Research article

Free boundary problem pricing defaultable corporate bonds with multiple credit rating migration risk and stochastic interest rate

  • Received: 25 August 2020 Accepted: 20 October 2020 Published: 22 October 2020
  • MSC : 91G40, 91G30, 35R35

  • In this paper, valuation for a defaultable corporate bond subject to multiple credit rating migration risk and stochastic volatility of interest rate is addressed in the structure framework through a free boundary problem, which is derived by a series of transformations. The existence, uniqueness and regularity of solution to the free boundary problem are obtained to verify the rationality of the bond pricing model. Furthermore, we show that the solution of the free boundary problem is convergent to a close form steady status, which may provide some information on the developing characteristics of the bond price. As the coexistence of stochastic interest rate and defaulting boundary, this convergence is achieved through an auxiliary free boundary problem and a Lyapunov argument. Interestingly, the converged steady status can be explicitly solved, which is not the case in the existing literatures on multiple credit rating migration. Finally, we present an explicit formula for valuating this defaultable bond with multiple credit rating migration risk and stochastic interest rate.

    Citation: Zhehao Huang, Yingting Miao, Zhenzhen Wang. Free boundary problem pricing defaultable corporate bonds with multiple credit rating migration risk and stochastic interest rate[J]. AIMS Mathematics, 2020, 5(6): 7746-7775. doi: 10.3934/math.2020495

    Related Papers:

    [1] Shuwei Zhu, Hao Liu, Meiji Cui . Efficient multi-omics clustering with bipartite graph subspace learning for cancer subtype prediction. Electronic Research Archive, 2024, 32(11): 6008-6031. doi: 10.3934/era.2024279
    [2] Peng Ren, Qunli Xia . Classification method for imbalanced LiDAR point cloud based on stack autoencoder. Electronic Research Archive, 2023, 31(6): 3453-3470. doi: 10.3934/era.2023175
    [3] Qianpeng Xiao, Changbin Shao, Sen Xu, Xibei Yang, Hualong Yu . CCkEL: Compensation-based correlated k-labelsets for classifying imbalanced multi-label data. Electronic Research Archive, 2024, 32(5): 3038-3058. doi: 10.3934/era.2024139
    [4] Yixin Sun, Lei Wu, Peng Chen, Feng Zhang, Lifeng Xu . Using deep learning in pathology image analysis: A novel active learning strategy based on latent representation. Electronic Research Archive, 2023, 31(9): 5340-5361. doi: 10.3934/era.2023271
    [5] Yutong Xie, Yingli Gao, Meijie Liao, Weiwei Tian . Study on the storage stability performance enhancement mechanism of graphene on rubber-modified asphalt based on size effect. Electronic Research Archive, 2023, 31(4): 2048-2070. doi: 10.3934/era.2023105
    [6] Yongmei Zhang, Zhirong Du, Lei Hu . A construction method of urban road risky vehicles based on dynamic knowledge graph. Electronic Research Archive, 2023, 31(7): 3776-3790. doi: 10.3934/era.2023192
    [7] Gongcai Wu . Intelligent recommendation algorithm for social networks based on improving a generalized regression neural network. Electronic Research Archive, 2024, 32(7): 4378-4397. doi: 10.3934/era.2024197
    [8] Wenrui Guan, Xun Wang . A Dual-channel Progressive Graph Convolutional Network via subgraph sampling. Electronic Research Archive, 2024, 32(7): 4398-4415. doi: 10.3934/era.2024198
    [9] Abdullah S. AL-Malaise AL-Ghamdi, Mahmoud Ragab . Tunicate swarm algorithm with deep convolutional neural network-driven colorectal cancer classification from histopathological imaging data. Electronic Research Archive, 2023, 31(5): 2793-2812. doi: 10.3934/era.2023141
    [10] Xinzheng Xu, Xiaoyang Zhao, Meng Wei, Zhongnian Li . A comprehensive review of graph convolutional networks: approaches and applications. Electronic Research Archive, 2023, 31(7): 4185-4215. doi: 10.3934/era.2023213
  • In this paper, valuation for a defaultable corporate bond subject to multiple credit rating migration risk and stochastic volatility of interest rate is addressed in the structure framework through a free boundary problem, which is derived by a series of transformations. The existence, uniqueness and regularity of solution to the free boundary problem are obtained to verify the rationality of the bond pricing model. Furthermore, we show that the solution of the free boundary problem is convergent to a close form steady status, which may provide some information on the developing characteristics of the bond price. As the coexistence of stochastic interest rate and defaulting boundary, this convergence is achieved through an auxiliary free boundary problem and a Lyapunov argument. Interestingly, the converged steady status can be explicitly solved, which is not the case in the existing literatures on multiple credit rating migration. Finally, we present an explicit formula for valuating this defaultable bond with multiple credit rating migration risk and stochastic interest rate.


