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...are the basis for people’s good faith on them. Until recently, market participants have been content to use the modern mathematical finance theory, pioneered by Black, Scholes and Merton in the 1970s =-=[11]-=-[85][106], to hedge and price derivatives. Experienced traders make estimates of their bid/ask prices starting from the bare-bones model prices. They also quantify their risk exposures in terms of mod...

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...ge as underlying processes, and treats securities issued by the firm such as bonds and stock as contingent claims on the firm’s assets. This stream of modeling begins with the seminal works of Merton =-=[86]-=- and Black and Cox [10], in which they assumed a geometric Brownian motion (GBM) for the firm asset process and modeled the firm default as the time the asset value hits a continuous or discrete barri...

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...dard normal under the null hypothesis that models i and j are equivalent in terms of likelihood function. Due to the serial correlation within the log-likelihood functions, Newey and West’s estimator =-=[90]-=- is used for ŝ. The Vuong test results are shown in 44 Table 2.2 and confirm that the Black-Cox model is consistently outperformed by the two TCBM models. Moreover, by this test, the EXP model shows ...

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...the basis for people’s good faith on them. Until recently, market participants have been content to use the modern mathematical finance theory, pioneered by Black, Scholes and Merton in the 1970s [11]=-=[85]-=-[106], to hedge and price derivatives. Experienced traders make estimates of their bid/ask prices starting from the bare-bones model prices. They also quantify their risk exposures in terms of model d...

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...rm current debt and long term non-current debt, equity into paid-in capital and retained earnings etc. [105]. A detailed balance sheet is very useful for static performance analysis of a firm. Altman =-=[1]-=- used seven quantities in a balance sheet to calculate the famous Z-score to determine firms’ survivability. Rating agencies rely heavily on balance sheets to estimate the default probability (PD), ex...

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...te these ideas by building mathematical models of capital structure based on the foundations laid down in the modern martingale approach to arbitrage pricing theory as developed by Harrison and Kreps =-=[48]-=-, Harrison and Pliska [49], and Delbaen and Schachermeyer [31]. Thus we propose to model the dynamics of the balance sheet parameters as stochastic processes within the filtered probability space (Ω,F...

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...icating that the transition density can be safely approximated by a gaussian density. The Kalman filter is convenient for calculating the weekly likelihood function, which is needed in the Vuong test =-=[103]-=-, a test to compare the relative performance of nested models. If X̄t and P̄t denote the ex-ante forecast and variance of time t values of the measurement series obtained from Kalman filtering, the we...

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...Wiener-Hopf factorization gives the Laplace transform of the joint characteristic function of a Lévy process and its running maximum, and has been used to price barrier options under Lévy processes =-=[29]-=-. Similarly, it can be used to calculate the first passage time density in the structural approach. Moreover, a first passage time of the second kind invented by Hurd [57] has a semi-closed form densi...

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...cture based on the foundations laid down in the modern martingale approach to arbitrage pricing theory as developed by Harrison and Kreps [48], Harrison and Pliska [49], and Delbaen and Schachermeyer =-=[31]-=-. Thus we propose to model the dynamics of the balance sheet parameters as stochastic processes within the filtered probability space (Ω,F ,Ft,Q), where Q is an equivalent martingale measure (EMM). Fr...

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...derived from market CDS rates. Nielsen et.al. [91], Briys et.al. [15], and Schobel [96] modeled stochastic interest rates that brought randomness to the default boundary. Leland [72], Leland and Toft =-=[73]-=-, and Mella-Barral and Perraudin [83] considered strategic default which is exercised by a firm’s management (i.e. the equity holders) to maximize its equity value. Vendor’s software, such as Moody’s ...

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...affine (SEA) model whose time change activity rate follows a mean reverting process with one sided pure Lévy jumps, termed background driving Lévy processes (BDLP) by Barndorff-Nielson and Shephard =-=[4]-=-. Thus we take the time change with an absolutely continuous part and a pure jump part: dGt = λtdt+mdJt, dλt = −bλtdt+ dJt where J is the nondecreasing pure jump Lévy diffusion with an exponential Le...

