[Paper Review] MVA: Initial Margin Valuation Adjustment by Replication and Regression

Andrew R. Green, Chris Kenyon|arXiv (Cornell University)|May 2, 2014

Fuzzy Logic and Control Systems19 citations

TL;DR

This paper introduces Margin Valuation Adjustment (MVA) to account for the funding cost of initial margin in derivative pricing, extending the semi-replication framework of Burgard and Kjaer (2013) and Green et al. (2014). It proposes a computationally efficient method using Longstaff-Schwartz Augmented Compression (LSAC) to estimate expected initial margin profiles via regression, reducing Monte Carlo computational burden; MVA is found to be approximately 50% of the FVA for unsecured portfolios.

ABSTRACT

Initial margin requirements are becoming an increasingly common feature of derivative markets. However, while the valuation of derivatives under collateralisation (Piterbarg 2010, Piterbarg2012), under counterparty risk with unsecured funding costs (FVA) (Burgard2011, Burgard2011, Burgard2013) and in the presence of regulatory capital (KVA) (Green2014) are established through valuation adjustments, hitherto initial margin has not been considered. This paper further extends the semi-replication framework of (Burgard2013a), itself later extended by (Green2014), to cover the cost of initial margin, leading to Margin Valuation Adjustment (MVA). Initial margin requirements are typically generated through the use of VAR or CVAR models. Given the form of MVA as an integral over the expected initial margin profile this would lead to excessive computational costs if a brute force calculation were to be used. Hence we also propose a computationally efficient approach to the calculation of MVA through the use of regression techniques, Longstaff-Schwartz Augmented Compression (LSAC).

Motivation & Objective

• To formally incorporate the funding cost of initial margin into derivative valuation, completing the set of valuation adjustments (FVA, KVA, MVA).
• To address the high computational cost of calculating expected initial margin profiles using brute-force Monte Carlo simulations within risk-neutral pricing.
• To develop a scalable, efficient method for computing MVA that remains accurate under large market shocks inherent in VAR/CVAR models.
• To quantify the relative magnitude of MVA compared to FVA in representative interest rate swap portfolios.
• To demonstrate the feasibility and performance of LSAC as a portfolio compression technique for initial margin estimation.

Proposed method

• Extends the semi-replication framework of Burgard and Kjaer (2013) and Green et al. (2014) to include initial margin funding as a new valuation adjustment (MVA).
• Models initial margin using historical VAR or CVAR (Expected Shortfall) approaches, requiring estimation of the expected initial margin profile over time.
• Applies Longstaff-Schwartz regression to approximate portfolio values and initial margin exposures using a basis of linear combinations of swaps and annuities.
• Introduces LSAC (Longstaff-Schwartz Augmented Compression) by augmenting the state space with additional shocks to improve regression accuracy under large market moves.
• Uses GPU-accelerated C++/CUDA implementations to enable fast brute-force comparisons for validation and performance benchmarking.
• Employs a fixed set of basis functions across all time points to maintain consistency and reduce computational overhead.

Experimental results

Research questions

• RQ1 How can the funding cost of initial margin be consistently incorporated into derivative valuation within a semi-replication framework?
• RQ2 What is the computational cost of calculating expected initial margin profiles using standard Monte Carlo methods, and can it be reduced?
• RQ3 Can regression-based methods like Longstaff-Schwartz be adapted to accurately estimate initial margin under VAR/CVAR models with large shocks?
• RQ4 How does the magnitude of MVA compare to FVA in typical interest rate swap portfolios?
• RQ5 What is the performance gain of LSAC over brute-force Monte Carlo in terms of speed and accuracy for MVA computation?

Key findings

• MVA is a significant valuation adjustment, amounting to approximately 50% of the FVA for an unsecured portfolio of interest rate swaps.
• For a portfolio of 10,000 swaps, the LSAC method achieves a computational speedup of up to 100× compared to brute-force Monte Carlo.
• Regression accuracy for portfolio valuation remains high even with a small number of basis functions, improving as the portfolio ages.
• The LSAC method maintains high accuracy in estimating VAR and IM funding costs, with errors decreasing as the number of basis functions increases.
• The expected positive exposure (EPE) and negative exposure (ENE) profiles show that MVA is most material in portfolios with high notional and long-dated maturity.
• The method is particularly effective for linear derivatives, with extensions via the 'early start' Monte Carlo approach applicable to complex instruments.