Bridging the Gap: Integrating Pricing Models and Risk Models in Financial Projections

_{*This article first appeared in the QuantMinds International newsletter and website in November 2024.}  

Financial modelling is a discipline wrought with intricacies, where demanding tasks such as projecting exposures and margin requirements typically rely on the use of advanced pricing models. Adding to this complexity is the use of sophisticated techniques like Least-Squares Monte Carlo (LSMC), which is commonly employed to help reduce costs when closed-form solutions are unavailable. The landscape becomes even more challenging when projections are based on a dedicated risk model. 

In numerous real-world applications it can be advantageous, or necessary, to project future values, exposures, and sensitivities based on historically consistent market scenarios. Banks use such projections to compute potential future exposure (PFE) and expected exposure (EE) metrics for limits and internal capital requirements. Similarly, insurance companies project post-hedge profit and loss (P&L) for annuity blocks to determine reserve requirements, while asset managers and pension funds employing derivative overlays also project post-hedge P&L for risk management purposes. 

This article explores the challenges and methodologies involved in integrating risk models and pricing models, highlighting the nuances and innovations that drive this important aspect of risk measurement. 

Dichotomy of pricing models and risk models

Pricing models and risk models serve distinct purposes in the quantitative finance landscape. Pricing models, such as the Black-Scholes, Heston, and Hull-White models, are primarily used for pricing, hedging, and revaluing derivatives. These models are formulated in continuous time and are typically specified under the pricing measure (Q). The dynamics of these models aim to capture important empirical features while balancing against concerns such as tractability and ease of calibration, which are critical in pricing or hedging applications. 

On the other hand, risk models are designed to measure risk for derivatives portfolios, enforce limits, and determine capital requirements. Models like the GARCH, PCA, and Diebold-Li models place a strong emphasis on reproducing empirical features and facilitating statistical estimation. These models often operate in discrete time and may be specified in terms of entire systems, reflecting the data-generating process (DGP) under the physical measure (P). 

Of course, pricing models can and often are used for risk measurement purposes. In some instances, risk scenarios might be generated from the physical measure or P dynamics of a given model. 

LSMC-style methods for pricing models in risk applications

One of the key techniques used to reduce the cost of computing future values or margin requirements is the Least-Squares Monte Carlo (LSMC) method. Originally adapted from Longstaff-Schwartz (2001) and Carriere (1996) among others, LSMC has been widely used in the context of counterparty credit risk (CCR) and valuation adjustments (XVA). The basic idea is to project contract cash flows onto earlier state variables of the model to estimate conditional expectations. 

Traditionally the ‘outer’ scenarios used to measure risk – and the ‘inner’ scenarios used to evaluate cashflows – coincide and are generated from a pricing model. The situation becomes more complex when the outer scenarios come from a dedicated risk model with dynamics and risk factors that differ from the pricing model. 

Accommodating risk model-generated scenarios in LSMC

At a high level, when combining risk models and pricing models to compute risk metrics, the financial quantities generated from the risk model, such as curves or volatility surfaces, should be translated into the pricing model scenario by scenario. They will manifest in the state variables and model parameters. Thus, both the state variables and the model parameters vary by scenario, and parameters are typically high-dimensional elements, such as curves. At first glance, it may appear that we need to project cashflows onto these high-dimensional elements, and a natural next step is to try to apply dimension reduction techniques. Fortunately, we can sidestep this entirely. 

First, we have already noted that the state variables and model parameters of the inner pricing model will be a function of the financial quantities generated by the outer risk model. Second, the risk model is typically a factor model represented in terms of its own state variables, and these are what ultimately drive the financial quantities it generates. In such cases therefore, the conditioning information is fully encoded in the states of the risk model, and they can be used for LSMC. 

One complication lies in the fact that we do require the parameters of the pricing model along a given scenario, and this can involve calibration, such as when the risk model is generating volatility surfaces. Calibration can of course be a computationally expensive task, but we can take advantage of the fact that volatility surfaces generated using a low-dimensional (e.g. factor) model will live in a low-dimensional space, and thus so will the calibrated parameters. This simplifies the relationship between the states of the risk model and the parameters of the pricing model to one which can be learned in a manageable way. 

Navigating pricing and risk with confidence

Integrating pricing models and risk models enables institutions to project exposures, margin requirements, and other key metrics using models which better reflect the historical behaviors of the risk factors driving their portfolios. By leveraging these tools, financial institutions can better navigate the intricate landscape of risk and pricing, ensuring robust and reliable projections for a wide range of applications.