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
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 nonfinancial enterprises. 2
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. 3
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. 4
Dealing with term structure of interest rates 5
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 6
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
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 9
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. 10
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 11
Convenient compounding convention Continuous compounding: dt () = exp[ tyt ()] Instantaneous forward rates: 1 t yt () = rtdt () t 0
Snapshots (static fitting) 13
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: n Pk d( ti) Fi, k i= 1
The Problem Zero-coupon bonds: d( t) = e r( t) t P = N d( t ) k k k Coupon-bearing bonds: P k n i = Fi kd t 0, ( i ) P Fd d = d t ) ( i The system is typically underdetermined. Discount function values are to be found in intermediate points. Mathematically the problem is ill-posed i 16
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). 17
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
Problem statement ensuring positive forward fates Select the solution in the form: t 2 dt () = exp f ( τ) dτ 0 Select the degree of non-smoothness: T f 0 2 τ dτ ( ) min Minimize conditionally the residual: N n 2 ti T 2 2 k 0 i, k k 0 k= 1 i= 1 w exp f () τ dτ F P + α f () τ dτ min f 22
The semi-analitical solution Smirnov & Zakharov (2003): f(t) is a spline of the following form: C1exp{ λk( t tk 1)} + C2exp{ λk( t tk 1)}, λk > 0 f( t) = C1sin( λk( t tk 1)) + C2cos( λk( t tk 1)), λk < 0, C1( t tk 1) + C2, λk = 0 t [ tk 1; tk] f( tk 0) = f( tk + 0), f ( tk 0) = f ( tk + 0) 23
History can be useful Consider to consecutive days, when short term bonds are not traded (or quoted the second day:
Working with missing data 25
Approach Data history accumulation Filtration. Data augmentation. Application of fitting algorithms. 26
2 ˆ x T = 1 ξ 2 0 Filters Value level filter. Value change filter. Relative position of trade price compared to market quotes. Liquidity filter (number of deals, turnover). 27
Relative position of trade price compared to market quotes 28
Missing data density 29
Overall trades volume 30
Bayesian estimation approach Prior density Posterior density - Multivariate parameter (random) - Observable data Likelihood function 31
Markov Chain Monte-Carlo X all X = ( X, X ) X all obs mis obs X mis - Complete dataset (observable +unobservable) - observable data - unobservable data p( θ X ) obs - Complex distribution p( θ X, X ) obs mis - Simple distribution 32
Markov Chain Monte-Carlo Imputation Step Generating missing data X ~ p( X X, θ ) θ () t ( t 1) mis mis obs Posterior Step Generating posterior distribution parameters ~ p( X, X ) () t θ () t obs mis Markov Chain (1) (1) (2) (2) ( X, θ ),(, ),... d mis X θ mis p( Xmis, θ Xobs ) 33
ЕМ algorithm (industry standard) Filling missing data with conditional expectations y old ij = yij, E n i= 1 o yij yi, o o 0, yij yi и yik yk, old o old u c = yij y, θ, yij yi ; ijk o old ( ) cov yij, yik y, θ, в Recompute posterior distribution modes n new 1 old µ j = yij, j = 1,..., d n i= 1 n new 1 old old old new new σ jk = yij yik + cijk µ j µ k, j, k = 1,..., d n i= 1 противном случае. Modes of joint posterior distribution of parameters and missing data 34
Parameters evaluation results (correlation coefficients) 35
Stochastic Evolution 36
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. x x 0 1 τt 2 x τt rt( x) βt βte = + + βt e τ t
Consistency Problems - II Nearly all arbitrage-free dynamic models are primitive. All such models are affine (Bjork, Christensen, 2001, Filipovic, Teichmann, 2004). N r( x) h ( x) Y( x) λ = + t 0 i it, i= 1
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 couponbearing 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 i f (, t t ) = f (0, t ) + α( u, t, ω) du + σ ( u, t, ω) dw ( u), 0 0 i= 1 0 t t τ. t current time, t maturity time, Brace, Musiela (1994): One infinite-dimensional equation. rt ( x) = f( t, x+ t), x, t +. Filipovic (1999): Infinite Brownian motions. Currently: credit risks and stochastic volatility. n t i
The Model Based on the infinite-dimensional (Filipovic,1999) extension of the HJM framework. In Musiela parametrization: j j t = t + α + % t σt βt j= 1 dr ( Dr ) dt d. No-arbitrage condition: j j α( x) % σ ( x) % σ ( τ) dτ. = j= 1 0 x
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. rx ( ) = const for x T Realistic. j j % σ (, t ω, h)( x) = σ ( x) h( x).
Model Specification Ito SDE in Sobolev space. j j j j j j drt = Drt + rtσ I( rtσ ) rtσtγ dt + rtσt βt, j= 1 j= 1 j= 1 ( If)( x) = f( τ) dτ, γ j j σ 0 x market price of risk, volatility parameters.
Observations Need a way to incorporate the stream of new information. Let qk () r be the price of the k-th bond with respect to the true forward rate curve r. Let the observed prices distribution p p k ~ Nq ( ( r), w) k k k be random with is of order of the bid-ask spread. w k
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 () i p at time t i dp dp rt i () i α J( r) N( qr ( t ) p, diag( w )). i k e r t i 0
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: T d r( ) () r t τ J = 0 dτ Conditional on 1 observation: all prices observed at the same time (snapshot) and using flat priors: log ( ) ( ) min. N 2 T 1 d r() τ Pr = wk ( qk r Pk) + α dτ 0 This problem k= 1 formulation leads to da τ known nonparametric model: Smirnov, Zakharov (2003). 2 dτ 2
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 K of zero-coupon bonds tends to infinity. Times to maturity uniformly distributed on [0,T]. Number of observations M tends to infinity. Time between observations Δt tends to 0 MΔ t = L - observation period (e.g. 60 trading days). The true forward rate r t has a bounded derivative.
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.