Sobol Sequences: Business Cases for Pricing and Risk Management

For more than 20 years, Monte Carlo methods have been used in various ways by financial institutions—from pricing financial derivatives to running complex risk management calculations. The Monte Carlo approach is easy to use but requires a large number of scenarios to achieve accuracy. Several variance reduction techniques are available that reduce the number of simulations required. The Quasi-Monte Carlo (QMC) method, the most efficient and common variance reduction approach, employs highly uniform low-discrepancy sequences (like Sobol, the most popular and efficient sequence) that enable a higher rate of convergence compared to the standard Monte Carlo. Together, QMC and Sobol offer a cost-effective solution for improving computer capabilities resulting in a significant reduction of computational time.

Option Pricing & Sensitivities: Business Benefits of Sobol

Monte Carlo is at the center of pricing engines because analytical formulas cannot be applied to many exotic financial products with complicated features. There are key business benefits of using Sobol for option pricing and sensitivities’ calculations:

• Increased accuracy of computation of option prices and sensitivities.
• Reduced computational time due to reduced number of required paths.
• More accurate hedging strategies and reduced hardware costs.

Scrambled Sobol sequence generator for error estimates

The need for statistical methods to analyze error estimates, in the context of QMC based on Sobol sequences, can be satisfied by Randomized Quasi-Monte Carlo methods, which use scrambling and other techniques to randomize sequences. Error estimates for Quasi-Monte Carlo sequences are then based on treating each scrambled sequence as a different and independent random sample. Scrambled Sobol has become the most popular choice in Risk Management because of its simplicity and efficiency.

SOBOL for PFE and XVA, including Wrong Way Risk

Emerging from the credit crisis that began in 2007, many financial institutions recognized the need to better manage their Counterparty Credit Risk (CCR). As a result, they have begun to quantify, price and manage their CCR.

By pricing credit valuation adjustment (CVA) into trades at deal time, firms are enhancing their existing CCR systems or designing new ones. The aim is to develop capabilities such as incremental calculations of CVA, real-time computations, incorporation of right/wrong way risk and expansion of product coverage to include all trade types—including exotics.

The importance of speed

Given the high number of scenarios and steps required for calculations, speed is as important as accuracy. While the use of Monte Carlo is common practice for the CCR and CVA calculations, the QMC approach must become the preferred approach because it significantly reduces scenarios and hardware configuration during the computation of potential future exposures (PFE) and x-value adjustments (XVA).

The following table provides the approximate number of QMC + Brownian Bridge paths needed to produce CVA and CVA sensitivities, with errors roughly equivalent to classical Monte Carlo, with 10,000 paths for far in the money portfolios of various sizes (one 10-year fixed rate payer swap in each currency).

For banks implementing the Monte Carlo approach, Targeted Review of Internal Models (TRIM) guidelines propose applying additional methods to measure the simulation error of the effective expected positive exposure. For the Quasi-Monte Carlo approach, this instruction does not apply.  

Wrong Way Risk calculations

The incorporation of Wrong Way Risk (WWR) in CCR and CVA adds another layer of complexity in the calculations for PFEs and XVAs. WWR refers to an adverse relationship between the exposure of a derivative(s) and the credit quality of the counterparty to the trade(s). There are several proposed methodologies to include Wrong Way measurements in CCR and CVA calculations. Scrambled Sobol is considered the best approach for efficient WWR calculations.

Sobol for IMA DRC

In the upcoming new Market risk rules (FRTB), the calculation of issuer default risk under the internal model (IMA DRC) is another typical case, where efficiency and accuracy of the estimations can be an “expensive” simulation task for banks. The use of QMC brings significant computational gains in these calculations, especially through the use of Scrambled Sobol.

A similar business case to IMA DRC and WWR is credit portfolio management for buy-side institutions. The joint calculation of market and credit risks is complex, and the use of QMC is an important tool for practitioners to efficiently capture the high tails of the distribution.

Conclusion

Financial institutions can realize key business benefits from the use of Sobol, including an increased accuracy of computation of option prices and sensitivities, reduced computational time, more accurate hedging strategies and reduced hardware costs.

To learn more, view our on-demand webinar featuring Scotiabank and Dr. Sergei Kucherenko, Managing Director, BRODA Ltd., Increasing Speed and Accuracy in the Front and Middle Office with Sobol Sequences Generators or contact us.