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...ecurities on the firm, and possibly its balance sheets as well. For formal descriptions of probability theory and stochastic processes, one can refer to standard probability textbooks such as [3] and =-=[65]-=-. [99] also provides mathematical finance interpretations of probability theory. 91.3 Literature Review 1.3.1 Understanding a Firm’s Capital Structure Balance Sheet Analysis The straightforward way to...

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... was able to work because of the special nature of the underlying process. This difficulty with predictable defaults was the original motivation for replacing structural models by reduced form models =-=[64]-=- and incomplete information models [38]. Recently, a class of “hybrid” reduced form models that include the stock price and a default hazard process have been developed. These model equity and debt pr...

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...general asset models outside the Gaussian world. Such models require slow algorithms such as Monte Carlo simulation, numerical integration, PDE/PIDE or Fourier transforms. In line with Carr and Madan =-=[23]-=-, Dempster and Hong [32] derived spread option prices as Fourier transform in the log strike variable. In particular, they derived tight lower and upper bounds for spread option prices and found the l...

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... mathematical models of capital structure based on the foundations laid down in the modern martingale approach to arbitrage pricing theory as developed by Harrison and Kreps [48], Harrison and Pliska =-=[49]-=-, and Delbaen and Schachermeyer [31]. Thus we propose to model the dynamics of the balance sheet parameters as stochastic processes within the filtered probability space (Ω,F ,Ft,Q), where Q is an equ...

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...d by one dimensional numerical integrations. 2. Introduce uncertainties into the current observation of a firm’s asset or debt so that the firm’s default event becomes unpredictable. Duffie and Lando =-=[38]-=- assumed that a firm’s asset is observed with accounting noises, only at certain points in time. Giesecke [46] assumed that a firm’s default threshold has some prior distribution known to investors, a...

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...arithmetic Brownian motion, and in this case (4.1) has a Black-Scholes type formula for any T,K. In the special case K = 0 when St is geometric Brownian motion, (4.1) is given by the Margrabe formula =-=[82]-=-. In the basic case where St is geometric Brownian motion and K > 0, no explicit pricing formula is known. Instead there is a long history of approximation methods for this problem. Numerical integrat...

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...traded securities on the firm, and possibly its balance sheets as well. For formal descriptions of probability theory and stochastic processes, one can refer to standard probability textbooks such as =-=[3]-=- and [65]. [99] also provides mathematical finance interpretations of probability theory. 91.3 Literature Review 1.3.1 Understanding a Firm’s Capital Structure Balance Sheet Analysis The straightforwa...

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...ressed, it still has some chance of revival by accessing liquidity or bank lines of credit (LOC) before default is triggered. The performance of several structural models has been tested by Eom et.al.=-=[42]-=-. While the Merton and Black-Cox models have been widely accepted due to their consistency with financial intuition and tractability in mathematical treatment, their shortcomings are also well known. ...

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...s such as traded options. When some investors gain access to a larger filtration, either from superior information channels or better modeling, they may be able to create a socalled dynamic arbitrage =-=[19]-=- to risklessly cash-out at the expense of others. In this thesis, we work with probability space. Our market observables include stock prices, implied volatility (IV) surfaces and CDS term structures ...

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... in the volatility (hence a rise in spreads) and an increased likelihood of defaults at that instant. 50 Beyond SEA model and TCBM one can consider the more sophisticated time changed Lévy processes =-=[25]-=- usually used in option pricing. While these models provide broader generality, their first passage time of second kind may not have explicit density function and their transition density function is ...

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...n et.al. [91], Briys et.al. [15], and Schobel [96] modeled stochastic interest rates that brought randomness to the default boundary. Leland [72], Leland and Toft [73], and Mella-Barral and Perraudin =-=[83]-=- considered strategic default which is exercised by a firm’s management (i.e. the equity holders) to maximize its equity value. Vendor’s software, such as Moody’s KMV (MKMV) [30], stemmed from the Mer...

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... Merton model has a time inconsistency problem when it comes to derive the PD term structure. For more detailed discussion of these models, one can refer to standard credit risk books for example [39]=-=[69]-=-[97]. Within the structural approach the majority of works that succeed in producing flexible term structures start from one of two fundamental ideas: 1. Introduce jumps into the firm asset, debt or l...