    Nowadays, the miniaturization of bioelectronic devices in order to improve their performance, the design of more efficient compounds to exert their function accurately, the fabrication of durable and green-friendly packaging items minimizing the content of scarce based-fossil fuel materials or the precise knowledge of the entire mechanisms which govern biomolecules and cellular systems require the progress of high-throughput techniques. From its discovery [1], atomic force microscopy (AFM) has emerged as promising technique to interrogate multitude of properties from samples with different nature. AFM presents some advantages in comparison to other nanoscale microscopies. AFM does not request the addition of external contrast agents to measure carbon-sourced biological samples like scanning electron microscopy (SEM) or conduct the experiments at ultra-low temperatures as cryo-transmission electron microscopy (cryo-TEM). Furthermore, AFM is capable to devote the measurements at liquid media mimicking the near-physiological conditions inner the living cell. AFM consists of a flexible cantilever where a laser beam is reflected towards a photodetector sensor device. The flexible cantilever ends with a sharp tip which interacts with the external sample surface. The close-loop feedback maintain the setpoint parameter under controlled by modifying the proportional-integral-derivative (PID) settings [2]. Finally, the sample is mounted on a piezoelectric tube made by ceramic materials which are deformed under applied voltages. This fact enables to deliver excellent vertical resolution. AFM treasures multitude of operational modes specialized to acquire one specific property of the sample of interest.

    AFM imaging consists to raster the sample with an AFM tip in order to collect topology maps. Today it exists many operational modes to gather the topography information of soft matter systems. Contact mode (CM-AFM) sets the cantilever bending as setpoint and the user can define the grade of interaction between the AFM tip and the scanned sample. Tip-sample interaction is settled by the constant applied force between both surfaces. Alternatively, tapping mode (TM-AFM) controls the oscillation amplitude, the frequency shift or the phase variance as feedback. Here, the tip-sample interaction is based on the AFM cantilever oscillation damping. TM-AFM is considered a semi-intermittent mode that minimizes the non-desirable frictional forces when the AFM probe oscillates with the sufficient amplitude. The main drawbacks experienced for CM-AFM and TM-AFM are the possibility to drag the imaged features during data acquisition in combination with potential sample damage due to the high-lateral forces exerted by the AFM tip and the lack of control of absolute vertical forces that could drive the soft samples compression, respectively. Suppliers have developed other operational modes in order to overcome the aforementioned limitations like peak-force tapping (Bruker) or QI mode (JPK-Bruker). The first mentioned technology monitors the applied AFM tip load forces on the sample surface. This fact enables to preserve the integrity of soft matter systems recording high-quality topography information. Peak force tapping mode has pave the way to evolve peak-force quantitative nanomechanics (PF-QNM) to address the nanomechanical properties of soft samples (See section titled nanoindentation). On the other hand, QI-mode makes negligible the sample invasiveness by efficient algorithms that allows automated tip-sample interaction rendering thus, nondestructive imaging straightforward. All the above described modes display low temporal resolutions (from tens of seconds to minutes scale). High-speed AFM (HS-AFM) through the design of ultra-short cantilevers [3] achieved to work at high-resonance frequencies that leads microseconds acquisition times [4]. Independently of the selected operational mode, imaging always is advisable to choose AFM cantilevers with low spring constants (range of 0.01–0.1 N/m) to gather topology information of soft matter systems.

