kevin shea, cfa kevin.shea@mathworks€¦ · calibration approaches –market data: lsqnonlin,...
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1© 2014 The MathWorks, Inc.
Calibration and Simulation of Interest Rate Models
Kevin Shea, CFA
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Agenda
Calibration to Market Data
Calibration to Historical Data
Simulation and Valuation
Counterparty Credit Risk Analysis
Questions and Answers
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Interest Rate Models
𝑟 𝑡 = 𝑥 𝑡 + 𝑦 𝑡 + 𝜑 𝑡𝑑𝑥 𝑡 = −𝑎𝑥(𝑡)𝑑𝑡 + 𝜎𝑑𝑊1(𝑡)𝑑𝑦 𝑡 = −𝑏𝑦 𝑡 𝑑𝑡 + 𝜂𝑑𝑊2 𝑡
𝑑𝑊1 𝑡 𝑑𝑊2 𝑡 = 𝜌𝑑𝑡
G2++
𝑑𝑟(𝑡) = 𝜃 𝑡 − 𝑎𝑟 𝑑𝑡 + 𝜎𝑑𝑊(𝑡)
Hull-White
𝑑𝑟(𝑡) = 𝑎 𝑏 − 𝑟 𝑑𝑡 + 𝜎√𝑟𝑑𝑊(𝑡)
Cox-Ingersoll-Ross
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Calibrate to Market Data
Choose a set of liquid
calibration instruments –
typically caps, floors,
swaptions.
Find the set of model
parameters that matches as
closely as possible the
observed prices.
𝑘=0
𝑛
(𝑃𝑖 − 𝑃𝑖(𝜃))2
𝑃𝑖: Market Price 𝑃𝑖: Model Price
𝜃: Model Parameters
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Calibrate CIR Model using MLE of Transition Density
𝑑𝑟(𝑡) = 𝑎 𝑏 − 𝑟 𝑑𝑡 + 𝜎√𝑟𝑑𝑊(𝑡)
𝑎: mean reversion speed
𝜎: volatility of the short rate
𝑏 : level
Aït‐Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. The
Journal of Finance, 54(4), 1361-1395.
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Stochastic Differential Equation Models
Suite of models including : bm, gbm, cir, hwv,
heston, cev
Simulate methods
Framework for creating
custom models
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Calibrate using Kalman Filter
Formulate models as
state space systems.
Use Kalman filter to
estimate parameters.
Estimate parameters
from historical yield
curves.
Park, F.C., “Implementing Interest Rate Models: A Practical Guide.” Capital Markets & Portfolio
Research, Inc. white paper, 2004
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State Space formulation for G2++ Model
Park, F.C., “Implementing Interest Rate Models: A Practical Guide.” Capital Markets & Portfolio
Research, Inc. white paper, 2004
𝑥𝑡 = 𝐴𝑥𝑡−1 + 𝐵𝜇𝑦𝑡 = 𝐶𝑥𝑡 + 𝐷𝜖
𝐴 = 𝑒−𝑎∆𝑡 00 𝑒−𝑏∆𝑡
𝐵 =
𝜎1 − 𝑒−2𝑎∆𝑡
2𝑎0
0 𝜂1 − 𝑒−2𝑏∆𝑡
2𝑏
Transition Equation
Measurement Equation
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State Space Model
Supports time-invariant and
time-varying, linear state-
space models.
Perform univariate and
multivariate time-series data
analysis.
Functionality to: estimate,
filter, smooth,
simulate, forecast
New state space model, ssm
in Econometrics Toolbox™.
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Interest Rate Model Simulation
Support for Hull-White, G2++
and LIBOR Market Model.
simTermStructs simulates
entire term structure.
Specify models and simulate
entire term structure
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Swap Portfolio
Store data in a MATLAB Table.
Easy to read in data.
Tabular display.
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Valuing the Portfolio
Value portfolio using swapbyzero
Use parfor to loop
over simulation
dates.
Tenors
D
a
t
e
s
>> Values = zeros(nDates,nSwaps,nTrials);
>> parfor dateidx=1:nDates
Values(dateidx,:,:) = swapbyzero(...)
end
Instruments
D
a
t
e
s
Interest
Rates Portfolio
Values
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Counterparty Credit Risk Functions
Support for computing credit
exposures.
Support for computing various
credit exposure profiles,
including potential future
exposure and expected
exposure.
Compute exposures and CCR profiles
>> Exposures = creditexposures(Values);
>> Profiles = exposureprofiles(SimDates,Exposures);
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Computing Credit Valuation Adjustment
Compute exposure from exposureprofiles
Compute default probabilities from cdsbootstrap Simulation Time
Default Probability Density
Simulation Time
Discounted Exposure
𝐶𝑉𝐴 = (1 − 𝑅) 0
𝑇
𝐷𝑖𝑠𝑐𝐸𝑥𝑝 𝑡 𝑑𝑃𝐷(𝑡)
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Summary
Calibration Approaches
– Market Data: lsqnonlin, simulannealbnd
– Historical Data: mle, ssm
Monte Carlo Simulation in MATLAB
– cir
– HullWhite1F, LinearGaussian2F
Counterparty Credit Risk
– creditexposures, exposureprofiles
– cdsbootstrap
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