Economic Scenario Generator: Applications in Enterprise Risk Management. Ping Sun Executive Director, Financial Engineering Numerix LLC

Economic Scenario Generator: Applications in Enterprise Risk Management Ping Sun Executive Director, Financial Engineering Numerix LLC

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Outline Economic Capital and Solvency II: The Role of Economic Scenario Generator (ESG) Risk-Neutral (RN) vs Real World (RW): Bridging the Gap Algorithmic Exposures for Advanced Risk Measures

Economic Capital and Solvency II Basel II motivated, solvency capital requirement of EU insurance companies Pillar 1 Quantitative Measures Technical Provisions Minimal Capital Requirement (MCR) Solvency Capital Requirement (SCR) Pillar 2 Qualitative Processes Corporate Governance Risk Management Supervisory Interaction Capital Add-ons Pillar 3 Reporting & Disclosure Annual Solvency Reports Public Disclosure

Economic Capital and Solvency II ESG Requirements Model Choice Capturing the Entire Volatility Surface Describing Volatility Stochasticity and Clustering Handling Tail Distribution Specific Models to Different Asset Classes (interest rate, equity, credit, foreign exchange, inflation, etc.) Model Calibration Risk Neutral Calibrating to implied volatility surface Calibrating to variance swap, etc. Real World Time Series Analysis Based Methods (MLE, etc.) Calibrating to user specified projection Hybrid Model Joint Calibration of Models Across Different Asset Classes Correlation Input and Calibrating to User Projection Martingale Property Fast Model Calibration

ESG Example Cliquet Option : PV & Future Greeks

Outline Economic Capital and Solvency II: The Role of Economic Scenario Generator (ESG) Risk-Neutral (RN) vs Real World (RW): Bridging the Gap Algorithmic Exposures for Advanced Risk Measures

Real World Modeling Risk Neutral vs. Real Word Arbitrage Free Assumption under Risk Neutral (RN) Theory The values of all assets grow at the same instantaneous rate equal the risk-free rate of interest Deviation of the Real World (RW) Asset Dynamics Due to asset exposure to systematic risks and fundamental economic parameters such as risk preferences of investors 30 years of S&P500 accumulation (with dividend reinvestment) versus 30 years of rolling over at the risk-free (Fed Funds) rate (initialised at 1).

Real World Modeling Risk Neutral vs. Real Word RN pricing models often follows normal or log-normal distribution In RW historically realized distribution has fatter tail, different from those projected by the RN models. SPX returns distribution compared to Gaussian Source: J. Gatheral, 2006. The Volatility Surface: A Practitioner s Guide, Wiley, NJ.

Real World Modeling Risk Neutral vs. Real Word The presence of Variance Premium Consider the VIX and 22-day ahead S&P500 return variance. If there is no volatility premium, the VIX should be an unbiased redictor of future S&P500 return variance. Mean VIX=21.9% Mean RV=18.1% CBOE VIX index against 22-day ahead variance of S&P500 returns over the period Jan 1990-Jun 2013. Figures are reported as annualised volatilities.

Real World Modeling RW Heston Model with Risk Premium ds S = r + φv dt + VdW 1 dv = κ θ V dt + ξ V dw 2 with < dw 1, dw 2 >= ρ Equity premium φv Volatility premium θ and κ RN vs. RW Heston 20000 Simulated Real-World and Risk-Neutral Scenarios 15000 10000 5000 Risk-Neutral Path Real-World Path 0

Real World Modeling RW Heston Model with Risk Premium ds S = r + φv dt + VdW 1 dv = κ θ V dt + ξ V dw 2 with < dw 1, dw 2 >= ρ Equity premium φv Volatility premium θ and κ RN vs. RW Heston 40.00% 35.00% 30.00% 25.00% 20.00% 15.00% 10.00% 5.00% 0.00% Real-World and Risk-Neutral Volatility Paths Real-World Vol Risk-Neutral Vol

