Keywords: Time change, default intensity, credit risk, CDS options
Stochastic time-change offers a parsimonious and economically well-grounded device for introducing stochastic volatility to simpler constant volatility models. The constant volatility model is assumed to apply in a latent "business time." The speed of business time with respect to calendar time is stochastic, and reflects the varying rate of arrival of news to the markets. Most applications of stochastic time-change in the finance literature have focused on the pricing of stock options. Log stock prices are naturally modeled as Lévy processes, and it is well known that any Lévy process subordinated by a Lévy time-change is also a Lévy process. The variance gamma (Madan and Seneta, 1990; Madan et al., 1998) and normal inverse Gaussian (Barndorff-Nielsen, 1998) models are well-known early examples. To allow for volatility clustering, Carr, Geman, Madan, and Yor (2003) introduce a class of models in which the background Lévy process is subordinated by the time-integral of a mean-reverting CIR activity-rate process, and solve for the Laplace transform of the time-changed process. Carr and Wu (2004) extend this framework to accommodate dependence of a general form between the activity rate and background processes, as well as a wider class of activity rate processes.
In this paper, we generalize the basic model in complementary directions. We discard the assumption that the background process is Lévy, and assume instead that the background process () has a known Laplace transform, . Maintaining the requirement that the business clock () and the background process are independent, we develop two different series solutions for the Laplace transform of the time-changed process given by . In fact, our methods apply generically to a very wide class of smooth functions of time, and in no way require to be the Laplace transform of a stochastic process. Henceforth, for notational parsimony, we drop the auxilliary parameter from .
Our two series solution are complementary to one another in the sense that the restrictions imposed by the two methods on and on differ substantively. The first method requires that be a Lévy process, but imposes fairly mild restrictions on . The second method imposes fairly stringent restrictions on , but very weak restrictions on . In particular, the second method allows for volatility clustering through serial dependence in the activity rate. Thus, the two methods may be useful in different sorts of applications.
Our application is to modeling credit risk. Despite the extensive literature on stochastic volatility in stock returns, the theoretical and empirical literature on stochastic volatility in credit risk models is sparse. Empirical evidence of stochastic volatility in models of corporate bond and credit default swap spreads is provided by Jacobs and Li (2008), Alexander and Kaeck (2008), Zhang et al. (2009) and Gordy and Willemann (2012). To introduce stochastic volatility to the class of default intensity models pioneered by Jarrow and Turnbull (1995) and Duffie and Singleton (1999), Jacobs and Li (2008) replace the widely-used single-factor CIR specification for the intensity with a two-factor specification in which a second CIR process controls the volatility of the intensity process. The model is formally equivalent to the Fong and Vasicek (1991) model of stochastic volatility in interest rates. An important limitation of this two-factor model is that there is no region of the parameter space for which the default intensity is bounded nonnegative (unless the volatility of volatility is zero).4
In this paper, we introduce stochastic volatility to the default intensity framework by time-changing the firm's default time. Let denote the calendar default time, and let be the corresponding time under the business clock. Define the background process as the time-integral (or "compensator") of the default intensity and as the business-time survival probability function . If we impose independence between and , as we do throughout this paper, then time-changing the default time is equivalent to time-changing , and the calendar-time survival probability function is
The idea of time-changing default times appears to have first been used by Joshi and Stacey (2006). Their model is intended for pricing collateralized debt obligations, so makes the simplifying assumption that firm default intensities are deterministic.6 Mendoza-Arriaga et al. (2010) apply time-change to a credit-equity hybrid model. If we strip out the equity component of their model, the credit component is essentially a time-changed default intensity model. Unlike our model, however, their model does not nest the CIR specification of the default intensity, which is by far the most widely used specification in the literature and in practice. Most closely related to our paper is the time-changed intensity model of Mendoza-Arriaga and Linetsky (2012).7 They obtain a spectral decomposition of the subordinate semigroups, and from this obtain a series solution to the survival probability function. As in our paper, the primary application in their paper is to the evolution of survival probabilities in a model with a CIR intensity in business time and a tempered stable subordinator. When that CIR process is stationary, their solution coincides with that of our second solution method. However, our method can be applied in the non-stationary case as well and generalizes easily when the CIR process is replaced by a basic affine process. Empirically, the default intensity process is indeed non-stationary under the risk-neutral measure for the typical firm (Jacobs and Li, 2008; Duffee, 1999).
