Random factor loading

Consider Equation (4.3.8) in Section 4.3. Using the arguments of Andersen and Sidenius (2005), calibration to the CDO market requires manipulation of the parameters of quantities involved in Equation (4.3.8) so that we are close to the value of

E[f(L

t

)]

implied by the

market. Recall

f(L

t

) = (L

t

−k)

+, for some

k∈(0,1]

and

(L

t

)

t≥0 is the credit portfolio loss

process.

In this calibration exercise we typically consider, simultaneously, the fit of multiple CDOs, perhaps constrained to the same maturity, but attempting to re-price as much traded tranches as possible (see Andersen and Sidenius (2005)). To obtain a good fit under the factor framework one must look to develop the density function,

ψ(t,z)

, or the conditional survival function,

q

i

(t,z)

, as defined in Equation (4.3.6). The OFGC model can change the

function

q

i

(t,z)

only through a constant parameter,

ρ. The degree to which this is a highly

limited model comes through Theorem 16 or Proposition 17 and the fact that the model leads to a correlation curve. In this section we introduce a model with more free parame- ters in

q

i

(t,z)

in order to obtain better market fits.

Model setup

The Random Factor Loading (RFL) model attempts to extend the OFGC model yet retaining broadly the structure of the OFGC model. The idea is to take the parameter of the OFGC,

ρ, and make it a function of the systemic variable,Z. This notion is similar to that of local

volatility in equities, where the volatility parameter is a function of the underlying. Empir- ically both ideas are well founded; in our case firms are more correlated in bad economic times than in good times, i.e. when

Z

is low

ρ

should be high and when

Z

is high

ρ

should be lower.

Hence the RFL model makes correlation functionally dependent on the systemic factor and it is suggested that correlation should be decreasing as a function of

Z.

Following Andersen and Sidenius (2004), we start generally by defining for all obligors,

i∈ {1, . . . , n}

:

X

i

=ρ

i

(Z) +v

i

i

+m

i

,

(4.8.1)

where

v

i

:=

p

1−_V[ρ

i

(Z)]

and

m

i

:=

−E[ρ(Z)]

are chosen so that

X

i has zero mean and

unit variance. We note that:

v

i

=

s

1−

Z

Rd

ρ

i

(z)

2

ψ(z)dz+m

2i

;

m

i

=

−

Z

Rd

ρ

i

(z)ψ(z)dz;

p

i

(z)

=

G

i

c

i

−ρ

i

(z)−m

i

v

i

,

(4.8.2) also we have:

p

i

=Q(τ

i

< t) =Q(X

i

< c

i

)

=

E

Q(

i

≤

c

i

−ρ

i

(z)−m

i

v

i

)|Z

=

Z

Rd

G

_i

c

i

−ρ

i

(z)−m

i

v

i

dψ(z)(z),

(4.8.3) where

ψ(z)

is the distribution function for

Z

and we have suppressed dependence on time.

G

_i is the cumulative distribution of

i. In this specific form of the RFL model we:

• Let

Z

and

i have the standard normal distribution.

• Make

ρ

i

(Z)

homogeneous and equal to

ρ(Z)

for all

i∈ {1, . . . , n}

.

• Take

ρ(Z)

to be a two point distribution so that:

ρ(Z) =











αZ

if

Z < θ,

βZ

if

Z

≥θ.

(4.8.4)

From our discussion we want

β < α

to reflect empirical observations. With these restric- tions we recover the following:4

v

=

p1−[α

2

_{(Φ(θ)−θφ(θ)) +β}

2

_{(θφ(θ) + 1−Φ(θ))−(βφ(θ)−αφ(θ))}

2

_],

m

=

(αφ(θ)−βφ(θ),

p

i

(Z)

=

Φ(

c

i

−ρ(Z)−m

v

),

where

φ

is the standard normal density. We also have that the default probabilities (sup- pressing dependence on obligors

i∈ {1, . . . , n}

) are:

p

=

Q(τ < t)

=

Q(X < c)

=

Φ2

pc−m

(v

2

_+α

2

, θ,

α

p

(v

2

_+α

2

!

+ Φ

pc−m

(v

2

_+β

2

!