    Table 1 recaps the capabilities of all mentioned AFM imaging modes. AFM imaging succeed to study the biomolecular morphological changes upon ligand catalysis [5][8] which is relevant to know those conditions where the biomolecule of interest renders the best catalytic performance. Next generation of bioreactors strongly depends on the insights achieved on this regard [9]. In addition, AFM imaging is capable to discern the number of units involved in hepatotoxic strand formation under the presence of certain metal ions [10] or the effect of coniferyl alcohol polymerization by Fenton reaction on bioinspired lignin films [11]. This latest study is relevant to create novel antibacterial films based on plant sourced polymers [12]. The record of biomolecular surface charges [13] or conformational dynamics of ATP synthases [14] and rhodopsin dimers from the G-protein-coupled photoreceptor family [15] under the presence of protons and light, respectively are additional relevant cases that showcase AFM as core technique in this field. The final example of AFM imaging studies is the assessment of the interplay between proteins and DNA chains [16], [17] which can be substantial to devise biomolecular scaffolds [18]. Hence, AFM is a powerful tool to elicit the morphological properties of soft matter systems [19].

    Table 1.  Comparison between available AFM imaging modes according to the feedback and the main advantages and weaknesses depicted.
    Operational mode Setpoint Main advantage Main weakness
    CM-AFM Cantilever bending High stability High lateral forces
    TM-AFM Resonance amplitude Frequency shift Phase difference Low lateral forces No control vertical force
    PF-TM Applied force (>50 pN) Constant feedback Low temp. resolution (~ms)
    QI-mode Deflection of cantilever down to the height of zero force Constant feedback Low temp. resolution (~ms)
    HS-AFM Resonance amplitude High temp. resolution (~µs) Low scan range

     | Show Table
    DownLoad: CSV

    Then, AFM based force spectroscopy (AFM-FS) determines the intra- or intermolecular interactions between biomolecules [20],[21]. The classical example of multidomain protein unfolding is the case of titin [22]. The role of redox conditions can lead to prevent and better diagnose cardiac diseases [23]. On the other hand, big efforts have been made to decipher the intermolecular interactions of DNA strands [24], protein-protein [25], cell-cell [26], small molecules like rotaxanes [27], and lignocellulosic polymers with cellulose nanocrystals functionalized AFM levers [28]. The potential industrial applications of the latest described scientific work is the optimization of composite materials by introducing sustainable plant fibers as fillers inside matrices [29]. AFM-FS experiments require proper functionalized strategies [30] by orienting the partner towards the soft matter sample of interest [31]. AFM-FS can become to dynamic force spectroscopy (DFS) by varying the retraction velocity of the AFM cantilever.

    Through the Ritchie-Evans equation the dissociation parameters of the system studied can be obtained [32] (1):

    F*=(kBTxβ)+ln(RxβkoffkBT)

    where, F* is the most probable rupture force of one individual analyzed complex, kB is Boltzmann's contant, T is the absolute temperature, xβ is the distance between the bound state and the energetic maximum, R is the loading rate and koff is the dissociation rate at equilibrium. Thus, xβ and koff dissociation parameters are obtained when the F* is plotted respect to the neperian logarithm of R. The dissociation parameters have been obtained by DFS for flavoenzimes [33], enzyme-cofactor [34], proteins [35] or peptide-cell membrane receptors [36] complexes, among others. The Ritchie-Evans equation is rooted in the Bell model [37] which in turn is based on the Arrhenius equation or Kramers rate theory (2) [38]:

    k=k0eΔU/kT

    where, ΔU and k0 parameters are the variation of the single reaction coordinate and the intrinsic rate, respectively. Kramers rate theory states the free energy barrier (ΔG) proportionally decreases when external load forces are exerted. Subsequently, these load forces also cause the increase of the bond rupture rate. Conventional AFM setups (except HS-AFM) only provides small range of loading rates which are low. At this regime of loading rates, the mean values of F* obtained in (1) result from Arrhenius equation of kinetic rate [39]. Thus, it is demonstrated the loading rate dependence and stiffness of the AFM apparatus on the single-dimensional energy landscapes from the chemical bonds of interest [40],[41]. Experimental measurements have been carried out to demonstrate this hypothesis. The rupture force of native bonds formed by β1 and β5 polypeptide chains from ubiquitin protein increases with the loading device stiffness when varies from 10 to 1000 pN/m at the same tested AFM lever loading rate [42]. Therefore, Ritchie-Evans equation only is accepted for low regimes of loading rate. When the plot renders one single linear trend means that only exists one energetic barrier when the complex decouples, whereas two linear regimes are related to two different energy landscapes between the bound and the dissociate complexes. Free energies can be also obtained by physical simulations [43]. Recently, some applications have been found in this field like the exploration of the energy landscape of DNA respect to antiviral drugs [44] which will significantly aid to advance in the creation of more effective therapies against human diseases. In addition to binary transient biomolecular complexes, DFS can also be devoted to reconstruct energy landscapes for intramolecular folding events observed in multidomain proteins or nucleic acids [45].

