Computing survival probabilities based on stochastic differential models

We develop a new numerical method to compute survival probabilities based on stochastic differential models, a matter of great importance in several areas of science, such as finance, biology, medicine and geophysics. This novel approach is based on polynomial differential quadrature, which is combined with a high-order time discretization scheme. Numerical experiments are presented showing that the proposed method performs extremely well and is more efficient than the approaches recently developed in Costabile et?al. (2013) and Guarin et?al. (2011). We propose a very efficient method to compute survival probabilities.We combine polynomial differential quadrature with high-order time-stepping.We consider a reduced-form model and a structural model that arise from finance and insurance.The method is model independent and could also be extended to other stochastic processes.Numerical comparison with other recent approaches is provided.

[1]  Enrico Biffis,et al.  Affine Processes for Dynamic Mortality and Actuarial Valuations , 2004 .

[2]  Alfio Quarteroni,et al.  Numerical Mathematics (Texts in Applied Mathematics) , 2006 .

[3]  F. Black,et al.  Bond and Option Pricing when Short Rates are Lognormal , 1991 .

[4]  H. Gerber,et al.  On the Time Value of Ruin , 1997 .

[5]  Raviraj S. Adve,et al.  Molecular Communication Using Brownian Motion With Drift , 2010, IEEE Transactions on NanoBioscience.

[6]  D. Dickson,et al.  Ruin probabilities with compounding assets , 1999 .

[7]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[8]  Elisa Luciano,et al.  Mortality risk via affine stochastic intensities: calibration and empirical relevance , 2008 .

[9]  D. Dickson,et al.  Approximations to ruin probability in the presence of an upper absorbing barrier , 1984 .

[10]  D. Madan,et al.  Pricing Equity Default Swaps under an approximation to the CGMY L\'{e}% vy Model , 2007, 0711.2807.

[11]  D. Lemmens,et al.  Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model , 2010, 1011.1175.

[12]  C. Lefèvre,et al.  Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities , 2009 .

[13]  S. Löw,et al.  Estimation of volcanic hazards based on Cox stochastic processes , 2000 .

[14]  Recursive calculation of finite time ruin probabilities under interest force , 2003 .

[15]  Agostino Capponi,et al.  Arbitrage‐Free Bilateral Counterparty Risk Valuation Under Collateralization and Application to Credit Default Swaps , 2014 .

[16]  R. C. Merton,et al.  On the Pricing of Corporate Debt: The Risk Structure of Interest Rates , 1974, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[17]  Xiao-Jing Wang,et al.  A Biophysically Based Neural Model of Matching Law Behavior: Melioration by Stochastic Synapses , 2006, The Journal of Neuroscience.

[18]  Wing Lon Ng,et al.  Enhancing credit default swap valuation with meshfree methods , 2011, Eur. J. Oper. Res..

[19]  F. Black,et al.  VALUING CORPORATE SECURITIES: SOME EFFECTS OF BOND INDENTURE PROVISIONS , 1976 .

[20]  V. Young.  Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio , 2007, 0705.1297.

[21]  Warner Marzocchi,et al.  Probabilistic eruption forecasting at short and long time scales , 2012, Bulletin of Volcanology.

[22]  Chunsheng Zhou,et al.  The term structure of credit spreads with jump risk , 2001 .

[23]  K. Singleton,et al.  How Sovereign is Sovereign Credit Risk? , 2007 .

[24]  K. Singleton,et al.  Default and Recovery Implicit in the Term Structure of Sovereign CDS Spreads , 2008 .

[25]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[26]  M. Goovaerts,et al.  Explicit finite-time and infinite-time ruin probabilities in the continuous case , 1999 .

[27]  Damiano Brigo,et al.  A Stochastic Processes Toolkit for Risk Management , 2007, 0812.4210.

[28]  M. Costabile,et al.  Computing finite-time survival probabilities using multinomial approximations of risk models , 2015 .

[29]  Ghodrat Karami,et al.  Polynomial and harmonic differential quadrature methods for free vibration of variable thickness thick skew plates , 2005 .

[30]  M. Dahl,et al.  Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts , 2004 .

[31]  Y. Kwok,et al.  Credit Default Swap Valuation with Counterparty Risk , 2005 .

[32]  An operator splitting harmonic differential quadrature approach to solve Young’s model for life insurance risk , 2012 .

[33]  Damiano Brigo,et al.  A stochastic processes toolkit for risk management: Geometric Brownian motion, jumps, GARCH and variance gamma models , 2009 .

[34]  David A. Stanford,et al.  Erlangian approximation to finite time ruin probabilities in perturbed risk models , 2007, Numerical Methods for Structured Markov Chains.

[35]  S. Asmussen,et al.  Ruin probabilities for a regenerative Poisson gap generated risk process , 2011 .

[36]  G A Whitmore,et al.  Assessing lung cancer risk in railroad workers using a first hitting time regression model , 2004, Environmetrics.

[37]  D. Brigo,et al.  Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit , 2001 .

[38]  O. Aalen,et al.  Survival Models Based on the Ornstein-Uhlenbeck Process , 2004, Lifetime data analysis.

[39]  David Lando,et al.  On cox processes and credit risky securities , 1998 .

[40]  Claude Lefèvre,et al.  The probability of ruin in finite time with discrete claim size distribution , 1997 .

[41]  Damiano Brigo,et al.  A deterministic–shift extension of analytically–tractable and time–homogeneous short–rate models , 2001, Finance Stochastics.

[42]  P. Forsyth,et al.  PDE methods for pricing barrier options , 2000 .

[43]  Olivier Faugeras,et al.  The spikes trains probability distributions: A stochastic calculus approach , 2007, Journal of Physiology-Paris.