Hybrid Model Framework Fixed Income Hybrid Model Framework Select any Numerix model, with deterministic or stochastic components Inflation Define IR correlations IR HW1F Joint calibration (S)BK1F IR HW2F IR CIR 2F IR SV-LMM Credit INF JY (HW) INF JY (BK) Equity CR CTM HYBRID MODEL Foreign Exchange EQ BS EQ Dupire EQ Heston EQ Bates EQ LSV Commodities FX BS FX Heston Hybrids CMDTY Black CDMTY S1F CMDTY GS2F CMDTY Heston

Hybrid Model Framework Extended to Real World Nuemrix RW Hybrid Model Framework Joint Calibration Martingale Test RW Models in Individual Asset Classes Models with Variance Premium Equity & Foreign Exchange Black Model Heston Model Bates Model (Heston with Jumps) * Interest and Fixed Income Calibrated to IR projection CIR 2F HW 2F * Inflation Jarrow-Yildirim (JY) Model * Commodity Gabillon Model * Credit Transition Model (CTM) Transition probability implies the survival probability * Next Release

Outline Economic Capital and Solvency II: The Role of Economic Scenario Generator (ESG) Risk-Neutral (RN) vs Real World (RW): Bridging the Gap Algorithmic Exposures for Advanced Risk Measures

Risk Measure Based on Monte Carlo Simulation Risk Measures Market Risk Monte Carlo VaR Expected Shortfall Counter Party Risk Counterparty Credit Exposure (CCE) Potential Future Exposure (PFE) Expected Positive Exposure (EPE) Credit Valuation Adjustment (CVA) Exposure Distribution of Prices, Rates, Indexes on Future Dates Model Probability Measure Pricing Present Value (PV) of any instrument does not depends on the measure, guarantees by the arbitrage free theory CVA is measure independent Distribution Measure Dependent All exposure quantities are measure dependent

Option Value in the Future S t S 0 0 t 1 t 2 t 3 t 4 t n = T t

Option Value in the Future S t S t2 S 0 0 t 1 t 2 t 3 t 4 t n = T t

World of Black Scholes S t S t2 S 0 t S 0 e 2 r d 2 t Wt 0 t 1 t 2 t 3 t 4 t n = T t

Potential Future Exposure (PFE) PFE for European Call Option with Confidence Level S t S t2 S 0 0 t 1 t 2 t 3 t 4 t n = T t

Potential Future Exposure (PFE) PFE q inf x : P V, t, t t x PFE for European Call Option with Confidence Level x 0 - dx 2 e x 2 / 2 97.5% x t 0 1.96 PFE d t, S ln t,0 S 0 e 2 2 r d / t x0 2 S / K r d / 2 T t t,0 T t T t r T t S N d e KN d, t e t,0 t, dt,

Exposure of Generic Instruments What if the pricing of a generic instrument requires MC? S t S t2 S 0 0 t 1 t 2 t 3 t 4 t n = T t

Challenges of Scenario Based Approach Scenario Based Approach (Brute Force) Methodology Generate scenarios Calibrate / Build a model associated to each scenario Price portfolio along each scenario If the portfolio contains plain vanilla instruments we can evaluate the price directly from the generated market Drawbacks Scenario generation There is a large variety of theoretical and phenomenological approaches to the scenario generation which becomes rather ambiguous for large cross-asset systems. Future Market Generation Scenarios of Underlying, Volatility, etc. Model Calibration Performance Nested Monte Carlo

Algorithmic Exposure Backward Pricing Backward Pricing with American (Least Square) Monte Carlo F. A. Longstaff and E. S. Schwartz, Valuing American Options by Simulation: A Simple Least-Square Approach Backward MC Pricing Procedure of American Option Generate MC paths, S T i Roll backward in time, determine the optimal exercise time 1. At time T i compute the option value along each MC paths O T i 2. At time T i 1 and along each MC path compute the conditional expectation of the option price via regression Assume the option price is a continuous function of the current state variables, defined through an expansion via certain basis functions Conduct least square fit of the option price along each MC path and determine the expansion coefficients Compute the option prices at time T i 1 as the continuation value from those at the previous time slide T i, V T i 1 3. Determine if it is optimal to exercise the option at T i 1, rather than later, V T i Update the option price if the answer is yes V T i 1 = max V T i 1, O T i 1 4. Roll back to time T i 2, T i 3,, T 0, repeat step 2 and 3 Option Price As a Continuous Function of the Current State Variables Stochastic process is Markovian Certain path-dependency requires extra state variable E.g. Asian or Lookback option

