Credit Risk Modeling A.Y. 2016/17 M.A. in Finance

A.Y. 2016/17 M.A. in Finance

Luca Regis¹

1DEPS, University of Siena

Agenda

1 Introduction: measuring and managing credit risk.

2 Default modeling: intensity-based approach.

3 Default modeling: firm-value or structural approach.

4 Pricing defaultable assets (also credit derivatives).

Classes are held on: Wed 16-18 room 12; Thu 12-14 room 12.

Useful references

Duffie, D. and K. J. Singleton, Credit Risk. Pricing, measurement and Management, 2003, Princeton University Press.

Brigo, D. and F. Mercurio, Interest Rate Models - Theory and Practice with Smile, Inflation and Credit, 2006, Springer Finance.

Brigo, D. and Morini, M. and A. Pallavicini, Counterparty Credit Risk, Collateral and Funding, 2013, Wiley Finance.

What is credit risk?

Definition

Credit risk is the risk associated to the default or to changes in the market value of a credit or of an asset held in a portfolio following a credit-linked event happening to issuers or counterparties.

Why modeling credit risk?

Measure the risks inherent a portfolio, for internal risk management needs or for capital requirements;

Price instruments that can be affected by such events,

“defaultable”.

Defaultable assets:

1 Corporate bonds;

2 Government bonds (at least some of them);

3 Credit derivatives;

4 Any Over-the-Counter derivative (counterparty default risk).

Default and credit-linked events

Credit-linked events:

Change in the quality of the credit;

Default of a debtor.

This course will be focusing on the risk associated to a default event.

Default: failure to fulfill some important obligation.

Payment associated to credit derivatives,f.i., can be triggered by these credit-linked events:

1 Bankruptcy: a court declares it;

2 Failure to pay: missed payment (coupons, bills, ...);

3 Restructuring: debt of a company is redefined (amount, maturity,...);

4 Repudiation/moratorium: bankruptcy/restructuring of sovereign debt;

5 Obligation and acceleration default: violation of covenants in the debt contract.

Peculiarities in the Credit Risk Economics

Asymmetric information between the borrower, that knows more about its own quality, and the lender. As a

consequence:

Adverse selection: if the interest charged to lenders is too high, none or only the poor quality ones will demand credit.

Possible solution: limit the access to credit, to incentivize the selection of good quality borrowers.

Moral hazard: the larger the borrowing, the higher the incentive for borrowers to undertake risky behaviors.

Main issues

Credit risk is an hot topic: credit crisis was at the basis of the crisis started in 2007/8;

Credit derivatives market: boomed before the crisis, now it is still alive although it declined in size.

Increasing attention on loss estimation and regulatory requirements.

Lots of technical issues:

1 Modeling the default event;

2 Simulating, forecasting portfolio losses under real-world measure P; pricing under a risk-neutral measure Q;

3 Assessing exposures, i.e. losses in case of default;

4 Relationship with other risks (f.i. interest-rate risk);

5 Single-name vs. multi-name credit derivatives pricing (dependence in multiple defaults).

6 Adjusting prices and fair values to account for counterparty default risk: Credit Value Adjustment (CVA);

Size of the derivatives market (Source: BIS)

Traditional approach to credit risk: Credit Ratings

Rating agencies (Moody’s, Standard & Poor’s, Fitch) are specialized companies that provide assessments of the credit quality of issuers of debt instruments.

Credit ratings measure the worthiness of debtors, using accounting data, historical data about default frequencies, expert judgement... they assign a ”class” to an issuer:

AAA, AA, A,....Aaa..

Main distinction: investment vs. speculative grade They provide a ”naive” assessment of the probability of defaults;

Are intended to provide measures of credit quality that are not affected by the business cycle;

evidence of momentum: a downgrading is more likely to be followed by another downgrading.

Default probabilities per rating class do vary over time.

Default Rates

The use of ratings to measure default probabilities

One goal of credit risk measurement is to assess the likelihood of default of an issuer or a reference entity.

One possibility: take the average frequency at which similar entities, i.e. subjects with same credit rating, have defaulted.

Be careful: historical default probabilities! They can be used to simulate default arrivals and actual portfolio losses, NOT for pricing!

Multi-year solvency probabilities

The table below provides the cumulative average default rates by credit rating....

... BUT an issuer can change rating over time!

One year vs. multi-year default rates

Consider an A-rated bond, where the average default rate of A-rated firms is p_A.

The survival probability over one year is thus 1 − p_A. The default probability over two years, assuming pA stays constant, is NOT

1 − (1 − p_A)²,

because the firm can change rating at the end of the first year.

It is necessary to model rates of migration across ratings:

transition matrices allow to do this, modeling the probabilities that firms rated x at the beginning of one year will have rating y at the end of that year.

One-year transition matrix: example

Credit spreads/1

Consider:

1 A default-free zero-coupon bond: it pays 100 with certainty at time 1;

2 A defaultable zero-coupon bond: it pays

if there is no default between current time 0 and 1: 100 if there is default between current time 0 and 1:

Rec = 100(1 − L)

Rec is the recovery, the complement to 100 of L, the Loss Given Default;

Assume default occurring with probability q.

Assume a risk-free rate r.

The fair promised interest rate of the defaultable bond y must be:

100

1 + y = [(1 − q)100 + q100(1 − L)]

1 + r

Credit Spreads/2

It follows that:

1 + r = (1 + y) [1 − q + q − qL]

1 + r = (1 + y) [1 − qL]

y = r

1 − qL

| {z }

>r

+ pL 1 − qL

| {z }

>0

> r.

y − r is the credit spread, the extra-remuneration due to the presence of default.

y − r is somewhat close to Lq, the expected loss rate.

If L = 0 or q = 0: y = r.

If L = 1: y = ^r+q_1−q, y − r is close to q.

