[1] Blanchet-Scalliet, C., and Jeanblanc, M. Hazard rate for credit risk and hedging defaultable contingent claims, Working Paper, 2002.
[2] Delbaen F., and Schachermayer, W. A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300:463–520 (1994).
[3] Samuelson, P. Rational theory of warrant pricing. Industrial Management Review 6: 13–31 (1965).
[4] Merton, R.C. The theory of rational option pricing. Bell Journal of Economics and Management Science 4:141–183 (1973).
[5] Black, F., and Scholes, M. The pricing of options and corporate liabilities. Journal of Political Economy, 81: 637–59 (1973).
[6] Balland, P. Deterministic implied volatility models. Quantitative Finance 2: 31–44 (2002).
[7] Kellerer, H. Markov-komposition und eine anwendung auf martingale (in German). Mathematische Annalen 198: 217–229 (1972).
[8] Cox, J. The constant elasticity of variance option pricing model. Journal of Portfolio Management (December 1998).
[9] Föllmer, H., and Schied. Stochastic Finance. 2nd edition. Berlin: de Gruyter, 2004.
[10] Madan, D., and Yor, M. Making Markov martingales meet marginals: with explicit constructions. Working Paper 2002.
[11] Dupire, B. Pricing with a smile. Risk 7 (1):18–20, 1996.
[12] Gyöngy, I. Mimicking the one-dimensional marginal distributions of processes having an Ito differential”. Probability Theory and Related Field 71:501–516, 1986.
[13] Revuz, D., and Yor, M. Continuous Martingales and Brownian Motion, 3rd ed. Heidelberg: Springer, 1999.
[14] Merton, R.C. Option pricing with discontinuous returns. Bell Journal of Financial Economics 3: 145–166 (1976).
[15] Protter, P. Stochastic Integration and Differential Equations, 2nd edition. Heidelberg: Springer, 2004.
[16] Buehler, H. Expensive martingales. Quantitative Finance (April 2006).
[17] Glasserman, P. Monte Carlo Methods in Financial Engineering. Heidelberg: Springer, 2004.
[18] Buehler, H. Volatility markets: Consistent modelling, hedging and practical implementation. PhD thesis, 2006.
[19] Heston, S. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies (1993).
[20] Aït-Sahalia, Y., and Kimmel, R. Maximum likelihood estimation of stochastic volatility models. NBER Working Paper No. 10579, June 2004.
[21] Andersen, L., Piterbarg, V. Moment explosions in stochastic volatility models. Working Paper, April 15, 2004. http://ssrn.com/abstract=559481.
[22] Carr, P., and Madan, D. Towards a theory of volatility trading. In Robert Jarrow (ed.). Volatility. London: Risk, (2002), 417–427.
[23] Lewis, A. Option Valuation under Stochastic Volatility. Newport Beach, CA: Finance Press, 2000.
[24] Hagan, P., Kumar, D., Lesniewski, A., Woodward, D. “Managing smile risk,” Wilmott, pp. 84–108 (September 2002).
[25] Jourdain, B. Loss of martingality in asset price models with lognormal stochastic volatility. Working Paper, 2004. http://cermics.enpc.fr/reports/CERMICS-2004/CERMICS-2004-267.pdf
[26] Hagan, P., Lesniewski, A., and Woodward, D. Probability distribution in the SABR model of stochastic volatility. Working Paper, March 22, 2005.
[27] Henry-Labordère, P. “A general asymptotic implied volatility for stochastic volatility models”. April 2005 http://ssrn.com/abstract=698601
[28] Bourgade, P., Croissant, O. Heat kernel expansion for a family of stochastic volatility models: δ-geometry. Working Paper, 2005. http://arxiv.org/abs/cs.CE/0511024.
[29] Scott, L. Option pricing when the variance changes randomly: theory, estimation and an application. Journal of Financial and Quantitative Analysis 22:419–438 (1987).
