Valuing Credit Risk - Variance Reduction Techniques for Monte Carlo Methods

Abstract

This paper deals with the valuation of credit risk derivatives on the basis of Monte Carlo simulation methods with the main viewpoint on variance reduction techniques. Therefore, first an overview on credit risk derivatives like credit default swaps and first to default baskets is given. It turns out that modelling of the joint distribution of dependent credit default times proves to be the crucial element. Once obtained, any credit derivative can be valued. A convenient way of achieving this is by use of the copula concept, which migrates marginal distributions of credit default times obtained from a credit curve into a joint distribution incorporating any kind of desired depen- dency structure. A section devoted to this concept provides the necessary background and properties. Next, the general Monte Carlo concept is introduced in detail and care- fully adapted to the valuation of credit derivatives, following the path of constructing dependent uniform random variables from dependent normal random variables. At the same time, first insight is gained in the field of variance reduction which is intensified in chapter four, where a series of techniques including antithetic sampling and control variates is presented. The main focus shall lie from there on on importance sampling. In order to increase the efficiency of Monte Carlo methods, sampling is restricted to the region of importance where the function to be evaluated - here: the indicator function of the credit default times - does not vanish. This technique is applied and examined in detail in the final chapter for the one- and multi-credit case. Exponential as well as normal importance sampling densities are derived.

List of Abbreviations

aka also known as

a.s.

AV

BDS

CC

CDF

CDO

CDS

Corr

Cov

CV

DF

E

FTDS

iid

IS

LIBOR

MC

PV

R

RV

SLLN

SS

Var

VaR

VR

wrt

3

Mathematical Notation

1 A indicator function of A: 1 A = 1 if A holds, 0 otherwise

d = equality with respect to distribution

C(u 1 , . . . , u n ) copula function

Corr(X, Y ) correlation of (X, Y ) : Cov(X, Y ) = Corr(X, Y )Var(X)Var(Y )

Cov(X, Y ) covariance of (X, Y ) : E(XY ) − E(X)E(Y )

exp(λ)

E f

E(X|Y ) conditional distribution of X given Y

F (−1) (x)

