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The First Global Financial Crisis of the 21st Century – Part 2 (June-December 2008)

The First Global Financial Crisis of the 21st Century – Part 1

From left – Dr. Volf Frishling (NAB Capital), Joe Maisano (TTA), Michael Reznik (MathRidge) and Dr. Alex Radchik (TTA, Seated)

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“Statistics are like a bikini. What they reveal is suggestive, but what they conceal is vital” – Aaron Levenstein

On the 29 of March 1900 at the Sorbonne in Paris a young Ph.D. student was defending his doctoral dissertation titled “The Theory of Speculation”. It was a mathematical treatment of the observed fluctuations of French government bonds and their options. The main conclusions of the study were that:

1. there are no discernible trends in the market
2. the market has absolutely no memory of what it has done in the past
3. the difference of the logged returns are normally distributed

To this date these results still stand as the null hypothesis of modern finance and economics.

In recognition of the great work done the thesis was awarded the insulting grade of {mention honorable} instead of the usual {mention tres honorable}. The student, Louis Bachelier.

In retrospect, perhaps it is not too surprising that Bachelier was treated so shamefully by the academic intelligentsia. After all, at the start of the 20th century everyone still believed that we live in a world that is perfectly predictable and explainable. For Bachelier to come out and make the assertion that financial markets were totally random was close to heresy. It took Einstein, Heisenberg, Schroedinger, Dirac and many other great minds and the development of Quantum Mechanics to finally rid us of the deterministic Newtonian viewpoint.

To the first approximation Bachelier’s theory of market returns fits the observed data reasonably well. However upon closer inspection it becomes obvious that the market marches to a different drummer. Even to this day some people argue that there appears to be some evidence suggesting trends and business cycles, in at least, some segments of the market. Furthermore, Benoit Mendelbrot presented strong evidence that the differenced logged returns are not exactly normally distributed but have rather fat tails. As the recent Asian currency crisis showed us: the market might, or might not, have a long memory, but it certainly has a short temper.

On a quiet afternoon at the University of Chicago in the early 1950′s a Ph.D. student got an idea: could investors target risk and return levels by taking the partial derivatives of the correlation coefficients between the various asset classes. Consistent, systematic application of such procedures could provide investors with a portfolio that that would generate similar levels of return to their existing portfolio, but at a lower level of risk. Initially the idea was received with little enthusiasm. After all, who would be prepared to sit down with pencil and paper and do all the required calculations. Nevertheless, after some early hesitation as to the usefulness of such a model, the student: Harry M.Markowitz, was finally awarded a Ph.D in 1954.

Without the computer revolution that has accrued in the last few decades Markowitz’s mean/variance model would have languished in relative obscurity. However, due to the ubiquitous availability of today’s inexpensive computers with phenomenal processing power and sophisticated software, even the largest monster portfolio problem can be reduced to a tame and purring kitten.

“One of the greatest pieces of economic Wisdom is to know what you do not know” – John Kenneth Gaibraith

The development of Modern Portfolio Theory, the Theory of Efficient Markets, the understanding of risk/return relationships and the importance of portfolio diversification stands as one of the most valuable analytic tools in finance. On Wednesday, October 17, 1990 Markowitz shared the Noble prize in Economics with Merton Miller and William Sharpe. Not bad for an idea that was worked out in one afternoon.

Dated October 1970, a paper titled “A Theoretical Valuation Formula for Option, Warrants and Other Securities” was rejected for publication by the Journal of Political Economy. Soon afterwards the same paper publication was rejected for publication by The Review of Economics and Statistics. Neither journal even bothered to have the paper reviewed. After another three years (May/June 1973) and some minor revisions, the paper was finally published titled “The Pricing of Options and Corporate Liabilties”, authors: Black, F. and Scholes M.

