5.3 Leverage with recourse credit linked notes
5.3.2 Numerical Example
In this numerical example we take a reference obligor and a counterparty that both have constant CDS spreads of 400bps across maturity. In addition we assume they have market recoveries of 25% each.
Example 5 (LCLN with recourse). The specific trade we consider has the following fea- tures:
1. Maturity (
T
) = 3y.2. Notional (
N
) = USD 10,000,0003.
k= 1000
bps4.
s˜
R(0, T) = 400
bps (The initial swap spread of the reference obligor). 5.s
C(0, T) = 400
bps (The initial swap spread of the counterparty). 6.F
= 5
.Table 5.2 and Figure 5.4 shows an important relationship between correlation and volatility: • The table and figure provide values for the fair LCLN spread for various spot volatili-
ties, speed of mean reversions and correlations.
• Note that LCLN fair spread levels above 20% are produced. At first this seems wrong since one may expect the maximum the fair spread can be is 5 times the fair CDS swap spread which is
5×4% = 20%
. However this can be explained in the following manner:• Recall that the fair swap spread of a CDS is
s˜(t, T) =
D˜˜(t,T)P(t,T).
– With an LCLN the value of the default leg, call itD
˜
lev(t, T)
, is unlikely to be more than 5 times (the leverage factor) of the un-leveraged CDS default leg.– Yet the premium leg of a CDS (which is its risky duration),P(
˜
t, T)
, is larger than the premium leg of an equivalent maturity LCLN, call itP˜
lev(t, T)
.– The reason for this is that the premium leg of a CDS is only extinguished upon de- fault, whereas the premium leg of an LCLN is extinguished upon a default event or a trigger event. Hence the trigger event is effectively creating an additional dampening impact to the risky duration of an LCLN making it less than that of a comparable CDS.
– This gives rise to the possiblity of the fair margin being more than 20%.
• Consider Figure 5.4 and the graph with 90% default correlation. We note that as volatility increases the fair LCLN spread increases up to 5 times of the fair CDS swap spread (which is 400bps).
• Hence the effect of volatility seems to be like a de-correlator. This can be understood from the following perspective: when two obligors are highly correlated if we add volatility it creates spread dispersion. This makes the event of joint defaults less likely, which subsequently means the counterparty can receive a higher spread premium. • There also seems to be little or no variation in fair LCLN spreads for correlations
below 60% (we see a variation in premium from 17% to 23%). Again this seems to be because of the de-correlating effects of volatility.
Spot volatility(σ) Correlation 20% 40% 60% 90% 0% 19.30% 18.37% 16.99% 8.04% 1% 19.57% 18.76% 17.51% 10.71% 2% 19.62% 19.01% 17.90% 12.94% 3% 19.74% 19.71% 18.85% 15.19% 4% 20.33% 20.73% 20.16% 16.79% 5% 20.97% 20.87% 20.97% 18.72% 6% 21.59% 21.96% 21.25% 20.76% 7% 22.51% 22.29% 22.10% 22.02% 8% 22.58% 22.57% 22.62% 21.77%
Table 5.2: Leverage with recourse spread premium above the bench mark interest rate, Libor, as a function of varying correlation and extended Vasicek spot volatilities with speed of mean reversion fixed at 0. We use a hypothetical flat CDS spread of 400bps for both counterparty and underlying.
0.00% 2.00% 4.00% 6.00% 8.00% 18.00% 20.00% 22.00% 24.00% 0 1 2 3 Spot volatility Fair spread
Speed of mean reversion
Fair leverage spread with correlation 20%
0.00% 2.00% 4.00% 6.00% 8.00% 18.00% 20.00% 22.00% 24.00% 0 1 2 3 Spot volatility Fair spread
Speed of mean reversion
Fair leverage spread with correlation 40%
0.00% 2.00% 4.00% 6.00% 8.00% 15.00% 17.00% 19.00% 21.00% 23.00% 25.00% 0 1 2 3 Spot volatility Fair spread
Speed of mean reversion
Fair leverage spread with correlation 60%
0.00% 2.00% 4.00% 6.00% 8.00% 8.00% 10.00% 12.00% 14.00% 16.00% 18.00% 20.00% 22.00% 24.00% 0 1 2 3 Spot volatility Fair spread
Speed of mean reversion
Fair leverage spread with correlation 90%
Figure 5.4: Leverage with recourse spread premium above the bench mark interest rate, Li- bor, as a function of varying speed of mean reversion and extended Vasicek spot volatilities with correlation fixed at 20%, 40%, 60% and 90% respectively.
