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Financial Architecture Series, Part 8: Models and Models
By Russell Goyder PhD | November 21, 2013

The word "model" is used a lot in quantitative finance. We routinely say "the Black-Scholes model" or "the SABR model" and so on.

At thefreedictionary.com/model I see a list of 8 definitions of the word model, covering a variety of contexts from model railways to individuals on catwalks who could use a few square meals. The relevant entry for our purposes is:

A schematic description of a system, theory, or phenomenon that accounts for its known or inferred properties

In my experience, people usually use the word model in too narrow a sense. Yes, the SABR model is a model, but only for the evolution of the distribution of the par rate of a given swap. Yes, Black-Scholes model is a model, but only for the evolution of the distribution of the price of some asset. These specific variables are almost never enough to do anything useful like price a derivative. For example, if we are pricing an option or swaption, how is the forward curve modelled? What assumptions do we make about repo rates and dividends? How will we discount?  In other words, how do we model the time-value of money?

It gets worse. What about a portfolio with multiple types of asset in multiple currencies? For each underlying asset, be it equity, currency, inflation level or anything else, and each interest rate in each currency, we will likely assign a well-known model for the dynamics. Quadratic Gaussian for the USD rates smile, LSV for exchange rate X and so on. But in the minds and vernaculars of many, this remains a collection of distinct models - it's as if models need names to be models!

In fact, the full set of modelling assumptions we make about the universe, or at least the small part of it that is relevant to financial valuations, conforms to the definition above and forms our complete model for the given valuation. The sum total of everything we assume is our model. This encompasses the assignment of specific stochastic processes to underlyings as we did just now, but goes much further.

For example, how is the discount curve built in each currency? Are we using discount curves consistently across the portfolio? What about different curves for different collateral agreements? What about Libors and other curves? What is the structure of the FX rates used to connect each currency? Which discount curves are used to calculate forward FX rates? How are the distributions of individual underlyings correlated - or do we assume independence, either explicitly or implicitly by valuing different parts of the portfolio separately? Even if it is a linear portfolio, what about counterparty exposure, CVA, FVA etc?

Of course, people have discount curves and use them, as they do Libor curves, FX rates, correlations and much more. But such modelling assumptions are often made implicitly - they are rarely considered as aspects of the total model for everything relevant to the portfolio's valuation. This in turn can lead to the use of inconsistent models and mistakes in valuation. In contrast, by calling such assumptions out explicitly and managing them in one place, we can scale up to solve big, portfolio problems, and remain self-consistent in the face of the complexity that goes with such territory.

For this reason, we promote this idea of a full set of relevant modelling assumptions to first class status within any valuation architecture, and name the concept Model, with a capital M:

Model: The set of modelling assumptions required for a valuation. 