inverse function of the distribution function F if it exists

F X (x) distribution function of the RV X

f (x|y) conditional probability density of X given Y

F (k) , k ≥ 1

kth mixed partial derivative of F

Γ α,β Gamma distribution with parameters α, β

ϕ(x) probability density function of the standard normal distribution

N (x) standard normal cumulative distribution function evaluated at x

N (µ, σ 2 ) normal distribution with expectation µ and variance σ 2

N n (µ, Σ) n-dimensional normal distribution with expectation vector µ

P f probability with respect to the probability density f

supp(f ) support of function f : the joint set where f does not vanish

r interest rate

transpose of vector x

[x] the largest y from the set of whole numbers smaller or equal to x

4

Contents

1  Introduction: Credit Risk and Credit Derivatives  6

2  Modelling Joint Defaults  9

2.1  The Copula Function Approach  9

2.2  Default Correlation  13

3  Monte Carlo Simulation  21

3.1  General Principles and Theoretical Background  21

3.2  Monte Carlo Approach for Credit Derivatives  24

4  Variance Reduction Techniques for Monte Carlo Methods  29

4.1  Antithetic Sampling  29

4.2  Variate Recycling  30

4.3  Control Variates  31

4.4  Stratified Sampling  32

4.5  Conditional Expectation  32

4.6  Importance Sampling  33

5  Application to Credit Risk  37

5.1  Basic Variance Reduction Techniques  37

5.2  Importance Sampling Techniques  39

5.2.1  The One Credit Case  39

5.2.2  The Two Credit Case  45

5.2.3  The n Credit Case  47

6  Conclusion  57

  Bibliography  58

5  NA  

Chapter 1

Introduction: Credit Risk and

Credit Derivatives

Credit risk refers to the risk of incurring losses due to changes in a counterparty‘s credit quality. The over the recent years ever more popular and successful credit derivatives allow to isolate and actively manage that risk by providing a payoff upon a credit event arrival, be it a rating downgrade or default in form of failure to pay or bankruptcy of the reference credit. The basic building block for more complex derivatives is the plain vanilla credit default swap (CDS), which offers protection against default of a specific underlying over a specified time horizon 1 . A premium is paid on a regular basis on a notional amount N as an insurance fee against the losses from default of a risky position of notional N . The premium often is a spread over the plain vanilla non-credit swap rate. The spread s achieving a CDS present value of zero is called fair CDS spread. CDS spreads are equal to asset swap spreads and par spreads. The payment of this premium stops either at maturity of the CDS or at the time of default, whichever comes first. At the time of default before maturity, the protection buyer receives the payment N (1 − R) from the protection seller, with R being the recovery rate of the underlying credit risky instrument. Observe that a CDS serves not only as an insurance against default but also as an insurance against changes in the rating of the underlying since the market value of a CDS changes if the rating of the underlying changes 2 . Hence, two different pricing problems arise: First, at origination fix the CDS premium s so that the value of the CDS is zero. This is crucial for making markets. Second, after origination with changing credit quality and market interest rates, the current value of the CDS, expressed in terms of the difference between the current fair spread s t and the fair spread s 0 at the time of engagement in the transaction, has to be determined. This is crucial for hedging.

More sophisticated credit derivatives are linked to several underlyings and include bas- ket default swaps (BDS) like kth to default swaps or collateralized debt obligations (CDO). A BDS similarly to a CDS offers protection against the event of the kth de- fault among a basket of n ≥ k underlying credit names against payment of a spread s kth . Most popular are first to default swaps (FTDS) (k = 1) which offer highly attrac-

vgl. Schmidt (2001) S.6

6

tive spreads (premiums) to a credit investor (protection seller). If the n credits inside a BDS are assumed to be independent, a FTDS is approximately equivalent to n CDS and therefore the spread s f irst is close to the sum of the n CDS spreads, provided the term structure of credit spreads s i is flat for each credit in the basket. This is due to the likelihood of multiple defaults being at the minimum now. The other extreme, total dependence (correlation ρ ij = 1 for all 1 ≤ i, j ≤ n ) yields s f irst = max(s 1 , . . . , s n ). The FTDS spread thus is the worst of all the CDS spreads since in the case of perfect positive correlation the basket is dominated by the name with the worst spread 3 . This is due to the fact that all state variables in the respective normal copula are identical as we will see in the next chapter. The truth however lies somewhere between those two extremes. Based on the dependence of issuers on several general economic factors or direct firm inter linkages, credit quality changes of several issuers are often (not perfectly) correlated. Hence

n

max(s 1 , . . . , s n ) ≤ s f irst ≤ The higher the diversification rate of the basket, the higher the spread s f irst which on the other hand means that for increasing positive default correlation the spread is decreasing since the probability of multiple defaults increases and the degree of de- fault protection provided by the FTDS is diminished. Investors holding positions with numerous counterparties therefore are exposed to the aggregated risk of losses due to correlated credit events arrivals. That is why efficient modelling of default correlation becomes the most important part of credit risk valuation.

A cash flow CDO is a securitization of corporate obligations in which assets are sold into a special purpose vehicle which uses them as collateral to issue notes 4 . CDOs are issued as multiple classes of equity and debt that are tranched with respect to seniority. They are based upon a pool of n default-risky underlyings which often is generated synthetically via single name CDS. Credit investors receive a spread for the risk (of the respective tranche) taken. The change in value of a tranche is highly dependent on a large number of factors such as pool composition, leverage, minimum overcollateralization ratios, reinvestment period and yield curve environment. The lowest tranche (equity) takes the responsibility for the first losses due to default up until a certain barrier thus acting as a buffer protecting upper tranches (mezzanine, senior) from losses due to defaults. There are two main reasons for the issuance of CDOs. First there is the desire to transfer the risk of assets held by commercial banks in order to shrink their balance sheets and reduce regulatory capital. Second, equity tranche investors hope to achieve a leveraged return between the realized yield on collateral assets and the financing costs of the upper tranches. These are called Arbitrage CDOs. The main reason investors buy CDOs is that they offer custom risk exposure that cannot be achieved in any other way. Investment-grade investors also seek a diversification benefit by gaining exposure to asset classes that would otherwise be outside their investment mandate. Regarding the valuation of tranches, they depend upon the credit quality (spreads) of the names and from the dependency structure among the credits inside the pool. Hence, the estimation of aggregated loss distributions