Default probabilities and credit spreads

Since a typical recovery rate Rec/100 is 50%, a typical credit spread should be around half the annual default probability, but indeed in reality spreads are much higher, due to risk premia, cyclicality, liquidity,....

Source: Huang and Huang (2002)

Regis Credit Risk Modeling A.Y. 2016/17 M.A. in Finance 18/105

Risk premium

We said that

y = r

1 − qL + qL 1 − qL.

Q: what is the correct probability measure under which the q in the above formula is computed?

A: It is the risk-neutral probability Q obtained adjusting the real-world or historical probability P by the price of default risk, i.e. for the presence of a risk premium associated to the default risk.

Indeed, abstracting from recovery, i.e. setting L = 0, the price of a 1-year Ba-rated bond (see table in slide before) will not be (1 − 1.29%)/(1 + r), but will be equal to 1 − q^∗ where q^∗ is the risk neutral default probability.

Risk-neutral vs. actual probabilities

Let us take the opposite perspective: we observe the market quote of a defaultable coupon bond, traded at par and paying 108 euros in 1 year. Historical default rates suggest p = 0.99. How can we recover p^∗, the risk-neutral survival probability?

Assume L = 50% of the par value, r = 6%. Then, p^∗ is obtained solving

100 = 1

1.06[p^∗108 + 50%(1 − p^∗)108] , p^∗ = 0.965.

The difference between p and p^∗ can be rationalized in terms of the presence of a default risk premium arising from agents’ risk aversion.

Cyclicality

There is documented evidence (H.Chen, JF, 2010) that both default rates and recovery rates are cyclical, i.e. are respectively higher and lower in recessions. This, for instance, lowers the value of corporate debt, amplifying spreads.

A more technical view on the pricing of a defaultable zero coupon bond

When there is default risk, the time-t value of a bond paying 1 at a future date T, ¯P (t, T ) is the risk neutral expectation of the discounted payoff 1:

1{τ >t}P (t, T ) = E¯ D(t, T )1_{{τ >T }}+ RecD(t, tR)1{τ ≤T }|G_t , where:

E is an expectation under the risk-neutral measure;

1_{A} is an indicator function, =1 if condition A is satisfied, 0 otherwise;

τ is the time of default;

Gtis the filtration at time t comprehensive of the “market”

information Ftand of the information on whether the default has happened, σ({τ < u}, u ≤ t) ;

D(t, s), s ≥ t can be either deterministic or stochastic;

Rec, can be stochastic, t_R= τ or T .

A more technical view on the pricing of a defaultable zero coupon bond/2

Notice that, denoting with P (t, T ) the price of a bond with maturity T when there is no default risk, i.e. the time-t value of a bond paying 1 at a future date T,

P (t, T ) = E [D(t, T )1|Gt] = Eh

e^−RtTr(s)ds|G_ti , and assuming that the Recovery is paid at maturity t_R= T , the price of the defaultable bond can be seen as:

1_{{τ >t}}P (t, T )¯ = ED(t, T )(1 − 1_{{τ ≤T }})

+ (1 − LGD)D(t, T )1{τ ≤T }|G_t =

= P (t, T )

| {z }

Default-free bond

− ELGD × D(t, T )1_{{τ ≤T }}|G_t

| {z }

Disc. Risk-neutral expected Loss

.

Defaultable coupon bonds

Similarly, the discounted payoff of a coupon bond issued by a company C is

n

X

i=1

ciD(t, Ti)1_{{τ >Ti}}+ D(t, Tn)1_{{τ >Tn}}+ RecD(t, TR)1_{{τ ≤T }}.

T_R is the time at which the recovery Rec is paid.

Its time-t price can be evaluated discounting the payoff under the risk-neutral measure as for the zero-coupon bond.

Types of credit derivatives / Single-name

Payoffs of Single-name products are linked to the credit events of one reference entity.

Credit Default Option: put option gives the right to the buyer to reimbursed for losses if a credit-event occurs before maturity;

Credit Default Swaps (CDS): periodic premium vs.

protection;

Credit-linked notes, step-up bonds: bonds whose payments are linked to credit events associated to a reference entity.

Floating rate notes: structured as interest rate swaps (IRS) in which the fixed payer holds a bond paying coupons, that is paid to the payer of the floating rate. The floating payer pays LIBOR rate+ a spread.

Types of credit derivatives / Multi-Name

Payoffs of Multi-Name products are linked to credit events of more than one reference entity.

Basket default swap: first-to default, k-th to default, last-to-default. Like CDS, but a set of reference entities is defined and protection is paid when the first, k-th or last reference entity experiences default. More risk than CDS:

leverage for protection seller; lower rate, partial hedging of multiple defaults by buyer.

Multi-name products/2

Index CDS’s (iTraxx, CDX): protection from default of many entities, each with equal notional; protection leg pays loss increment occurring at each default in exchange for a periodic premium.

Collateralized Debt Obligations: collection of CDS’s with same maturity on different names; total loss is tranched into attachment points (e.g. A and B) and the incremental difference between B and total loss is paid, when this exceeds A. Notice that the rate R is reduced when a default occurs.

Why using credit derivatives?

Who are the...

Protection Sellers? investment banks, insurers, asset management companies and reinsurers;

Buyers? mostly banks and asset management companies.

Purpose:

Modify or customize credit exposures (e.g. targeted to a selected maturity);

Modify credit concentrations (by industry, geographical area,...);

Go short on defaultable bonds (easier or cheaper than on bonds market);

Credit Default Swaps

Credit Default Swaps (CDS) are protection contracts, in which two parties, the protection buyer (B) and the protection seller (S), agree on the following:

If a reference entity (E) defaults at a certain time Ta< τE < Tb, S pays B a certain deterministic cash

amount LGD (protection leg). In exchange, B pays S a rate R at regular intervals (premium leg), T_a+1, ..., until T_b, or until default τE.

LGD is the protection provided to B against default, and it is the notional.

