Towards a non-parametric Towards a non-parametric stochastic framework: a consistent approach of integrated modelling of credit integrated modelling of credit and interest rate risk Sergey Smirnov Sergey Smirnov Nickolay Andreev, Victor Lapshin Marat Kurbangaleev, Polina Tarasova
Basics of Reduced-Form Model of Credit Risk • Modeling of an unpredictable time of default. • The key y element is the default intensity y rate) , which is conditional default (hazard density. y • Model is to be calibrated to market price data . Corporate fundamentals are NOT explicitly taken Corporate fundamentals are NOT explicitly taken into account. • Applied • Applied to to credit credit risk risk sensitive sensitive pricing pricing instruments. 2
Pros & Cons of Default Intensity Models Pros : Cons : Relative tractability Treat default as absolutely � � unpredictable event Results are highly Don’t demand much data � � dependent on specification of default Allow modeling interest � intensity intensity rate and credit risk in a joint framework 3
Interest Rate and Credit Risk Analogies Interest Rate and Credit Risk Analogies • • Discount Function d(t,s) Survival Probability Function P(t,s) d(t,s1)>d(t,s2) if s1<s2 P(t,s1)>=P(t,s2) if s1<s2 1. 1. d(t,s)>0 P(t,s)>0 2. 2. d(t,0)=1 P(t,s)=1 3. 3. d(t,s) d(t,s) → 0 with s →∞ 0 with s P(t,s) P(t,s) → 0 with s →∞ (no one lives 0 with s (no one lives 4. 4. 4. 4. forever) • Default Intensity Function • Instantaneous Forward Rate Nonnegative Function � N Nonnegative ti � 4
Default Intensity Specifications (Deterministic) • The simplest example is a time homogeneous default model g • Default intensity is a positive constant λ t • Survival probability: S i l b bilit − λ = s t e e P P ( ( t t , s s ) ) where t – current date, s – term. • In moment t market participants forecast credit quality to remain the same over the q y time 5
Default Intensity Specifications (Deterministic) • The next example is a time inhomogeneous default model • Default intensity term structure is deterministic but • Default intensity term structure is deterministic, but non-constant ( polynomial, exponential, etc. ) • Survival probability: p y s ∫ ∫ − λ τ τ = ( ) d t e P ( t , s ) 0 λ λ • The form of depend on current perceptions of f f ( ( s ) ) f t market participant about future credit quality of obligor • The following slide illustrates how assumed hazard The following slide illustrates how assumed hazard rate term structure effects default probability function 6
Default Intensity vs Probability Function Default Intensity vs Probability Function 7
Default Intensity vs Probability Density Default Intensity vs Probability Density 8
Implementation of Models with Deterministic Specification • The trivial use of deterministic specification of default intensity is the p y extraction of risk-neutral hazard rate function bootstrapping bond or CDS data function bootstrapping bond or CDS data. Assumed form of hazard rate curve is piecewise constant . i i t t • Bootstrapping methodology is discussed pp g gy further below. 9
Default Intensity Specifications (Stochastic) • Default intensity assumed to be a stochastic process • The market is assumed to be arbitrage-free and complete, therefore unique risk-neutral probability measure exists, under which default- sensitive assets are priced • Risk-neutral survival probability: ⎛ ⎛ ⎞ ⎞ s ∫ ∫ ⎜ ⎜ ⎟ ⎟ − λ λ τ τ τ τ = ( ( ) ) d d t e P ( t , s ) E ⎜ ⎟ Q 0 ⎝ ⎠ • What process should be chosen? 10
Parametric Default Intensity Specifications • The widest class of parametric specifications of default intensity is affine p y model class. • Affine models are rather tractable and • Affine models are rather tractable and have quite simple analytical expressions • Examples: � Vasicek � Cox-Ingersoll-Ross � Hull-White 11
Parametric Default Intensity Specifications (continuation) Vasicek (1977) : • Hazard rate follows mean-reverting Brownian g motion process . ( ( ) ) λ λ = = κ κ μ μ − λ λ + + σ σ d d dt dt dW dW t t t • Mean reversion level is constant, thus credit quality does not change in long run quality does not change in long run. • Hazard rate volatility is constant as well and independent of hazard rate level. i d d t f h d t l l • Model doesn’t avoid appearance of negative hazard rate levels 12
