of Leveraged and Inverse Exchange-Traded Funds
April 8, 2009

Check it out.

The Error

They begin from a fairly standard starting point

However, they proceed to state that since

“holds for any period”, then it follows that

where is the ETF NAV and is the leverage factor.

Unfortunately, that is not correct. The problem is that

only holds when is 1 day. Otherwise, we could let be 1 year and this would say that the 1-year return of the ETF is times the 1-year return of the index, which we already know is not true.

This should have also been obvious by plugging into their final expression

which violates the relation defining leveraged ETFs they started with. As a result of this error, their discussion of return dynamics in Section 4 must be re-examined

The Solution

The correct way to look at this is to let

and

If is 1 day, then

so that

If we assume is a geometric Brownian motion (as they do), then

where . With a slight abuse of notation, we can drop the indices and let

so that

This allows us to rewrite (using the definition of the binomial coefficient)

Noting that

and

we arrive at a disappointingly simple, yet important, expression

The expression above governing leveraged ETFs is the starting point for further analysis. We will come back to this in a subsequent post.

To be continued…

My last post:

Why do scientists go into finance?

started out as an introduction to this post, but grew a life of its own.

I am quite fed up with all the quant bashing we’ve seen in the media since the current crisis began. A recent article by Felix Salmon appeared in Wired Magazine:

Recipe for Disaster: The Formula That Killed Wall Street

I actually thought the article was quite good. I did take a small exception to the last paragraph though:

In the world of finance, too many quants see only the numbers before them and forget about the concrete reality the figures are supposed to represent. They think they can model just a few years’ worth of data and come up with probabilities for things that may happen only once every 10,000 years. Then people invest on the basis of those probabilities, without stopping to wonder whether the numbers make any sense at all.

I think this statement exaggerates the situation. I took it as a minor deviation from an otherwise good article. Unfortunately, he gleefully followed up his article with a reference to Paul Wilmott in:

When Quants Don’t Think

Wilmott is quite right that quants need to stop being so passive. But he also knows full well that they won’t be. It’s far too easy for them to go along with what everybody else is doing — and that’s exactly why the copula function turned out to be so disastrous.

As I alluded to in the last post, it really has nothing to do with going “along with what everybody else is doing”. It is more about doing what your boss tells you to do. It is wrong to point the finger at the quants when the real culprit was upper management. The quants had absolutely no authority and no incentive to reel in excessive risk taking. As I’ve said, the quants know quite well the strengths and the weaknesses of the models they employ. If you asked them if the models satisfactorily represented the risks in CDOs, they would quite plainly say “no”. Did anyone ask? Of course not. They were paid to build models and that is what they did. When the models expressed risk beyond what upper management thought it should be, the quants were forced to dumb down the models. Period. They were making too much money. In fact, as I’ve said in

CDOs and Risk Management,

CDOs were great because they represented a blind spot in existing risk management systems based on VaR. VaR essentially defines what a “tail event” would be given a particular financial model of a portfolio. Typically, this “tail event” is the loss at which you are not likely to exceed on average 1 out of 100 times. One out of 100 times, a CDO is not likely to lose ANYTHING. Hence, it’s risks do not show up on any risk system. This is what I refer to as “risk management arbitrage”. What these models do not tell you is the loss you can expect “if a tail event occurs”. As we now know, if a tail event occurs, a CDO can lose EVERYTHING.

Talking heads like Nassim Taleb are extremely unhelpful and are, in fact, counterproductive. He has been on the war path for years telling us that securities do not follow a normal distribution. HELLO!! We know that already. There is not one risk model I have ever seen (except some third party models that are not worth the disk space they use up) that is based on the assumptions that Nassim Taleb is attacking. This whole idea that quants assume normal distributions is completely bogus. Taleb has always been irritating to quants for his completely content-free rants and his straw man arguments. Now that we see him on Bloomberg almost every day, it is time someone puts him in his place.

It is true that the Black-Scholes model assumes returns are normally distributed, but that is pretty irrelevant. What the Black-Scholes model allows you to do is to convert a price into another number called “implied volatility”. The implied volatility shares a purpose similar to the yield of a bond. It helps you compare related securities. If two similar companies have outstanding bond issues with similar coupons and similar maturities, the yield of the bonds help determine which is the better value. Do people go on rants about how meaningless the “yield” is? No. The yield on a bond is based on a flimsier model than Black-Scholes and no one seems to care. To compute yield, you must assume the issuer is going to make every interest payment and that you can reinvest that interest payment at the same rate throughout the life of the bond. The yield is then the discount rate under this pretty bogus model that matches the theoretical price to the market price of the bond. When you really think about it, the yield of a bond is a pretty bogus concept. However, it does help make informed decisions about the relative valuation of bonds. The implied volatility serves precisely the same purpose. A fairly bogus model spits out a number that helps make informed decisions about the relative value of derivatives.

For Nassim Taleb to go on the media circuit complaining about black swans and the fallacy of normal distributions, it is as if a former bond trader suddenly decided to go on multi-decade crusade against the use of bond yields. Ok. We get it already.