Our two expansion methods are developed for a general function and wide classes of time-change processes in Sections 2 and 3. An application to credit risk modeling is presented in Section 4. The properties of the resulting model are explored with numerical examples in Section 5. In Section 6, we show that stochastic time-change has a very large effect on the pricing of deep out-of-the-money options on credit default swaps. In Section 7, we demonstrate that our expansion methods can be extended to a much wider class of multi-factor affine jump-diffusion business time models.
The method of this section imposes weak regularity conditions on , but places somewhat strong restrictions on . Throughout this section, we assume
We assume that is a smooth function of time. Imposing Assumption 1, we expand as a formal series and integrate:
Defining the constants
To obtain a generating function for the constants , we substitute for and then divide each side by .
To guarantee that the series expansion in equation (2.5) is convergent, we would require rather strong conditions. The function must be entire, the coefficients in equation (2.3) must decay faster than geometrically, and the coefficients must vanish at a geometric rate in . In application, it may be that none of these assumptions hold. If is analytic but non-entire, then , so geometric behavior in the would not be sufficient for convergence. Furthermore, we will provide a practically relevant specification below in which the are increasing in for fixed . Even if the series expansion in equation (2.5) is, in general, divergent, we will see that it may nonetheless be computationally effective.
We now provide an alternative justification for equation (2.5) to clarify the convergence behavior of our expansion. We introduce a regularity condition on :
Assumption 2 implies
Let be the remainder function from the generating equation (2.6), that is,
Since can be arbitrarily large, Proposition 1 provides a rigorous meaning for equation (2.5). However, it does not by itself explain why we should expect to provide a good approximation to . Equation (2.8) shows that the divergent sum (2.5) comes from the Laplace transform of the locally convergent sum in (2.6) (with replaced by ). It has been known for a long time that a divergent power series obtained by Laplace transforming a locally convergent sum is computationally very effective when truncated close to the numerically least term. In recent years, this classical method of "summation to the least term" has been justified rigorously in quite some generality for various classes of problems.10 The analysis of Costin and Kruskal (1999) is in the setting of differential equations, but their method of proof extends to much more general problems. Although the series in our analysis is not a usual power series, the procedure is conceptually similar and therefore expected to yield comparably good results.
For an interesting class of processes for , the sequence takes a convenient form. Let be the scaling parameter of the process, and let be the precision parameter. We introduce the assumption
This solution is especially convenient for two reasons. First, when the precision parameter is large, the expansion will yield accurate results in few terms. The variance is inversely proportional to , so converges in probability to as . This implies that for large . Since the expansion constructs as plus successive correction terms, it is well-structured for the case in which is not too volatile. The same remark applies to the more general case of Proposition 1, but the logic is more transparent when a single parameter controls the scaled higher cumulants. Second, in the special case of Assumption 3, the coefficients depend only on the chosen family of processes for and not on its parameters . In econometric applications, there can be millions of calls to the function , so the ability to pre-calculate the can result in significant efficiencies.
The three-parameter tempered stable subordinator is a flexible and widely-used family of subordinators. We can reparameterize the standard form of the Laplace exponent given by Cont and Tankov (2004, 4.2.2) in terms of our precision and scale parameters ( and ) and a stability parameter with . We obtain
Two well-known examples of the tempered stable subordinator are the gamma subordinator ( ) and the inverse Gaussian subordinator ( ). For the gamma subordinator, the constants simplify to , so
In the inverse Gaussian case ( ), the parameters are
The method of this section relaxes the assumption that is a Lévy process, but is more restrictive on . In the simplest case, we require that
Let denote the moment generating function for . We assume
Assumption 5 can accommodate non-Lévy specifications as well. Since volatility spikes are often clustered in time, it may be desirable to allow for serial dependence in the rate of time change. Following Carr and Wu (2004) and Mendoza-Arriaga et al. (2010), we let a positive process be the instantaneous activity rate of business time, so that
When is a Laplace transform of the time-integral of a nonnegative diffusion and when is a Lévy subordinator, Proposition 3 is equivalent to the eigenfunction expansion of Mendoza-Arriaga and Linetsky (2012).11 However, because our approach is agnostic with respect to the interpretation of , it can be applied in situations when spectral decomposition is unavailable, e.g., when the background process is a time-integral of a process containing jumps. All that is needed is that has a convergent Taylor series expansion as specified in Assumption 4. Moreover, our approach makes clear that need not be a Lévy subordinator.