[30] Fouque, J-P., Papanicolaou, G., and Sircar, K. Derivatives in Financial Markets with Stochastic Volatility. New York: Cambridge University Press: 2000.
[31] Overhaus, M., Ferraris, A., Knudsen, T., Milward, R., Nguyen-Ngoc, L., and Schindlmayr, G. Equity Derivatives—Theory and Applications. Hoboken, NJ: Wiley, 2002.
[32] Schoutens, W. Levy Processes in Finance. Hoboken, NJ: Wiley, 2003.
[33] Cont, R., and Tankov, P. Financial Modelling with Jump Processes. Boca Raton, FL: CRC Press, 2003.
[34] Schoebel, R., and Zhu, J. Stochastic volatility with an Ornstein-Uhlenbeck process: An Extension. European Finance Review 3:2346 (1999).
[35] Bates, D. Jumps and stochastic volatility: Exchange rate process implicit in deutschemark options. Review of Financial Studies 9:69–107 (1996).
[36] Brace, A., Goldys, B., Klebaner, F., and Womersley, R. Market model of stochastic implied volatility with application to the BGM model. Working Paper, 2001. http://www.maths.unsw.edu.au/~rsw/Finance/svol.pdf.
[37] Schönbucher, P.J. A market model for stochastic implied volatility. Philosophical Transactions of the Royal Society A 357:2071–2092 (1999).
[38] Cont, R., da Fonseca, J., and Durrleman, V. Stochastic models of implied volatility surfaces. Economic Notes 31(2): 361–377 (2002).
[39] Haffner, R. Stochastic Implied Volatility. Heidelberg: Springer, 2004.
[40] Derman, E., and Kani, I. Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility/International Journal of Theoretical and Applied Finance 1(1):61–110 (1998).
[41] Barndorff-Nielsen, O., Graversen, S., Jacod, J., Podolskij, M., and Shephard, N. A central limit theorem for realised power and bipower variations of continuous semimartingales. Working Power 2004. http://www.nuff.ox.ac.uk/economics/papers/2004/W29/BN-G-J-P-Sfest.pdf.
[42] Demeterfi, K., Derman, E., Kamal, M., and Zou, J. More than you ever wanted to know about volatility swaps. Journal of Derivatives 6(4):9–32 (1999).
[43] Carr, P., and Madan, D. Towards a theory of volatility trading. In: Robert Jarrow, ed., Volatility. Risk Publications, pp. 417–427 (2002).
[44] Carr, P., and Lewis, K. Corridor variance swaps. Risk (February 2004).
[45] El Karoui, N., Jeanblanc-Picquè, M., and Shreve, S.E. Robustness of the Black and Scholes formula. Mathematical Finance 8:93 (April 1998).
[46] Carr, P., and Lee, R. Robust replication of volatility derivatives. Working Paper, April 2003. http://math.uchicago.edu/rl/voltrading.pdf.
[47] Heath, D., Jarrow, R. and Morton, A. Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 60(1992).
[48] Bergomi, L. Smile dynamics II. Risk (September 2005).
[49] Buehler, H. Consistent variance curves Finance and Stochastics (2006).
[50] Musiela, M. Stochastic PDEs and term structure models. Journées Internationales de France, IGR-AFFI, La Baule (1993).
[51] Björk, T., and Svensson, L. On the existence of finite dimensional realizations for nonlinear forward rate models. Mathematical Finance, 11(2): 205–243(2001).
[52] Filipovic, D. Consistency Problems for Heath-Jarrow-Morton Interest Rate Models (Lecture Notes in Mathematics 1760). Heidelberg: Springer, 2001
[53] Filipovic, D., and Teichmann, J. On the geometry of the term structure of interest rates. Proceedings of the Royal Society London A 460: 129–167 (2004).
[54] Buehler, H. Volatility markets: Consistent modeling, hedging and practical implementation. PhD thesis TU Berlin, to be submitted 2006.
[55] Björk, T., and Christensen, B.J. Interest rate dynamics and consistent forward curves. Mathematical Finance 9(4): 323–348 (1999).
