Monte Carlo Simulation Number Of by jessifer

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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



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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.




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