Numerical schemes for option pricing in regime-switching jump diffusion models. (English) Zbl 1290.91180
Summary: In this paper, we present algorithms to solve a complex system of partial integro-differential equations (PIDE’s) of parabolic type. The system is motivated by applications in finance where the solution of the system gives the price of European options in a regime-switching jump diffusion model. The new algorithms are based on theoretical analysis in [I. Florescu et al., Electron. J. Differ. Equ. 2012, Paper No. 231, 12 p. (2012; Zbl 1294.35171)] where the proof of convergence of the algorithms is carried out. The problems are also solved using a more traditional approach, where the integral terms (but not the derivative terms) are treated explicitly. Another contribution of this work details a novel type of jump distribution. Empirical evidence suggests that this type of distribution may be more appropriate to model jumps as it makes them more clearly distinguishable from the signal variability.
MSC:
91G60 | Numerical methods (including Monte Carlo methods) |
91G20 | Derivative securities (option pricing, hedging, etc.) |
60J75 | Jump processes (MSC2010) |
65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
45K05 | Integro-partial differential equations |
Keywords:
numerical algorithms; system of partial integro-differential equations; regime-switching jump diffusion; option pricing; implicit and explicit finite element methodsCitations:
Zbl 1294.35171References:
[1] | DOI: 10.1016/j.apnum.2004.08.037 · Zbl 1117.91028 · doi:10.1016/j.apnum.2004.08.037 |
[2] | DOI: 10.1023/A:1011354913068 · Zbl 1274.91398 · doi:10.1023/A:1011354913068 |
[3] | DOI: 10.1007/s00186-008-0282-1 · Zbl 1178.91189 · doi:10.1007/s00186-008-0282-1 |
[4] | DOI: 10.1142/S0219024911006620 · Zbl 1233.91257 · doi:10.1142/S0219024911006620 |
[5] | DOI: 10.1016/j.insmatheco.2006.05.001 · Zbl 1141.91420 · doi:10.1016/j.insmatheco.2006.05.001 |
[6] | DOI: 10.1007/s10092-007-0128-x · Zbl 1150.65033 · doi:10.1007/s10092-007-0128-x |
[7] | DOI: 10.1080/13518470701201512 · doi:10.1080/13518470701201512 |
[8] | DOI: 10.1137/S0036142903436186 · Zbl 1101.47059 · doi:10.1137/S0036142903436186 |
[9] | DOI: 10.1007/s00211-003-0511-8 · Zbl 1126.91036 · doi:10.1007/s00211-003-0511-8 |
[10] | DOI: 10.1093/imanum/drh011 · Zbl 1134.91405 · doi:10.1093/imanum/drh011 |
[11] | DOI: 10.1080/07362990701420118 · Zbl 1155.91380 · doi:10.1080/07362990701420118 |
[12] | Florescu I., Electron. J. Differ. Equ. 2012 pp 1– |
[13] | Florescu I., Electron. J. Differ. Equ. 2010 pp 1– |
[14] | Florescu I., Quant. Finance pp 1– |
[15] | DOI: 10.1080/713665550 · doi:10.1080/713665550 |
[16] | DOI: 10.1137/110820920 · Zbl 1232.91707 · doi:10.1137/110820920 |
[17] | DOI: 10.1002/num.21714 · Zbl 1282.91377 · doi:10.1002/num.21714 |
[18] | DOI: 10.1142/S0219024909005245 · Zbl 1204.91127 · doi:10.1142/S0219024909005245 |
[19] | DOI: 10.1287/mnsc.48.8.1086.166 · Zbl 1216.91039 · doi:10.1287/mnsc.48.8.1086.166 |
[20] | DOI: 10.1137/090777529 · Zbl 1232.91712 · doi:10.1137/090777529 |
[21] | DOI: 10.1137/100806552 · Zbl 1227.91034 · doi:10.1137/100806552 |
[22] | DOI: 10.1142/S0219024910005863 · Zbl 1233.91284 · doi:10.1142/S0219024910005863 |
[23] | Liu R., Journal of Applied Mathametics and Stochastic Analysis 2006 pp 1– |
[24] | DOI: 10.1016/0304-405X(76)90022-2 · Zbl 1131.91344 · doi:10.1016/0304-405X(76)90022-2 |
[25] | DOI: 10.2307/3318541 · Zbl 0991.60045 · doi:10.2307/3318541 |
[26] | DOI: 10.1016/j.apnum.2011.02.002 · Zbl 1213.91164 · doi:10.1016/j.apnum.2011.02.002 |
[27] | DOI: 10.1080/00207160.2012.669475 · Zbl 1255.91410 · doi:10.1080/00207160.2012.669475 |
[28] | DOI: 10.1002/0471742015 · Zbl 1089.65053 · doi:10.1002/0471742015 |
[29] | DOI: 10.1155/2008/474623 · Zbl 1141.91386 · doi:10.1155/2008/474623 |
[30] | DOI: 10.1137/060674697 · Zbl 1178.35225 · doi:10.1137/060674697 |
[31] | Toivanen J., J. Comput. Finance 13 pp 61– |
[32] | Topper J., Financial Engineering with Finite Elements (2005) |
[33] | DOI: 10.2143/AST.39.2.2044646 · Zbl 1180.91298 · doi:10.2143/AST.39.2.2044646 |
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