As we will see in the next section, there are situations in which Assumption 4 does not hold, so neither Proposition 3 nor the corresponding eigenfunction expansion of Mendoza-Arriaga and Linetsky (2012) pertains. However, our method can be adapted so long as has a suitable expansion in powers of an affine function of . We will make use of this alternative in particular:
Although Assumption 4' is not a sufficient condition for Assumption 2, it is sufficient for purposes of approximating by the expansion in derivatives of Section 2. Let denote the approximation to given by the finite expansion
We now apply the two expansion methods to the widely-used default intensity class of models for pricing credit-sensitive corporate bonds and credit default swaps. In these models, a firm's default occurs at the first event of a non-explosive counting process. Under the business-time clock, the intensity of the counting process is . The intuition driving the model is that is the probability of default before business time , conditional on survival to business time . We define as the time-integral of , which is also known as the compensator.
Let denote the calendar default time, and let be the corresponding time under the business clock. Current time is normalized to zero under both clocks. The probability of survival to future business time is
When is a Lévy subordinator, the process is not differentiable, so the change of variable cannot be applied. We have so far fixed the current time to zero to minimize notation. To accommodate analysis of time-series behavior, let us define
We acknowledge that the assumption of independence between and may be strong. In the empirical literature on stochastic volatility in stock returns, there is strong evidence for dependence between the volatility factor and stock returns (e.g., Jones, 2003; Andersen et al., 2002; Jacquier et al., 2004). In the credit risk literature, however, the evidence is less compelling. Across the firms in their sample, Jacobs and Li (2008) find a median correlation of around 1% between the default intensity diffusion and the volatility factor. Nonetheless, for a significant share of the firms, the correlation appears to be material. We hope to relax the independence assumption in future work.
We re-introduce the basic affine process, which we earlier defined in Section 3. Under the business clock, follows the stochastic differential equation
We digress briefly to consider whether the method of Carr and Wu (2004) can be applied in this setting. The compensator is not Lévy, but can be expressed as a time-changed time-integral of a constant intensity, where the time change in this case is the time-integral of the basic affine process in (4.2). Thus, we can write as where is trivially a Lévy process and is a compound time change. However, this approach leads nowhere, because is equivalent to . Put another way, we are still left with the problem of solving the Laplace transform for .
When and in the absence of jumps ( ), the expansion in (ii) is equivalent to the eigenfunction expansion in Davydov and Linetsky (2003, 4.3).13Therefore, the associated solution to under a Lévy subordinator is identical to the solution in Mendoza-Arriaga and Linetsky (2012). Our result is more general in that it permits non-stationarity (i.e., ) in expansion (i) and accommodates the presence of jumps in the intensity process in expansions (i) and (ii). Furthermore, it is clear in our analysis that our expansions can be applied to non-Lévy specifications of time-change as well, such as the mean-reverting activity rate model in (3.2).
Subject to the technical caveat at the end of Section 3, Assumptions 1 and 4' are together sufficient for application of expansion in derivatives without restrictions on . To implement, we need an efficient algorithm to obtain derivatives of . Let be the family of functions defined by
We explore the effect of time-change on the behavior of the model, as well as the efficacy of our two series solutions. To fix a benchmark model, we assume that follows a CIR process in business time with parameters , and . This calibration is consistent in a stylized fashion with median parameter values under the physical measure as reported by Duffee (1999). Our benchmark specification adopts inverse Gaussian time change. In all the examples discussed below, the behavior under gamma time-change is quite similar.
The survival probability function is falling monotonically and almost linearly, so is not scaled well for our exercises. Instead, following the presentation in Duffie and Singleton (2003, 3), we work with the forward default rate, .14 In our benchmark calibration, we set starting condition well below its long-run mean in order to give reasonable variation across the term structure in the forward default rate. Both and are scale-invariant processes, so we fix the scale parameter with no loss of generality.