[56] Duffie, D., Pan, J., and Singleton, K. “Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68: 1343–1376 (2000).
[57] Dupire, B. Arbitrage pricing with stochastic volatility. In Carr, P., Derivatives Pricing: The Classic Collection pp. 197–215, London: Risk, 2004.
[58] Sin, C. Complications with stochastic volatility models. Advances in Applied Probability 30: 256–268 (1998).
[59] Brace, A., Gatarek, D., and Musiela, M. The market model of interest rate dynamics. Mathematical Finance 7: 127–154 (1997).
[60] Heath, D., Jarrow, R., and Morton, A. Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 61(1): 77–105 (1992).
[61] Hull J., and White, A. One-factor interest-rate models and the valuation of interest rate derivatives. Journal of Financial and Quantitative Analysis 28: 235–254 (1993).
[62] Vasicek, O. An equilibrium charecterisation of the term structure. Journal of Financial Economics 5: 177–188 (1997).
[63] Black, F., and Karasinski P. Bond and option pricing when short rates are lognormal. Financial Analysts Journal (July–August 1991): 52–59.
[64] Cox, J.C., Ingersoll, J.E., and Ross, S.A., A theory of the term structure of interest rates. Econometrica 53: 385–407 (1985).
[65] Black, F., Derman, E., and Toy, W. A one-factor model of interest rates and its application to Treasury bond options. Financial Analysts Journal (July–August 1990): 52–59.
[66] Jamshidian, F. Forward induction and the construction of yield curve diffusion models. Journal of Fixed Income 1:62–74 (1991).
[67] Hull, J., and White, A. Numerical procedures for implementing term structure models I: Single-factor models. Journal of Derivatives 2: 7–16 (1994).
[68] Jamshidian, F. Bond and option evaluation in the Gaussian interest rate model. Research in Finance 9:131–70 (1991).
[69] Kloeden, P., and Platen, E. Numerical Solution of Stochastic Differential Equations, 3rd ed. Heidelberg: Springer, 1999.
[70] Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. Numerical Recipes in C, 2nd ed., Cambridge: Cambridge University Press, 1993.
[71] Gill, P.E., Murray, W., and Wright, M.H., Practical Optimization. San Diego: Academic Press, 1981.
[72] Black, F., and Scholes, M. The pricing of options and corporate liabilities. Journal of Political Economy 81:637–654 (1973).
[73] Merton, R.C. Theory of rational option pricing. Bell Journal of Economics and Management Science 4:141–183 (Spring 1973).
[74] Black, F., and Cox, J. Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance 351–367 (1976).
[75] Merton, R.C. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance 29:449–470, 1974.
[76] Geske, R. The valuation of corporate liabilities as compound options. Journal of Financial and Quantitative Analysis 12:541–552 (1977).
[77] Hull, J., and White, A. The impact of default risk on the prices of options and other derivatives securities. Journal of Banking and Finance 19(2):299–322 (1995).
[78] Nielsen, L.T., Saa-Requejo, J., and Santa-Clara, P. Default risk and interest rate risk: The term structure of default spreads. Working Paper, INSEAD, 1993.
[79] Schonbucher, P.J. Valuation of securities subject to credit risk. Working paper, University of Bonn, February 1996.
[80] Zhou, C. A jump-diffusion approach to modeling credit risk and valuing defaultable securities. Finance and Economics Discussion Series, Federal Reserve Board 15 (1997).
[81] Longstaff, F.A., and Schwartz, E.S. A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 29:789–819 (1995).
[82] Briys, E. and Varenne, F. Valuing risky fixed rate debt: An extension. Journal of Financial and Quantitative Analysis 32(2):239–248 (1997).
[83] Ramaswamy, K., and Sundaresan, S.M. The valuation of floating rate instruments, theory and evidence. Journal of Financial Economics 17:251–272 (1986).
[84] Jarrow, R.A., and Turnbull, S.M. Pricing derivatives on financial securities subject to credit risk. Journal of Finance 50:53–85, 1995.