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...as a formula for the value of equity. Beyond the Merton and Black-Cox Models Along the lines of Merton and Black-Cox, there have been various extended versions in credit risk modeling. Hull and White =-=[55]-=- revised the Black-Cox model by using piecewise constant default barriers and were able to fit perfectly a given term structure of default rates derived from market CDS rates. Nielsen et.al. [91], Bri...

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...pace of probability densities. Others 17 [79] have been able to obtain higher accuracy for the spread distribution by using a Gram-Charlier density function as pioneered in finance by Jarrow and Rudd =-=[63]-=-. The Gram-Charlier density function approximates the terminal spread distribution in terms of a series expansion related to its higher order moments. This higher order expansion is able to capture mo...

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...ctor loadings. The popularity of the structural approach has reached various multi-name problems, such as economic capital (EC)[30][47], credit valuation adjustment (CVA)[55][14][12][75], CDO pricing =-=[56]-=-[54]. Due to the fact that a broad class of Lévy processes is subordinated Brownian motion [57][29], some far-reaching research has studied correlation for this class of Lévy processes. In spite of ...

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...ton model has a time inconsistency problem when it comes to derive the PD term structure. For more detailed discussion of these models, one can refer to standard credit risk books for example [39][69]=-=[97]-=-. Within the structural approach the majority of works that succeed in producing flexible term structures start from one of two fundamental ideas: 1. Introduce jumps into the firm asset, debt or lever...

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...edit spreads must have a zero short spread and a particular hump shape that is unrealistic. At least for investment grade credits there is evidence against humps and for increasing credit spreads [76]=-=[95]-=-[50]. Moreover, the Merton model has a time inconsistency problem when it comes to derive the PD term structure. For more detailed discussion of these models, one can refer to standard credit risk boo...

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...developed to improve tractability. Zhou [111] is the first to use Monte Carlo simulation in this kind of structural models to make the connection between jumps and non-zero short spread. Kou and Wang =-=[68]-=- used Kou’s earlier asset price model and took advantage of the “memoriless” feature of the exponential jumps to derive an explicit Laplace transform of the first passage time, leading to the formulas...

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...’s asset or debt so that the firm’s default event becomes unpredictable. Duffie and Lando [38] assumed that a firm’s asset is observed with accounting noises, only at certain points in time. Giesecke =-=[46]-=- assumed that a firm’s default threshold has some prior distribution known to investors, and which can be updated with new information from bond markets. In the vendor software CreditGrades [45], the ...

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...a firm. In Bensoussan et.al [7], the asset volatility of a firm is analytically derived from the equity volatility. The interpretation of the leverage effect in equity markets is also addressed. Duan =-=[37]-=- and Ericsson and Reneby [43] used maximum-likelihood estimation (MLE) to obtain a firm’s asset value from equity data. The best-known vendor software of this kind Moody’s KMV uses a proprietary struc...

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...f credit spreads must have a zero short spread and a particular hump shape that is unrealistic. At least for investment grade credits there is evidence against humps and for increasing credit spreads =-=[76]-=-[95][50]. Moreover, the Merton model has a time inconsistency problem when it comes to derive the PD term structure. For more detailed discussion of these models, one can refer to standard credit risk...

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...trike variable. In particular, they derived tight lower and upper bounds for spread option prices and found the lower bounds to be more accurate than their Monte Carlo benchmark. In the line of Lewis =-=[74]-=-, Leentvaar and Oosterlee [71] derived general basket option prices as Fourier transforms in the log asset variables. A general framework using Lewis type Fourier transforms in log asset variables is ...

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...hat we have used the fact that e−iNηx(`)/2 = (−1)n for all ` ∈ Z. The selection of suitable values for N and η in the above FFT approximation of (2.8) is determined via general error bounds proved in =-=[70]-=-. In rough terms, the pure truncation error, defined by taking η → 0, N → ∞ keeping ū = Nη/2 fixed, can be made small if the integrand of (2.4) is small and decaying outside the square [−ū, ū]. Sim...

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...hite [55] revised the Black-Cox model by using piecewise constant default barriers and were able to fit perfectly a given term structure of default rates derived from market CDS rates. Nielsen et.al. =-=[91]-=-, Briys et.al. [15], and Schobel [96] modeled stochastic interest rates that brought randomness to the default boundary. Leland [72], Leland and Toft [73], and Mella-Barral and Perraudin [83] consider...

