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
•ܴܹ‫ܣ‬௖௥௘ௗ௜௧ ൌ ‫ ܦܣܧ‬ ൈ ‫ܭ‬ ܲ‫,ܦ‬ ‫,ܦܩܮ‬ ‫ܯ‬ 8%⁄ , where under IMM:
‫ܦܣܧ‬ ൌ ∝ ∙ max ‫ܧܲܧ ݁ݒ݅ݐ݂݂ܿ݁ܧ‬ଵ௒ , ‫ܧܲܧ ݁ݒ݅ݐ݂݂ܿ݁ܧ‬ଵ௒
௦௧௥௘௦௦௘ௗ
, i.e. a 1Y horizon is considered
EPE model backtested by validating the risk factor distributions up to 1Y
‫ܯ‬ ൌ min
∑ ாாாೖ∙ఋ௧ೖ∙ௗ௙ೖ ೟ರభೊ ା ∑ ாாೖ∙ఋ௧ೖ∙ௗ௙ೖ ೟ಭభೊ
∑ ாாாೖ∙ఋ௧ೖ∙ௗ௙ೖ ೟ರభೊ
, 5ܻ , i.e. the full exposure profile is taken into
account
•CVA risk capital charge : aVaR-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 toVaR
• 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

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
WhichYield 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 NewYorkTimes - 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

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 forVaR) 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: improvedVariance 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 ‫ݐ‬௜ knowing the
values observed at ‫ݐ‬௜ି૚.
• Supposing the parameters ܽ and ߪ are known, we calculate the likelihood of having a specific
series of data: product of the conditional probabilities of each step.
• Noting ܴ௜ሺ߬ሻ ൌ ܴሺ‫ݐ‬௜, ‫ݐ‬௜ ൅ ߬ሻ zero coupon rate at time ‫ݐ‬௜ for the pillar ߬, the likelihood function
ࣦ is the probability density ݂௔,ఙ of the realization of the path ሺܴሻ௜
ࣦ ܽ, ߪ|ܴଵ, . . , ܴே ൌ ෑ ݂௔,ఙሺܴ௜ାଵሺ߬ሻ|ܴ௜ሺ߬ሻ
ே
ଵ
ሻ
• In practice we use the logarithm of the likelihood function
݈ࣦ݊ ܽ, ߪ|ܴଵ, . . , ܴே ൌ ෑ ݂௔,ఙሺܴ௜ାଵሺ߬ሻ|ܴ௜ሺ߬ሻ
ே
ଵ
ሻ
• The estimated parameters ܽ∗
and ߪ∗
are: (ܽ∗
,ߪ∗
)=ܽ‫ݔܽ݉݃ݎ‬௔,ఙࣦ ܽ, ߪ
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 ܴ ‫,ݐ‬ ‫ ݐ‬ ൅ ߬ |ܴ ‫,ݏ‬ ‫ ݏ‬ ൅ ߬ is normal with known mean and standard deviation function of mean
reversion and volatility.
• HW 1F : ݀‫ݎ‬ ‫ݐ‬ ൌ a ߠ ‫ݐ‬ െ ‫ݎ‬ ‫ݐ‬ ݀‫ݐ‬ ൅ ߪܹ݀ሺ‫ݐ‬ሻ , where :
− ‫ݎ‬ : short rate
− ܽ: mean reversion
− ߪ: volatility of the short term rate
− ߠ ‫ݐ‬ : deterministic function chosen to fit the initial term structure of interest rates.
For a given pillar ߬ the theoretical variance of the variation
‫ܦ‬௜ ߬ ൌ ܴ௜ାଵ ߬ െ ܴ௜ ߬ is a function of volatility and mean reversion :
ܸܽ‫ݎ‬்௛௘௢ ‫ܦ‬ ߬ ൌ ∆
ߪ 1 െ ݁ି௔ఛ
ܽ߬
ଶ
cf. [1] Park (2004) & [6] Schmidt (1997)
Estimate model’s parameters (mean reversion, volatility ) to fit theoretical variance
curve to observed variance curve:
ܸܽ‫ݎ‬ை௕௦ ‫ܦ‬ ߬ ൌ
ଵ
ே
∑ ‫ܦ‬௜ ߬ െ ‫ܧ‬ሺ‫ܦ‬ሺ߬ሻ ሻଶே
ଵ with ‫ܧ‬ ‫ܦ‬ሺ߬ሻ ൌ
ଵ
ே
∑ ‫ܦ‬௜ ߬ே
ଵ