Parametric Default Intensity Specifications (continuation) Cox-Ingersoll-Ross (1985): • Sometimes called “Exponential • Sometimes called Exponential Vasicek”. ( ( ) ) λ = κ μ − λ + σ λ d dt dW t t t t • Hazard rate volatility is proportionate to Hazard rate volatility is proportionate to square root of current hazard rate level. • Avoid negative hazard rates. A id ti h d t 13
Parametric Default Intensity Specifications (continuation) Hull-White (1990): • Similar to Vasicek but mean reversion • Similar to Vasicek, but mean reversion level is function of time. ( ( ) ) λ = κ θ − αλ + σ λ d dt dW t t t t t • Model can fit forward curve but it is not Model can fit forward curve, but it is not feasible and requires recalibration on a daily basis daily basis. 14
Heath – Jarrow –Morton Framework(1992) • Entire forward curve depends on single (or single range of) stochastic shock, but each instantaneous forward rate has its own sensitivity to this shock. • HJM framework may be developed to infinite HJM framework may be developed to infinite dimension extension which is equivalent to non-parametric specification. non parametric specification. 15
Problems of Joint Estimation of Interest Rate and Credit Risk 1. What specification should be used for interest rates and what for hazard rate? 2. 2 H How do interest rates and hazard rates interact? d i t t t d h d t i t t? 3. For instrument of single type (bonds for example) interest rate and credit risk can NOT be separated in interest rate and credit risk can NOT be separated in reduced-form model. In order to separate them we have to use several instruments, for example bonds a e o use se e a s u e s, o e a p e bo ds and CDS. 4. Liquidity has a significant impact on bonds prices, q y g p p therefore ignoring liquidity factor causes errors in interest rates and default probabilities estimates. ( see Buhler-Trapp(2006,2008) ) (2006 2008) ) B hl T 16
Dataset Description Dataset Description • Eurozone sovereign bonds price data : • Market price • • Bid & Ask Bid & Ask • Source: Bloomberg • Eurozone sovereign CDS price data: • Conventional spreads of par spreads • S Source: Reuters R t • Issuers : Germany France Italy Spain Ireland Greece Issuers : Germany, France, Italy, Spain, Ireland, Greece, Portugal • Time period : March 2010 – June 2011 Time period : March 2010 June 2011 17
How to Get Default-Free Zero- Coupon Yield Curve • Use one from a “trusted source” such as Bloomberg or Use one from a “trusted source” such as Bloomberg or Reuters. • Obtain one from market data using one of the following • Obtain one from market data using one of the following snapshot methods: 1. Bootstrapping – too rough and sensitive to errors in data and data pp g g volume; 2. Parametric (Nelson-Siegel (1987), Svenson (1994)) – produce curves with limited class of forms; curves with limited class of forms; 3. Splines (Smirnov, Zakharov (2003)) – sensitive to errors in data (filtration is needed) • Constructed curve is highly dependent on used data (government bonds and interest swaps) • Large errors are introduced at this step. 18
Hazard Rate Term Structure: Bootstrapping • Use the obtained zero-coupon yield curve to bootstrap default intensities. • General methodology of bootstrapping hazard rate from G l th d l f b t t i h d t f CDS data: 1 Get CDS spreads (or up-fronts) on particular entity for all available 1. Get CDS spreads (or up-fronts) on particular entity for all available tenors and get default-free zero-coupon yield curve; 2. Calculate implied hazard rate for the shortest tenor assuming it b i being constant until CDS maturity; t t til CDS t it 3. Moving to the next longer tenor, find its implied constant hazard rate for terms between its term to maturity and the term to maturity of y y previous CDS, assuming hazard rate for shorter terms being obtained on the previous step; 4 Recursively calculate entire term structure of hazard rate moving to 4. Recursively calculate entire term structure of hazard rate moving to longer tenors 19
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