Towards a non-parametric Towards a non-parametric stochastic - - PowerPoint PPT Presentation

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 element is the default intensity y y (hazard rate), which is conditional default 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

to pricing credit risk sensitive

• Applied

to pricing credit risk sensitive instruments.

2

Pros & Cons of Default Intensity Models

Pros:

• Relative tractability

Cons:

• Treat default as absolutely
• Don’t demand much data

unpredictable event

• Results are highly
• Allow modeling interest

dependent on specification of default intensity rate and credit risk in a joint framework intensity

3

Interest Rate and Credit Risk Analogies Interest Rate and Credit Risk Analogies

• Discount Function d(t,s)

1.

d(t,s1)>d(t,s2) if s1<s2

• Survival Probability Function P(t,s)

1.

P(t,s1)>=P(t,s2) if s1<s2

2.

d(t,s)>0

3.

d(t,0)=1

4.

d(t,s)→0 with s→∞

2.

P(t,s)>0

3.

P(t,s)=1

4.

P(t,s)→0 with s→∞ (no one lives

4.

d(t,s) 0 with s

4.

P(t,s) 0 with s (no one lives forever)

• Instantaneous Forward Rate

Function N ti

• Default Intensity Function
• Nonnegative
• Nonnegative

4

Default Intensity Specifications (Deterministic)

• The simplest example is a time

homogeneous default model g

• Default intensity is a positive constant λt

S i l b bilit

• Survival probability:

e

s

t

s t P

λ −

= ) (

where t – current date, s – term.

e

s t P ) , (

• 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:

s

∫

p y f f f

e

d

t

s t P

τ τ λ ) (

) , ( ∫ =

−

) ( λ

• The form of depend on current perceptions of

market participant about future credit quality of obligor

• The following slide illustrates how assumed hazard

) (s

t

λ

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

• f

deterministic specification of default intensity is the p y extraction

• f

risk-neutral hazard rate function bootstrapping bond or CDS data function bootstrapping bond or CDS data. Assumed form of hazard rate curve is i i t t piecewise constant.

• 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

d ) ( τ τ λ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ =

−

e

t

d Q

E s t P

) (

) , (

τ τ λ

• 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 .

( )

dW dt d σ λ μ κ λ + − =

• Mean reversion level is constant, thus credit

quality does not change in long run

( )

t t t

dW dt d σ λ μ κ λ + =

quality does not change in long run.

• Hazard rate volatility is constant as well and

i d d t f h d t l l independent of hazard rate level.

• 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”.

( )

• Hazard rate volatility is proportionate to

( )

t t t t

dW dt d λ σ λ μ κ λ + − =

Hazard rate volatility is proportionate to square root of current hazard rate level. A id ti h d t

• Avoid negative hazard rates.

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.

( )

• Model can fit forward curve but it is not

( )

t t t t t

dW dt d λ σ αλ θ κ λ + − =

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 H d i t t t d h d t i t t? 2. How do interest rates and hazard rates interact? 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

B hl T (2006 2008)) Buhler-Trapp(2006,2008))

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 R t

• Source: Reuters
• 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)

18

• Large errors are introduced at this step.

Hazard Rate Term Structure: Bootstrapping

• Use the obtained zero-coupon yield curve to bootstrap

default intensities. G l th d l f b t t i h d t f

• General methodology of bootstrapping hazard rate from

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 t t til CDS t it being constant until CDS maturity;

• 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

• btained 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

Problems with Bootstrapping for Hazard Rate

• CDS

premiums are paid

• n

standard dates thus

• CDS

premiums are paid

• n

standard dates, thus payment dates for all CDS contracts are perfectly matched, but there are few tenors for particular entity, so p y in general CDS data is insufficient to get satisfactory hazard rate term structure using bootstrapping.

• Assuming piece-wise constant form of hazard rate term

structure we artificially increase volatility of hazard rates in nodes rates in nodes.

• For CDS written on debt of distressed entities (down-

ward sloping CDS spread curve) bootstrapping admits of ward sloping CDS spread curve) bootstrapping admits of negative hazard rates.

• We

illustrate this fact with hazard rate structures

20

We illustrate this fact with hazard rate structures bootstrapped from CDS on Greece.

Problems with Bootstrapping Problems with Bootstrapping

21

Cox-Ingersoll-Ross Equations for Hazard Rate

• We assume that the risk-free discount

function d(t) is known.

