long horizon simulations for counterparty risk

Challenges of long horizon simulations for Counterparty Risk modeling

Alexandre Bon, Murex Alexander Sokol, Compatibl

Amsterdam - December 4th, 2013

Challenges of long-horizon simulations for CCR

1. Introduction

2. Calibration in the Risk Neutral Measure

3. Calibration in the Real World Measure

4. Modeling the High Interest Rate Regime

5. Summary

Copyright © 2013 Murex S.A.S. All rights reserved3

Why model CCR exposures over long horizons?

Valuation adjustments: CVA, DVA, FVA

•CVA is the market value of counterparty credit risk for OTC derivatives (or the difference between the risk-free price and the mid-market price of the portfolio)

• Expectation over time of discounted future exposures weighted by default probabilities and recoveries

Risk control & economic capital

• Limits on PFE (potential future exposure profile)• EPE (expected positive exposure) as a loan equivalent exposure input to Economic Capital model

• Expected and peak exposure profile over the lifetime of the portfolio


Why model CCR exposures over long horizons?

Basel III, Regulatory Capital calculations

•�� , ��,� 8%⁄ , where under IMM:

� �� ∝∙ max �� !"��#$, �� !"��#$%��%%�� , i.e. a 1Y horizon is considered

� EPE model backtested by validating the risk factor distributions up to 1Y

� � min ∑ )))*∙+*∙�,*-./0 1∑ ))*∙+*∙�,*-2/0∑ )))*∙+*∙�,*-./0

, 54 , i.e. the full exposure profile is taken into account

•CVA risk capital charge : a VaR-like market risk charge covering the potential CVA losses induced by Credit Spreads volatility


Modeling approaches

CCR metrics are typically implemented via Monte Carlo simulation

• Evolve risk factors values along thousands of scenarios paths across hundreds of time points• Re-price all outstanding transactions on each point • Apply credit risk mitigants to derive exposure values and sample relevant statistics

CVA is a price : conceptually similar to pricing an exotic derivative contract through Monte Carlo simulation

• American option on the close-out value of the portfolio• Risk Neutral probabilities, discounted expectations

PFE is a risk metric: conceptually similar to VaR

• Look at quantile and expected shortfall measures• Typically Real World probabilities

Implementation approaches evolved from Market Risk or Derivatives Pricing techniques encounter specific challenges in the context of CCR measures.


Implementation considerations

Choice of risk factor evolution models

•Hybrid modeling framework required• Joint evolution of large number of risk factors across multiple time-points •Arbitrage-free and consistent with valuation model (at least for CVA)• Econometrically realistic (at least for PFE and EPE back-testing)• Simple vs. sophisticated models: ease of interpretation vs. ease of calibration•Appropriateness to the underlying portfolio and available calibration data

Sometimes, something gotta give…


Implementation considerations

Calibration

•Calibration frequency : reactivity (CVA hedging) vs. stability (PFE limits)•Choice of data inputs (cleansing, weighting, proxies…) •Method: speed of calibration, stability, convergence.

Path dependency

• Position valuations for exotics•Accurate deal aging with cash/delivery exercises, ATEs … •Collateral balances are path dependent

Path consistency


Risk-Neutral vs. Real-World probabilities

CVA: Risk Neutral probabilities are the natural choice

• Need to match hedging instruments prices & sensitivities• Fair-value / bilateral pricing: PDs are implied from CDS quotes, natural • Consistent distribution assumptions for scenarios generation and positions re-pricing

PFE & EPE: Real world probabilities are often preferred

• Rooted in actual distributions : suitable for quantile analysis (PFE limits, Backtesting)• Risk factors evolved in the physical measure, positions valued in risk-neutral measures

95% PFE : 5Y payer IRS


1. Introduction




5. Summary


Risk Neutral calibration

The challenges are similar to those met when pricing long-dated derivatives, only worse:

• Lack of liquid instruments (e.g. swaptions quotes on emerging markets): typically use proxy curves and surfaces from similar instruments / markets.

• Extrapolation beyond quoted horizons (volatilities)� E.g. German EU bank holds a 60Y IRS in GBP with a UK corporate. No issue with IR data, but since the risk is reported in EUR, but what is the 60Y EUR/GBP vol?

• Correlations : � Theoretically correct to use RW correlations in the RN measure, however vols and correlations are then sampled from different data sets

� Very few instruments can let one get implied correlations (typically FX cross pairs) : option to override selected points of a historical correlation matrix with implied data.

� Instantaneous or terminal correlations?