    Thereupon, nanoindentation measurements consist in penetrate some nanometers the external sample surface by the AFM tip through applying external force on the AFM cantilever. After recording the force-distance curve, the mechanical deformation underwent by the sample is obtained by the slope of the force profile. Higher slopes are related to stiffer samples, whereas smoother slopes correspond to soft materials [46]. Nanomechanical parameters like the Young's modulus or energy dissipation are obtained by nanoindentation studies. Many theoretical frameworks have been built up to address the Young's modulus of soft matter systems using the AFM tip as nanoindenter. The approaches most used in this field are rooted in Hertz and Derjaguin-Müller-Toporov (DMT) works.

    Hertz model hypothesized the nanoindentator geometry is an ideal sphere exerting perpendicular penetration on the studied surface [47]. Hertz model neglects the adhesion forces established between the AFM tip and the sample surface (3).

    d=(9F216E2effReff)1/3

    here, d is the indentation depth, F is the indentation force, Eeff is the effective Young's modulus and Reff is the effective curvature radius of the nanoindenter.

    The elastic modulus of the soft matter sample of interest can be obtained by (4):

    1Eeff=(1υTET)+(1υSEs)

    being, υ the Poisson's ratio and the subscripts T and S the AFM tip and sample, respectively. Then, DMT model takes into consideration long-range AFM tip-sample adhesion forces [48]. The Young's modulus of the soft matter system can be achieved by (5):

    E=(RFa3)+(F+2πRw)

    where, a is the contact radius between the surfaces of the AFM tip and the indented sample, F is the load force applied by the nanoindenter and w is the required energy to separate the unit area of both surfaces. Commonly, it is recommended the use of Hertz model when the radius of the AFM tip employed is large, whereas DMT model is broadly conducted for sharp AFM tips. Young's modulus of cells [49], virus capsids [50], cancer cells [51], lignocellulosic films at different relative humidities [52] or hydrogels [53] have been revealed. It has been reported that changes on cellular membrane rigidity affects mechanotransduction processes [54] and proliferation [55] which impacts on the growth of human diseases [56]. On the other hand, revealing the nanomechanical properties of biopolymer films, composites, hydrogels or biotissues serve to acquaint their wettability, rigidity and elastic performance which is crucial to use them for many industrial applications [57].

    AFM-nanoscale infrared spectroscopy (AFM-nanoIR) is based on a pulsed, tunable IR laser light aligned to the same scanned area by the commercial AFM tip. The photothermal expansion induced by sample absorption causes the vibration of the chemical bonds involved in the scanned sample surface area being thus, chemically characterized [58]. Two different modes to acquire chemical information of soft matter systems by AFM-nanoIR are available. First, AFM-nanoIR can perform chemical mapping at one defined wavenumber value. This aspect is of enormous importance when the user aims to discriminate between surfaces with different chemistry and it exists previous knowledge about the expected chemical bonds that serve as target. One illustrative example is the characterization of plant cell sections embedded by resins with benchmark wavenumber of 1710 cm−1 typical of carboxylic acids coming from the tested resin agents [59]. Chemical mapping offers excellent lateral resolution below 10 nm. The second approach is the “point and shoot” method based on obtaining conventional full IR spectral at specific soft matter sample regions. It is not unusual to observe slightly shifts of the wavenumbers found on the maximum of the IR spectra peaks between nanoscale and bulk measurements. This fact is attributed to the changes underwent by chemical bond lengths that eventually can take place when differences in electronegativity of neighboring atoms are experienced [60]. AFM nanoIR technique has been exploited to locate cell membrane receptors [61], the interrogation of chemical changes observed by the interaction of amyloidogenic proteins involved in neurodegenerative disorders with peptide inhibitors [62] or polymeric blends [63] for the manufacturing of sustainable food packaging or other enveloping items.