[85] Duffie, D., and Singleton, K. Econometric modelling of term structures of defaultable bonds. Working Paper, Stanford University, 1994.
[86] Duffie, D., and Singleton, K. An econometric model of the term structure of interest rate swap yields. Journal of Finance 52(4):1287–1321 (1997).
[87] Duffie, D., and Singleton, K.J. Modeling term structure of defaultable bonds. Review of Financial Studies 12:687–720 (1999).
[88] Lando, D. On Cox processes and credit risky bonds. Review of Derivatives Research 2(2/3):99–120 (1998).
[89] Ingersoll, J.E. A contingent claim valuation of convertible securities. Journal of Financial Economics 4:289–322 (1977).
[90] Brennan, M.J., and Schwartz, E.S. Convertible bonds: Valuation and optimal strategies for call and conversion. Journal of Finance 32:1699–1715 (1977).
[91] Brennan, M.J., and Schwartz, E.S. Analysing convertible bonds. Journal of Financial and Quantitative Analysis 15(4):907–929 (1980).
[92] Nyborg, K.G. The use and pricing of convertible bonds. Applied Mathematical Finance 3:167–190 (1996).
[93] Carayannopoulos, P. Valuing convertible bonds under the assumption of stochastic interest rates: An empirical investigation. Quarterly Journal of Business and Economics 35(3):17–31 (summer 1996).
[94] Cox, J., Ingersoll, J., and Ross, S. A theory of the term structure of interest rates. Econometrica 53:385–467 (1985).
[95] Zhu, Y.-I., and Sun, Y. The singularity separating method for two factor convertible bonds. Journal of Computational Finance 3(1):91–110 (1999).
[96] Epstein, D., Haber, R., and Wilmott, P. Pricing and hedging convertible bonds under non-probabilistic interest rates. Journal of Derivatives, Summer 2000, 31–40 (2000).
[97] Barone-Adesi, G., Bermúdez, A., and Hatgioannides, J. Two-factor convertible bonds valuation using the method of characteristics/finite elements. Journal of Economic Dynamics and Control 27(10):1801–1831 (2003).
[98] Nogueiras, M.R. 2005. Numerical analysis of second order Lagrange-Galerkin schemes. Application to option pricing problems. Ph.D. thesis, Department of Applied Mathematics, Universidad de Santiago de Compostela, Spain.
[99] McConnell, J.J., and Schwartz, E.S. LYON taming. The Journal of Finance 41(3):561–577 (July 1986).
[100] Cheung, W., and Nelken, I. Costing the converts. Risk 7(7):47–49 (1994).
[101] Ho, T.S.Y., and Pfteffer, D.M. Convertible bonds: Model, value, attribution and analytics. Financial Analyst Journal, (September–October 1996):35–44.
[102] Schonbucher, P.J. Credit Derivatives Pricing Models: Models, Pricing and Implementation. Hoboken, NJ: Wiley, 2003.
[103] Arvanitis, A., and Gregory, J. Credit: The Complete Guide to Pricing, Hedging and Risk Management. London: Risk Books, 2001.
[104] Bermúdez, A., and Webber, N. An asset based model of defaultable convertible bonds with endogenised recovery. Working Paper, Cass Business School, London, 2004.
[105] Vasicek, O.A. An equilibrium characterisation of the term structure. Journal of Financial Economics, 5:177–188 (1977).
[106] Hull, J.C., and White, A. Pricing interest rate derivative securities. Review of Financial Studies 3:573–592 (1990).
[107] Bermúdez, A., and Nogueiras, M.R. Numerical solution of two-factor models for valuation of financial derivatives. Mathematical Models and Methods in Applied Sciences 14(2):295–327 (February 2004).
[108] Davis, M., and Lischka, F. Convertible bonds with market risk and credit risk. Studies in Advanced Mathematics. Somerville, MA: American Mathematical Society/International Press, 2002:45–58.