• Inconsistent market views : � Caps vs. Swaptions volatilities can imply different short rate volatilities in practice

� Which Yield Curve ? LIBOR vs. OIS, FX vs. IR desk data…


Choice of models in a low rates environment

Classical view: negative rates do not exist (or only as a temporary aberration in short-term or fwd-fwd rates)

• HJM models (Vasicek, Hull & White 1F / 2F) � Parsimonious and analytically tractable (closed-form solutions for calibration and revaluation of vanilla products)

� Could be used as long significantly negative rates are occurring with very low probabilities

• Other models can prevent negative rates, but with a computational cost:� A multi-currency BGM is common option today (but also: Black-Karasinsky, CIR++, linear Cheyette…)

� Mechanistic models like Nelson-Siegel usually discarded for CVA and evolution in the risk neutral measure (non-arbitrage free)

Over recent years, very low yields have changed market practices

• VaR models : switch from geometric to arithmetic returns for yield curves• Apparition of 0-strike floors and negative strike swaptions markets• Rush to fix models to allow for negative rates, but not too negative (e.g. shifted lognormal models -SABR, move from lognormal to normal vols quotations)

“Assuring people that they can get a positive rate of return on safe assets means promising them something the market doesn’t want to deliver – it’s like farm price supports, except for rentiers.” Paul Krugman - The New York Times - Nov 16th, 2013.


Rates behaving badly : example with JPY rates

Can hardly get the best of both worlds in a multi-factor settings:

• Impact on EPE & CVA may be relatively minor• Raise significant issues for PFE

� Implied vols can be large w.r.t low rates values

� HW diffusion: the PFE of receiver IRS can comes from scenarios with unrealistically negative rates

� BGM diffusion: the PFE of payer IRS is larger (generated by unrealistically large rates)

20Y payer IRS PFE & PFL : HW vs. BGM

EE & EL profiles


Rates behaving badly : example with JPY rates

Risk factors diffusion under Hull & White & BGM diffusion models:

• JPY 6M Zero Coupon at 15Y point• Calibration to swaption volatilities (Oct, 2013)

BGM

HW


1. Introduction




5. Summary


Estimation from historical data series

Selecting the calibration method and data points

• Challenges: relevant data series shorter than horizon considered, typically consider daily time-fractions (as for VaR) that may not capture adequately long term dynamics

• The risk factor model applies simplified dynamics, calibration method should allow reverse-engineering non-model-compliant variations as « noise »

• Ideal method should: converge efficiently, not be over-sensitive to data cleansing or initializing values, remain stable for different regimes, not be a black box

Example: estimation of vol & MR for Hull & White interest rate models (1F / 2F)

• Efficient/General Method of Moments• Maximum likelihood (with / without Kalman filters)•We found both methods to perform unsatisfactorily, especially w.r.t the Mean Reversion term

We propose a new calibration algorithm: improved Variance Estimation Method

Validation :

• Accurately recover inputs from model generated data (generate 1000 series of 800 daily returns)• Test the algorithms on empirical data


Maximum likelihood method

The method consists in finding model parameters maximizing the product of the conditional probabilities of observing actual values at � knowing the values observed at �56.• Supposing the parameters 8and 9are known, we calculate the likelihood of having a specific series of data: product of the conditional probabilities of each step.

• Noting ��:;< �: � , � = ;< zero coupon rate at time � for the pillar ;, the likelihood function > is the probability density �?,@ of the realization of the path :�<�

> 8, 9|�#, . . , �B C�?,@:��1#:;<|��:;<B

#<

• In practice we use the logarithm of the likelihood function

DE> 8, 9|�#, . . , �B C�?,@:��1#:;<|��:;<B

#<

• The estimated parameters 8∗ and 9∗ are: (8∗,9∗)=8GHI8J?,@> 8, 9

We apply this method to find Hull White parameters based on rates curves historical time series


Variance Estimation Method

Hull & White 1F model reminder:

• The process � , = ; |� K, K = ; is normal with known mean and standard deviation function of mean reversion and volatility.

• HW 1F : LG a M N G L = 9L�: < , where :

− G : short rate− 8: mean reversion − 9: volatility of the short term rate− M : deterministic function chosen to fit the initial term structure of interest rates.