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...he Black-Cox model by using piecewise constant default barriers and were able to fit perfectly a given term structure of default rates derived from market CDS rates. Nielsen et.al. [91], Briys et.al. =-=[15]-=-, and Schobel [96] modeled stochastic interest rates that brought randomness to the default boundary. Leland [72], Leland and Toft [73], and Mella-Barral and Perraudin [83] considered strategic defaul...

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... dynamics of credit spreads. TCBMs have been used in other credit risk models, for example [87], [44], [5] and [84]. One model we study is an adaptation of the variance gamma (VG) model introduced by =-=[81]-=- in the study of equity derivatives, and remaining very popular since then. We will see that this infinite activity pure jump exponential Lévy model adapts easily to the structural credit context, an...

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...imensional integral of conditional Black-Scholes call option prices with respect to a Gaussian Kernel, and can be accurately computed using numerical recipes such as quadrature rules. As discussed in =-=[17]-=-, numerical algorithms have weaknesses compared to analytical algorithms. Notably, the former does not evaluate prices and the Greeks as rapidly as the latter, making it less favored in trades that re...

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...equity market and credit market are intrinsically connected. The empirical observations supporting this connection and thereafter financial modeling interpreting this connection can be found in [84], =-=[26]-=- and their references. Finally, we mention that a stable model estimation over a 78 week period typically involved about 120 evaluations of the function ρ(Y,Θ), and took around one minute on a standar...

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...ce-to-default (DtD) that serves as a normalized measure of the margin between the asset and debt. Other authors have incorporated more financial intuition into the definition of default. Cetin et.al. =-=[27]-=- defined default as the time when a firm’s cash balance hits a secondary barrier after staying below the primary barrier of zero (financial stress) for an extended period of time. Yildirim [108] defin...

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...ncial innovation. Market makers who provide financial market liquidity have tailored existing derivatives or invented entirely new ones to meet customers’ financial needs. For example, variance swaps =-=[21]-=- provide investors with opportunities to hedge and speculate on market volatilities. Credit Default Swap (CDS) contracts provide investors with default protection on sovereign, municipal and corporate...

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... loadings. The popularity of the structural approach has reached various multi-name problems, such as economic capital (EC)[30][47], credit valuation adjustment (CVA)[55][14][12][75], CDO pricing [56]=-=[54]-=-. Due to the fact that a broad class of Lévy processes is subordinated Brownian motion [57][29], some far-reaching research has studied correlation for this class of Lévy processes. In spite of the ...

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...propagated into US capital markets and eventually eroded the global economy. Among many other causes of the crisis is OTC derivatives’ role in transferring risks, that can lead to systemic risks (See =-=[28]-=- for a good illustration of so-called contagion and systemic risks). A panacea might be to go back to the old, simple days when there was no room for complex, toxic securities. However, this view miss...

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...nd Wu [26] take a stochastic volatility model for the stock price and assume that the default arrival rate is driven by the volatility and another independent credit risk factor. In Carr and Linetsky =-=[22]-=-, the stock price has a local volatility with constant elasticity of variance, and the default intensity is specified as an affine function of the instantaneous variance of the stock. [84] obtain even...

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...secke [46] assumed that a firm’s default threshold has some prior distribution known to investors, and which can be updated with new information from bond markets. In the vendor software CreditGrades =-=[45]-=-, the default threshold is a log-normally distributed random variable drawn at time zero. There are several notable merits of the structural approach. First of all, it has a clear economic interpretat...

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...t short spreads are observed to be positive even for investment grade firms. The natural way to overcome this deficiency is to introduce jumps into the asset process. A number of authors, notably [5],=-=[16]-=-, have successfully implemented jump diffusion and pure jump versions of the Merton model. However they share that model’s unrealistically simple debt structure. Similar extensions to the Black-Cox fi...

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... truncated domains, but unlike earlier work using FFT, they apparently do not rely on knowing the analytic Fourier transform of the payoff function or integrability of the payoff function. Lord et.al =-=[78]-=- provide error analysis that explains their observation that errors decay as a negative power of the size N of the grid used in computing the FFT, provided the truncation is taken large enough. Ooster...