vgl. Mina S.10 ff

7

in credit risk measurement and the valuation of multi-name credit derivatives and CDOs

requires a model for the joint default behavior of numerous credit-risky securities such

as bonds or loans. This is what we will investigate now.

8

Chapter 2

Modelling Joint Defaults

2.1 The Copula Function Approach

Before we will be able to handle default correlation models in the next section in full detail, the concept, definition and properties of copulas have to be introduced. First, observe that given a joint distribution of random variables (RVs) the marginal distributions and the correlation structure between the RVs can be extracted but in general not vice versa. An exception is the multivariate normal distribution which can be fully described knowing only the marginal distributions and the correlation structure. This is one reason why multivariate normals are appealing, another one is that margins of multivariate normals are (univariate) normal as well. Now there are many different techniques and ways how to specify a joint distribution of RVs - which is by no means unique - with given marginal distributions and a given correlation structure. One possibility is to develop multivariate distributions as immediate extensions of univariate ones (e.g. the bivariate Pareto or gamma). The drawbacks are that a different family is needed for each marginal distribution and extensions above the bivariate case often are not clear 1 . Among the multivariate distribution construction techniques, the copula approach is a simple and convenient one. We start with

Definition 1 (Copula) Given uniform random variables (RVs) U 1 , . . . , U m , the joint distribution function of (U 1 , . . . , U m )

C(u 1 , u 2 , . . . , u m , ρ) := P (U 1 ≤ u 1 , . . . , U m ≤ u m ) (2.1)

is called a copula function C : [0, 1] m → [0, 1] with correlation structure ρ 2 .

Note that in the special case of independence among the (U 1 , . . . , U m ) it holds

m

C(u 1 , . . . , u m ) = and in the special case of perfect correlation it holds 3

vgl. J¨ ackel (2002) S. 45

9

C(u, v) = min 1≤j≤m (u j )

Hence, for m = 2 an example of a copula function would be the mixture copula of the form 4

C(u, v) = (1 − ρ)uv + ρmin(u, v) (2.2)

An immediate application of this definition links arbitrary marginal distribution func- tions F j (x j ), j = 1, . . . , m with the joint distribution function F (x 1 , . . . , x m ) via 5

C(F 1 (x 1 ), . . . , F m (x m )) = P (U 1 ≤ F 1 (x 1 ), . . . , U m ≤ F m (x m )) (2.3)

(−1) = P (F

1 = P (X 1 ≤ x 1 , . . . , X m ≤ x m ) = F (x 1 , . . . , x m )

where we have used the simple, but very important 6

Lemma 2 Consider U ∼ U [0,1] uniform, F strictly monotonous 7 then F (−1) (U ) ∼ F .

P (F (−1) (U ) ≤ x) = P (U ≤ F (x)) = F (x) Proof 2

Hence, the copula does not constrain the choice of marginal distributions. There are two principal ways of using the copula idea. Copulas can be extracted from well-known multivariate distribution functions via

(−1)

(u 1 ), . . . , F (−1)

C(u 1 , . . . , u m ) = F (F

1 C is then called the copula of F which remains invariant under strictly increasing com- ponentwise transformations - a property which is not shared by the correlation matrix. On the other hand, new multivariate distribution functions can be created by joining arbitrary marginal distributions together with copulas. This is summarized in 8

Theorem 3 (Sklar‘s Theorem) Let F be a joint distribution function with contin- uous margins F 1 , . . . F m . Then there exists a unique copula C : [0, 1] m → [0, 1] such that

F (x 1 , . . . , x m ) = C(F 1 (x 1 ), . . . , F m (x m )) (2.5)

Conversely, if C is a copula and F 1 , . . . , F m are distribution functions, then F given by (2.5) is a joint distribution function with margins F 1 , . . . , F m .