CDS’s are the basic derivative used to mitigate default risk, e.g. to protect B when it holds a bond issued by E.

From a modeling point of view, two types of needs:

1 models that can price correctly CDS’s;

2 models that can be calibrated using CDS’s.

Price of a CDS/1

Given the rate R, with α_i the distance between two successive payments (typical: 3 months), the price of a CDS at t < T_a is given by

PCDS (t, R, LGD) = E^t





D(t, τ )(τ − Tβ(τ )−1)R1{T_a<τ <T_b}

| {z }

Accrued rate at default

+

+

b

X

i=a+1

D(t, Ti)αiR1{τ ≥T_i}

| {z }

CDS Payments if no default

− 1{T_a≤τ ≤T_b}D(t, τ )LGD

| {z }

Protection payment at default











 .

Assumption: payment of last R is not at default, but postponed to Ti and interest accrual period equal to either zero or accrued for the whole period; T_{β(τ )}is the first payment time T_i after default.

Price of a CDS/2

Pricing a CDS amounts to establishing the rate R, i.e. the payments of the fixed leg.

R is set at contract start t = Ta= 0 s.t. the contract is fair:

P V (Premium Leg) = P V (Protection Leg).

Leaving aside accrued interest:

Et

" _b X

i=1

D(0, Ti)αiR1{τ >T_i}

#

= EtLGD × D(0, τ )1_{{0<τ ≤Tb}} . Hence, if R is constant, when T_a= 0 and t=0:

R = R_0,b(0) = EtLGD × D(0, τ )1_{{0≤τ ≤Tb}} αiEt

hPb

i=1D(0, Ti)1_{{τ >Ti}} i .

Modeling prices of credit derivatives

Prices depend on:

1 Estimated exposures: losses in the event of default;

2 Dynamics of interest rates;

3 The way default time(s) is (are) modeled;

4 Dependence (or lack of) between default time τ and interest rates.

However, we can start by providing some model-free valuation formulas, under independence between default arrival and discount factors.

Model-independent Price of CDS

Assume D(s, t) independent of τ for any 0 ≤ s ≤ t. Leaving accrual aside, the time-0 value of the premium leg is

Et

" _b X

i=a+1

D(0, Ti)αi1_{{τ >Ti}}

#

=

=

b

X

i=a+1

αiR Et[D(0, Ti)]

| {z }

P (0,Ti)

Et[1{τ ≥T_i}]

| {z }

Q(τ ≥Ti)

=

= R

b

X

i=a+1

αiP (0, Ti)Q[τ ≥ Ti]

| {z }

“DV01” or unit premium leg

=

= P remLeg_0,b(P (0, ·), Q(τ > ·)

Model-independent Price of CDS/2

Now, assuming also a constant LGD, the value of the Protection Leg P rotLeg_a,b(LGD; P (0, ·), Q(τ > ·) is

EtLGD × D(0, τ )1_{{Ta≤τ ≤Tb}} =

= LGDEt

Z +∞

0

1{T_a≤t≤T_b}D(0, t)1{τ ∈[t,t+dt)}



=

= LGD

Z _Tb

Ta

ED(0, t)1{τ ∈[t,t+dt)} =

= LGD

Z _Tb

Ta

E [D(0, t)] E1{τ ∈[t,t+dt)} =

= LGD

Z _Tb

Ta

P (0, t)Q(τ ∈ [t, t + dt)) =

= −LGD

Z _Tb

Ta

P (0, t)d_tQ(τ ≥ t).

Model-independent Survival Probabilities from CDS Prices

The above formulas are model-independent, i.e. they are valid for all the modeling choices (interest rate dynamics, default arrival modeling,...).

They involve only the initial default-free zero-coupon curve and the survival probabilities at time 0.

When T_a= 0, t = 0 the market quotes the fair price R^mkt_0,b (0) for different maturities Tb.

Solving for the different T_b, starting from the shortest one, the equation

P rotLeg_0,b(LGD; P (0, ·), Q(τ > ·)) =

= P remLeg_0,b(R^mkt_0,b (0); P (0, ·), Q(τ > ·)) for Q(τ > ·), it is possible to find the model-independent market implied survival curve : “stripping”.

Counterparty credit risk in derivatives transactions

Consider a transaction for an OTC derivative product, where a default-free party enters a transaction with a counterparty (C).

Such derivative offers a discounted payoff Π(s, t), with t final maturity, when C is default-free as well.

If we instead allow for the possibility that C defaults:

Π(s, t) = 1¯ _{τC>t}Π(s, t) + 1_{τC<t}[Π(s, τ_C)+

+ RecD(s, τ_C)EτCΠ(τ_C, t))⁺− D(s, τ_C)(−EτCΠ(τ, t))⁺ . If no default: original payoff.

if default: payoffs until default + recovery of residual NPV if positive - residual NPV if negative.

Counterparty credit risk in derivatives transactions/2

The payoff

Π(s, t) = 1¯ _{τC>t}Π(s, t) + 1_{τC<t}[Π(s, τ_C)+

+ RecD(s, τ_C)Eτ_CΠ(τ_C, t))⁺− D(s, τ_C)(−Eτ_CΠ(τ, t))⁺ . is equal to:

Π(s, t) = Π(0, t) − LGD1¯ _{τC≤t}(EτCΠ(τ_C, t))⁺. Original payoff - LGD * a call option with zero strike on the residual NPV at default.

Example: counterparty risk on equity options

Suppose the buyer (A) of an equity option on a firm B is default free. When he enters the transaction, he buys the option from a counterparty C. Assuming no counterparty risk, the payoff of the option is

Π(0, T ) = D(0, T )(S^B_T − K)⁺. If instead C can default, the payoff is modified:

Π(0, T )¯ = D(0, T )(S_T^B− K)⁺

− LGD1_{{τC<T }}D(0, τC)(EτCD(τ_C, T )(S_T^B− K)⁺) =

= D(0, T )1 − LGD1_{{τC<T }} (S_T^B− K)⁺) =

= D(0, T )Rec1_{{τC<T }}+ 1_{{τC>T }} (S_T^B− K)⁺.

Two approaches to default arrival modeling

In this course, we will deal mostly with the modelling of default and its effect on pricing.

Two main approaches to credit risk modeling, that differ, from a technical point of view, in the way default arrival, i.e. the time τ at which the default of a company occurs is modeled:

1 Reduced-form approach: main idea: disaggregate the value of cash flows in a default-free versus a cum-default environment; the intensity at which default occurs has a given assumed form;

2 Structural approach: main idea: default occurs because the firm experiences financial deficiencies. Default arrival is connected to the changes in the fundamentals of the firm, i.e. when firm value hits some boundary.

Default intensity: definition

The main idea underlying reduced-form models of default arrival is that the intensity at which a default occurs is directly modeled.

Default time τ is the first arrival time of a jump process.

Probability of default before time t + ∆t, conditional on survival until t is

P rob(τ ∈ [t, t + ∆t]|τ > t, Ft) = λ(t)∆t.

λ is the intensity or hazard rate:

λ = lim

∆t→0

1

∆tP rob(τ ∈ [t, t + ∆t]|τ > t, F_t).

Λ(t) =Rt

0λ(s)ds is the cumulated intensity or hazard rate, or Hazard function.

Default intensity: modeling choices

Intensity can be:

1 Constant. Easiest modeling, τ is the first jump time of a time-homogeneous Poisson Process;

2 Deterministic. τ is the first jump time of a

time-inhomogeneous Poisson Process. This choice allows to capture the term structure of credit spreads; no uncertainty in the intensity and hence no volatility in credit spreads.

3 Stochastic. τ is the first jump time of a doubly stochastic (Cox) process. This choice allows to capture the time varying nature, i.e. the term structure of credit spreads;

uncertainty in the intensity and hence volatility in credit spreads is captured. Which process for intensity?

Mean-reverting, CIR, Affine, HJM-like...

Time homogeneous Poisson process

Definition

A time homogeneous Poisson Process M(t) is a

right-continuous, unit-increasing jump process, with stationary independent increments and M₀ = 0.

Increments are distributed as a Poisson random variable:

Q(Mt− M_s = k) ∼ P oisson(λ(t − s)).

Time-to-default τ , i.e. the first jump time of M is

exponentially distributed (with parameter λ): there exists a positive constant λ s.t.

Q(Mt= 0) = Q(τ > t) = S(0, t) = e^−λt =⇒ Q(τ ≤ t) = 1−e^−λt. It follows that the expected time to default is _λ¹.

Time homogeneous Poisson process/2

Survival probabilities behave as discount rates: we will exploit this analogy in the following more complex contexts.

The default intensity can be interpreted as a credit spread.

The probability of default over a time period of length ∆t is approximately equal to λ∆t:

Q(τ ∈ [t, t + ∆t))

Q(τ > t) = Q(τ > t) − Q(τ > t + ∆t) Q(τ > t) =

= e^−λt− e^{−λ(t+∆t)}

e^−λt ≈ λ∆t.

Default is unpredictable: arrival is independent of time, default time is inaccessible.

Time homogeneous PP: example

Suppose a constant default intensity λ = 0.04.

Then, default probability in 1 year is:

Q(τ ≤ 1) = 1 − S(0, 1) = 1 − e^−0.04 = 3.9211%.

The expected time to default is _0.04¹ = 25 years.

0 2 4 6 8 10 12 14 16 18 20

T 0.4

0.5 0.6 0.7 0.8 0.9 1

S(0,T)

Survival probabilities

Time homogeneous PP: example/2

Suppose now also a constant risk-free interest rate r = 0.03 and a deterministic term structure of interest rates

determined by such constant r: D(0, t) = e^−rt.

The price of a zero-coupon defaultable bond with maturity T = 1 year is then

E[D(0, T )1{τ >T }|G_t] = e^−re^−λ = e^−(r+λ)

= 0.9704

| {z }

P (0,1)

∗0.9608 = 0.9324

| {z }

P (0,1)<P (0,1)¯

.

Notice that: constant λ implies constant yield to maturity;

constant spread with varying maturity.

Stripping with a constant intensity

Sometimes, for simplicity, it is convenient, when stripping survival probabilities, to assume an hazard function.

The simplest one assumes a constant default intensity:

Q(τ > t) = e^−λt Q(τ ∈ [t, t + dt)|τ ≥ t) = λdt.

d_tQ(τ ≥ t) = −λe^−λtdt = −λQ(τ ≥ t)dt.

If R is paid continuously, the CDS price is equal to

−LGD

Z Tb

Ta

P (0, t)λQ(τ ≥ t)

 + R

Z Tb

Ta

P (0, t)Q(τ ≥ t)

 . Now, if we take the market price R_0,b^mkt(0) and plug it in the above equation, the CDS present value should be zero:

λ = R^mkt_0,b (0) LGD .

Time inhomogeneous PP

Definition

A time inhomogeneous Poisson process Nt is a Poisson process whose intensity is time varying. Defining the hazard rate as

Λ(t) = Z t

0

λ(s)ds,

the process is obtained as a time-changed Poisson process:

Nt= M_Λ(t). N_t:

is unit jump-increasing;

has independent increments;

has non-i.i.d. distributed increments

Time inhomogeneous PP/2

Since Nt= M_Λ(t) it follows that the first jump time τ of N occurs when M jumps at Λ(τ ).

Recall that in a homogeneous Poisson Process τ ∼ Exp(λ) and Λ(τ ) = λτ ∼ Exp(1). This implies that in a

time-inhomogeneous Poisson Process Λ(τ ) := ξ ∼ Exp(1).

