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Value-at-Risk Introducing a Performance-Based Monte Carlo Method Background Monte Carlo simulation is becoming recognized as the optimal quantitative methodology for measuring “Value-at-Risk” (VaR). While analytical methods have been developed and successfully implemented for vanilla portfolios, including those with standard option positions, market practitioners principally agree that Monte Carlo is the only viable VaR method for portfolios containing short-dated and exotic option positions. As a result, many market participants have implemented Monte Carlo-based VaR methods over the last several years. However, while innovations in Monte Carlo methods have enabled risk managers to greatly reduce the computation time, remaining performance constraints continue to limit the use of Monte Carlo in calculating VaR on a intra-day basis. Summary of Summit Monte Carlo The purpose of this article is to outline a methodology developed by Summit Systems, Inc. (Summit) which provides a theoretically-sound and practical Monte Carlo method that can be readily employed as an intra-day VaR tool. The key to the approach is using an analytic, non-linear method in conjunction with Monte Carlo in order to limit the required randomly-generated rate paths that must be applied to a portfolio. Under this method, an estimate of the true distribution of a portfolio’s mark-to-market is first generated using a Gamma/Vega method developed by Summit. The distribution results are then used to stream-line the Monte Carlo simulation process by limiting the application of randomly-generated rate paths to those that are statistically relevant given the portfolio’s anticipated risk profile. Modeling Approach Summit’s Gamma/Vega method involves three critical and sequential calculation steps: 1) a second order Taylor series expansion of the mark-to-market relative to price (to capture gamma) and a first order volatility measure; 2) a representation of the independent option risk components as the sum of chi-squared distributions; and 3) the use of Fast Fourier Transform to recover the true mark-to-market profile. The portfolio distribution generated using this methodology properly captures nonlinear option payoffs, generally to a high accuracy level. Once calculated, the gamma/vega-based portfolio distribution serves as an input to the simulation process to limit the required revaluations to rate paths expected to result in losses within a prescribed confidence interval. When initiated, the Summit Monte Carlo engine will automatically generate a user-defined number of random rate paths (e.g., ten thousand) based on applicable volatility/correlation input parameters. However, rather than applying each rate scenario to the entire contents of a portfolio (which is a computationally-intensive process), the Summit model restricts the revaluation process to rate paths where loss exposure is expected to fall within a parametrically estimated VaR interval. In addition to restricting the rate paths, the Monte Carlo simulation is limited to trades involving optionality; previously obtained parametric VaR measures are relied on for non-option positions. The user will have the ability to define portfolio subsets and identify whether parametric or Monte Carlo VaR measures will be calculated for each segment. Based on portfolio sensitivities (i.e., deltas, gammas and vegas) obtained during the Gamma/Vega calculation, a statistically-based mark-to-market estimate may be obtained for the portfolio. From this single estimate, a range can be determined to represent the expected variability of the actual mark-to-market due to anticipated model error. The mark-to-market range estimate should be predicated on historical analysis of the accuracy of the Gamma/Vega method against actual portfolio return distributions. Using the known portfolio sensitivities, the randomly-generated rate scenarios resulting in such a mark-to-market range can be identified (i.e., with a Taylor expansion). Once identified, each relevant scenario is separately applied to generate a mark-to-market calculation for each transaction in the subject portfolio. When this process has been completed, VaR can be obtained by first ordering the results (i.e., by calculating the number of scenarios falling below the range, the ordered number of scenarios falling within the range and the number of scenarios falling above the range) and then selecting the appropriate mark-to-market based on the desired confidence level. A summary of the specific calculation steps is provided below: CALCULATION STEP 1. Apply perturbation hedge to relevant market variables (e.g., interest rates, FX rates and implied volatilities) and calculate sensitivities (deltas, gammas, vegas, etc.) 2. Apply Gamma/Vega Method to estimate mark-to-market distribution and resulting VaR 3. Determine upper and lower bounds relative to Gamma/Vega VaR accuracy based on empirical evidence 4. Generate random scenarios for the same market variables used in the Gamma/Vega VaR calculation 5. Using the sensitivities (i.e., greeks) determined in Step #1, estimate resulting mark-to-market under each random scenario generated in Step #4 (i.e., with Taylor Expansion) 6. Determine if mark-to-market generated in Step #5 is within the upper and lower bound determined in Step #3 – on a scenario-by-scenario basis. 7. For all scenarios resulting in MTMs within the upper and lower bounds, revalue the entire portfolio and order each MTM result 8. Count the number of scenarios falling below the MTM lower bound and the number of scenarios falling above the MTM upper bound 9. Calculate VaR by looking at the nth percentile MTM (e.g., 10th percentile will give us a 90% confidence-level VAR). Critical Assumptions and Model Parameters 2 VaR estimates are often moved through time (generally be adjusting linear VaR measures as a square root of time). However, to properly model risk stemming from exotic option portfolios as a function of time, it is essential to capture multiple-step rate paths. By generating such paths, it is possible to properly capture the reset and exercise effects and the risk of crossing potential option barrier levels when calculating VaR. Once multi-step risk has been calculated, it is necessary to provide risk managers with proper hedging tools to ensure that certain risk factors are neutralized when VaR measures adjust through time. This is necessary to create a portfolio environment that accurately reflects actual steps that would be executed by traders to minimize risk. In Summit’s initial Monte Carlo implementation, an n-step rate path will be introduced and risk managers will be able to ensure delta neutrality by automatically applying hedges with one or two liquid instruments (e.g., the appropriate near and far futures contracts). In subsequent implementations, hedges to neutralize both gamma and vega risk will be employed and the number of available hedging vehicles will be expanded. The above-detailed Summit method eliminates performance problems caused by having to revalue individual transactions thousands of times by tailoring the application of rate paths based on inherent position risk profiles. When using Monte Carlo to calculate VaR for portfolios containing options priced using lattice models, additional and often significant performance bottlenecks ineluctably result when the portfolio is modeled through time. This is due to the need to re-calibrate models due to the passage of time and changes in market variables. The calibration process can be extremely time consuming from a system resource perspective and can be exacerbated when calculating VaR for portfolios with transactions priced with multiple lattice approaches (e.g., Hull & White, HJM and BDT.) To alleviate such performance concerns, Summit’s initial Monte Carlo model will rely on a single lattice approach (Summit’s proprietary H&W trinomial tree) to price path dependent options captured in the VaR measure. The adjustment in the calibration to reflect the passage of time will be based on the assumption that it is a linear function of market prices. Additionally, Summit will provide an API to enable clients to employ Monte Carlo based on pricing/calibration with proprietary trees (such as HJM). Conclusion Implementing Summit’s Gamma/Vega method in conjunction with a discerning Monte Carlo simulation approach, enables users to limit the revaluation process to a small enough number of rate paths to allow the Monte Carlo process to run on a quasi-real-time basis without compromising the integrity of reported data. By using techniques such as distributed/parallel processing, Monte Carlo calculation speeds can be further reduced. The result is a measurement tool that enables risk managers to quantify VaR throughout the trading day relative to established limits and take corrective measures to reduce risk on a dynamic basis. At the same time, the method yields robust VaR results for even the most complex exotic option positions. 3 4