• We assume the following SDE for the spot

default intensity process (risk-neutral): default intensity process (risk neutral): The s r i al probabilities to time t are kno n

d ( )d d

t t t t

t Z λ κ μ λ ν λ = − +

• The survival probabilities to time t are known

to equal

t

⎤ ⎡

[ ]

) ( exp ) ( exp ) ( λ τ λτ t B t A d E T P

t

− = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− =

∫

22

were A(t) and B(t) have simple analytical expressions.

CDS Pricing Formula CDS Pricing Formula

• CDS are priced in terms of par spread:

( )

LGD ( )d 1 ( )

T

d Q

∫

( ) ( )

0 LGD· ( )d 1

( ) ( ) ( ) ( ) ( ) ( )d 1 ( )

N T

d Q R d T T T Q T d T Q τ τ − =

∫ ∑ ∫

( )

1 ( ) 1

( ) ( , ) ( ) ( ) ( , )d 1 ( )

i i i i I i

d T T T Q T d T Q

τ

α τ α τ τ

− =

+ −

∑ ∫

• where R is the CDS par spread, d(t) is the discount

function, is the year fraction between t₁ and t₂,

1 2

( , ) t t α

function, is the year fraction between t₁ and t₂, and

1 2

( , ) t t α

( : 1,..., , ) max{ }

i i

I i N T T τ τ = < =

23

CDS Pricing Assumptions CDS Pricing Assumptions

• LGD is deterministic and constant;
• Liquidity is ignored;
• Liquidity is ignored;
• Often interest rates and default

probabilities are assumed to be independent; p ;

• Counterparty risk is NOT taken into

account (introduction of CCP); account (introduction of CCP);

• In case of credit event CDS is settled at

the moment when credit event occurs.

24

The Fitting Procedure The Fitting Procedure

W lib t 4 t λ

• We calibrate 4 parameters: μ, κ, ν, λ

to the observed data via the CDS pricing ti ( d t) equation (par spread concept):

( )

LGD· ( )d 1 ( )

T

d Q τ τ −

∫

( ) ( )

1 ( ) 1

( ) ( ) ( ) ( , ) ( ) ( ) ( , )d 1 ( )

N T i i i i I i

Q R d T T T Q T d T Q

τ

α τ α τ τ

−

= + −

∫ ∑ ∫

where R is the CDS par spread, d(t) is the discount function is the year fraction

1 i=

( ) t t α

discount function, is the year fraction between t and t , and

1 2

( , ) t t α

( : 1,..., , ) max{ }

i i

I i N T T τ τ = < =

25

Joint Framework for zero-coupon and hazard rate term structure

• Use a unified model for joint (possibly

correlated) dynamics of spot interest rate and spot default intensity. I.e. CIR-like model. Snapshot fitting possibilities of dynamic spot g y rate models are very limited. Systematic errors are introduced from using an Systematic errors are introduced from using an inappropriate model for fitting, resulting in misestimating the default probability. misestimating the default probability.

26

Joint Framework for zero-coupon and hazard rate term structure (cont.)

• Joint stochastic dynamics (see Brigo Mercurio (2006)):

Joint stochastic dynamics (see Brigo, Mercurio (2006)):

) d d ( ( d d d d )

t t t t

r t r r k W Z W t θ σ ρ ⎧ = ⎪ = ⎨ − + E , ( d d ( ) d d d d )

t t t t t t

W Z t t Z ρ λ κ μ λ ν λ = ⎨ = ⎪ ⎩ − + E

• If then CDS pricing is complicated due to

the correlation term:

t

⎡ ⎤

ρ ≠ If then this is just a double CIR model

( )

p ( ? ) d ex

t t

Q t r

τ

λ τ = ⎡ ⎤ − = ⎢ ⎥ ⎣ ⎦ +

∫

E

• If then this is just a double CIR model.

ρ =

27

Problems with CIR model Problems with CIR model

Let instantaneous interest rate dynamics be Let instantaneous interest rate dynamics be described by CIR model (unconstrained C ) CIR method):

dW dt r k dr σ θ + − = ) (

t t t

dW dt r k dr σ θ + ) (

Snapshot fitting for Eurozone government bonds is used and provides visibly good results. But The same model-parameters lead to degenerate instant rate dynamics (see next slide).

28

Problems with CIR model Problems with CIR model

29

Problems with CIR model

• An interest rate of zero is precluded if the

Problems with CIR model

• An interest rate of zero is precluded if the

degeneracy condition

1 2

2

≥ − = θ α k

g y is satisfied (for proper CIR method).

