We first show that the CDS spread can be represented as survival probability of the underlying entity. Recall expression (4.5),
%Jm,n(t )= 1
1
From section 2 we know that DBk B(t) is equivalent to the conditional survival probability at time t under Tk-forward default measure. The above formula exhibits the linkage between CDS spread and default probability, which can be attained by using market CDS quotes and the riskfree rate curve. For simplicity, we set
m= 0 and n = i in the spread of a CDS contract starting at Tm and terminating at Tn, namely,
Sm,n ≡ S0,i ≡ Si, for i= 1,2,...
A general relationship between CDS market quote and the survival probability is derived as probability, riskfree interest rate curve up to Tn and the recovery rate, we can obtain the Tn- survival probability. A survival curve from T1 to Tn can be extracted by a recurrence procedure. In the following we give an example about how to plot a survival curve of a legal entity by the above procedure.
Examples:
Besides extracting the survival curve of an entity from the CDS quote, we will also illustrate how the dynamics of survival probability and CDS spread term structure is affected when CDS spread has the possibility to jump. Since the hazard rate HBkB(t) is considered as a dynamic process, we can forecast the hazard rate term structure as well as default probabilities and predict the widening of CDS spreads provided that the parameters are appropriately estimated. In the first example we
derive the default probabilities of different terms from the CDS spread of IBM quoted on 1/20/2006, compare those to the default probabilities calculated by Bloomberg, and plot the default probability curve. In the second example, in which the specified reference entity is the British Airway, we plot the current CDS spread curve and use input different jump rates to simulate the corresponding CDS spread curve 0.5 year after that date to see how the CDS spread term structure would shift when the underlying CDS spread has different jump rates. A reasonable guess is that the default probability curve as well as the CDS spread curve has a greater up ward movement as the jump rate increases. The input CDS spreads of British Airways is quoted on 4/11/2006 and the chosen parameters are: σkL = 0.25 and σkH = 0.35 for all k, mY = 0.5, s= 0.2 . In both examples we use EURIBOR curve as the reference riskfree forward curve and set the recovery rate R be 0.4. Spreads data is listed in table 5.1.1. The results of example 1 and example 2 are respectively illustrated in figure 5.1.1 and figure 5.1.2.
Table 5.1.1: CDS spreads and default probabilities of IBM on 1/20/2006
Year Spread Model DP Bloomberg DP
0.5 6.576 0.0005 0.0005
1 6.576 0.0011 0.0011
2 10.230 0.0034 0.0035
3 13.915 0.0071 0.0071
4 16.748 0.0114 0.0115
5 19.581 0.0167 0.0169
7 27.608 0.0333 0.0339
10 39.642 0.0681 0.0707
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Figure 5.1.1 The default probability curve of IBM on 1/20/2006 derived by this model and that calculated by Bloomberg.
Table 5.1.2 CDS mid spreads of British Airway (4/11/2006)
1y 2y 3y 4y 5y 6y 7y 25.0 40.0 62.0 99.0 125.5 139.0 152.5
8y 9y 10y
CDS Spread
(in bps) 166.3 180.2 194.0
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Figure 5.1.3 Widening of CDS spreads with different jump rates for the underlying CDS spread.
5.2 Implied Volatilities of Credit Default Swaptions
In this subsection we investigate the implied volatility structure of credit default swaption. Glasserman and Kou (2003) indicate the importance of jumps and stochastic volatility in implied volatility curve. Das and Sundarum (1999) investigate the shape of implied volatility in jump-diffusion and implied volatility models. We shall discuss how implied volatilities are related to jumps and discover how jump parameters of CDS spreads affect volatility structures of credit default swaption.
Consider the case of payer swaptions. Different implied volatility structures are demonstrated with respect to different inputs of jump parameters, which are jump rate, mean value of jump size and log-volatility of jump size of the corresponding CDS spread. The numerical experiment sets the jump intensity (or jump rate) of the underlying CDS spread to be λm,n =λ, the initial CDS spread to be 500 basis points, swaption time to maturity ( T ) to be two years with tenor-length (δ) six months.We also fix the riskfree forward rate (Lk) at 4% and assume a constant CDS spread volatility γm,n to be 0.25. The strike spread is ranged from 200 bps to 800 bps increasing by 50 bps. The following three different cases are considered.
In the first case we input different jump rates (λ) and hold other parameters fixed. The implied volatility tends to be higher as jump rate grows (see Table 5.2.1).
Larger jump rate means the underlying CDS spread jumps more frequently, which in turn will push up swaption price as well as implied volatility.
Secondly, we allow the log-volatility of the jump size in CDS spread (s) to vary, with other parameters fixed, to examine how the implied volatility curve would fluctuate according to various inputs of s. The results are listed in Table 5.2.2. The climbing log-volatility of jump makes the implied volatility of CSO shift up. Since the more dispersed the underlying CDS spread is, the higher possibility big jumps would occur, which makes the CSO more hazardous and implied volatility larger.
In the last case, different values of mean jump size of CDS spread (mY) are input and other parameters are held fixed. Table 5.2.3 demonstrates the numerical results. It shows the various shapes of implied volatility curves with different mY. When the mean jump size is zero, it yields a volatility “smile”; while the nonzero mean jump size brings into a volatility “skew”, with an upward skew if mY > 0 and downward skew if mY < 0. The skew and smile phenomenon can be interpreted in terms of market’s expectation of jumps. In the case of positive mean jump size, an upward sloping volatility curve is the result of the possible large positive jumps which is concerned by the market participants. A downward skew volatility curve, where the mean jump size is negative, comes from the fear of large negative jumps. In the case
of mY = 0, the market has a symmetric anticipation for jumps, so the volatility curve is shaped as a smile. As is shown in Table 5.2.3 and Figure 5.2.3, the implied volatility curve of CSO whose underlying mean jump size equals to zero falls below that of CSO whose underlying mean jump size is greater or less than zero.
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Figure 5.2.1 Implied Volatility Curve of a credit default swaption with different jump rates and fixed jump size mean and log-volatility, where mY = 0, s= 0.3
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Figure 5.2.2 Implied Volatility Curve of a two-year credit default swaption with different jump size log-volatility and fixed jump size mean and jump rate, where
λ = 0.5 , mY = 0.3.
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Figure 5.2.3 Implied Volatility Curve of a two-year credit default swaption with different mean jump sizes and fixed jump rate and jump size log-volatility, where λ= 0.5 , = 0.25s