“A man will fight harder for his interest than for his rights” – Napoleon

Armed with their newly formed option pricing formula it took them no time in finding a suitable candidate for their first field test: National General’s new warrants. Using historical data for their parameter estimations it appeared that the warrants were considerably underpriced. Without any further delay they bought a considerable parcel of these warrants, only to watch their prices tumble.

Two facts came out of the exercise. The first being that when estimating volatility it is not a good idea to use historical data, rather one should use the market’s implied volatility. The second, and more important fact: even if one does everything correctly, the market is still a harsh mistress.

Tuesday, October 14 1997 Professor Myron Scholes is formally notified that he is to share this years Nobel prize in Economics with Professor Robert Merton for their contribution to the $50+ trillion derivative market. Although overjoyed by the news, both Professors expressed their sadness that Fisher Black could not have shared their triumph. Black died in 1995 from throat cancer.

What conclusions can be drawn? Perhaps it is that great ideas, like great wines, take years to be truly recognised.

This article was written by Dr. Adam Kucera while a senior quantitative analyst at Commonwealth Bank of Australia.

The difference with a CMS is that the two rates are on such a different basis, they are not even in the same market. A common CMS would be to swap a quarterly or semi-annual BBSW rate against a two, three or five-year interest rate swap. The BBSW rate would be reset on each swap roll, as would the swap. The difference is in the tenors of the instruments – while the BBSW swap rate applies only until the next rateset date, the swap rate still may have years to run. It is this difference in the underlying rates that turns a very simple concept (from a product point of view) into a reasonably involved modeling exercise.

It is because of this mix of short-term resetting on long-term rates that the CMS is a useful instrument. It gives investors the ability to place bets on the shape of the yield curve over time – if the yield curve steepens, rates on longer swaps will increase, thus increasing the spread to BBSW. Portfolio managers can also use the instruments to hedge a floating rate debt without introducing duration risk from the hedging instrument. This is due to the fact that the fixed ‘coupon’ on a CMS is periodically reset, so the duration on the fixed rate stays constant.

When pricing the swap, we can follow the same high-level approach as with a basis or fixed/floating swap. The fair value will be the value of the short-rate side plus the value of the long-rate side (where one is paying and the other receiving). The only hard part is coming up with the rates on the long-rate side.

Let’s say, for example, we have a three-year CMS where we are are paying semi-annual BBSW plus a margin and receiving the 3 year semi-annual swap rate. Over the 3 year life of this trade, we will have 6 swap rolls, or rate reset dates. For the short-rate (BBSW) side, we can use implied forward rates from a yield curve. For the receiver side, we need a three step process to come up with the rates.

Firstly, we start with par swap rates for each roll – the rate being the fixed swap rate for a theoretical swap starting at the roll date and maturing 3 years after. It would be tempting to think that we could use these rates for the pricing, and indeed, it would not necessarily be a bad approximation. To obtain accuracy, though, we must take into account the different maturities of the respective rates – to compare apples with apples, we must adjust the swap rates to be comparable with the shorter rates.

The first adjustment we need to make is called a ‘convexity adjustment’. A simple interest rate and a swap rate have different sensitivities to underlying curve movements, and therefore we cannot directly compare one rate with another. Consider a six month bill. The relationship between the rate and price of the bill is non-linear and, if you were to plot price according to rate, the shape would be ‘convex’. Similarly, the relationship between a swap rate and price is also non-linear, and also ‘convex’. However, because the rate of curvature in the two relationships differ, we need to make the convexity adjustment. As the adjustment relates to curvature, we need to use a little calculus to find the first and second derivatives of the swap, then also take into account the volatility of the swap rate. Because of these requirements, the pricing of constant maturity swaps becomes a non-trivial exercise.

The second adjustment we need to make is a ‘timing adjustment’. We need to perform this step whenever we are dealing with in-arrears CMS. Again, this adjustment relates to the difference in nature of the two rates. A vanilla swap will take into account paying interest in arrears, as each leg covers the life of the rate for that leg. Calculating a payoff between a short rate and a swap rate, on the other hand, does not adequately reflect the lag in payments throughout the life of the longer rate. In order to make a timing adjustment, we need to know the correlation between the two rates and the volatilities of both the short and long rates.

