Valuing Credit Risk - Variance Reduction Techniques for Monte Carlo Methods

Valuing Credit Risk
Variance Reduction Techniques for
Monte Carlo Methods

Hochschule für Bankwirtschaft
Private Hochschule der Bankakademie

Master-Thesis zur Erlangung des Titels ”Master of Arts Banking and Finance”

Ralph Karels

Wintersemester 2003

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 dependency structure. A section devoted to this concept provides the necessary background and properties. Next, the general Monte Carlo concept is introduced in detail and carefully 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 e±ciency 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.

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

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 horizon¹. 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 changes². 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 st and the fair spread s0 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 basket 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 default 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 attractive 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 sfirst is close to the sum of the n CDS spreads, provided the term structure of credit spreads si 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 p_ij = 1 for all 1 <= i; j <= n ) yields s^first = max(s1, . . . . , sn). 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 spread3. 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.

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1 vgl. Schmidt (2001) S.2 ff

2 vgl. Schmidt (2001) S.6