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...ctions of financial models diverge dramatically from empirical observations, and consequently much criticism has been cast on quantitative modeling in general, and sometimes even a particular formula =-=[94]-=-[102]. The basis of this negative voice is that quantitative analysts (“quants”) failed to build “right” models to capture dangerous risks that ultimately pushed the financial system to the brink. To ...

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...duced number of factor loadings. The popularity of the structural approach has reached various multi-name problems, such as economic capital (EC)[30][47], credit valuation adjustment (CVA)[55][14][12]=-=[75]-=-, CDO pricing [56][54]. Due to the fact that a broad class of Lévy processes is subordinated Brownian motion [57][29], some far-reaching research has studied correlation for this class of Lévy proce...

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...bination of one asset and the strike. The motivation for this stems from avoiding modeling individual assets and correlations which can be volatile in some markets. In the Bachelier appoximation [104]=-=[98]-=-[92], the spread of the two assets follows an arithmetic Brownian motion, leading to an analytic Bachelier formula for pricing spread options similar to pricing plain-vanilla options. Ultimately, this...

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...the total current assets plus half of the total long-term assets as the asset, and the current liabilities as the debt. . . . . . . . . . . . . 79 4.1 Benchmark prices for the two-factor GBM model of =-=[32]-=- and relative errors for the FFT method with different choices of N . The parameter values are the same as Figure 4.1 except we fix S10 = 100, S20 = 96, ū = 40. The interpolation is based on a matrix...

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...qual share of drafting and finalizing of the paper. The third paper “A Fourier Transform Method for Spread Option Pricing” in chapter 4 has been published in the SIAM Journal of Financial Mathematics =-=[59]-=-1. This paper introduces a new formula for general spread option pricing based on Fourier analysis of the payoff function. This method is found to be easy to implement, stable, efficient and applicabl...

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...nd show that with careful parameter estimation, TCBM models can do a very good job of explaining the observed dynamics of credit spreads. TCBMs have been used in other credit risk models, for example =-=[87]-=-, [44], [5] and [84]. One model we study is an adaptation of the variance gamma (VG) model introduced by [81] in the study of equity derivatives, and remaining very popular since then. We will see tha...

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...nd Oosterlee [71] derived general basket option prices as Fourier transforms in the log asset variables. A general framework using Lewis type Fourier transforms in log asset variables is presented in =-=[62]-=-, which they demonstrated in the pricing of different financial derivatives. The Fourier transform method is well suited for a wide range of asset models, as long as the characteristic functions of th...

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...l determines that fixing the log leverage ratio alone does not fully capture the CDS term structure, rather, a second source of randomness, the activity rate can still add degree of freedom to it. In =-=[20]-=- some inhomogenous Lévy processes (called Sato process herein) have been shown to successfully calibrate options. So the SEA model is expected to attain even better performance. Second of all, VG and...

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...a for pricing spread options similar to pricing plain-vanilla options. Ultimately, this becomes a moment matching technique, and is commonly used for pricing more general basket options. Brigo et.al. =-=[13]-=- has tested the accuracy of such distributional approximations using notions of distance on the space of probability densities. Others 17 [79] have been able to obtain higher accuracy for the spread d...

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...reful parameter estimation, TCBM models can do a very good job of explaining the observed dynamics of credit spreads. TCBMs have been used in other credit risk models, for example [87], [44], [5] and =-=[84]-=-. One model we study is an adaptation of the variance gamma (VG) model introduced by [81] in the study of equity derivatives, and remaining very popular since then. We will see that this infinite acti...

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... with careful parameter estimation, TCBM models can do a very good job of explaining the observed dynamics of credit spreads. TCBMs have been used in other credit risk models, for example [87], [44], =-=[5]-=- and [84]. One model we study is an adaptation of the variance gamma (VG) model introduced by [81] in the study of equity derivatives, and remaining very popular since then. We will see that this infi...

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... by a reduced number of factor loadings. The popularity of the structural approach has reached various multi-name problems, such as economic capital (EC)[30][47], credit valuation adjustment (CVA)[55]=-=[14]-=-[12][75], CDO pricing [56][54]. Due to the fact that a broad class of Lévy processes is subordinated Brownian motion [57][29], some far-reaching research has studied correlation for this class of Lé...