During the description of the adjustments above, we mentioned the volatility of the rates. Any use of volatility when modelling an instrument will invariably lead to discussions as to the accuracy and appropriateness of the method. Since volatility, like value, is intangible, we cannot expect to find accurate numbers on any vendor screen. Those volatilities that are widely available are subjective, often averaged, and easily manipulated. One way of coming up with the volatility we need is to construct an at-the-money volatility surface based on inputs implied from the prices of exchange traded options, then interpolate from this surface using the tenor and frequency of the swap. This method is very suitable for buy-side and back-office use as it requires no subjective input such as skews or margins. There are of course, many variations and alternatives including incorporating volatility smiles, etc.

While the convexity and timing adjustments obviously depend on a number of factors, including volatility, steepness of the yield curve and correlation, it would not be unreasonable to expect a combined adjustment in the order of 2 – 10 basis points on each rate adjusted. This makes a significant contribution to the pricing of the CMS.

Once we have adjusted the swap rates, we can price the CMS as an ordinary vanilla swap, but using our new adjusted rates in place of BBSW for one of the legs. Typically, for quoting a CMS, you would solve for a margin above BBSW (usually after adding a margin to the swap rate that you are setting from) such that the fixed and floating sides had equal value. This would be the fair-value margin to use in a quote to clients. Buy-side clients who are revaluing a CMS would enter the swap rate and BBSW margins, and solve for a market value using a current yield curve, volatility surface and correlation figures.

The non-quant guide aims to be a cursory overview of financial markets concepts and not an in-depth analysis. This article contains many omissions regarding the mathematical details of pricing constant maturity swaps.

I recently had the good fortune to work on an especially challenging and interesting client engagement in the field of Energy Risk Management. Our client was an electricity retailer, not the largest in the state, but not the smallest either. The challenges that this client faced appear to be shared amongst the majority, if not all, of the energy retailers in New South Wales. With Full Retail Contestability (FRC) now in place, and the Enron memories (and lawsuits) still fresh in everyone’s mind, Energy Risk Management is increasingly being placed in the spotlight. With such a high political profile and large financial stakes, electricity supply and sales must be insulated from disasters arising from poor risk management. Any problems with systems, methodologies and/or procedures had to be identified and plans put in place to address such problems.

The first thing that some electricity industry people will tell you about is the volatility of the pool price. Prices that are a steady $40 that jump to $4000 in less than half and hour and lose (or make) institutions tens of millions of dollars in a day or two. If you get it very wrong in the financial markets you can generally say good-bye to bonuses and perhaps even the desk head. If you get power markets very wrong it can be time to switch off the lights and hire the lawyers.

Listening to some of the anecdotal evidence, it all sounds a bit like the ‘wild west’. A bunch of refugee traders from the financial markets turning their hand to speculating with energy derivatives, risk managers who can’t accurately price the derivatives because the models aren’t there, governments changing the rules every so often while freak bushfires send prices screaming into the stratosphere. Saddle up and enjoy the ride! Fortunately, it is not nearly as gung-ho as some people would have you believe.

While it is true that many traders were not long ago in the employ of banks or brokers, there are also some very seasoned energy people calling the shots. Also, there is very little speculation in electricity trading. In fact, the word ‘trading’ is a bit of a misnomer. ‘Hedging’ would be a far better word to describe what these people do. Alas, if the energy trading desks were renamed as hedging desks, it would probably become more difficult to attract new talent.

The trading environment of energy derivatives is also very different to that of the financial markets. Probably the biggest difference is the extraordinary lack of liquidity in the market. Energy retailers need to negotiate and document an International Swaps and Derivatives Association agreement (an ‘ISDA’) with each counterparty before they can trade with them.

