Stochastic modelling of term structure of interest rates,

realistic with no arbitrage Prof. Dr. Sergey Smirnov

Head of the Department of Risk Management and Insurance Director of the Financial Engineering and Risk Management Lab

Higher School of Economics, Moscow

ETH Zürich, October 7, 2010

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About Lab •  The Financial Engineering & Risk Management Lab

(FERMLab) was founded in March 2007 within the Department of Risk Management and Insurance of State University – Higher School of Economics.

•  The Lab’s mission is to promote the studies in modern financial engineering, risk management and actuarial methods both in financial institutions, such as banks, asset managers and insurance companies, and in non-financial enterprises.

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Lab Team •  Smirnov Sergey, PhD, Director of FERMLab; Head of Department of

Risk-Management and Insurance; PRMIA Cofounder, Member of Education Committee of PRMIA; Vice-Chairman of European Bond Commission; Member of Advisory Panel of International Association of Deposit Insurers.

•  Sholomitski Alexey, PhD, Deputy Director of Laboratory for Financial Engineering and Risk Management; Deputy Head of Department of Risk-Management and Insurance. Actuarial science.

•  Kosyanenko Anton, MS,MA. Data filtering and augmentation.. •  Lapshin Victor, MS. Term structure of interest rate modelling . •  Naumenko Vladimir, MA. Market microstructure and liquidity

research.

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Key Lab Activities •  Accumulation of financial and economics data required for

empirical studies. •  Empirical studies of the financial markets’ microstructure. •  Development of structured financial products pricing and

contingent liability hedging models. •  Risk evaluation and management based on quantitative

models •  Actuarial studies for insurance and pensions applications. •  Familiarization with and practical application of data

analysis and modeling software.

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Dealing with term structure of interest rates

Instruments

•  Term structure of interest rates can be constructed for different market instruments: bonds, interest rate swaps, FRA, etc.

•  In this presentation we consider only bond market as a source of information

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Classification

•  By information used: – Snapshot methods. – Dynamic methods.

•  By a priori assumptions: – Parametric methods. – Nonparametric (spline) methods.

Some Approaches •  Static methods - yield curve fitting

–  Parametric methods (Nelson-Siegel, Svensson)

–  Spline methods (Vasicek-Fong, Sinusoidal-Exponential splines)

•  Dynamic methods (examples) –  General affine term structure model –  HJM Markov evolution of forward rates

9

A Priori Assumptions

•  Incomplete and poor available data cannot be treated without additional assumptions.

•  Different assumptions lead to different problem statements and therefore to different results.

•  Often assumptions are chosen ad hoc, without economic interpretation

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The Data

•  Available data: bonds, their prices, possibly bid-ask quotes.

•  Difficulties with observed data: – coupon-bearing bonds; –  few traded bonds, – different credit quality and liquidity.

Term structure descriptions

•  Discount function

•  After a convention of mapping discount function to interest rates for different maturities is fixed: – Zero coupon yield curve –  Instantaneous forward curve

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Convenient compounding convention

•  Continuous compounding:

•  Instantaneous forward rates:

( ) exp[ ( )]d t t y t= −

0

1( ) ( )t

y t r t dtt

= ∫

13

Snapshots (static fitting)

Usual assumptions •  Bonds to be used for determining yield

curve are of the same credit quality, •  All instruments have approximately the

same liquidity. •  A method using bonds of different credit

quality for construction of risk-free zero coupon in Euro zone was propose by Smirnov et al (2006), see

www.effas-ebc.org 14

Data treatment

•  Bond prices at the given moment. •  Bid/Ask quotes at the given moment. •  Other parameters: volumes, frequencies

etc. •  Bond price is assumed to be

approximately equal to present value of promised cash flows:

,1( )

n

k i i ki

P d t F=

≈∑

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The Problem •  Zero-coupon bonds:

•  Coupon-bearing bonds:

•  The system is typically underdetermined. •  Discount function values are to be found in

intermediate points. •  Mathematically the problem is ill-posed

( )k k kP N d t=

∑ =≈ n

i ikik tdFP0 , )(

FdP ≈ )( ii td=d

ttretd )()( −=

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Different Approaches

•  Assumptions on a specific parametric form of the yield curve (parametric methods)

•  Assumptions on the degree of the smoothness (in some sense) of the yield curve (spline methods).

The term structure

of interest rates estimation by different central banks

BIS Papers No 25 October 2005 “Zero-coupon yield curves: technical

documentation” Mainly parametric methods are used

Parametric methods of yield curve fitting

Svensson ( 6 parameters) Instantaneous forward rate is assumed to have the following form:

Nelson-Siegel (4 parameters) is a special case of Svensson with

Assuming specific functional form for yield curve is arbitrary and has no economic ground

Nonparametric methods •  Usually splines

•  Flexibility •  Sensibility •  Possibility of smoothness/accuracy

control

The preconditions 1.  Approximate discount function should be decreasing

(i.e. forward rates should be non-negative) with initial value equal to one, and positive.