    Finally, magnetic force microscopy (MFM) measures the magnetic response provided by the sample upon AFM tips coated with cobalt-chromium. Distance between AFM tip and the external sample surface area is fixed at several hundred of nanometers in order to prevent the non-desirable short-range Van der Waals interactions which can interfere during the data acquisition. Repulsive and attractive magnetic forces coming from the sample cause a bend of the flexible cantilever which will be recorded by the AFM electronic system. Thus, MFM detects local magnetic fields with the classical AFM spatial resolution [64]. Magnetic properties of magnetosome nanoparticles coming from magnetotactic bacteria [65] which could be exploited as microswimmer robots under induced flow fields [66], proteins functionalized with magnetic nanoparticles [67] or magnetic nanoparticles embedded in polymer matrices [68] have been measured by MFM. The main applications of the most relevant outcomes found by MFM are based on hyperthermia [69], drug delivery [70] treatments, development of energy storage devices [71], or emulsion separation in petroleum industry [72], among others. To conclude, AFM is presented as suitable alternative to assess the physico-chemical properties of soft materials. Five different operational modes have been presented to illustrate the potential of this technique (Figure 1) and their potential applications (Figure 2).

    Figure 1.  Schematic illustration of atomic force microscopy (AFM) technique highlighted in black circle. AFM imaging (TM-AFM), AFM force spectroscopy (AFM-FS), nanoindentation, AFM-nanoscale infrared spectroscopy (AFM-nanoIR) and magnetic force microscopy (MFM) are remarked in blue, purple, brown, red and orange rectangles, respectively. The adhesion force and young's modulus values are obtained from the regions depicted in red color from their respective force-distance curves.
    Figure 2.  Some relevant applications of the five addressed AFM operational modes (AFM imaging, AFM-FS, nanoindentation, AFM-nanoIR and MFM, respectively) for Research & Development and Industry fields.

    Promising future perspectives are expected in the use of AFM in this field. Volumetric analyses have been developed by defining a mask threshold on topography images obtained by AFM imaging [73]. This aspect facilitate the statistical study of the observed features providing less scattering results. The main limitation to conduct imaging with conventional AFM apparatus is the poor temporal resolution achieved. HS-AFM allows to acquire ultra-fast images that allows to monitor in real-time the change biomolecular dynamic conformations in relevant conditions. The design and fabrication of smaller cantilevers in combination to high-response Z-scanners with the subsequent optimization of hardware and software components will be the basis of the next-generation of HS-AFMs. In this framework, the “only trace imaging” (OTI) mode has been evolved by discharging the record of the backward scan period [74]. OTI mode enables scan rates ranging from 8 frames per second (fps) to 30 fps improving nearby 2.5 times the data acquisition from previous HS-AFMs. Moreover, HS-AFM has recently used in the field of AFM-FS to determine the dissociation parameters of streptavidin:biotin [75]. Streptavidin:biotin system serves as proof-of-concept being the strongest non-covalent complex known in nature. HS-AFM succeed to obtain the information related to the complex rupture at loading rates more similar than the taken place in the living life organisms. Alternately, the optimization of AFM-FS experiments drives molecular recognition imaging (MRI) studies [76]. MRI simultaneously acquires the topography map and the maximum AFM-tip adhesion force for each pixel of the scanned area being possible to quantitatively correlate the visualized features with their adhesion properties. Moreover, the combination of AFM-FS with Kelvin probe force microscopy can open new avenues in the recognition of soft matter systems by combining specific intermolecular interactions with surface potentials [77]. These quantitative MRI methods significantly improve previous qualitative MRI approaches like AFM-based simultaneous topography and recognition imaging (TREC) [78] where the functional principle is based on the contrast image created between the lower part of the AFM cantilever amplitude oscillation respect to the upper part. The lower and upper parts of the amplitude oscillation yield topography information and molecular recognition images, respectively. Quantitative MRI technologies may open a door in the development of ultrasensitive detection technologies [79]. Furthermore, MRI experiments have shown to map biomolecular alterations like DNA methylation [80] which is relevant to monitor DNA mechanism effects. This technology could be expandable to study the redox functions involved in aging and disease [81]. It is also expected the employment of alternative techniques like computational modeling in combination with AFM can shed light to the physico-chemical properties of soft matter systems [82]. Nevertheless, AFM-FS also needs to deal with challenges, such as the noise threshold limit during sensing soft matter inter-/intramolecular forces which is nearby 15–20 pN. Optical tweezers (OT) [83] and magnetic tweezers (MT) [84] tools have raised to solve this limitation. OT and MT can trap the soft matter sample (typically biomolecules or living cells) by a laser beam source or through magnetic microbeads, respectively. Other advantage presented by OT and MT is the lack of sample immobilization on solid surfaces eliminating detrimental lateral forces during data acquisition. We expect the crosstalk of the five presented AFM operational modes in combination with other complementary single molecule techniques will significantly aid to better understand the underlying mechanisms of soft matter systems at the nanoscale level. This aspect is relevant to assess information that can be hidden in bulk studies like transient phenomena, rare events, conformational changes or crowding effects or local heterogeneities, among others.