This can be an involved process, taking months. Further to this restriction, there are only a handful of potential counterparties out there. If this is not limiting enough, each counterparty will only be interested in doing deals that fit in with their business. In the financial markets, you have plenty of brokers who will offer you a price on almost anything – the price may vary with the brokers focus and position in any particular market, but a price will be offered. In energy markets, your potential counterparties are disparate organisations – generators, retailers and the ‘pool’ itself. A generator will happily offer you a swap agreement on electricity, as they are a producer. A retailer on the other hand will be less likely to do this, as they would be exposed to the pool price on the variable side of the swap. With the huge potential price spikes, any margin made generally will not cover the risk.

There are also some difficulties in obtaining a fair price for the derivative instrument that an energy retailer would like to trade. When trading swaps, as is in the case with interest rate swaps, the forward curve is critical. Aside from the structure of the deal itself, the curve is solely responsible for implying fair value. In electricity markets, there is a forward curve, but this curve is regarded as rather dubious by many in the industry.

Even more challenges exist for pricing option-based products such as caps and floors. This time there are three points of influence: the forward curve, volatility and the pricing model itself. In a very liquid market, volatility can be implied from the prices trading at any one moment. This can’t be done in electricity markets as it is a very illiquid market. Historical volatility is one option, but while historical volatility is generally unreliable for pricing in most markets, it is even worse in power markets due to periodic regulation changes. The upcoming introduction of FRC will no doubt affect future pool price volatility, and given that instruments with a three year maturity are commonly dealt, it will no doubt affect the pricing.

Which leads us to the pricing model itself, a very interesting topic with many points of view available. In most markets, including equities, foreign exchange and interest rate derivatives, there is a benchmark. While many models exist and each have their supporters, the Black-Scholes pricing model, or slight derivatives thereof, is regarded as benchmark pricing. Of course, this model is of little use for more complex derivatives such as ‘path dependent’ and ‘multi-factor’ instruments, but such markets have had the benefits of many great minds working on them for many years, and in general traders can rely on the models as accurate.

As a stark contrast, the electricity industry has as yet failed to adopt a standard pricing model. While price spikes can be huge and unpredictable, this isn’t really the problem. The problem is more about what factors are influencing these pricing movements, and the fact is that there are many. Electricity is a highly ‘seasonal’ market, with different price and volatility behaviour in summer as to winter (and different to ‘neither’, which is energy-speak for the combination of spring and autumn). There is also an interesting daily profile, with a much smaller amount of power being used before 6am and after 8pm than in the middle of the day.

The problem can be looked at as three-fold:

Firstly, there are financial constraints. While a large bank can make a great deal of ‘fee’ income from the margins on correctly priced derivatives, electricity retailers are operating with far tighter margins. Because of this, the retailers have not had their own teams of PhDs working on this for the last five years to come up with a solution. Up until now, the work done has largely come from academically-based projects.

Secondly, this is a far newer market than equities and foreign exchange. Even the relatively new interest rate derivative market has had a good twenty-five years to mature.

Thirdly, many of the trading and risk professionals working in energy markets started their careers in the financial markets. Energy market derivatives pose a different set of issues requiring a ‘back-to-basics’ approach, and it is often difficult to ‘forget what you know’.

So without a benchmark pricing model, it becomes difficult to asses ‘fair value’. Add to this the extraordinary lack of liquidity, and fair value itself becomes a rather academic point. What does this then mean for a Value at Risk (VaR) calculation, which essentially relies on a portfolio being priced at fair value?

As the Energy Market matures and grows, and as more work comes to fruition within the industry and academic groups, the current difficulties will be replaced by new challenges. An increase in liquidity will no doubt give rise to an increase in competition. More consistent pricing models may reduce margins and lead to a more commoditised market (as happens with all OTC markets eventually). Until such time however, there is a lot of work to be done and it is an exciting place to be.