Also it follows that:

τ = Λ⁻¹(ξ).

Conditional survival probabilities can be obtained as S(t, T ) = 1 − Q(τ ≤ T |τ > t) =

= 1 −Q(Λ(τ ) > Λ(t)) − Q(Λ(τ ) > Λ(T ) Q(Λ(τ ) > Λ(t)) =

= 1 − 1 +e^{−Λ(T )}

e^−Λ(t) = e^{−R0Tλ(s)ds} e^{−R0tλ(s)ds} = e⁻

RT t λ(s)ds.

Time inhomogeneous PP and credit spreads

Assume constant r and no recovery:

P (0, t) = e¯ ^−rtS(0, t) = e^−rte^{−R0tλ(s)ds}= e^{−(rt+R0tλ(s))ds}. Hence, time-varying spreads!

Time inhomogeneous PP: example

Two notable examples of models with time-varying intensity:

1 piecewise constant intensity.

2 piecewise linear intensity.

Piecewise constant intensity

Assume that

λ(t) = λi∈ R+, for t ∈ [Ti−1, Ti] . S(0, t) = Q(τ > t) = e^−Λ(t), where

Λ(t) = Z t

0

λ(s)ds =

β(t)−1

X

i=1

(T_i− T_i−1)λ_i+ λ_β(t)(t − T_β(t)−1), where β(t) denotes the first time-point at which the

intensity changes after t.

We can apply the formulas we derived previously (under independence between discount factors and default) to retrieve the Premium and Protection Leg of a CDS, with the objective of recovering hazard rates from CDS quotes.

Piecewise constant intensity/2

The premium leg at t = 0 will then be:

R

b

X

i=1

P (0, T_i)α_iQ(τ > Ti) =

= R

b

X

i=1

P (0, Ti)αie^−Λ(Ti) The protection leg at t = 0 will be:

LGD

b

X

i=a+1

Z T_b T_a

P (0, u)duQ(τ > t) = LGD

b

X

i=a+1

Z T_b T_a

P (0, u)due^−Λ(u)=

= −LGD

b

X

i=a+1

λi

Z Ti

Ti−1

e^{−Λi−1−λi(u−Ti−1)}P (0, u)du, where Λj =RTj

0 λ(s)ds =Pj

i=1λi(Ti− T_i−1).

Stripping with a piecewise constant intensity

Hence, when T_a= 0, with the market quoting R^mkt_0,T

b quotes for CDSs with different maturities Tb = {Tb1, ...Tbn} , we can use

R^mkt_0,Tb

b

X

i=1

P (0, Ti)αie^−Λ(Ti)=

= LGD

b

X

i=1

λi

Z Ti

Ti−1

e^{−Λi−1−λi(u−Ti−1)}P (0, u)du.

Consider the first maturity, Tb1. If Ti’s reset a larger frequency than this maturity, we will have to find the constant λ^b1= λ_i with i equal to the set of i’s for which Ti≤ T_b1. Then, after having computed λ^b1, we can compute λ^b2 and so on...

Stochastic intensity: Cox process

Intensity can be time varying and stochastic, thus allowing to introduce uncertainty on credit spreads.

Intensity λ(t) follows a stochastic process. Desirable properties:

1 Ft-adapted: λ(t) is known, given market information up to t;

2 right continuous;

3 λ(t) > 0 for every t.

When the intensity is stochastic, the process jumping at τ is called Cox process.

Cox process/2

Doubly stochastic process: default occurs at the first jump time of the process, τ := Λ⁻¹(ξ), where:

1 λ is stochastic: uncertainty in the jump intensity;

2 ξ is random: uncertainty in jump arrival (conditional on F^λ, i.e. on the filtration generated by the intensity, the process is a Poisson process with time-varying intensity).

ξ is independent of all quantities whose information is contained in F and λ.

The survival probability S(0, T ) conditional on all the information available at time 0 is obtained as

S(0, T ) = Q(Λ(τ ) ≥ Λ(T )) = Q(ξ ≥ Z s

0

λ(s)ds) =

E

 Q

 ξ ≥

Z T 0

λ(s)ds|F^λ



= E h

e⁻

RT 0 λ(s)dsi

.

Cox process and analogy with interest rates

Conditional on λ(t), Cox process is a time-inhomogeneous Poisson process.

Given all available information at time t the conditional probability of survival up to T , at time t is equal to

S(t, T ) = Et

h e⁻

RT t λ(s)dsi

.

depicts a clear analogy with the short rate proess r(t).

Analogously to the forward interest rate, the forward default intensity (rate) f (t) can be defined and is the mean arrival rate of default at t, conditioning on survival up to t (while intensity is conditional on all the available

information up to t).

Which models for default intensities?

Models with non-negative intensities and leading to closed-form solutions for the survival probabilities are the natural candidates.

In general, we will describe the intensity as the (jump-) diffusive stochastic process λ(t) following the SDE:

dλ(t) = µ(t, λ(t))dt + σ(t, λ(t))dW (t)(+dJ (t)).

Usual trade-off between complexity and tractability.

Most convenient choice: affine processes!

Affine processes

An affine processes X is a Markov processes taking values on a subset of R^d, for which the conditional characteristic function takes the form

E



e^iu·X(t)|X(s)

= eφ(t−s,iu)+ψ(t−s,iu)·X(s), with φ(·, iu) and ψ(·, iu) ”regular” coefficients.

Jump-diffusive processes X of the type

dXt= µ(Xt)dt + σ(Xt)dBt+ dJt

with µ(X_t), σ(X_t)σ(X_t)⁰ and the jump measure associated to J having affine dependence on the state Xt are affine processes.