    [1] H. Bessembinder, S. E. Jacobsen, W. F. Maxwell, K. Venkataraman, Capital Commitment and Illiquidity in Corporate Bonds, Journal of Finance, 73 (2017), 1615-1661.
    [2] F. Black, J. C. Cox, Valuing Corporate Securities: Some Effects of Bond Indenture Provisions, Journal of Finance, 31 (1976), 351-367.
    [3] E. Briys, F. de Varenne, Valuing risky fixed rate debt: An extension, The Journal of Financial and Quantitative Analysis, 32 (1997), 239-348.
    [4] T. Chellathurai, Probability density of recovery rate given default of a firm's debt and its constituent tranches, International Journal of Theoretical and Applied Finance, 20 (2017), 1750023.
    [5] G. d'Amico, G. Di Biase, J. Janssen, R. Manca, Semi-Markov Migration Models for Credit Risk, 1 Eds., Hoboken: John Wiley & Sons, 2017.
    [6] S. Das, P. Tufano, Pricing credit-sensitive debt when interest rates, credit ratings, and credit spreads are stochastic, Journal of Financial Engineering, 5 (1996), 161-198.
    [7] D. Duffie, K. J. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies, 12 (1999), 687-720.
    [8] F. Fei, A. Fuertes, E. Kalotychou, Credit Rating Migration Risk and Business Cycles, Journal of Business Finance & Accounting, 39 (2012), 229-263.
    [9] H. Frydman, T. Schuermann, Credit Rating Dynamics and Markov Mixture Models, Journal of Banking and Finance, 32 (2008), 1062-1075.
    [10] M. G. Garrori, J. L. Menaldi, Green Functions for Second Order Parabolic Integro-differential Problems, 1 Eds., New York: Longman Scientific & Technical, 1992.
    [11] B. Hu, J. Liang, Y. Wu, A free boundary problem for corporate bond with credit rating migration, J. Math. Anal. Appl., 428 (2015), 896-909.
    [12] Z. Huang, T. Jiang, Z. Wang, On a multiple credit rating migration model with stochastic interest rate, Math. Method. Appl. Sci., 43 (2020), 7106-7134.
    [13] J. C. Hull, Options, Futures and Other Derivatives, 8 Eds., New York: Pearson International Edition, 2012.
    [14] R. Jarrow, S. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-86.
    [15] R. A. Jarrow, D. Lando, S. M. Turnbull, A Markov Model for the Term Structure of Credit Risk Spreads, Review of Financial Studies, 10 (1997), 481-523.
    [16] L. Jiang, Mathematical Modeling and Methods of Option Pricing, 8 Eds., New Jersey: World Scientific, 2005.
    [17] D. Lando, On Cox Processes and Credit Risky Securities, Review of Derivatives Research, 2 (1998), 99-120.
    [18] H. E. Leland, K. B. Toft, Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads, Journal of Finance, 51 (1996), 987-1019.
    [19] S. Li, X. Zhao, C. Yin, Z. Huang, Stochastic interest model driven by compound Poisson process and Brownian motion with applications in life contingencies, Quantitative Finance and Economics, 2 (2018), 246-260.
    [20] J. Liang, Y. Zhao, X. Zhang, Utility indifference valuation of corporate bond with credit rating migration by structure approach, Economic Modelling, 54 (2016), 339-346.
    [21] J. Liang, Y. Wu, B. Hu, Asymptotic traveling wave solution for a credit rating migration problem, J. Differ. Equations, 261 (2016), 1017-1045.
    [22] J. Liang, H. M. Yin, X. Chen, Y. Wu, On a corporate bond pricing model with credit rating migration risks and stochastic interest rate, Qualitative Finance and Economics, 1 (2017), 300-319.
