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A stochastic delay model for pricing debt and equity: numerical techniques and applications. (English) Zbl 1304.91237

Summary: Delayed nonlinear models for pricing corporate liabilities and European options were recently developed. Using self-financed strategy and duplication we were able to derive a random partial differential equation (RPDE) whose solutions describe the evolution of debt and equity values of a corporate in the last delay period interval in the accompanied paper [E. Kemajou et al., “A stochastic delay model for pricing debt and loan guarantees: theoretical results”, Preprint (2012), arXiv:1210.0570]. In this paper, we provide robust numerical techniques to solve the delayed nonlinear model for the corporate value, along with the corresponding RPDEs modeling the debt and equity values of the corporate.Using financial data from some firms, we forecast and compare numerical solutions from both the nonlinear delayed model and classical Merton model with the real corporate data. From this comparison, it comes up that in corporate finance the past dependence of the firm value process may be an important feature and therefore should not be ignored.

MSC:

91G50 Corporate finance (dividends, real options, etc.)
91G80 Financial applications of other theories
37N40 Dynamical systems in optimization and economics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
91B55 Economic dynamics

Software:

phipm; Algorithm 919
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References:

[1] Gryglewicz, S., A theory of corporate financial decisions with liquidity and solvency concerns, J Financ Econ, 99, 365-384 (2011)
[2] Tambue, A.; Lord, G. J.; Geiger, S., An exponential integrator for advection-dominated reactive transport in heterogeneous porous media, J Comput Phys, 229, 10, 3957-3969 (2010) · Zbl 1423.76355
[3] Dumas, B.; Fleming, J.; Whaley, R. E., Implied volatility functions: empirical tests, J Financ, 53, 6, 2059-2106 (1998)
[5] Scott, L. O., Option pricing when the variance changes randomly: theory, estimation and an application, J Financ Quant Anal, 22, 419-438 (1987)
[6] Blattberg, R. C.; Gonedes, N. J., A comparison of the stable and student distributions as statistical models for stock prices, J Bus, 47, 244-280 (1974)
[7] Merton, R. C., On the pricing of corporate debt: the risk structure of interest rates, J Financ, 29, 449-470 (1974)
[8] Arriojas, M.; Hu, Y.; Mohammed, S.; Pap, G., A delayed Black and Scholes formula, J Stoch Anal Appl, 25, 2, 471-492 (2007) · Zbl 1119.60059
[9] Bensoussan, A.; Crouhy, M.; Galai, D., Stochastic equity volatility and the capital structure of the firm, Philos Trans R Soc London Ser A, 347, 449-598 (1994) · Zbl 0822.90012
[10] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J Polit Econ Dyn Control, 81, 3, 637-654 (1973) · Zbl 1092.91524
[13] Wilmott, P.; Dewynne, J.; Howison, S., Option pricing: mathematical models and computation (1993), Oxford Financial Press: Oxford Financial Press Oxford, UK · Zbl 0797.60051
[14] Mao, X.; Sabanis, S., Delay geometric Brownian motion in financial option valuation, Stoch Int J Probab Stoch Proces (2012)
[15] Hobson, D. G.; Rogers, L. C.G., Complete models with stochastic volatility, Math Financ, 8, 1, 27-48 (1998) · Zbl 0908.90012
[18] Geiger, G.; Lord, G. J.; Tambue, A., Exponential time integrators for stochastic partial differential equations in 3D reservoir simulation, Comput Geosci, 16, 2, 323-334 (2012) · Zbl 1348.76147
[19] Cen, Zhongdi; Le, Anbo; Xu, Aimin, Exponential time integration and second-order difference scheme for a generalized Black-Scholes equation, J Appl Math, 2012, 2012 (2012), Article ID 796814, http://dx.doi.org/10.1155/2012/796814 · Zbl 1235.91171
[20] Duffy, D. J., Finite difference methods in financial engineering: a partial differential equation approach (2006), John Wiley & Sons Ltd: John Wiley & Sons Ltd West Sussex, England · Zbl 1141.91002
[22] Niesen, J.; Wright, W. M., Algorithm 919: a Krylov subspace algorithm for evaluating the \(\varphi \)-functions appearing in exponential integrators, ACM Trans Math Softw, 38, 3 (2012), Article 22 · Zbl 1365.65185
[23] Tambue, A., Efficient numerical simulation of incompressible two-phase flow in heterogeneous porous media based on exponential Rosenbrock-Euler method and lower-order Rosenbrock-type method, J Porous Media, 16, 5 (2013)
[24] Tambue, A.; Berre, I.; Nordbotten, J. M., Efficient simulation of geothermal processes in heterogeneous porous media based on the exponential Rosenbrock-Euler and Rosenbrock-type methods, Adv Water Res, 53, 250-262 (2013)
[27] Mao, X., Stochastic differential equations and their applications (2007), Horwood Publishing Limited: Horwood Publishing Limited Chichester, UK
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