Affine processes/2

The important property that affine processes enjoy is that for any affine function R(X(u)), (net of some technical conditions):

Et

h e

RT

t −R(X(u))du+w·X(T )i

= eα(T −t)+β(T −t)·X(t), where α(·) and β(·) solve the generalized ODEs (see Duffie, Pan and Singleton, 2000):

β(t)˙ = ρ1− K₁^Tβ(t) − 1

2β(t)^TH1β(t) − l1(θ(β(t)) − 1) , α(t)˙ = ρ0− K₀· β(t) − 1

2β(t)^TH0β(t) − l0(θ(β(t)) − 1) , with α(T ) = 0, β(T ) = u, θ(c) =R e^czdν(z), where ν is the jump measure, and µ(x) = K₀+ K₁x, (σ(x)σ(x)^T)_ij = (H₀)_ij+ (H₁)_ij · x, γ(x) = l₀+ l₁· x, R(x) = ρ₀+ ρ₁· x.

CIR intensity process

The well-known (one-dimensional) CIR intensity process is an affine process:

dλt= k(θ − λt)dt + σp λtdBt. θ is the long-run intensity;

k is the speed of mean-reversion Et[λ_T] = θ + e^{−k(T −t)}(λ_t− θ).

It is non-negative and positive if the Feller condtion (2kθ > σ²) is met;

Its associated survival probability is thus S(t, T ) = eα(T −t)+β(T −t)λt,

with α(·) and β(·) solving the appropriate ODEs and available in closed form.

“Basic” affine process (Duffie and Garleanu, 2001)

Consider the jump-diffusive process dλt= k(θ − λt)dt + σp

λtdBt+ dJt, where:

B is a standard one-dimensional Brownian motion, J is a compound Poisson process with jump intensity ξ and jump size exponentially distributed with mean ψ.

S(t, T ) = E h

e^{−RtTλ(s)ds} i

= eα(T −t)+β(T −t)λ(t). The solution to the system of generalized ODEs

β(t)˙ = −kβ(t) +1

2σ²β(t)²− 1

˙

α(t) = kθβ(t) + ξ ψβ(t) 1 − ψβ(t), with α(0) = β(0) = 0 is available in closed form.

Basic affine process

Indeed, the system descends from the fact that:

µ(x) = kθ − kx; σ(x)σ(x)⁰ = σ²x;

γ(x) = ξ; R(x) = −x.

The “jump transform” is θ(β(t)) =

Z

e^β(t)zdv(z),

with R e^β(t)zdv(z) being the characteristic function of an exponential random variable with mean ψ,

φz(t) = Z

e^itxdν(z) = 1 1 − itψ Hence, substituting β(t) = it above,

θ(β(t)) − 1 = βψ 1 − β(t)ψ.

Continuous diffusions

Let us now have a digression on simulation, starting from reviewing the methods to generate paths from continuous purely diffusive processes.

We will consider sampling from solutions to Stochastic Differential Equations:

dX(t) = µ(t, X)dt + σ(t, X)dW (t), X(t₀) = X₀ First of all, generating sample paths amounts to discretizing the process, i.e. approximating it by considering its

realizations at a finite set of time points [t0, t1, ..., t_N] and then interpolate to produce a continuous-time trajectory.

We usually split the sampling interval into equally spaced sub intervals.

Methods for generating diffusions

Samples from continuous diffusions can be generated using:

1 the exact transition density between two consecutive time points p(t,x;s,y).

2 the closed-form solution of the SDE, i.e. the exact dynamics followed by the process;

3 a dynamics which approximates the original SDE.

1-Exact Transition density

If the transition density

p(t_i, x_i; t_i+1, x_i+1) = P (X(t_i+1) = x_i+1|X(t_i) = x_i) is known for any pair of consecutive times, then:

Algorithm

1 Initialise X₀= x₀ and ∆t = T /N , where [0,T] is our sampling interval and N is the number of time steps.

2 For i=1,...N, sample X_i, from the density p(t_i, ·; t_i−1, X_i−1);

3 {X_i}_i=0,...,N is a sample of process X on [0,T].

Example 1: Standard Brownian Motion

Consider the simple process dX(t) = dW (t). The standard brownian motion W(t) is a continuous stochastic process s.t.:

1 W(0)=0;

2 The increments W(t)-W(s) are independent;

3 W (t) − W (s) is distributed as a N (0, t − s) for any 0 ≤ s < t ≤ T .

Subsequent values of the process W(t) can be generated easily, as we know the transition density between two consecutive time points:

W (t_i) = W (t_i−1) +pt_i− t_i−1Z_i, i = 1, ....N, where Z₁, ..., Z_n are independent standard normal random variables.

MATLAB code

function W=b_m_sim(T,N) dt=T/N;

W=zeros(N+1,1);

W(1)=0;

Z=icdf(’norm’,rand(N,1));

for i=1:N

W(i+1)= W(i)+sqrt{dt}*Z(i);

end

t=[0:dt,T];

plot(t,W,’*-’) end

A more interesting example: the Vasicek model

The Vasicek (1977) model is a well-known interest rate model for the short rate:

dr(t) = a (b − r(t)) dt + σdW (t), r(0) = r₀. The Vasicek model falls into the domain of Gaussian models for the short rate and its transition density p(t, ·; s, y) is Normally distributed with mean b + e^−a(t−s)(y − b) and variance ^σ_2a² 1 − e^−2a(t−s)
.

MATLAB Simulation of Vasicek’s model

function X=Vasicek_sim(a,b,s,r0,T,N) X=zeros(N+1,1);

X(1)=r0;

dt=T/N;

for i=1:N

mu=b+exp(-a*dt)*(X(i)-b);

var=s^2*(1-exp(-2*a*dt))/(2*a);

X(i+1)=icdf(’norm’,rand,mu,sqrt(var));

end

t=[0:dt:T];

plot(t,X,’*’) end

2-Exact Solution

This method can be applied if the SDE can be solved explicitly, i.e. if there exists a functional of time t and the driving noise W up to t such that

X(t) = G(t, {W_i}_i=0,...t)

It consists in discretizing the underlying noise over a finite set of sampling times and apply the functional to obtain the value of the process X at those set of time points.