Here we assume positive correlation. Mixture copulas for negatively correlated RVs have a different form.

vgl. Laurent (2002) S.8

Remark 4 The uniqueness of the copula C stems from the continuity of the margins.

One way of constructing copulas is the method of compounding 9 :

Example 5 (Copula construction) Given a risk classification parameter γ, a RV X can be modelled exponential

P (X ≤ x|γ) = 1 − e −γx

If we assume γ ∼ Γ(α, β) Gamma then it can be shown that

x

F X (x) = β R Thus X has a Pareto distribution. Assuming that coming from the same risk class γ induces a dependency on two RVs X 1 and X 2 with distribution functions F 1 and F 2 which on the other hand are assumed to be independent, conditional on the risk class γ. Their joint distribution function F is

F (x 1 , x 2 ) = 1 − P (X 1 > x 1 ) − P (X 2 > x 2 ) + P (X 1 > x 1 , X 2 > x 2 ) −α −α x 1 x 2

= 1 − = 1 − = 1 − = F 1 (x 1 ) + F 2 (x 2 ) − 1 +

This yields the copula function

−α

(1 − u 1 ) − 1 α + (1 − u 2 ) − 1

C(u 1 , u 2 ) = u 1 + u 2 − 1 +

Applying this compounding technique, a number of important copula families, e.g. Hougaard‘s copula family, can be generated. Another major technique is the Archimedean approach which we will not discuss any further here.

Another important result is that a copula accounts for all the dependence between (X 1 , . . . , X m ) in the sense that given arbitrary but strictly increasing functions g 1 , . . . , g m over the range of X 1 , . . . , X m respectively, then

C(X 1 , . . . , X m ) = C(g 1 (X 1 ), . . . , g m (X m )) (2.6)

Thus, the manner in which X 1 , . . . , X m move together is captured by the copula, re- gardless of the scale in which each variable is measured. We will end this section with showing how copulas can be used to simulate outcomes from a multivariate distribu- tion, i.e. we construct an algorithm to generate X 1 , . . . , X m with known distribution function F given by the copula construction (2.4) which shall be of the form

vgl. Frees (1997) S.3 ff

C(u 1 , . . . , u m ) = Φ (−1) (Φ(u 1 ) + . . . + Φ(u m )) (2.7)

The idea 10 is to recursively simulate the conditional distribution of X k |X 1 , . . . , X k−1 k = 2, . . . , m under the assumption that the joint density of (X 1 , . . . , X m ) exists. Then we have 11

∂ k

Φ (−1)

f k (x 1 , . . . , x k ) = = (Φ (−1) ) (k)

Therefore the conditional density of X k |X 1 , . . . , X k−1 is given by

f k (x 1 , . . . , x k )

f k (x k |x 1 , . . . , x k−1 ) =

·

Now the conditional distribution of X k |X 1 . . . , X k−1 is

x k

F k (x k |x 1 . . . , x k ) =

−∞

= Φ(F 1 (x 1 )) + · · · + Φ(F k−1 (x k−1 ))

(Φ (−1) ) (k−1) (c k−1 + Φ(F k (x k ))) =:

a Hence, the algorithm known as the Archimedean construction denotes as follows

• Draw U 1 , . . . , U m ∼ U [0,1] iid uniform

(−1)

• Set X 1 = F (U 1 ) ∼ F 1 and c 0 = 0

1

• k = 2, . . . , m: Recursively compute X k as the solution of

(Φ (−1) ) (k−1) (c k−1 + Φ(F k (x k )))

10 vgl. Frees (1997) S.11

11 The superscript (k) means the kth mixed partial derivative.

12