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...elated to its higher order moments. This higher order expansion is able to capture more features of the true distribution, especially in its tails. With a different treatment but a similar idea, Kirk =-=[66]-=- combined the second asset with the fixed strike and assumed a log normal distribution for the sum of the two. He then used the Margrabe formula to price spread options. In [35] 5, it was pointed out ...

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...t a similar idea, Kirk [66] combined the second asset with the fixed strike and assumed a log normal distribution for the sum of the two. He then used the Margrabe formula to price spread options. In =-=[35]-=- 5, it was pointed out that the Kirk method is equivalent to a linearization of the nonlinear exercise boundary and they implemented a second-order boundary approximation and demonstrated an improved ...

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... spreads must have a zero short spread and a particular hump shape that is unrealistic. At least for investment grade credits there is evidence against humps and for increasing credit spreads [76][95]=-=[50]-=-. Moreover, the Merton model has a time inconsistency problem when it comes to derive the PD term structure. For more detailed discussion of these models, one can refer to standard credit risk books f...

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...e, a trader could lose everything in option trading with only a modest movement of the underlying stock. From a global perspective, the 2007-2010 financial crisis (for a description of the crisis see =-=[51]-=- and its references therein) would have been contained within the housing market where it originated if complex derivatives such as MBS, MBS CDO had not become so widespread (See [51][52]). The subpri...

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...). People have been able to implement reverse engineering of structural models using readily available equity and bond data to estimate unobserved asset and debt values of a firm. In Bensoussan et.al =-=[7]-=-, the asset volatility of a firm is analytically derived from the equity volatility. The interpretation of the leverage effect in equity markets is also addressed. Duan [37] and Ericsson and Reneby [4...

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...tion of one asset and the strike. The motivation for this stems from avoiding modeling individual assets and correlations which can be volatile in some markets. In the Bachelier appoximation [104][98]=-=[92]-=-, the spread of the two assets follows an arithmetic Brownian motion, leading to an analytic Bachelier formula for pricing spread options similar to pricing plain-vanilla options. Ultimately, this bec...

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...ell the model. 15 pool of public firms in its database. Its “GCorr” (Global Correlation) model provides asset correlation matrices calculated from these time series. Very recently, Eberlein and Madan =-=[41]-=- went further to use both traded equity options and balance sheet statements to evaluate a firm’s impact on the economy. They argued that limited liability applies to all of a firm’s stakeholders not ...

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...ates how common statistical inference, in particular maximum likelihood estimation (MLE) and Kalman filter (KF) can be efficiently implemented for a new class of credit risk models introduced by Hurd =-=[57]-=-. For this class of models, traditional estimation methods face challenges such as computational cost due to a non-explicit probability density function (PDF). My coauthor Professor Hurd and I have ma...

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...its capital structure. 1.1 How Good is Mathematical Modeling for Financial Derivatives? In financial markets, a derivative is a security whose value depends on other, more basic, underlying variables =-=[53]-=-. The global financial derivative markets have seen staggering growth during the last two decades. According to the Bank for International Settlements (BIS) in March 2011 [9] the notional outstanding ...

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... they derived tight lower and upper bounds for spread option prices and found the lower bounds to be more accurate than their Monte Carlo benchmark. In the line of Lewis [74], Leentvaar and Oosterlee =-=[71]-=- derived general basket option prices as Fourier transforms in the log asset variables. A general framework using Lewis type Fourier transforms in log asset variables is presented in [62], which they ...

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...ses, and treats securities issued by the firm such as bonds and stock as contingent claims on the firm’s assets. This stream of modeling begins with the seminal works of Merton [86] and Black and Cox =-=[10]-=-, in which they assumed a geometric Brownian motion (GBM) for the firm asset process and modeled the firm default as the time the asset value hits a continuous or discrete barrier equal to the debt va...

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...is equivalent to a linearization of the nonlinear exercise boundary and they implemented a second-order boundary approximation and demonstrated an improved accuracy. In a parallel paper, Deng et. al. =-=[34]-=- extended the two-asset setting to multi-asset setting. This was achieved by approximating the arithmetic average of asset prices by the corresponding geometric average, and again the moment matching ...