Algorithm

1 Set X₀ = x₀ and ∆t = T /N ;

2 For i=1,...,N, sample the brownian motion W (t_i) and set Xi = G(ti, {W (t1), ..., W (ti)})

3 (Xi)i=1,...,N is a sample of the process X on [0,T].

Example: Geometric Brownian Motion

The Geometric Brownian Motion is the most widely used model for stocks. It implies log-normal prices.

dS(t) = S(t) (rdt + σdW (t)) The solution of this SDE is

S(t) = S0exp



r −σ² 2



t + σW (t)

 . If we discretize it, we get:

Si= Si−1exp



r −σ² 2



∆t + σ

√

∆tZi



, i = 1, ..., N.

where Z_is are independent samples from a standard normal random variable.

MATLAB Simulation of GBM

function S=GBM(r,s,S0,T,N) S=zeros(N+1,1);

S(1)=S0;

for i=1:N

S(i+1)=S(i)*...

exp((r-s^2/2)*dt+s*sqrt(dt)*randn);

end

t=[0:dt:T];

plot(t,S,’*-’)

3-Approximating the dynamics

When the two previoulsy described methods are not applicable, it is possible to simulate the solution

approximating the dynamics of the system, i.e. by solving the stochastic difference equation associated to the SDE.

There are several ways of approximating, we will see the most common one, the Euler scheme.

The Euler scheme

The SDE

dX(t) = µ(t, X)dt + σ(t, X)dW (t), X(t₀) = X₀ can be discretized in the following way:

Xi+1= Xi+ µ(ti, Xi)∆t + σ(ti, Xi)

√

∆tZi, where {Zi}’s are i.i.d. standard normals.

Algorithm

1 Set X0 = x0 and ∆t = T /N

2 For i = 0, ..., N − 1, sample Z_i and compute X_i+1

3 X_i obtained this way is a sample of process X over [0,T]

Approximation Error

Being based on an approximation, the use of the Euler scheme entails an error.

Let X_EU be the approximate solution computed based on the Euler Scheme and X the exact one. Then

Esup_{t∈[0,T ]}|X_EU(t) − X(t)|² ≤ C∆t, with C constant.

The approximation error is smaller the smaller ∆t.

Example: Cox, Ingersoll, Ross (1985) process

The CIR process is used in the domain of interest rates. It is a square root process, which may never become negative:

dr(t) = a(b − X(t))dt + σp

r(t)dW (t).

We can discretize it using the Euler Scheme as:

r_t+1= r_t+ a(b − r_t)∆t + σ√ r_t√

∆tZ_i, where {Z_i}s are independent random samples from a standard normal distribution.

MATLAB Simulation of CIR process using Euler Scheme

function X=CIR(a,b,s,r0,T,N) X=zeros(N+1,1);

X(1)=r0;

dt=T/N;

for i=1:N

X(i+1)=X(i)+a*(b-X(i))*dt+s*sqrt(X(i))*sqrt(dt)*randn;

end

t=[0:dt:T];

plot(t,X,’*-’) end

Transition density for the CIR process

The CIR process has a known transition law:

r(t) = σ²(1 − e^−a(t−s))

4a ∗ χ²_d 4ae^−a(t−s)

σ²(1 − e^−a(t−s))r(s)

! , t > s,

where χ²_d denotes the pdf of the non-central chi-squared distribution with d = ^4ab_σ2 degrees of freedom.

Homework: Try to simulate the CIR process using its exact transition and compare the results with those obtained by using the Euler Scheme for different lengths of the time step dt.

Generating paths from jump processes

A jump process varies according to discontinuities only.

We consider a compound Poisson stochastic process

J (t) =

N (t)

X

j=1

Yj,

where

N describes the occurrence of jumps: for an event ω, for instance a jump trajectory, N (t, ω) counts the number of jumps between the initial time and current time t;

Yj represents the magnitude of the j-th jump.

Generating N and Y

In order to generate a path from a jump process it is necessary to simulate both N and Y.

N (t) is a counting process, i.e. a non-decreasing process that takes values in N and that defines the number of jumps occurred up to time t.

Y is generated by sampling from an assigned distribution.

Assume N and Y are independent.

Simulating a counting process

1 Generate the inter-arrival times, i.e. the time spans

between two consecutive jumps. They are random variables T₁, ...T_N.

2 Jump times, τ₁, .., τ_N i.e. the instants at which jumps occur, are obtained as the cumulative sums of interarrival times:

τ_j =

j

X

i=1

T_i

3 For each time instant t, N (t) counts the number of jumps that have occurred since the beginning and prior to t:

N (t) =

+∞

X

n=1

1_τn≤t.

Simulating a Poisson process

A homogeneous Poisson process is a unit-jump increasing process whose inter-arrival times are i.i.d. exponential.

Recall the density of the exponential distribution f (x) ∼ Exp(λ) = λe^−λx

N (t) is distributed according to a Poisson law P o(λt):

f_{N (t)}(n) = e^−λt(λt)ⁿ/n!

N (t) can be simulated via conditional simulation or via countdown simulation.

Simulating the counting process

Algorithm - Conditional simulation

The algorithm samples the number of jump occurrences first, and then their location. Conditional on N (t) = n jump times of a homogeneous Poisson process are uniformly distributed on [0, T ].

1 Simulate N (T ) ∼ P o(λT ).

2 Simulate the N (T ) jump times τ1, ..., τ_{N (T )} as uniform samples on [0, T ].

3 N (T ) and jump times are a sample form the Poisson process.

MATLAB code

function [NT,tau]=sim_Pp_cond(lambda,T) NT=icdf(’Poisson’,rand,lambda*T);

% alternatively, use NT=poissrnd(lambda*T);

tau=rand(NT,1).*T;

tau=sort(tau);

end

Countdown simulation

Algorithm

If λ is constant, N is a Poisson process and inter arrival times are exponentially distributed with parameter λ.