Figure 1 shows how the term structure of the forward default rate changes with the precision parameter . We see that lower values of flatten the term-structure, which accords with the intuition that the time-changed term-structure is a mixture across time of the business-time term-structure. Above , it becomes difficult to distinguish from the term structure for the CIR model without time-change., starting condition , and inverse Gaussian time-change with . When , the model is equivalent to the CIR model without time-change. Term-structures calculated with the series expansion in exponential functions of Proposition 3 with 12 terms.
Finding that time-change has negligible effect on the term structure for moderate values of does not imply that time-change has a small effect on the time-series behavior of the default intensity. For a given time-increment , we obtain by simulation the kurtosis of calendar time increments of the default intensity (that is, ) under the stationary law. For the limiting CIR model without time-change, moments for the increments have simple closed-form solutions provided by Gordy (2012). The kurtosis is equal to , which is invariant to the time-increment .
In Figure 2, we plot kurtosis as a function of on a log-log scale. Using the same baseline model specification as before, we plot separate curves for a one day horizon ( , assuming 250 trading days per year), a one month horizon ( ), and an annual horizon (). As we expect, kurtosis at all horizons tends to its asymptotic CIR limit (dotted line) as . For fixed , kurtosis also tends to its CIR limit as . This is because an unbiased trend stationary time-change has no effect on the distribution of a stationary process far into the future. For intermediate values of (say, between 1 and 10), we see that time-change has a modest impact on kurtosis beyond one year, but a material impact at a one month horizon, and a very large impact at a daily horizon., and inverse Gaussian time-change with . Dotted line plots the limiting CIR kurtosis. Both axes on log-scale. Moments of the calendar time increments are obtained by simulation with 5 million trials.
Next, we explore the convergence of the series expansion in exponentials. Let denote the estimated forward default rate using the first terms of the series for and the corresponding expansion for . Figure 3 shows that the convergence of to is quite rapid. We proxy the series solution with terms as the true forward default rate, and plot the error in basis points (bp). The error is decreasing in , as the series in Proposition 5 is an asymptotic expansion. With only terms, the error is 0.25bp at , which corresponds to a relative error under 0.25%. With terms, relative error is negligible (under 0.0005%) at ., starting condition , and inverse Gaussian time-change with and .
We turn now to the convergence of the expansion in derivatives. In Figure 4, we plot the error against the benchmark for terms in expansion (2.10). The benchmark curve is calculated, as before, using the series expansion in exponential functions with 12 terms. The magnitude of the relative error is generally largest at small values of . For , the forward default rate is off by nearly 0.5bp at . Observed bid-ask spreads in the credit default swap market are an order of magnitude larger, so this degree of accuracy is already likely to be sufficient for empirical application. For , the gap is never over 0.025bp at any ., starting condition , and inverse Gaussian time-change with .
In Figure 5, we hold fixed and explore how error varies with . As the expansion is in powers of , it is not surprising that error vanishes as grows, and is negligible (under 0.005bp in absolute magnitude) at . Experiments with other model parameters suggest that absolute relative error increases with and and decreases with ., starting condition , and inverse Gaussian time-change with . Number of terms in expansion is fixed to .
In the previous section, we observe that stochastic time-change has a negligible effect on the term structure of default probability for moderate values of , which implies that introducing time-change should have little impact on the term-structure of credit spreads on corporate bonds and credit default swaps (CDS). Nonetheless, time-change has a large effect on the forecast density of the default intensity at short horizon. Consequently, introducing time-change should have material impact on the pricing of short-dated options on credit instruments.15 In this section, we develop a pricing methodology for European options on single-name CDS in the time-changed CIR model.
At present, the CDS option market is dominated by index options. The market for single-name CDS options is less liquid, but trades do occur. A payer option gives the right to buy protection of maturity at a fixed spread (the "strike" or "pay premium") at a fixed expiry date . A payer option is in-the-money if the par CDS spread at date is greater than . A receiver option gives the right to sell protection. We focus here on the pricing of payer options, but all results extend in an obvious fashion to the pricing of receiver options. An important difference between the index and single-name option markets is that single-name options are sold with knock-out, i.e., the option expires worthless if the reference entity defaults before . As we will see, this complicates the analysis. Willemann and Bicer (2010) provide an overview of CDS option trading and its conventions.