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...7], the asset volatility of a firm is analytically derived from the equity volatility. The interpretation of the leverage effect in equity markets is also addressed. Duan [37] and Ericsson and Reneby =-=[43]-=- used maximum-likelihood estimation (MLE) to obtain a firm’s asset value from equity data. The best-known vendor software of this kind Moody’s KMV uses a proprietary structural approach 4 to estimate ...

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...e crisis see [51] and its references therein) would have been contained within the housing market where it originated if complex derivatives such as MBS, MBS CDO had not become so widespread (See [51]=-=[52]-=-). The subprime mortgage market only amounted to a small part of the US economy or even of the prime mortgage market. However, its impact swiftly propagated into US capital markets and eventually erod...

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... that one log leverage ratio value has an invariant CDS term structure. This characteristic for general Lévy processes also make them inadequate to price options across maturities as well as strikes =-=[67]-=-. The two-factor nature of the SEA model determines that fixing the log leverage ratio alone does not fully capture the CDS term structure, rather, a second source of randomness, the activity rate can...

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... by using piecewise constant default barriers and were able to fit perfectly a given term structure of default rates derived from market CDS rates. Nielsen et.al. [91], Briys et.al. [15], and Schobel =-=[96]-=- modeled stochastic interest rates that brought randomness to the default boundary. Leland [72], Leland and Toft [73], and Mella-Barral and Perraudin [83] considered strategic default which is exercis...

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... are the main focus of the present paper. Only a few authors have been able to make substantial headway in modeling actual observed capital structures by two factor models. Benos and Papanastasopouls =-=[6]-=- have extended the Merton model by modeling the asset and default barrier of a firm as independent geometric Brownian motion. They found that their model systematically outperformed the Merton model i...

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...:= −logE[e−uGt ] = t [ bu+ acu 1 + au ] (3.26) 68 and by choosing a = 1−b c the average speed of the time change is normalized to 1; 2. The second type of time change is a variance gamma (VG) process =-=[80]-=-, that is, G is a gamma process with drift having characteristics (b, 0, ν) where b ∈ (0, 1) and ν(z) = ce−z/a/z, a > 0 on (0,∞), the Lévy measure, has support on R+. The Laplace exponent of Gt is ψV...

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...e combination of one asset and the strike. The motivation for this stems from avoiding modeling individual assets and correlations which can be volatile in some markets. In the Bachelier appoximation =-=[104]-=-[98][92], the spread of the two assets follows an arithmetic Brownian motion, leading to an analytic Bachelier formula for pricing spread options similar to pricing plain-vanilla options. Ultimately, ...

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...{t|x+ σWt + βσ2t ≤ 0}. The corresponding stopped TCBM is X (2) t = x+ σWGt∧t∗ + βσ 2(Gt ∧ t∗) (2.3) and we note that X (2) t(2) = 0. The general relation between t(1) and t(2) is studied in detail in =-=[58]-=- where it is shown how the probability distribution of t(2) can approximate that of t(1). For the remainder of this paper, however, we consider t(2) to be the definition of the time of default. The fo...

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...-Barral and Perraudin [83] considered strategic default which is exercised by a firm’s management (i.e. the equity holders) to maximize its equity value. Vendor’s software, such as Moody’s KMV (MKMV) =-=[30]-=-, stemmed from the Merton model. Rather than 13 looking at asset and debt individually, Moody’s KMV models its proprietary accounting ratio called distance-to-default (DtD) that serves as a normalized...

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...ssue of computing the gamma function is not a real difficulty. Fast and accurate computation of the complex gamma function in for example, Matlab, is based on the Lanczos approximation popularized by =-=[93]-=-4. 4According to these authors, computing the gamma function becomes “not much more difficult than other built-in functions that we take for granted, such as sinx or ex”. 94 In this paper, we demonstr...

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...pute 2It came to our attention after the submission of our paper that the result of this Theorem has been simultaneously and independently stated in another working paper by A. Antonov and M. Arneguy =-=[2]-=-. 3Here and in rest of the paper, some variables such as u, , x are defined to be row vectors with components u = (u1, u2) etc. We use implied matrix multiplication so that ux′ = u1x1 +u2x2 where x′ ...