For a given pillar ; the theoretical variance of the variation�� ; ��1# ; N �� ; is a function of volatility and mean reversion :

O8GPQ�R � ; ∆ 9 1 N �5?U

8;V

cf. [1] Park (2004) & [6] Schmidt (1997)

Estimate model’s parameters (mean reversion, volatility ) to fit theoretical variance curve to observed variance curve:

O8GWX% � ; #B ∑ �� ; N �:�:;< <VB# with � �:;< #

B ∑ �� ;B#


Variance Estimation Method: HW1F

Find the parameters (MR, volatilities, correlation) minimizing the squared residual over all pillars

Y O8GWX% � ; NO8GPQ�R � ; Z?�?[��%V

U


Implementation of the Variance Estimation Method

Looking at historical data across multiple markets:

First intuition:

• HW 1F will not be flexible enough to satisfactorily fit all curves shapes, but HW 2F should.

Second intuition :

• Realised variance curves often exhibit characteristic shapes

•We can use this property to improve our calibration algorithm


Variance Estimation Method: HW2F

Use the variance method to calibrate HW 2F parameter with the same approach as for HW 1F.

� The theoretical variance for the HW2F:

O8GPQ�R � ; ∆U\ 9#]:8#, ;< V = 9V]:8V, ;< V = 2_]:8#, ;<]:8V, ;<

� Use least square method to estimate model’s parameters (mean reversions, volatilities and correlations):

Y O8GWX% � ; NO8GPQ�R � ; Z?�?[��%V

U

Example: HKD zero rates

Variance method fit with HW1F Variance method fit with HW2F


Variance estimation method: calibration algorithm

Analysis of the shape of variance curve profile:

• First analyze the curve of observed variance over the pillars: ; � O8GPQ�R � ;• Three classical profiles are considered: Downwards, Upwards, Humped

Hull White 1F:

• Good fit in monotonic profiles (Positive mean reversion when downward, negative mean reversion when upward)• Monotonic profiles are not the usual type of profiles, so we try to determine and fit only a significant maturity window where the variance profile is actually monotonic

0,04

0,54

1,04

1,54

0 10 20 30 40

vo

lati

liti

es

Pillars

Variance fit with HW 1F: USD curves 1998

ObservedVolatility

FittingVolatility

0,04

0,54

1,04

1,54

2,04

0 10 20 30 40

vo

lati

liti

es

pillars

Variance fit with HW 1F : USD curves 2001

ObservedVolatility

FittingVolatility


Variance estimation method: calibration algorithm

Hull White 2F

• Humped profile: � Requires a negative correlation (empirical tests have generally found very strong negative _)� The algorithm runs a least square routine with _ N1 to estimate the 4 parameters of the model (8#, 8V, 9# and 9V).

� Based on the 4 estimated parameters, the algorithm runs again a least square minimization to estimate the correlation _.

• Complex shapes (the profile is not a simple hump):� Iterate over correlation values in the range [-1,0], run least square minimization to estimate (8#, 8V, 9# and 9V), select the best set of parameters.

0,01

0,51

1,01

1,51

0 10 20 30 40

Vo

lati

liti

es

pillars

variance fit with HW 2F: USD curve 2010

ObservedVolatility

FittingVolatility

0,03

0,23

0,43

0,63

0,83

1,03

1,23

1,43

1,63

0 10 20 30 40

Vo

lati

liti

es

Pillars

Variance fit with HW 2F: USD curves 1992

ObservedVolatility

FittingVolatility


MLE vs. Variance Estimation : Results on HW1F

On real observations: EUR Zero coupon rates 1D and 3M pillars, sliding time window by 1 day :

Findings on MLE are consistent with [1] F.C. Park (2004)

Data series : EUR Zero coupon rate for 1D and 3M pillars


Results: maximum likelihood estimation

Satisfactory results for vol estimation, but

The mean reversion can hardly be estimated reliably !

• Generated zero coupon series: 1000 observations, 1Y pillar

• Actual time series: EUR 1D vs. 3M Zero coupon

Generated data Estimated MR Estimated Vol

MR=1% and Vol=0.1% 20.30% 0.11%

Generated data Estimated MR Estimated Vol

MR=5% and vol=3% -5.30% 2.92%

Pillar Estimated MR Estimated Vol

1D 27.40% 1.85%Pillar Estimated MR Estimated Vol

3M -5.30% 0.96%


Historical calibration: art & science

Variance method: Pillar choice (test on generated series)

• We test the impact of pillars of zero coupon curve used in the calibration • Tests are done on generated zero coupon rate series:1000 series of 800 daily observations.