−Φ2

pc−m

(v

2

_+β

2

, θ,

β

p

(v

2

_+β

2

!

.

(4.8.5) Numerical example

The last equality in Equation (4.8.5) is a function which is not easily invertible, compared to the OFGC case where we needed to invert a standard Gaussian distribution. Here the route to recovering the thresholds,

c, is to perform a root search. This can become time

consuming as we do the inversion for each obligor and at multiple time points. For example suppose we have a portfolio of 125 obligors, a CDO with a maturity of 10 years and we wanted to discretise the protection leg according to the coupon payment dates. Then we would need to do

125×10×4 = 5000

(where 4 is the coupon frequency for payments). We can reduce this computational problem by exploiting the monotonicity of the default threshold

c

as a function of time, i.e. the default thresholds increase monotonically with time as default probabilities also increase. We can then generate default thresholds,

c, for

fewer points than required up until the trade maturity and apply some interpolation. From this we can find the other thresholds,

c, for different times.

Figure 4.9 shows the implied loss distribution using the RFL model applied to index market data, Appendix G, for varying

α

and

β

with

θ

taken to be 0:

• Curve C1 corresponds to empirical observation, here

α

= 80%

and

β

= 20%

. By comparing the shape of the curve with that of Figure 4.7 in Section 4.7 curve C1 shows that the RFL model with such a parameterisation has a loss distribution which is qualitatively consistent with the existence of a base correlation curve.

• In curve C2 we take

α

=β

=√44.6%

. This reduces down to the OFGC model. Here 44.6% is the compound (and base) correlation of the equity tranche which we recov- ered in Section 4.5 (Table 4.1).

• Curve C3 is for completeness (α

= 20%

and

β

= 80%

) and is counter-empirical. The curve is counter-empirical as it is constructed by assuming that in good economic times the default of obligors are more correlated than in bad economic times.

RFL Loss distribution for different levels of alpha and beta

0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Loss P ro b a b il it y C1 C2 C3

Figure 4.9: RFL loss distributions.

4.9

Chapter conclusion

In this chapter we have detailed methods of pricing multi-obligor products. Moreover we provide a new algorithm to generate a portfolio’s loss distribution:

• We considered methods of generating default dependence. We provided a quantitative argument to show that correlating stochastic intensities that are driven by diffusion processes leads to a loss of control over spread dynamics.

• We demonstrated that under the conditionally independent framework one can have tractable results.

• We highlighted that the aim of multi-obligor models was to effectively account for the loss distribution of a credit portfolio and capture the dependence relationship between obligors.

• We reviewed the standard model under the factor framework, the one factor Gaussian copula (OFGC). Under this model it is necessary to construct correlation curves. • This is because the OFGC model does not re-price all traded tranches. Correlation

curves are used in conjunction with the OFGC model (but are exogenous to the model) in order to re-price all traded tranches.

• We considered the compound and base correlation curve construction approaches. The compound correlation method was shown to be weak. Base correlation was shown to be more useful. Nevertheless it contained significant instances of arbitrage. • We considered the random recovery model, which is an extension to the OFGC model.

The model contains economically relevant and empirically observed features and re- duces instances of arbitrage in comparison to the OFGC model.

• We demonstrated how to generate the loss distribution of a portfolio using the base correlation curve. Two methods were provided:

– The standard method developed by Turc et al. (2004).

– The loss algorithm, which we develop.

We established that our method is more efficient and that it is more broadly applicable than the method devised by Turc et al. (2004).

• We reviewed the random factor loading model. An extension of the model is consid- ered in Chapter 6.

• The loss algorithm enables us to generate a portfolio’s loss distribution as implied by the base correlation curve. We argued that the loss distribution generated should be representative of the real loss distribution of the credit portfolio, since the base correlation curve allows for exact re-pricing of traded tranches.

• By constructing a portfolio’s loss distribution we were able to show that the OFGC model produces arbitrage and that these instances of arbitrage are reduced with the random recovery model. Moreover from this work we were able to establish that the

RFL model qualitatively produces a loss distribution which is similar to the market implied loss distribution.

Chapter 5

Spread Dynamics With Default

Correlation

In this chapter we will examine methods of generating both spread dynamics and default dependence in order to value multi-obligor products that are influenced by spread dynam- ics.