Algorithmic Exposure Price vs. Exposure A. Antonov, S. Issakov, and S. Mechkov, Risk January 2015 Backward induction for future values Algorithmic Exposure Procedure On Observation Date t obs, set Exposure the same as price distribution, v t obs = V t obs When roll back in time, apply the exercise update, but skip the conditional expectation. Examples American Option Update Pricing V T i 1 = max V T i 1, O T i 1 Exposure v T i 1 = V T i 1 1 Um T >U n T + O T i 1 1 Um T U n T Roll Procedure Between Dates Pricing V t = E N t Exposure v t = N t v T j N T j The Final Exposure is v = v 0 N t obs Other Examples Includes Bermudan Swaption, Autocap, etc. V T N T F t

Algorithmic Exposure Example Algorithmic Exposure Example FX European Option PFE as a Function of Time Horizon : Backward MC Simulation vs. Analytic Formula 18 16 14 12 10 8 Numerix Simulation Analytic PFE 6 4 2

Algorithmic Exposure Example Algorithmic Exposure Example FX European Option ETL as a Function of Time Horizon : Backward MC Simulation vs. Analytic Formula 18 16 14 12 10 8 Numerix Simulation Analytic ETL 6 4 2

Algorithmic Exposure Example Algorithmic Exposure Example IR Swap with One Way Collateral Call CCE as a Function of Time Horizon :

Algorithmic Exposure Under Real World Measure The nested stochastic pattern: Outer-Loop: real-world dynamics Inner-Loop: risk-neutral dynamics

Algorithmic Exposure Under Real World Measure S t RW RN S tobs S 0 0 t 1 t obs t 3 t 4 t n = T t

Algorithmic Exposure Under Real World Measure Algorithmic Exposure under Real World Measure Generate both the RN and RW scenarios from Now Date RW provides the economical scenarios RN provides the pricing scenarios On Observation Date t obs, set Exposure the same as price distribution, v t obs = V t obs Resampling of the RN price Distribution Option price v t obs is a continuous function of the current state variables Current state variable distribution is from real world simulation When roll back in time, apply the exercise update, but skip the conditional expectation.

Algorithmic Exposure Under Real World Measure Algorithmic Exposure under Real World Measure Real World as Measure Change of Risk Neutral Apply the cross-currency analogy set our initial model as foreign one w.r.t. some FX rate process Set a domestic currency model. The initial model states will get a drift adjustment depending on FX vol and correlation (Quanto Drift) 6 Month LIBOR Averages for Different FX Volatilities (Different Measures)

Algorithmic Exposure Under Real World Measure Algorithmic Exposure under Real World Measure Real World as Measure Change of Risk Neutral Apply the cross-currency analogy set our initial model as foreign one w.r.t. some FX rate process Set a domestic currency model. The initial model states will get a drift adjustment depending on FX vol and correlation (Quanto Drift) PEF at 97.5% Confidence for Different FX Volatilities (Different Measures) 10Y cancelable swap on 1 EUR notional : Receiving semi-annually A 6M Libor Paying annually a fixed rate (= 2.57%) Owner has a right to cancel the swap annually from Year 4.

Conclusion ESG is One Key Tool in Determining SCR Required By Solvency II Role of Risk Premium : Risk Neutral and Real World Models are Different Not Only in Their Calibration Methods Algorithmic Exposure Provides a Powerful Tool for Computing Risk Factors in Both Risk Neutral and Real World Measures

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