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Stochastic Volatility E®ects on Defaultable Bonds

Issuance of Stability Bonds under joint and several guarantees would a priori lead to a situation where the prohibition on bailing out would be breached. In such a situation, a Member State would indeed be held liable irrespective of its 'regular' contributing key, should another Member State be unable to honour its financial commitments. In this case, an amendment to the Treaty would be necessary. This could be made under the simplified procedure if a euro area common debt management office were constructed under an inter- governmental framework, but would most likely require the use of the ordinary procedure if. | Stochastic Volatility Effects on Defaultable Bonds Jean-Pierre Fouque Ronnie Sircar Knut Splna December 2004 revised October 24 2005 Abstract We study the effect of introducing stochastic volatility in the first passage structural approach to default risk. We analyze the impact of volatility time scales on the yield spread curve. In particular we show that the presence of a short time scale in the volatility raises the yield spreads at short maturities. We argue that combining first passage default modeling with multiscale stochastic volatility produces more realistic yield spreads. Moreover this framework enables us to use perturbation techniques to derive explicit approximations which facilitate the complicated issue of calibration of parameters. Contents 1 Introduction 2 1.1 Defaultable Bonds. 3 1.2 Outline of the Paper. 3 2 The Constant Volatility Case 4 3 Stochastic Volatility 6 3.1 A Class of Models . 6 3.2 Stochastic Volatility Effects in Yield Spreads . 8 4 Fast Volatility Factor and Singular Perturbation 12 4.1 The European Case . 12 4.2 Barrier Options. 13 4.3 Pricing Defaultable Bonds. 15 Department of Mathematics NC State University Raleigh NC 27695-8205 fouque@math.ncsu.edu. Work supported by NSF grant DMS-0455982 Department of Operations Research Financial Engineering Princeton University E-Quad Princeton NJ 08544 sircar@princeton.edu. Work partially supported by NSF grants DMS-0306357 and DMS-0456195. Department of Mathematics University of California Irvine CA 92697 ksolna@math.uci.edu. 1 5 Slow Volatility Factor and Regular Perturbation 16 6 Models with Fast Slow Volatility Factors 20 6.1 The Combined Two Scale Stochastic Volatility Models. 20 6.2 The Combined Volatility Perturbations. 22 6.3 Accuracy of the Approximation. 23 6.3.1 Illustration from Numerical Simulations. 24 6.3.2 Convergence Result . 25 7 Calibration 26 7.1 Calibration Formulas. 26 7.2 Calibration Exercise. 27 A Fast Scale Correction Formulas 31 B Slow Scale Correction Formulas 32