To simplify the analysis and to keep the focus on default risk, we assume a constant risk free interest rate and a constant recovery rate . In the next section, we generalize our methods to accommodate a multi-factor model governing both the short rate and default intensity. The assumption of constant recovery can be relaxed by adopting the stochastic recovery model of Chen and Joslin (2012) in business time. We assume that follows a mean-reverting ( ) CIR process in business time, and that the clock is a Lévy process satisfying Assumption 1 with Laplace exponent . All probabilities and expectations are under the risk-neutral measure.
In the event of default at date , the receiver of CDS protection receives a single payment of at . Therefore, the value at date of the protection leg of a CDS of maturity is
To price the premium leg, we make the simplifying assumption that the spread is paid continuously until default or maturity. The value at date of the premium leg of a CDS of maturity is then
To simplify exposition, we assume that CDS are traded on a running spread basis.16 Let be the net value of the CDS for the buyer of protection at time given and the spread , i.e., is the difference in value between the protection leg and the premium leg. Note that does not depend directly on time . This simplifies to
The value at time 0 of the payer option is the expectation of expression (6.3) over the joint distribution of
under the risk-neutral measure:
This expectation is most easily obtained by Monte Carlo simulation. In each trial , we draw a single value of the business clock expiry date from the distribution of . Next, we draw from the noncentral chi-squared transition distribution for given and . The transition law for the CIR process is given by Broadie and Kaya (2006, Eq. 8). The option value is estimated by
Figure 6 depicts the effect of on the value of a one month payer option on a five year CDS. Model parameters are taken from the baseline values of Section 5. The riskfree rate is fixed at 3% and the recovery rate at 40%. Depending on the choice of , the par spread is in the range of 115-120bp (marked with circles). For deep out-of-the-money options, i.e., for , we see that option value is decreasing in . Stochastic time-change opens the possibility that the short horizon to option expiry will be greatly expanded in business time, and so increases the likelihood of extreme changes in the intensity. The effect is important even at large values of for which the term-structure of forward default rate would be visually indistinguishable from the CIR case in Figure 1. For example, at a strike spread of 200bp, the value of the option is nearly 700 times greater for the time-changed model with than for the CIR model without time-change.
Perhaps counterintuitively, the value of the option is increasing in for near-the-money options. Because the transition variance of is concave in , introducing stochastic time-change actually reduces the variance of the default intensity at option expiry even as it increases the higher moments. Relative to out-of-the-money options, near-the-money options are more sensitive to the variance and less sensitive to higher moments.) payer option on a 5 year CDS as function of strike spread. CIR model under business time with parameters , starting condition , and inverse Gaussian time-change with . Riskfree rate . Recovery . Circles mark par spread at date 0. When , the model is equivalent to the CIR model without time-change.
The effect of time to expiry on option value is depicted in Figure 7. The solid lines are for the model with stochastic time-change (), and the dashed lines are for the CIR model without time-change. Relative to the case of the short-dated (one month) option, stochastic time-change has a small effect on the value of the long-dated (one year) option. This is consistent with our observation in Figure 2 that the kurtosis of converges to that of as grows large. Because the Lévy subordinator lacks persistence, stochastic time-change simply washes out at long horizon., starting condition . Solid lines for model with inverse Gaussian time-change with and , and dashed line for the CIR model without time-change. Riskfree rate . Recovery . Circles mark par spread at date 0.
We have so far taken the business-time default intensity to be a single-factor basic affine process. In this section, we show that our methods of Sections 2 and 3 can be applied to a much wider class of multi-factor affine jump-diffusion models. For the sake of brevity, we limit our analysis here to stationary models.
Let be a -dimensional affine jump-diffusion, and let the default intensity at business time be given by an affine function . We now obtain a convergent series expansion of
By Proposition 1 of Duffie et al. (2000, 2), has exponential-affine solution , where denotes the inner product. Functions and satisfy complex-valued ODEs which we represent simply as
The following is a classical theorem due to Poincaré (see e.g., Ilyashenko and Yakovenko, 2008).
Let denote the common polydisk of analyticity of and .
Let be a ball in which is analytic and
We turn now to the second equation in the ODE system (7.1). Assume, without loss of generality, that is analytic in the same polydisk as , i.e., in , which implies that the expansion converges uniformly and absolutely inside . Imposing Assumption 7, we substitute (7.6) to obtain a uniformly and absolutely convergent expansion in , and integrate term-by-term to get
The absolute and uniform convergence of the expansions of and extends to the expansion of , which is the multi-factor extension of Proposition 5(ii). Thus, subject to Assumption 7 and , expansion in exponentials can be applied to the multi-factor model. Furthermore, the construction in (3.1) of a finite signed measure satisfying Assumption 2 extends naturally, so expansion in derivatives also applies.
