Just as statistics is really an elaborate form of a particular kind of probability activity over sufficiently large numbers, then so too am I beginning to see the volatility of equity derivatives and the credit spread of the fixed income world as two other distinct kinds of mathematical context within which you can find probability theory applied. And of course, probability is useful to us insofar as it can place a number on something ultimately unknowable. Albeit a known unknown ('risk' or measurable uncertainty, in the Knightian sense). My main point here is that credit spread and volatility, the two great inputs into fixed income and volatility modelling, are brothers.

## Sunday, 24 February 2013

### Sky, carpet, eye, rule-set, game-play

In my head, the analytics behind fixed income have always seemed a lot more certain than the uncertainty expressed in the volatility world - equity derivatives, convertibles, exotics, etc. This distinction had been a top-to-bottom one, since the fixed income depended on more established mathematics, largely based on traditional calculus and algebra, whereas the new derivatives mathematics is based on stochastic calculus which had only been discovered in the 1950s. On top of that, fixed income operated in a world of coupon cash flows, with only the chance of default as the ultimate

*deus ex machina*lurking in a corner somewhere. The pay-off diagrams were similarly predictable and had fewer dimensions of risk.
In reality there's a less clear distinction. Indeed, by dealing with the phenomena of fixed income in a mathematically straightforward way, there's an argument that it could have a tendency to deceive practitioners into believing their world was a lot less uncertain than it actually was. There's no danger of that with the world of so-called volatility products (equity options, exotics, volatility and variance swaps).

But in both cases the quantitative analyst undertakes to model a number of uncertainties via concrete finite numbers which encode some element of uncertainty about the world, and nowhere is this more important to realise than in the idea of the

*credit spread*.
Before elaborating, I'll generalise - what we're doing is taking a part of the world and, somewhat like scientists, trying to model it with concrete numbers. Those concrete numbers, in the context of the world of financial contracts, represent a theory about how the life of a financial contract will play out. Now clearly there are so many dimensions of uncertainty around two or more parties engaging in a financial contract that it ought to be always in mind just how many things can go wrong. In short, the sky could fall down on your head, the carpet could be pulled from below your feet. your eyes could be deceiving you, your counter party might start playing a different game to the one you started playing or you could discover you made a game-play tactical or strategic error. This is as useful a broad classification as I've seen. I will refer to them as the errors in the

**sky**,**carpet**,**eye**,**rule-set**and**game-play**. The classification is of course arbitrary and the probabilities associated with them vary from country to country, from time to time.
Examples of sky surprises - hyper-inflation blow out all the expectations you had when the contract was initiated as to the value of future cash flows. A major political revolution transforms the meaning of ownership.

Examples of carpet changes. Dramatic legislative changes in property or tax law undermine some implicit assumptions which went into the analysis which leads you to sign the financial contract. Governments introduce new policy which re-contextualises the value of your existing base of contracts.

Examples of rule-set changes. Liquidity in your contract dries up, the government manipulates the calculation methodology of a key price index, a run of scandals leads to certain classes of contract being structurally re-priced.

Examples of game-play changes. You realise your estimate of the likelihood of a future event is dramatically wrong. An M&A action causes little-read and little understood legal provisions to be interpreted in a dramatically unfavourable light. Corporate so-called

*agency*issues result in unexpected corporate behaviour. Events which had been uncorrelated at the time of the contract's inception have become inextricably linked. Your institution changes the magnitude of the risk premium attached to its ownership of your financial contract.
In short, the world is a complex and unpredictable place. It ought not be a great surprise to learn that a financial model of the contract, together with those parameters which act as inputs into the model, will at any point in the life of the contract, be more or less real, accurate, close to reality.

Of course, this knowledge is what causes model builders to try to tie their models to the always-moving markets from which elements of the state of the world can be approximated from the values of certain market prices. In other words, this knowledge of the scale of the uncertainty compels model builders to make them real-time.

There's a philosophical question lurking here. How do we evaluate a model's usefulness during the life of a contract? Is there a correspondence to reality - the degree of closeness in proportion to the model's usefulness? Or perhaps we are not justified in speaking of such a correspondence view of science (and financial engineering). Operationally, practitioners behave as if they're acting under just such a correspondence perspective; in which the name of their game is to get a model which is the closest statement to reality. Or is there no such simple correspondence with reality to be expected?

Leaving that aside for now, the more prosaic question concerns how to articulate a model such that it contains a bunch of parameters which can, in theory at least, be correlated with a moving set of market variables that can be implied by various market prices. But how can you turn uncertainty about the always-unfolding future into concrete numbers? Well, you just can. Probabilities do this, standard deviations (volatility) do it also, and

*credit spread*also does it.
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