• Two cases : � Case 1: 1 short-term pillar: 3M, 1Y, 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 15Y, 20Y, 30Y� Case 2: several short term pillars: 1D, 3M, 6M, 9M, 1Y, 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 15Y, 20Y, 30Y

• Bias of estimated mean reversion in %, for different values of mean reversions and volatilities:

• Indeed, the theoretical variance is far less sensitive to the mean reversion for short term pillars! O8GPQ�R � ; ∆ @ #5�`ab?U

V


Historical calibration: dark art & common senseVariance method: Pillar choice (test on real data)

• Test HW1F estimation on TWD zero coupon rates• Two cases :

� Case 1: many short term pillars: 2M; 3M; 9M; 1Y; 1Y6M; 2Y; 3Y; 4Y; 6Y; 7Y; 8Y; 9Y; 10Y.� Case 2: less short term pillar: 10M; 2Y; 3Y; 4Y; 6Y; 7Y; 8Y; 9Y; 10Y.


1. Introduction




• Reconciling Counterparty Risk Models with Macroeconomic Theory• Historical Evidence That High Interest Rate Regime is Different• Adding Nonlinear Reversion (+R) to Interest Rate Models

5. Summary


The standard interest rate models have been in use for 20+ years for valuation. Why is the counterparty risk different?

The problem of unrealistic risk factor values at high quantiles is well known in derivatives valuation modeling

When applied solely to valuation, an argument can be made that unrealistic risk factor values at high quantile scenarios can be tolerated, for these two reasons:

• Because we are looking at expectations and not high quantiles• Because in the presence of hedging, our model is just an interpolation – we give model risk to the flow traders and they give it to the market at large

Neither argument applies to counterparty risk models


A conversation between an economist and a quant ata risk management committee meeting

• Economist: I have noticed that our risk model predicts 5% probability of 10bn exposure for this counterparty, however the total notional of our exposure to them is 1bn. How is this possible?

• Quant: This is a scenario of our local currency collapse, making the foreign cashflowswe will have to pay very expensive.

• Economist: I understand! You included in your model the probability of runaway inflation in the long term as a long term consequence of our central bank’s current attempts to stimulate the economy. This is very impressive.

• Quant (blushing): Nothing like that, it is much simpler. Let me explain.


Conversation continued …

• In our 95% scenario, the normal models we use project the rate above 10% for our local currency, while we the foreign currency rate first goes below -2.5%, then hovers near zero and finally settles at 1.5%.

• In the absence of capital controls, this rate differential would make the carry trade highly profitable applying a strong pressure on the FX rate.

• In risk neutral models which the quants over at the trading desk use for valuation, this would cause an FX drift equal to the rate differential. In the real world, the difference in risk premiums and economic fundamentals may help the FX rate resist the large difference in nominal interest rates for a while.

• However, if a 10 % difference in rates will persist for a decade, the FX rate will most likely collapse and this is reflected in our model.


Conversation continued …

• Quant: Here are some charts so you can see what I mean

• Economist: Your argument that the carry trade cannot be hugely profitable for an extended period of time makes sense. But don’t you think our central bank will take action when our interest rate keeps going up, while the other rate is near zero.

• Quant: This makes sense. I will go and put the central bank in our model.


Understanding the effect of central bank policy onnominal interest rates and FX

Selected content from the Reuters article published Friday Nov 29, 2013, entitled "Yen under pressure from carry trades, hits 5-yr low vs euro".Note: newspapers are working under real world measure.

The yen slid to a six-month low against the dollar and a five-year trough against the euro on Friday as rising risk sentiment fanned speculation that more investors might borrow and sell the low-yielding yen to buy riskier assets.

This trading strategy - called the yen-carry trade - comes as investors expect the Bank of Japan will keep or even enhance its ultra-easy policy, to help meet Prime Minister ShinzoAbe’s goal of sustaining growth and conquering 15 years of deflation.


More from the same article...

Still, many analysts think Japan still has a long way to reach its inflation target of 2 percent and the BOJ may need to take more steps some time next year to achieve that goal.

By contrast, the U.S. Federal Reserve is looking to reduce its stimulus, although it has not done so yet even six months after Chairman Ben Bernanke signaled that possibility in May.

Sterling was again a notable outperformer, hitting a fresh 11-month high on the dollar after the Bank of England surprised by scaling back stimulus for the housing sector on Thursday.

Traders took the move as further confirmation of the BoE’s confidence in the economic outlook and of their expectations that the BoE is moving closer to raising interest rates from the current record low of 0.5 percent.