[109] Black, F. Derman, E., and Toy, W. A one factor model of interest rates and its application to Treasury bond options. Financial Analyst Journal 46:33–39 (1990).
[110] Zvan, R., Forsyth, P.A., and Vetzal, K.R. A general finite element approach for PDE option pricing model. Proceedings of Quantitative Finance 98, (1998).
[111] Yigitbasioglu, A.B. Pricing convertible bonds with interest rate, equity and FX risk. ISMA Center Discussion Papers in Finance, University of Reading (June 2002).
[112] Cheung, W., and Nelken, I. Costing the converts. In Over the Rainbow, vol. 46, London: Risk Publications, 1995: 313–317.
[113] Kalotay, A.J., Williams, G.O., and Fabozzi, F.J. A model for valuing bonds and embedded options. Financial Analyst Journal (May–June 1993): 35–46.
[114] Takahashi, A., Kobayahashi, T., and Nakagawa, N. Pricing convertible bonds with default risk: A Duffie-Singleton approach. Journal of Fixed Income, 11(3):20–29, (2001).
[115] Tseveriotis, K., and Fernandes, C. Valuing convertible bonds with credit risk. Journal of Fixed Income, 8(2):95–102 (September 1998).
[116] Ayache, E., Forsyth, P.A., and Vetzal, K.R. Next generation models for convertible bonds with credit risk. Wilmott Magazine 68–77 (December 2002).
[117] Ayache, E., Forsyth, P.A., and Vetzal, K.R. Valuation of convertible bonds with credit risk. Journal of Derivatives 11(1):9–29, (April 2003).
[118] Protter, P. Stochastic Integration and Differential Equations, vol. 21 of Applications of Mathematics, 3rd ed. Heidelberg: Springer-Verlag, 1995.
[119] Jacod, J., and Shiryaev, A.N. Limit Theorems for Stochastic Processes. Berlin: Springer, (1988).
[120] Olsen, L. Convertible bonds: A technical introduction. Research tutorial, Barclays Capital, 2002.
[121] Das, S.R., and Sundaram, R.K. A simple model for pricing securities with equity, interest-rate and default risk. Defaultrisk.com, 2004.
[122] Andersen, L., and Buffum, D. Calibration and implementation of convertible bond models. Journal of Computational Finance 7(2):1–34 (2003).
[123] Kiesel, R., Perraudin, W., and Taylor, A. Credit and interest rate risk. In M.H.A. Dempster, ed., Risk Management: Value at Risk and Beyond. New York: Cambridge University Press, 2002.
[124] Bielecki, T. and Rutkowski, M. Credit Risk: Modeling, Valuation and Hedging. Heidelberg: Springer Finance, 2002.
[125] Bakshi, G., Madan, D., and Zhang, F. Understanding the role of recovery in default risk models: Empirical comparisons and implied recovery rates. Working Paper, University of Maryland, November 2001.
[126] Unal, H., Madan, D., and Guntay, L. A simple approach to estimate recovery rates with APR violation from debt spreads. Working Paper, University of Maryland, February 2001.
[127] Hamilton, D.T., Gupton, G., and Berthault, A. Default and recovery rates of corporate bond issuers: 2000. Special comment, Moody's Investor Service, Global Credit Research, February 2000.
[128] Altman, E.I., Resti, A., and Sirone, A. Analysing and explaining default recovery rates. Report, ISDA, Stern School of Business, New York University, December 2001.
[129] Realdon, M. Convertible subordinated debt valuation and “conversion in distress.” Working Paper, Department of Economics and Related Studies, University of York, 2003.
[130] Overhaus et al., Modelling and Hedging Equity Derivatives, Risk Books, 1999.
[131] JeanBlanc, M. Modelling of Default Risk. Mathematical Tools, 2000.”
[132] Duffie Khan A yield factor model of interest rates, 1996.
[133] Nelson, R.B. An introduction to copulas. Mathematical Tools, 2000, p.91.