2

σ

• Fitting results for Eurozone countries show that

during the last 2 years degeneracy condition was satisfied in less than 10% of cases.

• Some constraints must be applied to fitting

Some constraints must be applied to fitting

• procedure. But constrained fitting will lead to

lesser quality of discount and yield curves. esse qua ty o d scou t a d y e d cu es

30

Problems with CIR model Problems with CIR model

31

Problems with CIR model Problems with CIR model

• Another problem: CIR model implies a

priori given form of discount and yield p g y curve (Cox-Ingersoll-Ross, 1985):

t

r t T k B

e t T k A T t d

) , , , (

) , , , ( ) , (

− −

− =

σ θ

σ θ

where have simple analytical expressions

) ( ), ( ⋅ ⋅ B A

expressions.

32

Problems with CIR model

• Due to this factor even the best snapshot fitting can

Problems with CIR model

• Due to this factor even the best snapshot fitting can

produce forecasts that will be rather inaccurate and have unstable dynamics (which is important especially for y ( p p y derivatives pricing).

• Two next slides show that even for a short horizon yield

curve dynamics can be non-typical: yield “jumps” up to 5 % in 4 days while typical change for such period almost never exceeds 2%. A lt th f t d d ’t di t l

• As a result, the forecasted curve doesn’t predict real

behavior of interest rate structure

33

Problems with CIR model Problems with CIR model

34

Problems with CIR model Problems with CIR model

CIR forecast can be inadequate even for short horizon:

35

The Proposed Method The Proposed Method

E l th i fi it di i l d i d l

• Employ the new infinite-dimensional dynamic model

which as a by-product yields a decent snapshot fitting method both for interest rates and default intensities method both for interest rates and default intensities. The implied snapshot fitting method is non-parametric, which allows to eliminate model- non parametric, which allows to eliminate model inflicted errors.

• The method is not ad hoc, it is based on a sensible and

, sufficiently rich stochastic dynamics model. The possibilities of a correlated dynamics are being investigated in our current research.

36

HJM Equations HJM Equations

• Infinite-dimensional dynamics à-la Filipović

(within Heath-Jarrow-Morton framework). ( )

• In Musiela parametrization:

( ) ( ) ( ) ( ) ( ) ( )

j t t j t t t t

d r dt r Dr dr β σ α

∑

∞

⋅ ⋅ + ⋅ ⋅ + ⋅ = ⋅

1

, ~ ] , [

• No arbitrage condition:

j =1

( ) ( ) ( ) ( ) ( ) ( )

∑ ∫

⋅ ⋅ = ⋅ , ~ , ~ ,

x

• j

j

d r x r x r τ τ σ σ α

37

∫

=1 j

Double HJM Equations Double HJM Equations

Th j i t l t d d l

• The joint uncorrelated model

( )

⎪ ⎧ + + =

∑

∞

~

j jd

dt Dr dr β σ α

( )

⎪ ⎪ ⎪ ⎨ + + =

∑

∞ =1 j t t t t t

d dt Dr dr β σ α

( )

⎪ ⎪ ⎩ + + =

∑

∞ =1

~ ~ ~

j j t j t t t t

d dt D d β ν α λ λ

yields a nonparametric approach to snapshot yield curve and default intensity

⎩

p y y fitting.

One of the main topics in our current

38

One of the main topics in our current research

Results of hazard rate modeling Results of hazard rate modeling

• We estimated hazard rate term
• We estimated hazard rate term

structure using CIR and non- parametric specifications

• Results are illustrated on following
• Results are illustrated on following

slide

39

Results for hazard rate modeling (cont ) Results for hazard rate modeling (cont.)

40

Results for hazard rate modeling

(cont.)

41

Results for hazard rate modeling (cont.)

• We compared obtained results to respective bond

spread curve over default-free zero-coupon yield t t t term structure

• Because government bonds are relatively liquid,

spread term curve and hazard rate curve are rather alike

• Such comparison allows us to roughly assess

accuracy of obtained results

• We can NOT rely on such comparison in case of

entities with high credit rating and in case of illiquid g g bonds

42

Results for hazard rate modeling (cont.)

43

Results for hazard rate modeling (cont.)

44

How accurate hazard rates are? How accurate hazard rates are?

• As shown above all considered models produce

different hazard rate curve and roughly assessed accuracy of results. But how accurate those estimates really are?