It would be less disturbing if we didn’t see Paul Volcker going on the circuit parroting things that Taleb says. Toward the end of the video I posted on

A Call to all Finance Bloggers

we see Volcker referring to “black swans” and goes on a rant about financial engineers. His words are clearly straight out of Taleb’s mouth. They probably had dinner together at Davos or something. So now that Volcker has been infected by these bogus ideas, that is worrisome because Volcker obviously has a line to President Obama. The last thing we need is to get distracted by going after the quants.

Before leaving, let me turn some attention to Paul Wilmott. His recent turn against his own, i.e. his recent quant bashing, is disturbing. Paul Wilmott wrote some decent books on mathematical finance and derivatives pricing. But what he is probably more known for these days is his quant forum at Wilmott.com. Currently, there are over 66,000 registered users signed up at Wilmott.com. Long ago, I was a member myself and I even published my first finance-related research paper there. At the time, it was a decent place populated by knowledgeable quants. I got my first job on Wall Street due to the help of a friend I met on that forum. Unfortunately, in the last 5 years or so, the place has turned into a zoo. It is impossible to have a decent conversation there anymore as there is no form of moderation. He doesn’t care, just as long as more eyes turn to his web site. It is little more than a PR campaign.

Both Nassim Taleb and Paul Wilmott are opportunistically spouting nonsense about quants that really do a disservice to some really quite intelligent, experienced, and knowledgeable people who happen to be friends of mine.

In the last article, I showed how the Black-Scholes PDE is equivalent to a Wick-rotated Schrodinger equation describing a charged particle in an electromagnetic field. Here, I will expand a little bit on that.

In quantum mechanics, the “momentum operator” for a point particle is

.

To account for the interaction of the (charged) particle with an electomagnetic field, the momentum operator is augmented by the vector potential. The effect of this vector potential is to deform the wave function by a “gauge” factor

for some curve .

The gauge connection for the Black-Scholes PDE is given by

.

Inserting the corresponding gauge factor

into the Black-Scholes PDE results in

,

which is simply the heat equation from physics!

Therefore, the quest to categorify Black-Scholes is effectively a quest to categorify the heat equation.

What we want to do I think is to contruct a “yield curve space” in analogy to “loop space” where a point in yield curve space corresponds to a yield curve in some base space. We want to study Brownian motion on yield curve space under constraints of no arbitrage, which can hopefully be formulated as a statement about curvature.

Update:

In a comment, Blake Stacey points out this book. Digging a little bit turns up these arxiv papers and the author’s web page. It now appears likely that the author has done what I set out to do, but perhaps did not quite make the connection to “categorification” which is just a minor point. I’m tempted to order the book, but will definitely have a look at the arxiv papers in the meantime.

In this post, I will perform some computations to demonstrate a relationship between the Black-Scholes PDE and the Schrodinger equation of quantum mechanics. This relationship is not new. You can, for example, find something similar discussed here. The purpose for writing the gory details here is that I hope we (that’s right.. you too!) can “categorify” the Black-Scholes framework.

The Black-Scholes PDE is derived in a million places. One of those places is a paper I wrote in May 2002 and published on the quantitative finance web site Wilmott.com.

Noncommutative Geometry and Stochastic Calculus
Applications in Mathematical Physics

I derived it in a slightly unusual way via noncommutative geometry. Regardless of how it is derived, the Black-Scholes PDE is given by

One day, for some reason that I forget, I was curious whether this could be written in a form that looked like Schrodinger’s equation

so I started chugging away with some algebraic gymnastics. The first thing is to rewrite the equation with the time derivative on one side and everything else on the other.

Comparing this to the Schrodinger equation

tempted me to complete the square in the BS PDE resulting in

.

Now letting

,

where and

we find that the Black-Scholes PDE can be written as

.

Observation 1

The Black-Scholes PDE is a “Wick rotated” Schrodinger equation for a charged particle in an electromagnetic field, where the risk-free rate plays the role of a gauge connection.

Observation 2

The volatility plays the role of Planck’s constant while satisfies the commutative relation

.

Now let the games begin!

Note: Thanks to naumovic who pointed out in the comments below an algebraic mistake in an earlier version of this article.

One of the bedrocks of mathematical finance is the Black-Scholes equation. This equation helps to evaluate the fair price of stock options and involves stochastic calculus.

The Black-Scholes equation can be mapped to the Schrodinger equation. I have a writeup somewhere (or maybe on some forum somewhere) showing the details, but it is fairly straightforward to work it out. The analogy I want to point out is that the Black-Scholes equation can be thought of as modeling the dynamics of “point prices” just as the Schrodinger equation models the dynamics of “point particles”.

There are two primary financial instruments that populate any traditional portfolio: stocks and bonds. Stocks are described by a “point price” and hence stock options are governed by the Black-Scholes equation. Bonds are more complicated because there is no 0-dimensional “point price” for bonds. Bonds depends on a 1-dimensional “price curve”. There are models to describe the dynamics of 1-d “price curves”, but nothing has had quite the impact that the original Black-Scholes model did.

It might sound silly, but just as string dynamics seems to relate to a categorification of point particle dynamics as described here, I suspect one could develop a bond option pricing theory (for “price curves”) based on a categorification of the Black-Scholes equation (for “point prices”).

The above content was first posted as a comment over on the n-Category Cafe, but wanted to reproduce it here as well.