1 Let τ₀= 0, i = 0

2 Set i = i + 1

3 Generate inter-arrival times from the exponential, for instance using the inverse transform method: generating U ∼ U [0, 1] and then T_i= −_λ¹log U .

4 Set τ_i= τ_i−1+ T_i

5 If τi≤ T , go to Step 2, otherwise return τ₁, ..., τi−1 and the Poisson sample N (T ) = i − 1.

MATLAB code

function [NT,tau]=sim_Pp_count(lambda,T) tau(1)=0;

i=0;

while (tau(i+1)<=T) i=i+1;

tau(i+1)=tau(i)-log(rand)/lambda;

% alternatively, tau(i+1)=tau(i)+exprnd(lambda);

end

tau=tau(2:end);

NT=i-1;

end

Inhomogenous Poisson Process

In many applications, for instance in life insurance or credit risk, it is necessary to make use of a Poisson process with non-constant intensity.

The previous algorithms can be easily adapted.

Algorithm: conditional simulation

1 Simulate N (T ) ∼ P o

RT

0 λ(s)ds



2 Generate N (T ) independent inter-arrival times samples with common distribution density

f_τ(t) = λ(t) RT

0 λ(s)ds

Cox Process

The previous algorithm can be easily adapted to simulate a Cox process as well, using a two step procedure:

Algorithm: conditional simulation given λ(t)

1 Simulate λ(t), for 0 ≤ t ≤ T and compute the corresponding Λ(t) =Rt

0λ(s)ds.

2 Conditional on the sampled λ(t), 0 ≤ t ≤ T , simulate a time-inhomogeneous Poisson Process:

Simulate N (T ) ∼ P o(Λ(T ))

Generate the N (T ) independent inter-arrival times samples that have common distribution density

f_τ(t) = λ(t) RT

0 λ(s)ds .

Default arrival simulation

Indeed, when simulating a default arrival, the issue is simpler, because we are interested only in the first jump time of the Poisson process. In all cases, we simulate the arrival conditional on λ(t). We have two options:

1 Inverse-CDF Simulation: when the survival probability S(t, T ) is available in closed form, simulate U and let τ be chosen as the solution to

S(t, τ ) = U ;

2 Compensator Simulation: exploiting the fact that Λ(τ ) is exponentially distributed with parameter 1, since S(0, τ ) = e^{−Λ(τ )}. Then, simulate a standard unit-mean exponential variable Z and let τ be the solution to

Λ(τ ) = Z =⇒ τ = Λ⁻¹(Z).

Issues with the simulation of default arrival from stripped intensities

Assume we have stripped a piecewise constant intensity from CDS quotes.

Then, we can use the compensator method(or, equivalently) the inverse-cdf method to simulate the jump time.

Two problems:

1 The hazard function Λ(t) is defined for 0 ≤ t ≤ T , where T is the last maturity for which CDS quotes are available.

Then, it is necessary to extend somehow the function Λ beyond Λ(T ), because if sampled exponential Z > Λ(T ) there is no solution to the equation.

2 To obtain an acceptable precision when using Monte-Carlo simulation to simulate default arrival it is necessary to use a large number of scenarios.

Jump-Diffusive Processes

Consider a jump-diffusive process

dX(t) = µ(t, X(t))dt + σ(t, X(t))dW (t) + η(t, X(t⁻))dJ (t), where

J is a compound Poisson process J (t) =PN (t) j=1 Y_j; X(t) = lim_s→t−X(s);

ξ(t, x) is the jump intensity.

The idea is to simulate the continuous part and the jump part of the process separately.

Simulating a jump-diffusive process

Algorithm

1 Simulate the random jump times τ₁, ...τ_m of the compound Poisson process;

2 Simulate the continuous part of the process at dates t₁, ..., t_n using the Euler scheme for instance, with t_i= i∆t.

3 Simulate jump sizes and add them to the values X(t_i) simulated at jump times.

Problem: τ_k’s need not coincide with t_i’s.

Solution: approximate τ_k with τ_k^∗ = t_j, where t_j is the t_i such that min |t_i− τ_k| = |t_j− τ_k|.

Pricing in intensity-based models

Modelling the intensity directly in the intensity-based framework allows us to write the zero-recovery defaultable bond paying F at maturity T as (see Lando, 1994):

P (t, T ) = E¯ t

h e⁻

RT

t r(s)dsF 1{τ >T }

i

= Et

h e⁻

RT

t r(s)+λ^∗(s)dsF i

, where E is an expectation under measure Q and λ^∗(t) is an intensity process under the measure Q and F is

F_T-measurable.

The above formula accomodates also for dependence between short rate and intensity processes and holds provided only that r and λ are bounded and τ is doubly stochastic under Q.

Independence between default and interest rates

If Q-independence holds, P (t, T )¯ = Et

h e⁻

RT

t r(s)dsF 1_{{τ >T }} i

= Et

h e⁻

RT

t r(s)+λ^∗(s)dsF i

=

= E

h

e^−RtTr(s)dse^{−RtTλ∗(s)ds}Fi

=

= F E h

e⁻

RT t r(s)dsi

E h

e⁻

RT

t λ^∗(s)dsi

=

= P (t, T )S(t, T )F.

Introducing dependence between default and interest rates

Technically, a Poisson process and a Brownian motion defined on the same space are independent.

Hence, if the intensity of the Poisson process is

deterministic, τ and the stochastic discount factor D(s, t) are independent.

Thus, ¯P (t, T ) = ED(0, T )1_{{τ >T }} = E [D(0, T )] E1_{{τ >T }} = P (0, T )Q(τ > T ).

To introduce dependence between stochastic interest rates and default time, we need to introduce correlation between dr and dλ, with λ stochastic!