Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Market Models for Forward Swap Rates and Credit Default Swap Spreads Marek Rutkowski School of Mathematics and Statistics University of New South Wales Sydney, Australia Joint work with Libo Li Workshop on Stochastic Analysis and Finance City University of Hong Kong June 29-July 03, 2009 Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Outline Forward Swaps 1 Admissible Sets of Forward Swap Rates 2 Market Models of Swap Rates 3 Forward CDS Spreads 4 Market Models of CDS Spreads 5 Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models References on Modelling of Swap Rates M. Davis and V. Mataix-Pastor: Negative Libor rates in the swap market model. Finance and Stochastics 11 (2007), 181–193. S. Galluccio, J.-M. Ly, Z. Huang, and O. Scaillet: Theory and calibration of swap market models. Mathematical Finance 17 (2007), 111–141. F. Jamshidian: LIBOR and swap market models and measures. Finance and Stochastics 1 (1997), 293–330. F. Jamshidian: Bivariate support of forward Libor and swap rates. Mathematical Finance 18 (2008), 427–443. F. Jamshidian: Trivariate support of flat-volatility forward Libor rates. Working paper, University of Twente, 2007. R. Pietersz and M. van Regenmortel: Generic market models. Finance and Stochastics 10 (2006), 507–528. Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models References on Modelling of CDS Spreads N. Bennani and D. Dahan: An extended market model for credit derivatives. Presented at the international conference Stochastic Finance , Lisbon, 2004. D. Brigo: Constant maturity credit default swap pricing with market models. Working paper, Banca IMI, 2004. D. Brigo and M. Morini: CDS market formulas and models. Working paper, Banca IMI, 2005. S.L. Ho and L. Wu: Arbitrage pricing of credit derivatives. Working paper, HKUST, 2007. F. Jamshidian: Valuation of credit default swaps and swaptions. Finance and Stochastics 8 (2004), 343–371. L. Schlögl: Note on CDS market models. Working paper, Lehman Brothers, 2007. Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Goals We would like to address the following issues. A unified approach to modelling of market rates. 1 A comparison of the top-down and bottom-up approaches. 2 Separate study of two related inverse problems. 3 Positivity of rates within market models. 4 Construction of default times consistent with CDS spreads. 5 Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Problems The following problems arise in the context of market models. The choice of admissible families of swaps and/or credit default swaps. 1 Computation of Radon-Nikodym densities for martingale measures. 2 Computation of dynamics of forward swap rates and CDS spreads under 3 swap/CDS measures. Existence and construction of default times consistent with CDS 4 spreads. Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Forward Swaps Market Models for CDS Spreads Inverse Problem for Bonds Towards Generic Swap Models Families of Forward Swaps Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Forward Swaps Market Models for CDS Spreads Inverse Problem for Bonds Towards Generic Swap Models Forward Swap Terminology and notation: Let 0 < T 0 < T 1 < · · · < T n − 1 < T n be a fixed sequence of dates 1 representing the tenor structure T . We denote a i = T i − T i − 1 for i = 1 , . . . , n . 2 Let B ( t , T i ) stand for the price of the zero-coupon bond maturing at T i . 3 Let S = { S 1 , . . . , S l } be any family of l distinct forward swaps associated 4 with the tenor structure T . Any reset or settlement date for any swap S j in S belongs to T . 5 Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Forward Swaps Market Models for CDS Spreads Inverse Problem for Bonds Towards Generic Swap Models Forward Swap Rates The forward swap rate κ j for the forward swap S j starting at T s j and maturing at T m j equals s j , m j = B ( t , T s j ) − B ( t , T m j ) = P s j , m j κ j t t = κ , ∀ t ∈ [ 0 , T s j ] . � m j t s j , m j i = s j + 1 a i B ( t , T i ) A t where we set s j , m j P = B ( t , T s j ) − B ( t , T m j ) , t ∈ [ 0 , T s j ] , t and m j � s j , m j A = a i B ( t , T i ) , t ∈ [ 0 , T s j + 1 ] . t i = s j + 1 Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Forward Swaps Market Models for CDS Spreads Inverse Problem for Bonds Towards Generic Swap Models Bond Numeraire Let us choose the bond maturing at T b as a bond numéraire . In terms of the deflated bond prices B b ( t , T i ) = B ( t , T i ) / B ( t , T b ) , i = 1 , . . . , n , we obtain b , s j , m j = B b ( t , T s j ) − B b ( t , T m j ) = P s j , m j t κ , ∀ t ∈ [ 0 , T s j ∧ T b ] . � m j t b , s j , m j i = s j + 1 a i B b ( t , T i ) A t We call the process A b , s j , m j , defined as m j � b , s j , m j a i B b ( t , T i ) , t ∈ [ 0 , T s j + 1 ∧ T b ] , A = t i = s j + 1 the deflated swap annuity or deflated swap numéraire . Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Forward Swaps Market Models for CDS Spreads Inverse Problem for Bonds Towards Generic Swap Models Modelling Issues We will deal with the following interrelated modelling issues. Under which assumptions the joint dynamics of a given family of swap 1 rates is supported by an arbitrage-free term structure model, where by a term structure model we mean the joint dynamics of deflated bond prices? How to specify the joint dynamics for a given family of forward swaps in 2 terms of “drifts” and “volatilities” under a single probability measure? Under which assumptions the swap rates and/or CDS spreads follow 3 positive processes and the default time can be constructed? Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Forward Swaps Market Models for CDS Spreads Inverse Problem for Bonds Towards Generic Swap Models Swap Equation and Inverse Problem Each forward swap corresponds to the linear equation in which deflated 1 bond prices are treated as “unknowns”. Specifically, we deal with the following swap equation associated with 2 the forward swap S j and the numéraire bond B ( t , T b ) m j − 1 � s j , m j s j , m j B b ( t , T s j ) − a i B b ( t , T i ) − ( 1 + κ ) a m j B b ( t , T m j ) = 0 . κ t t i = s j + 1 The following inverse problem is of interest: describe all families of 3 forward swaps such that the knowledge of the corresponding family of swap rates is sufficient to uniquely specify the associated family of non-zero (or positive) deflated bond prices for any choice of the numéraire bond. Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
Market Models for Forward Swaps Density Processes of Martingale Measures Forward Swaps Market Models for CDS Spreads Inverse Problem for Bonds Towards Generic Swap Models Notation Let x i stand for a generic value of the deflated bond price B b ( t , T i ) and 1 s j , m j let κ j be a generic value of the forward swap rate κ j t = κ . t Since x i and κ j are aimed to represent generic values of the 2 corresponding processes in some stochastic model we have that ( x 0 , . . . , x n ) ∈ R n + 1 and ( κ 1 , . . . , κ l ) ∈ R l . However, if the bond B ( t , T b ) is chosen to be the numéraire bond then, 3 by the definition of the deflated bond price, the variable x b satisfies x b = 1 and thus it is more adequate to consider a generic value ( x 0 , . . . , x b − 1 , x b + 1 , . . . , x n ) ∈ R n . Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads
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