2.  Approximate discount function should be sufficiently smooth.

3.  Corresponding residual with respect to observed bond prices should be reasonably small.

4.  The market liquidity is taken into account to determine the reasonable accuracy, e.g. the residual can be related to the size of bid-ask spreads.

21

22

•  Select the solution in the form:

•  Select the degree of non-smoothness:

•  Minimize conditionally the residual:

Problem statement ensuring positive forward fates

2

0( ) exp ( )

td t f dτ τ⎡ ⎤= −⎢ ⎥⎣ ⎦∫

2

0( ) min

Tf dτ τ′ →∫

22 2

,0 01 1

( ) ( ) miniN n t T

k i k k fk iw exp f d F P f dτ τ α τ τ

= =

⎛ ⎞⎡ ⎤ ′− − + →⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠∑ ∑ ∫ ∫

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The semi-analitical solution Smirnov & Zakharov (2003): f(t) is a spline of the following form:

1 1 2 1

1 1 2 1

1 1 2

exp{ ( )} exp{ ( )}, 0

( ) sin( ( )) cos( ( )), 0,( ) , 0

k k k k k

k k k k k

k k

C t t C t t

f t C t t C t tC t t C

λ λ λλ λ λ

λ

− −

− −

−

⎧ − + − − >⎪⎪= − − + − − <⎨⎪ − + =⎪⎩

1[ ; ] ( 0) ( 0), ( 0) ( 0)k k k k k kt t t f t f t f t f t− ′ ′∈ − = + − = +

History can be useful •  Consider to consecutive days, when short

term bonds are not traded (or quoted the second day:

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Working with missing data

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Approach •  Data history accumulation

•  Filtration.

•  Data augmentation.

•  Application of fitting algorithms.

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Filters

•  Value level filter.

•  Value change filter.

•  Relative position of trade price compared to market quotes.

•  Liquidity filter (number of deals, turnover).

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2ˆ1

ξxT −=

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Relative position of trade price compared to market quotes

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“Missing” data density

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Overall trades volume

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Prior density

Likelihood function

Posterior density

- Multivariate parameter (random) - Observable data

Bayesian estimation approach

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Markov Chain Monte-Carlo

( , )all obs misX X X= obsX

misX

allX - Complete dataset (observable+unobservable) - observable data

- unobservable data

( | )obsp Xθ - “Complex” distribution

( | , )obs misp X Xθ - “Simple” distribution

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Imputation Step

Posterior Step

Generating missing data

( ) ( 1)~ ( | , )t tmis mis obsX p X X θ −

( ) ( )~ ( | , )t tobs misp X Xθ θ

Generating posterior distribution parameters

Markov Chain (1) (1) (2) (2)( , ), ( , ),... ( , | )dmis mis mis obsX X p X Xθ θ θ⎯⎯→

Markov Chain Monte-Carlo

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Filling missing data with conditional expectations

Recompute posterior distribution modes

Modes of joint posterior distribution of parameters and missing data

ЕМ algorithm (industry standard)

⎪⎩

⎪⎨

⎧

∈⎟⎠⎞⎜

⎝⎛

∈=

∑=

;,,

,,

1

uiij

oldon

iij

oiijij

oldij yyyyE

yyyy

θ ( )⎪⎩

⎪⎨⎧ ∈∈

=.,,,cov

,,0

случаепротивномвyyy

yyиyyc

oldoikij

okik

oiijold

ijk θ

∑=

==n

i

oldij

newj djy

n 1

,...,1,1µ

dkjcyyn

newk

newj

n

i

oldijk

oldik

oldij

newjk ,...,1,,1

1

=−⎟⎠⎞⎜

⎝⎛ +⋅= ∑

=

µµσ

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Parameters evaluation results (correlation coefficients)

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Stochastic Evolution

Low-Dimensional Models

•  Dynamics of several given variables (usually instantaneous rate and some others).

•  Low-dimensional dynamic models imply non-realistic zero-coupon yield curves: negative or infinite.

Consistency Problems

•  Very few static models may be embedded into a stochastic dynamic model in an arbitrage-free manner (Bjork, Christensen, 1999, Filipovic, 1999).

•  Nelson-Siegel model allows arbitrage with every non-deterministic parameter dynamics.

0 1 2( ) t t

x x

t t t tt

xr x e eτ τβ β βτ

− −= + +

Consistency Problems - II

•  Nearly all arbitrage-free dynamic models are primitive.

•  All such models are affine (Bjork, Christensen, 2001, Filipovic, Teichmann, 2004).

01

,( ) ( ) ( )N

t i i ti

r x h x Y x λ=

= +∑

Goals

•  Construction of an arbitrage-free nonparametric dynamic model, allowing for sensible snapshot zero-coupon yield curves, thesis of Lapshin (2010).

•  Peculiarities of data: –  Incompleteness: only several coupon-

bearing bonds are observed. – Unreliability: price data may be subject to

errors and non-market issues.

Heath-Jarrow-Morton (1992) Approach

•  Modelling all forward rates at once:

•  t – current time, t’ – maturity time, •  Brace, Musiela (1994): One infinite-dimensional

equation. •  Filipovic (1999): Infinite Brownian motions. •  Currently: credit risks and stochastic volatility.

