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Mathematicians have an ability to think clearly and precisely that is rare among finance professionals. We’re excellently placed to provide model validation consulting services. Learn how I found a critical conceptual error in risk modelling work by one of the largest financial consulting firms in the world.

There are two main kinds of models that quantitative analysts are called on to validate in the financial services: derivative pricing models, and risk models.

#### Validating derivative pricing models

Much of derivative pricing theory is now pretty standard and well-worn. However, there are some choices to be made when validating the appropriateness of the choice of model.

Firstly, there’s the choice of whether to use a computationally slower but more accurate numerical model (such as Monte Carlo, local volatility or stochastic volatility), vs a fast but approximate analytical model. This choice arises with Asian option, where a fast analytic method is known (method of moments), but makes the assumption that the sum of lognormal distributions is lognormal (which is not actually true). Similarly, there exist analytic Black-Scholes formulae for pricing barrier options. However, these models assume that volatility and interest rates are constant. Since volatility term structure has a huge impact on the valuation of barrier options, these models sacrifice a lot of accuracy for speed and simplicity. Whether the trade off is worth it can depend on whether the model is being used to risk purposes (such as a market risk VaR calculation), or front office pricing.

Another issue that arises is the choice of volatility input. Since exotic options are typically not liquid enough to allow for the construction of an implied volatility surface, the use of the European volatility surface must be justified somehow.

Once a model is chosen, there is often no question, in principle, of *how* to price the derivative. Validating derivative pricing models is thus often mainly about checking the correctness of the coding implementation. A standard way to do this is to build a second, independent model against which to compare the output of the original model. Since it’s impossible to run the two models with all possible inputs, usually one would try to generate a set of test parameters which cover every significant discrete case, such as each possible ordering of date parameters and date coincidence. Another important step is to compare the behaviour of the model to the product description, as just because the two models agree does not necessarily mean they are correctly implementing the intent in the product description. Another important step is to check boundary cases, such as pricing very close to a barrier, very far from a barrier, or after knock-out/knock-in (in the case of barrier options).

An important step is checking the model under stressed scenarios, including very low or very high volatility, and near-zero or negative rates.

However, not all derivatives can be priced with a well-known and standard method. Monte Carlo and other numerical models can require careful work to ensure the model is converging correctly under all circumstances. Custom derivatives can arise which require some ingenuity to price. Examples like high-dimensional derivatives with a large number of underlying assets can require novel mathematics to price, as standard methods are simply not fast enough on current computer hardware. In some cases, pricing early exercise optionality is mathematically non-trivial and/or computationally challenging. As mathematicians, we’re excellently placed to help you price these bespoke derivatives.

See also our derivative pricing consulting services.

#### Validating risk models

We can build and validate financial risk models including operational risk, market risk and credit risk.

In some cases such as market risk, there are industry standard methodologies (see also our market risk consulting services). However, there are still key choices to be made such as whether to use filtered historical simulation, where data may be weighted by recentness, or adjusted for volatility. One must also decide whether to use absolute or relative shifts, what historical period to use for shift generation, and what time horizon to use for shifts (eg 1 day or 10 day).

For market risk calculations for fixed income products, conceptual pitfalls can arise around calculating shifts in credit spreads (eg bond Z spread). These kinds of subtleties are often missed by the major financial consulting firms, who lack the rigorous mathematical thinking required to detect these errors.

In other cases, such as operational risk, there is no standard approach and much more room for creativity.

Looking for an external model validation consultant? Please get in touch to discuss how we can meet your needs.