1© 2014 The MathWorks, Inc.

Calibration and Simulation of Interest Rate Models

Kevin Shea, CFA

kevin.shea@mathworks.com

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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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