Recall in Chapter 1 we provided an overview of the literature. In that chapter we discussed the development of single obligor models and multi-obligor models. We argued that:

• The literature for single obligor models has looked to capture spread dynamics. • The literature for multi-obligor models has placed a focus on accounting for default

dependence.

In this thesis, so far, we have examined methods of capturing spread dynamics and default dependence separately:

• In Chapter 3 we discussed the valuation of single obligor credit products. We demon- strated by valuing LCLNs without recourse the significant influence of spread dynam- ics. In that chapter the emphasis was on putting in place appropriate dynamics on the intensity process.

• In Chapter 4 we detailed methods of pricing multi-obligor products. We demonstrated the importance of being able to capture the distribution of losses in a portfolio and the large effects of default correlation. Arguments were provided to show the difficulty in extending the single obligor framework by correlating stochastic intensities.

• Moreover in Chapter 4 we demonstrated how dependence could be generated using the factor framework setting, under which a default time copula arises naturally. Un- der the factor framework, a factor (exogenous to the intensity process) is introduced in order to generate default dependence. In its classic setup the framework is a one period model and hence one cannot introduce spread dynamics easily into the frame- work.

5.0.1

Summary of sections in this chapter

This chapter is split into four sections:

• In Section 5.1 we detail how a dynamic multi-obligor modelling framework can be constructed when we have

n >1

obligors:

– The model considers

F−adapted

continuous stochastic processes,

(˜λ

it

)

t≥0 with

i

∈ {1, . . . , n}

, which Schubert and Schönbucher (2001) call pseudo intensities of the obligors. Default dependence is generated by comparing the value of the integrated

F−adapted

pseudo intensities with exogenously generated correlated

uniform variables,

ζ

1

, . . . , ζ

n.

– We define two filtrations: thepartial filtration and thefull filtration. The former contains information about the default (or not) of only one obligor, the latter contains information about the default (or not) of all obligors.

– Under the partial filtration

(˜λ

i_t

)

t≥0 is the intensity of obligor

i∈ {1, . . . , n}

. Under

the full filtration the intensity of obligor

i

is influenced by quantities related to the other obligors.

– We define two copulas, thesurvival copula and thethreshold copula. The survival copula is the copula function of default times and the threshold copula is the copula function of the exogenous uniform random variables,

ζ

1

, . . . , ζ

n, related to

obligors

i

∈ {1, . . . , n}

. The survival copula is the conditional expectation of the threshold copula.

• In Section 5.2 we consider securitised loans. These are loans given to a borrower who is asked to provide collateral in order to mitigate losses that may occur because the borrower fails to pay:

– The collateral provided by the borrower are debt instruments issued by a third party reference obligor. Therefore there is a need to account for default correla- tion between the borrower and the third party reference obligor.

– These types of agreements are often linked to a dynamic form of risk mitigation called margin calling. Margin calling requires the borrower to provide more collateral whenever the value of the collateral she had originally posted falls in value by a pre-specified amount. The impact of this is to ensure in a worsening market the lender has collateral of sufficient value to reduce any potential losses from a borrower default. This means we need to consider the dynamics of the price of the collateral.

– Hence such products require that we account for default correlation as well as spread dynamics.

– By an application of the dynamic modelling framework detailed in Section 5.1 we show how we can value secured loan products.

• In Section 5.3 we consider the valuation of leverage credit linked notes with recourse:

– Recall in Chapter 3 (Section 3.4) we valued leverage credit linked notes without recourse. By considering recourse, where the issuer of an LCLN has recourse to the note investor (the counterparty), we need to account for default correlation.

– Default correlation between the counterparty and the reference obligor (the obligor which the LCLN refers to) is important as on a default of the reference obligor the counterparty may have to make additional payments to the issuer.

– Since we are valuing an LCLN which has an embedded American digital option, accounting for spread dynamics is also important.

– Some significant conclusions are reached in the section. In particular we find that when volatility is high it acts like a dampener on default correlation reducing instances of joint defaults.

• In Section 5.4 we provide a conclusion.

In document Credit Modelling: Generating Spread Dynamics with Intensities and Creating Dependence with Copulas (Page 158-166)