Empirical Evaluation of Hybrid Defaultable Bond Pricing ... - risklab

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10 7,5 5 2,5 1 Year 3 Years 5 Years 7 Years 10 Years 0 01.10.1993 25.11.1994 19.01.1996 14.03.1997 08.05.1998 02.07.1999 25.08.2000 19.10.2001 13.12.2002 Figure 1: 1- to 10-Year US Treasury Strips in %. Time Period: 1993 — 2003. Source: Bloomberg. 1,8 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 0 01.10.1993 16.09.1994 01.09.1995 16.08.1996 01.08.1997 17.07.1998 02.07.1999 16.06.2000 01.06.2001 17.05.2002 02.05.2003 1 Year 3 Years 5 Years 7 Years 10 Years Figure 2: 1- To 10-Year Credit Spreads of US Industrials A2 in %. Time Period: 1993 — 2003. Source: Bloomberg. 7
2,5 2 1,5 1 0,5 1 Year 3 Years 5 Years 7 Years 10 Years 0 01.10.1993 16.09.1994 01.09.1995 16.08.1996 01.08.1997 17.07.1998 02.07.1999 16.06.2000 01.06.2001 17.05.2002 02.05.2003 Figure 3: 1- To 10-Year Credit Spreads of US Industrials BBB1 in %. Time Period: 1993 — 2003. Source: Bloomberg. 3 The Model of Schmid and Zagst In the following section we give a short overview of the Schmid and Zagst three-factor defaultable term structure model. As a typical hybrid model it combines elements of structural and reduced-form models. The underlying nondefaultable short rate is assumed to either follow a mean reverting Hull-White process or a mean-reverting square root process with time-dependent mean reversion level. Therefore, the dynamics of the non-defaultable short rate are given by the following stochastic differential equation (SDE): dr (t) =[θ r (t) − a r r (t)] dt + σ r r (t) β dW r (t) , 0 ≤ t ≤ T ∗ , (3) where a r , σ r > 0 are positive constants, β =0or 1 2 , and θ r is a non—negative valued deterministic function. This speciÞcation implies that the current rate r (t) is pulled towards θr(t) a r with a speed of adjustment a r ,andifβ = 1 2 the instantaneous variance of the change in the rate is proportional to its level. One of the factors that determine the credit spread is the so called uncertainty index which can be understood as an aggregation of all information on the quality of the Þrm currently available: The greater the value of the uncertainty process the lower the quality of the Þrm. The uncertainty or signaling process is assumed to follow a mean reverting square root process. The development of 8
Page 1 and 2: working paper Empirical Evaluation
Page 3 and 4: using a compound options approach a
Page 5 and 6: Þrst introduced by Cathcart & El-J
Page 7: We assume that there exists a measu
Page 11 and 12: where with ( 1 A(t,T )−B(t,T )r P
Page 13 and 14: a r , b r , σ r , a w , σ w , a u
Page 15 and 16: with B(t, T ) = 1 ³ −âr(T 1 −
Page 17 and 18: mean reversion level of 5.21%. Thes
Page 19 and 20: where P d (t, T, r, w, u) =e Ad (t,
Page 21 and 22: 6 Comparison of the Model Performan
Page 23 and 24: Table 10: Average R 2 values (out
Page 25 and 26: D(0,T)/T*mr% US Industrials A2 US I
Page 27 and 28: 0,8% 0,7% 0,6% US Industrials BBB1
Page 29 and 30: Schmid, Zagst 2000 Schmid, Zagst 20
Page 31 and 32: Hull, J. & White, A. (1990). Pricin
Page 33 and 34: A DeÞnition of v (t, T ) v (t, T )
Page 35 and 36: solution: B(t, T ) = e −âr(T −
Page 37 and 38: D Results of Regression Analyses Ta
Page 39 and 40: Table 17: Test of out—of-sample e

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