Variance Estimation Method: HW1F
Find the parameters (MR, volatilities, correlation) minimizing the squared
residual over all pillars
෍ ܸܽ‫ݎ‬ை௕௦ ‫ܦ‬ ߬ െܸܽ‫ݎ‬்௛௘௢ ‫ܦ‬ ߬ ௣௔௥௔௠௘௧௘௥௦
ଶ
ఛ

Implementation of theVariance 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:
ܸܽ‫ݎ‬்௛௘௢ ‫ܦ‬ ߬ ൌ
∆
ఛమ ߪଵ‫ܤ‬ሺܽଵ, ߬ሻ ଶ
൅ ߪଶ‫ܤ‬ሺܽଶ, ߬ሻ ଶ
൅ 2ߩ‫ܤ‬ሺܽଵ, ߬ሻ‫ܤ‬ሺܽଶ, ߬ሻ
Use least square method to estimate model’s parameters (mean reversions, volatilities and correlations):
෍ ܸܽ‫ݎ‬ை௕௦ ‫ܦ‬ ߬ െܸܽ‫ݎ‬்௛௘௢ ‫ܦ‬ ߬ ௣௔௥௔௠௘௧௘௥௦
ଶ
ఛ
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: ߬ ܸܽ‫ݎ‬்௛௘௢ ‫ܦ‬ ߬
• 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
volatilities
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
volatilities
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 ߩ)
The algorithm runs a least square routine with ߩ ൌ െ1 to estimate the 4 parameters of the model (ܽଵ, ܽଶ, ߪଵ and ߪଶ).
Based on the 4 estimated parameters, the algorithm runs again a least square minimization to estimate the correlation ߩ.
• Complex shapes (the profile is not a simple hump):
Iterate over correlation values in the range [-1,0], run least square minimization to estimate (ܽଵ, ܽଶ, ߪଵ and ߪଶ), select the best set
of parameters.
0,01
0,51
1,01
1,51
0 10 20 30 40
Volatilities
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
Volatilities
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! ܸܽ‫ݎ‬்௛௘௢ ‫ܦ‬ ߬ ൌ ∆
ఙ ଵି௘షೌഓ
௔ఛ
ଶ

Historical calibration: dark art & common sense
Variance method: Pillar choice (test on real data)
• Test HW1F estimation onTWD 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.

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
• 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 at
a 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 cashflows
we 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 on
nominal 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 Shinzo Abe’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 onThursday.
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 nominal
interest 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 EvidenceThat 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 sample
currencies
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 reversion
dependent 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 ࣆሺ࢚, ࢘ ࢚ ሻ
There are several choices of parameterization of ߤሺ‫,ݐ‬ ‫ݎ‬ ‫ݐ‬ ሻ , 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:
ߤ ‫,ݐ‬ ‫ݎ‬ ‫ݐ‬ ൌ െ݂ሺ‫ݎ‬ሺ‫ݐ‬ሻ െ ߠሺ‫ݐ‬ሻሻ
where ݂ሺ‫ݔ‬ሻ is a function which is constant ܽ in several popular IR models and
increases rapidly for high rates in the proposed model.
We could choose ݂ ‫ݔ‬ ൌ ‫ݔ‬ ∝
with ∝ being different for positive and negative values of
‫.ݔ‬
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 of
departure 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 of
departure from the high interest rate regime
Original data
10% floor
15% floor
20% floor

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 theYield 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 »

Long horizon simulations for counterparty risk

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Andere mochten auch

Andere mochten auch (20)

Ähnlich wie Long horizon simulations for counterparty risk

Ähnlich wie Long horizon simulations for counterparty risk (20)

Kürzlich hochgeladen

Kürzlich hochgeladen (20)

Long horizon simulations for counterparty risk