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...w that with careful parameter estimation, TCBM models can do a very good job of explaining the observed dynamics of credit spreads. TCBMs have been used in other credit risk models, for example [87], =-=[44]-=-, [5] and [84]. One model we study is an adaptation of the variance gamma (VG) model introduced by [81] in the study of equity derivatives, and remaining very popular since then. We will see that this...

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...els in which correlations are parameterized by a reduced number of factor loadings. The popularity of the structural approach has reached various multi-name problems, such as economic capital (EC)[30]=-=[47]-=-, credit valuation adjustment (CVA)[55][14][12][75], CDO pricing [56][54]. Due to the fact that a broad class of Lévy processes is subordinated Brownian motion [57][29], some far-reaching research ha...

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...els to capture dangerous risks that ultimately pushed the financial system to the brink. To provide an opposite view, several renowned researchers wrote on this subject in their columns or papers [36]=-=[77]-=-[100][107]. These authors contrasted financial models with physics models and illustrated why financial models have yet to achieve the level of accomplishment of physics models. In Lo and Mueller’s Ta...

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...f the exponential jumps to derive an explicit Laplace transform of the first passage time, leading to the formulas for the first passage time density and 14 credit spreads involving Laplace inversion =-=[8]-=-. More generally, the Wiener-Hopf factorization gives the Laplace transform of the joint characteristic function of a Lévy process and its running maximum, and has been used to price barrier options ...

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...ic, underlying variables [53]. The global financial derivative markets have seen staggering growth during the last two decades. According to the Bank for International Settlements (BIS) in March 2011 =-=[9]-=- the notional outstanding amounts of Over-the-Counter (OTC) derivatives reached $583 trillion with gross market values of $25 trillion in June 2010. The notional outstanding amounts of exchange traded...

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...a reduced number of factor loadings. The popularity of the structural approach has reached various multi-name problems, such as economic capital (EC)[30][47], credit valuation adjustment (CVA)[55][14]=-=[12]-=-[75], CDO pricing [56][54]. Due to the fact that a broad class of Lévy processes is subordinated Brownian motion [57][29], some far-reaching research has studied correlation for this class of Lévy p...

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... models to capture dangerous risks that ultimately pushed the financial system to the brink. To provide an opposite view, several renowned researchers wrote on this subject in their columns or papers =-=[36]-=-[77][100][107]. These authors contrasted financial models with physics models and illustrated why financial models have yet to achieve the level of accomplishment of physics models. In Lo and Mueller’...

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...analysis; 3. An equal share of drafting and finalizing of the paper. The second paper “Two-Factor Capital Structure Models for Equity and Credit” in chapter 3 is ready to be submitted for peer review =-=[61]-=-. This paper extends the classical Black-Cox (BC) model in credit risk to a model of the capital structure of a firm that incorporates both credit risk and equity risk. In contrast to other extensions...

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...sed for pricing more general basket options. Brigo et.al. [13] has tested the accuracy of such distributional approximations using notions of distance on the space of probability densities. Others 17 =-=[79]-=- have been able to obtain higher accuracy for the spread distribution by using a Gram-Charlier density function as pioneered in finance by Jarrow and Rudd [63]. The Gram-Charlier density function appr...

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...ies on the firm, and possibly its balance sheets as well. For formal descriptions of probability theory and stochastic processes, one can refer to standard probability textbooks such as [3] and [65]. =-=[99]-=- also provides mathematical finance interpretations of probability theory. 91.3 Literature Review 1.3.1 Understanding a Firm’s Capital Structure Balance Sheet Analysis The straightforward way to under...

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...to capture dangerous risks that ultimately pushed the financial system to the brink. To provide an opposite view, several renowned researchers wrote on this subject in their columns or papers [36][77]=-=[100]-=-[107]. These authors contrasted financial models with physics models and illustrated why financial models have yet to achieve the level of accomplishment of physics models. In Lo and Mueller’s Taxonom...

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...ns of financial models diverge dramatically from empirical observations, and consequently much criticism has been cast on quantitative modeling in general, and sometimes even a particular formula [94]=-=[102]-=-. The basis of this negative voice is that quantitative analysts (“quants”) failed to build “right” models to capture dangerous risks that ultimately pushed the financial system to the brink. To provi...