0 01

( , ) (0, ) ( , , ) ( , , ) ( ),

0 .

nt t ii

if t t f t u t du u t dW u

t t

α ω σ ω

τ=

′ ′ ′ ′= + +

′≤ ≤ ≤

∑∫ ∫

( ) ( , ) ., ,tr x f t x t x t += + ∈°

The Model •  Based on the infinite-dimensional

(Filipovic,1999) extension of the HJM framework.

•  In Musiela parametrization:

•  No-arbitrage condition:

01

( ) ( ) ( ) .xj j

jx x dα σ σ τ τ

∞

=

=∑ ∫% %

1( ) .j j

t t t t tj

dr Dr dt dα σ β∞

=

= + +∑ %

The Simplest Possible Model

•  Linear local volatility: •  Objective dynamics required. •  Market price of risk is constant for each

stochastic factor. •  Finite horizon:

– Observations only up to a known T. –  – Realistic.

( , , )( ) ( ) ( ).j jt h x x h xσ ω σ=%

( ) for r x Tconst x= ≥

Model Specification

•  Ito SDE in Sobolev space.

1 11

0

( ) ,

( )( ) ( ) ,

market price of risk,volatility parameters.

j j j j j jt t t t t t t t t

j jj

x

j

j

dr Dr r I r dt r

I f x f d

rσ σ σ γ σ β

τ τ

γσ

∞∞ ∞

= ==

⎛ ⎞= + − +⎜ ⎟⎜ ⎟⎝ ⎠

=

−

−

∑ ∑∑

∫

Observations

•  Need a way to incorporate the stream of new information.

•  Let be the price of the k-th bond with respect to the true forward rate curve r.

•  Let the observed prices be random with distribution

•  is of order of the bid-ask spread.

( )kq r

kp~ ( ( ), )k k kN q r wp

kw

Credibility

•  Credibility is a degree of reliability of a piece of information (logical interpretation of probability).

•  Standard deviation of the observation error is assumed to be directly dependent on the credibility.

•  Factors affecting credibility: –  Bid-ask spread. –  Deal volume. –  Any other factors.

Smoothness

•  The yield curves used by market participants to determine the deal price are sufficiently smooth.

•  Non-smoothness functional J(r) has to be chosen

•  Each observation is conditionally independent. •  Bayesian approach: conditional on observation

0

( ) ( )( , ( )) ( · .)ti

i

ti

r i J rt k

r

dPN p diag w e

Pq r

dα

−

−∝ −

( ) at time iip t

Uncertainty and incompleteness

•  Sources : – Stochastic dynamics for all matrities needed. – Limited number of observed (coupon-bearing

bonds) – Credibility of observations.

Snapshot Case •  Choose a special non-smoothness functional:

•  Conditional on 1 observation: all prices observed at the same time (snapshot) and using flat priors:

•  This problem formulation leads to a known non-parametric model: Smirnov, Zakharov (2003).

2

0

( )( )

T td rdr

dJ

ττ

τ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠∫

( )2

21

01

( )log ( ) ( ) min.

N T

k k kk

d rP r q r Pw d

dτα τ

τ−

=

⎛ ⎞− = − + →⎜ ⎟⎜ ⎟⎝ ⎠

∑ ∫

Parameter Estimation

•  Estimating volatility parameters requires advanced techniques.

•  Markov Chain Monte-Carlo algorithm. – Parallel processing.

Asymptotic Consistency of the Method

•  The constructed estimate is consistent if: – The number of zero-coupon bonds tends

to infinity. Times to maturity uniformly distributed on [0,T].

– Number of observations tends to infinity. – Time between observations –  - observation period (e.g. 60

trading days). – The true forward rate has a bounded

derivative.

K

M tends to 0tΔ

M t LΔ =

tr

Approximate Algorithms

•  Linearization allows for Kalman-type filter for zero-coupon yield curve given known parameters.

•  Fast volatility calibration given volatility term structure up to an unknown multiplier.

Complete Algorithm

•  Once a week (month) – full volatility structure estimation.

•  Intraday volatility multipliers estimation. •  Intraday zero-coupon yield curve

estimation via maximum likelihood. •  Approximate method for real-time

response.

Model Validation •  3 time spans, Russian market, MICEX

data, snapshots 3 times per day. – 10 jan 2006 – 14 apr 2006, normal market. – 1 aug 2007 – 28 sep 2007, early crisis. – 26 sep 2008 – 30 dec 2008, full crisis.

•  In the normal market conditions the model is not rejected with 95% confidence level.

•  Works reasonably on the crisis data.

Complexity

•  Market data are limited. •  Only enough to identify models with

effective dimension = 2,3. •  More complex models are not identifiable. •  Tikhonov principle: the best model is the

simplest one providing the acceptable accuracy.

Main Results

•  First arbitrage-free nonparametric dynamic yield curve model, providing: – Plausible and variable snapshot curves. – A good snapshot method as a special case. – Positive spot forward rates. – Liquidity consideration: inaccuracy and

incompleteness in observations. •  Numerical algorithms and implementation

tested on the real market data.