5.4
Chapter conclusion
In this chapter we have considered a framework that contains both default dependence and spread dynamics:
• The modelling framework is efficient because it allows one to calibrate separately the dynamics of individual obligors and the dependence between obligors.
• This means the work done in Chapter 3 can be directly applied. In that chapter we constructed a calibration routine for an intensity process driven by the extended Va- sicek model.
• We considered two products:
– Securitised loans.
– Leverage credit linked notes.
Both products require we account for default correlation and credit spread dynamics. • We have found:
– Higher default correlation increases counterparty risk.
– Embedding thresholds in products that exhibit counterparty risk is important. Thresholds enable margin to be called whenever collateral posted by a counter- party falls in value. This significantly reduces the credit risk a lender takes in transactions that exhibit counterparty risk.
– If all things are kept equal increasing the volatility parameter of the model damp- ens the effects of default correlation. This is because higher volatility reduces in- stances where both the reference obligor and counterparty have similar spreads. This reduces the likelihood that (even with high default correlation) the condition for a joint default to occur (detailed in Note 3 (Chapter 4)) will hold.
Part of the working assumptions of this thesis has been that trading in credit markets does not occur in liquid and transparent markets (recall in Chapter 1 we explained that Duffie and Singleton (2003) had made this evaluation about the credit markets). The implications of this is that models have to be clear and interpretable, since a black box calibration routine will mean little without the products and liquidity to calibrate them to.
In this chapter we have considered two products modelled with an intensity model driven by the extended Vasicek model. The model is parsimonious and allows for the calibration to the term structure of credit spreads and credit default swaptions.
In order to create default dependence we are using a Gaussian copula for the threshold copula. Again the Gaussian copula is parsimonious in that it accounts for dependence via a single parameter,
ρ.
Arguments may still be put forward that the intensity model should be more developed, for instance perhaps by introducing a stochastic volatility model or adding a jump component, etc. Moreover it may be asserted that having a more complex dependence structure (such as the ones considered in Chapter 4) would be better. However what we give up in this process is interpretability. What we gain are more parameters to fit without the products to fit them to4.
In Chapter 6, however, we will consider products which require more developed depen- dence structures and spread dynamics to be accounted for. We will develop explicit meth- ods which advance the Schubert and Schönbucher (2001) framework.
4
Even though the CDO market displays heterogeneity in default correlation across time and capital structure (see Schönbucher (2006)), those CDOs are for a specific portfolio. Here we are trying to describe the dynamics and default correlation between a counterparty and a reference obligor.
Chapter 6
Future Research
In this chapter we consider two models which will require future research. These models attempt to develop the multi-obligor framework by accounting for spread dynamics via intensity modelling.
6.0.1
Summary of sections in this chapter
This chapter is split into three sections:
• In Section 6.1 we detail the framework developed by Andersen (2006), which intro- duces the notion of inter-temporal dynamics. By this we mean correlating the sys- temic common factor in the factor framework over multiple time horizons. This en- ables the level of common factors at earlier maturities to impact common factors at later maturities. From this we develop the notion of thestochastic liquidity threshold approach. This method makes the liquidity threshold in the factor framework setting (Chapter 4 (Sub-section 4.3.2)) stochastic.
• In Section 6.2 we consider the valuation of credit default index swaptions. We detail three distinct evolutions in the pricing of credit default index swaptions. Further, by:
1. A redefinition of the enlarged filtration,
G
.2. A consideration of pre-collapse quantities (which reduce instances of arbitrage). 3. And by assuming portfolio losses occur on a discretised set independently of the
CDIS swap spread process,
we develop, using intensity dynamics, a semi-analytical valuation of credit default in- dex swaptions. By introducing intensity dynamics we can account for the term struc-
ture of credit spreads and develop a model which will be able to calibrate to multiple credit default index swaptions.
• In Section 6.3 we provide a conclusion.