    [23] J. Liang, H. Zou, Valuation of Credit Contingent Interest Rate Swap with Credit Rating Migration, Int. J. Comput. Math., (2020), 1-15.
    [24] Y. Liang, C. Xu, An efficient conditional Monte Carlo method for European option pricing with stochastic volatility and stochastic interest rate, Int. J. Comput. Math., 97 (2020), 638-655.
    [25] F. Longstaff, E. Schwartz, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance, 50 (1995), 789-819.
    [26] D. G. McMillan, The information content of the stock and bond return correlation, Quantitative Finance and Economics, 2 (2018), 757-775.
    [27] F. Mehrdoust, A. R. Najafi, A short memory version of the Vasicek model and evaluating European options on zero-coupon bonds, J. Comput. Appl. Math., 375 (2020), 112796.
    [28] R. C. Merton, ON The pricing of corporate debt: the risk structure of interest rates, Journal of Finance, 29 (1974), 449-470.
    [29] H. Pedersen, N. R. Swanson, A survey of dynamic Nelson-Siegel models, diffusion indexes, and big data methods for predicting interest rates, Quantatitive Finance and Economics, 3 (2019), 22-45.
    [30] L. C. Thomas, D. E. Allen, N. Morkelkingsbury, A Hidden Markov Chain Model for the Term Structure of Bond Credit Risk Spreads, International Review of Financial Analysis, 11 (2002), 311-329.
    [31] P. Valta, Strategic Default, Debt Structure, and Stock Returns, Journal of Financial and Quantitative Analysis, 51 (2016), 197-229.
    [32] Z. Wang, Z. Liu, T. Jiang, Z. Huang, Asymptotic traveling wave for a pricing model with multiple credit rating migration risk, Commun. Math. Sci., 17 (2019), 1975-2004.
    [33] Y. Wu, X. Liang, Vasicek model with mixed-exponential jumps and its applications in finance and insurance, Adv. Differ. Equ-NY, 2018 (2018), 138.
    [34] Y. Wu, J. Liang, A new model and its numerical method to identify multi credit migration boundaries, Int. J. Comput. Math., 95 (2018), 1688-1702.
    [35] Y. Wu, J. Liang, B. Hu, A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior, Discrete & Continuous Dynamical System-B, 25 (2020), 1043-1058.
    [36] Y. Wu, J. Liang, Free boundaries of credit rating migration in switching macro regions, Math. Control Relat. F., 10 (2020), 257-274.
    [37] H. M. Yin, J. Liang, Y. Wu, On a New Corporate Bond Pricing Model with Potential Credit Rating Change and Stochastic Interest Rate, Journal of Risk and Financial Management, 11 (2018), 87.
  • This article has been cited by:

    1. Alessandro Magazzù, Carlos Marcuello, Investigation of Soft Matter Nanomechanics by Atomic Force Microscopy and Optical Tweezers: A Comprehensive Review, 2023, 13, 2079-4991, 963, 10.3390/nano13060963
    2. Anabel Lostao, KeeSiang Lim, María Carmen Pallarés, Arkadiusz Ptak, Carlos Marcuello, Recent advances in sensing the inter-biomolecular interactions at the nanoscale – A comprehensive review of AFM-based force spectroscopy, 2023, 01418130, 124089, 10.1016/j.ijbiomac.2023.124089
    3. Robert Winkler, Miguel Ciria, Margaret Ahmad, Harald Plank, Carlos Marcuello, A Review of the Current State of Magnetic Force Microscopy to Unravel the Magnetic Properties of Nanomaterials Applied in Biological Systems and Future Directions for Quantum Technologies, 2023, 13, 2079-4991, 2585, 10.3390/nano13182585
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3634) PDF downloads(127) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog