Harvey J. Stein

FIXING RISK NEUTRAL RISK MEASURES

In line with regulations and common risk management practice, the credit risk of a portfolio is managed via its potential future exposures (PFEs), expected exposures (EEs), and related measures, the expected positive exposure (EPE), effective expected exposure (EEE), and the effective expected positive exposure (EEPE). Notably, firms use these exposures to set economic and regulatory capital levels. Their values have a big impact on the capital that firms need to hold to manage their risks.Due to the growth of credit valuation adjustment (CVA) computations, and the similarity of CVA computations to exposure computations, firms find it expedient to compute these exposures under the risk neutral measure.Here, we show that exposures computed under the risk neutral measure are essentially arbitrary. They depend on the choice of numeraire, and can be manipulated by choosing a different numeraire. The numeraire can even be chosen in such a way as to pass backtests. Even when restricting attention to commonly used numeraires, exposures can vary by a factor of two or more. As such, it is critical that these calculations be carried out under the real world measure, not the risk neutral measure. To help rectify the situation, we show how to exploit measure changes to efficiently compute real world exposures in a risk neutral framework, even when there is no change of measure from the risk neutral measure to the real world measure. We also develop a canonical risk neutral measure that can be used as an alternative approach to risk calculations.

Joining Risks and Rewards

The dichotomy between risk analytics and pricing is well known amongst financial practitioners and researchers. For risk analysis, such as computing value at risk and credit exposures, expectations of future values must be computed under the real world measure. For pricing, expectations are computed under a risk neutral measure. This means that for calculations such as value at risk (VaR) and credit exposures (EEs, EPEs, etc) on derivative portfolios, risk factors are evolved to the horizon date under the real world measure, at which point the portfolio is repriced under a risk neutral measure.Simulation under the real world measure followed by repricing under a risk neutral measure is computationally intensive, especially when the repricing requires Monte Carlo. Because of the computational effort involved, shortcuts are often taken. One common shortcut is to assume the real world measure is the risk neutral measure, so that calculations can be done under one measure. This particular shortcut is especially problematic as it leads to results varying wildly depending on the numeraire chosen.Here we detail methods of avoiding this problem by combining the real world measure with the risk neutral measure. We present an application to risk analytics which speeds up the calculations by orders of magnitude, changing O(n^2) calculations to O(n) with a far smaller scaling constant. This allows the computation of exposures under the real world measure, obviating the need for the dangerous practice of computing exposures under a risk neutral measure.

FX Market Behavior and Valuation

Lecture notes for a short course on FX option valuation. Includes: - Mathematical framework for FX valuation - Handling the smile and term structure for vanilla options (calls and puts): --- Interpolation issues and techniques --- Handling business time --- Handling market conventions - Pricing of barrier options: --- Attention to the joints along with the marginals --- Barrier option pricing models ------ Black-Scholes ------ Vanna-volga ------ Semi-static hedging ------ Stochastic volatility - the Heston model ------ Local volatility ------ Stochastic local volatility ------ Random risk reversal model - Hedging performance as a measure of model quality.

Analytic Solution to the Two Dimension Merton Model

We present an exact analytic solution to the two-dimensional correlated default structural Merton model in the form of a local volatility problem using a conformal square-root transformation of the exact solution to a 2D hybrid barrier problem. We also give an approximation and evaluate it numerically to give an example. Finally we give an exact simpler solution for the zero correlation case.

Stable Reduced-Noise 'Macro' SSA - Based Correlations for Long-Term Counterparty Risk Management

We introduce a methodology from geophysics, Singular Spectrum Analysis (SSA), to obtain stable, noise-cleaned correlations for long term risk (e.g. counterparty risk). SSA is applied to time series to smooth them in a robust manner. The SSA-smoothed time series are then used to obtain the correlations. We call these “macro�? correlations because they are determined with macroscopic time scales. Stable correlations are desirable to suppress noise from short time scales that make risk measures unstable. If correlations move around, risk measures also move around, making business decisions difficult. SSA-based correlations ameliorate this business problem.

Valuation of Exotic Interest Rate Derivatives - Bermudans and Range Accruals

Exotic interest rate derivatives are hard to value. Care must be taken to make sure that sources of volatility that impact the contingent claim are properly modeled, and that appropriate relationships are maintained between the underlying rates involved.In this presentation, we outline the issues involved in valuing exotics. We review valuation issues for interest rate derivatives in general, and for caps, floors and swaptions. We outline a pricing methodology and apply it to Bermudan swaptions, range accruals, callable range accruals, spread options and callable spread range accruals.Outline: - Review of interest rate modeling - Handling of vanilla options - - Forward Libor and swap rates - - Caps and Floors - - Swaptions - - Cap stripping - - Smile lifting - Bermudan valuation - - Hedging Bermudans - - LGM model specification of the HW model - - Pricing cashflows and options under the LGM model - - Model calibration - - Numerical methods - Digital options - - Pricing via vanillas. - Range accruals - - Pricing as a portfolio of digitals - - Convexity adjustment - Change of measure and approximation - Callable range accruals - - Pricing under the one factor LGM model - - - Model calibration. - - - Use of control variates (adjusters). - - Calibration and pricing under the two factor LGM model - - - Model calibration. - Spread range accruals - - Pricing under the two factor LGM model.