The multi-factor extension can be applied to a joint affine model of the riskfree rate and default intensity. Let be the short rate in business time and let be the recovery rate as a fraction of market value at . We assume that is an affine function of the affine jump-diffusion , and solve for the business-time default-adjusted discount function . Subject to the regularity conditions in Assumption 7, we can thereby introduce stochastic time-change to the class of models studied by Duffie and Singleton (1999) and estimated by Duffee (1999). The possibility of handling stochastic interest rates in our framework is also recognized by Mendoza-Arriaga and Linetsky (2012, Remark 4.1).
We have derived and demonstrated two new methods for obtaining the Laplace transform of a stochastic process subjected to a stochastic time change. Each method provides a simple way to extend a wide variety of constant volatility models to allow for stochastic volatility. More generally, we can abstract from the background process, and view our methods simply as ways to calculate the expectation of a function of stochastic time. The two methods are complements in their domains of application. Expansion in derivatives imposes strictly weaker conditions on the function, whereas expansion in exponentials imposes strictly weaker conditions on the stochastic clock. We have found both methods to be straightforward to implement and computationally efficient.
Relative to the earlier literature, the primary advantage of our approach is that the background process need not be Lévy or even Markov. Thus, our methods are especially well-suited to application to default intensity models of credit risk. Both of our methods apply to the survival probability function under the ubiquitous basic affine specification of the default intensity. The forward default rate is easily calculated as well. Therefore, we can easily price both corporate bonds and credit default swaps in the time-changed model. In a separate paper, a time-changed default intensity model is estimated on panels of CDS spreads (across maturity and observation time) using Bayesian MCMC methods.
In contrast to the direct approach of modeling time-varying volatility as a second factor, stochastic time-change naturally preserves important properties of the background model. In particular, so long as the default intensity is bounded nonnegative in the background model, it will be bounded nonnegative in the time-changed model. In numerical examples in which the business-time default intensity is a CIR process, we find that introducing a moderate volatility in the stochastic clock has hardly any impact on the term-structure of credit spreads, yet a very large impact on the intertemporal variation of spreads. Consequently, the model preserves the cross-sectional behavior of the standard CIR model in pricing bonds and CDS at a fixed point in time, but allows for much greater flexibility in capturing kurtosis in the distribution of changes in spreads across time. The model also has a first-order effect on the pricing of deep out-of-the-money CDS options.
Bell polynomial identities arise frequently in our analysis, so we gather the important results together here for reference. In this appendix, and are scalar constants, and and are infinite sequences and . Unless otherwise noted, results are drawn from Comtet (1974, 3.3), in some cases with slight rearrangement.
We begin with the incomplete Bell polynomials, . The homogeneity rule is
For , the complete Bell polynomials, , are obtained from the incomplete Bell polynomials as
In this appendix, we set forth the closed-form solution to functions in the generalized transform
We follow the presentation in Duffie (2005, Appendix D.4), but with slightly modified notation. All functions and parameters associated with the generalized transform are written in Fraktur script. Let
We next define for
The discount function is
. To obtain these, let
be the value of
when and for , and similarly define
, etc. These simplify to
Using the same change of variable, the function has expansion
We combine these results to obtain
Since the composition of two analytic functions is analytic, a series expansion of in powers of is absolutely convergent for (equivalently, ). Thus, Proposition 5 holds with
Using the same change of variable, the function has expansion
We combine these results to obtain
Here we provide analytical expressions for . As in the previous appendix, the process is assumed to follow a basic affine process with parameters . Recall that
The functions and appear to be quite tedious (and the higher order and presumably even more so), as they depend on partial derivatives of , , and so on. Fortunately, these derivatives simplify dramatically when evaluated at . Define
Perhaps surprisingly, there are no further complications for and for . Proceeding along the same lines, we find
As in the previous appendix, the process is assumed to follow a basic affine process with parameters . Let us define
We interpret as the expected jump in conditional on a jump in at time . Let be the jump at time . Noting that is distributed exponential with parameter , we have