The relationship between inflation, the nominalinterest rates, and FX

FX markets are driven by interest rates and inflation expectations in short term and long term

The central bank will take action if the interest rates stay too high or too low relative to other reserve currencies for an extended period of time

In this presentation, we will focus primarily on the high rates regime because of its greater effect on counterparty risk via long term FX scenarios


Historical Evidence That High Interest Rate Regime is Different

Historical from OECD Stats website

• Available at no cost from stats.oecd.org• Covers time period from 1950 to present• Data from 40+ countries including some non-members of OECD


Historical time series for two very different samplecurrencies

USD - throwing money from the helicopter

• In a near zero interest rate regime today• Had previous periods of very low rates (e.g. after the dotcom crisis)

AUD - keeping money under lock and key

• Central bank is comfortable with high interest rates• Short rate never below 4% for the past 40 years

Not only for USD but also for AUD the rates exited the > 10% territory much more rapidly than one would expect given the typical reversion speed of 5% -10% assumed in interest rate modeling


Prior research on regime change in interest rates

The distinct nature of high interest rate regime has been the subject of extensive research in economic literature (STAR/CSTAR/ESTAR models) since the 1990s

• Modeling the conditional distribution of interest rates as a regime-switching, process, [6] Gray (1996)

• Contemporaneous threshold autoregressive models: Estimation, Testing and Forecasting, [7] Duekera, Solab, Spagnolo (2006)

Statistical evidence on the mean reversion of interest rates, [8] Van den End (2013)

This research has not been reflected in counterparty risk models


CEV model for the IR component with mean reversiondependent on rate level

The proposed approach will be demonstrated on the specific example(one factor short rate SDE), however it can be applied to other model types

We start from SDE of the following form

Here is the drift term which plays the role of mean reversion

First, is calibrated such that the reversion is accelerating in the high rate environment. The calibration target is historical record of high interest rates, across currencies in similar economies

As usual, can be calibrated to the option skew

A general form of mean reversion dependent on both time and rate level was discussed in several publications in the 1990s but did not become part of market practice.


Parameterization of c:d, e d <There are several choices of parameterization of f: , G < , all resulting in similar model properties

The model properties are primarily driven by the drift (reversion speed) in the two limits: very high rates and for near zero rates (negative shadow rates)

One possible choice is:

f , G N�:G: < N M: <<where �:J< is a function which is constant 8 in several popular IR models and increases rapidly for high rates in the proposed model.

We could choose � J J ∝ with ∝being different for positive and negative values of J.

Models modified by making mean reversion nonlinear and accelerating for high rates can be referred to as "+R" models, for example HW+R, CEV+R, CIR++R etc.


Cross-sectional calibration to the historical rate ofdeparture from the high interest rate regime

• Start from the short rate history in OECD data• Find the maximum rate• Find the rate 5y after the maximum was reached• Plot rate change in 5y after the maximum vs. the maximum rate


Cross-sectional calibration to the historical rate ofdeparture from the high interest rate regime

Original data

10% floor

15% floor

20% floor


1. Introduction




5. Summary


Summary

The presentation describes historical calibration of volatility and mean reversion of interest rates in real world measure for counterparty risk.

• Avoiding unrealistic scenarios for the interest rates is especially important because of the effect on FX projection and counterparty risk

A stable method of volatility and mean reversion calibration is proposed:

• Fitting the realized variances of Zero Coupons increments to logical theoretical variance curve shapes• The method is more stable and accurate than traditional methods

Estimating reliably the mean reversion remains a challenge regardless of the estimation procedure, especially given the available data series windows

• We attribute this effect to the different rate dynamics observed historically in high and low interest rate regimes

A nonlinear mean reversion term (+R) which rapidly accelerates for high rates is proposed to model the historically observed speed of departure from a high rate regime.

• The model permits cross sectional calibration based on OECD data for 40+ currencies.


Misc.

References

• [1] F.C. Park – March 2004« Implementing Interest Rate Models : a practical guide »

• [2] G.R. Duffee & R.H. Stanton – March 2004« Estimation of Dynamic Term Structure Models »

• [3] Murex Analytics – July 2013« Model Parameters Estimation from Historical Data »

• [4] M. Fisher – March 2001« Forces That Shape the Yield Curve »

• [5] W.M. Schmidt – 1997« On a general class of one-factor models for the term structure of interest rates »

• [6] S.F. Gray – 1996« Modeling the conditional distribution of interest rates as a regime-switching process »

• [7] M. Duekera, M. Sola, F. Spagnolo – 2006« Contemporaneous Threshold Autoregressive Models:Estimation, Testing and Forecasting »

• [8] J. Van den End – 2013« Statistical evidence on the mean reversion of interest rates »

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