• But how accurate those estimates really are?
• We propose the following procedure to estimate
• We propose the following procedure to estimate

accuracy of hazard rate estimates: 1 E t ti f h d t t t t 1.Extraction of hazard rate term structure 2.Bond model prices calculation and accuracy test

45

The first step: extraction of hazard rate

Collecting CDS, bond and default‐ free rate data free rate data Specification of Specification of default intensity process Deterministic Stochastic process à‐la Filipović à la Filipović with HJM framework Cox‐Ingersoll ‐Ross Piecewise

46

framework Cox Ingersoll Ross constant

The first step: extraction of hazard rate (continuation)

Cox‐Ingersoll ‐ Ross Piecewise constant à‐la Filipović with HJM framework Bootstrapping Fitting Obt i i ti t f d f lt i t it Obtaining an estimate of default intensity process from CDS data

47

The second step: bond model prices calculation and accuracy test

LGD assumption for bonds Loss of market value Loss of face value Bond model prices calculation and risky yield curve p y y construction Testing accuracy of default intensity process estimate Comparison with bond market Comparison with bond bid and ask quotes “Feasibility band” approach

48

prices

Our Fitting Method Our Fitting Method

A h t j ti f d i

• As a snapshot projection of our dynamic

model one is required to search for a function f(t) ti f i f(t) satisfying

2 2

( ) ( )d ( ) min,

N n T k i k g i k

g F d t P α τ τ ω ⎛ ⎞ ′ + − → ⎜ ⎟ ⎝ ⎠

∑ ∑ ∫

, (·) 1 1 2

( ) ( ) ( ) ( ) , ( )d d

k i k g i k g k i t

g t

= =

⎜ ⎟ ⎝ ⎠ ⎡ ⎤

∑ ∑ ∫ ∫

where P is the price of the kth bond F is the

2

( ) exp ( )d dg t g τ τ ⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦

∫

where Pk is the price of the kth bond, Fi,k is the cash flow on the kth bond at time ti.

49

Our Fitting Method Our Fitting Method

• The solution is an exponential-sinusoidal

spline with knots at every cash flow time: p y

1 1 1

( ) ( ) ( ), [ , ),

i i

i i i i i i

f p t p t t t

λ λ

τ φ τ φ τ τ

− − −

= − + − ∈

1

sinh , 0; sinh ( )

i i i i i

t t λ τ λ λ ⎧ > ⎪ − ⎪

1

( ) sin ( ) , 0; sin ( )

i

i i i i i

t t

λ

λ τ φ τ λ λ

−

⎪ ⎪ − ⎪ = < ⎨ − − ⎪

1 1

sin ( ) , 0.

i i i i i i

t t t t λ τ λ

−

⎪ ⎪ ⎪ = − ⎪ ⎩

50

1 i i

t t − ⎪ ⎩

Our Fitting Method Our Fitting Method

E iti t f d t

• Ensures positive spot forward rates.
• Ensures continuous and differentiable spot

forward rates forward rates.

• Takes liquidity (e.g. bid-ask spreads) into

account via weighing coefficients w account via weighing coefficients wk.

• Can be fine-tuned to exhibit any desired

proportion between smoothness of the proportion between smoothness of the forward rate curve and accuracy of replicating bond prices. p

• Does not introduce model-inflicted errors

in estimation.

51

The Feasibility Band The Feasibility Band

• In order to assess the accuracy of fitting,

we employ the notion of a feasibility band. p y y

• It transfers bid-ask bounds to the interest

rate domain rate domain.

1

) ln max min; ( ; d

i

i i t

d t r = − → ⎧ ⎪ ⎪

1

max,min; d 1 ... 0;

i N N

d d → ≥ ≥ ≥ ≥ ⎪ ⎪ ⎨ ⎪ ⎪

, 1

.

k i k i k i

F d b a i sk d

=

≥ ≥ ⎪ ⎪ ⎩

∑

52

Yield Curve Modeling Yield Curve Modeling

• Next slides show comparative advantages
• f non-parametric framework for modeling

p g term structure for Eurozone government bonds bonds.

• Due to the small number of parameters

popular frameworks such as CIR and popular frameworks such as CIR and Nelson-Siegel approach often fail to match specific form of the real term structure specific form of the real term structure.

53

Yield Curve Modeling (Examples) Yield Curve Modeling (Examples)

54

Yield Curve Modeling (Examples) Yield Curve Modeling (Examples)

55

Thank you for your i !!! attention!!!

ssmirnov@hse.ru

56