[134] Black, F., and Scholes, M. The pricing of options on corporate liabilities. Journal of Political Economy 81: 637–659 (1973).
[135] Heston, S.L. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6: 327–343 (1993).
[136] Dupire, B. Pricing with a smile. Risk 7: 18–20 (January 1994).
[137] Revuz, D., and Yor, M. Continuous Martingales and Brownian Motion, 3rd ed. Heidelberg: Springer, 1998:375 (theorem (2.1)).
[138] Gyöngy, L. Mimicking the one-dimensional marginal distributions of processes having an Ito differential. Probability Theory and Related Fields 71: 501–516 (1986).
[139] Clewlow, L., and Strickland, C. Implementing Derivative Models. Hoboken, NJ: Wiley, 1998.
[140] Wilmott, P., Dewynne, J., and Howison, J. Option Pricing: Mathematical Models and Computation. New York: Oxford Financial Press, 1993.
[141] Vázquez, C. An upwind numerical approach for an American and European option pricing model. Applied Mathematics and Computation 273–286 (1998).
[142] Bermúdez, A., and Moreno, C. Duality methods for solving variational inequalities. Computer Mathematics with Applications, 7:43–58 (1981).
[143] Hull, J.C., and White, A. Efficient procedures for valuing European and American path dependent options. Journal of Derivatives 1:21–31 (Fall 1993).
[144] Ewing, R.E., and Wang, H. A summary of numerical methods for time-dependent advection-dominated partial differential equations. Journal of Computational and Applied Mathematics 128:423–445 (2001).
[145] Pironneau, O. On the transport-diffusion algorithm and its application to the navier-stokes equations. Journal of Numerical Mathematics 38(3):309–332 (1982).
[146] Douglas, J., and Russell, T. Numerical methods for convection dominated diffusion problems based on combining methods of characteristics with finite element methods or finite differences. SIAM Journal on Numerical Analysis 19(5):871 (1982).
[147] Baker, M.D., Suli, E., and Ware, A.F. Stability and convergence of the spectral Lagrange-Galerkin method for mixed periodic/non-periodic convection dominated diffusion problems. IMA Journal of Numerical Analysis, 19:637–663 (1999).
[148] Baranger, D., Esslaoui, D., and Machmoum, A. Error estimate for convection problem with characteristics method. Numerical Algorithms, 21:49–56 (1999).
[149] Baranger, J., and Machmoum, A. A “natural” norm for the method of charactersistics using discontinuous finite elements: 2d and 3d case. Mathematical Modeling and Numerical Analysis 33:1223–1240 (1999).
[150] Boukir, K., Maday, Y., Metivet, B., and Razanfindrakoto, E. A high order characteristics/finite element method for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids 25:1421–1454 (1997).
[151] Rui, H., and Tabata, M. A second order characteristic finite element scheme for convection-diffusion problems. Journal of Numerical Mathematics 92:161–177 (2002).
[152] Bermúdez, A., Nogueiras, M.R., and Vázquez, C. 2006. Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part I: Time discretization. To appear in Siam Journal on Numerical Analysis.
[153] Ciarlet, P.G., and Lions, J.L. eds. Handbook of Numerical Analysis, vol. 1 of North-Holland. Amsterdam: Elsevier Science, 1989.
[154] Kangro, R., and Nicolaides, R. Far field boundary conditions for Black-Scholes equations. SIAM Journal on Numerical Analysis 38(4):1357–1368 (2000).
[155] Barles, G., and Souganidis, P.E. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotyc Analysis 4(4):271–283 (1991).
[156] Duvaut, G., and Lions, J.L. Les inéquations en mécanique et en physique. In Travaux et Recherches Mathématiques, vol. 21. Paris: Dunod, 1972.
[157] Glowinski, R., Lions, J.L., and Trémolières, R. Analyse Numérique Des Inéquations Variationnelles. Paris: Dunod, 1973.
[158] Bensoussan, A. and Lions, J.L. Applications Des Inéquations Variationneles En Contrôle Stochastique. Paris: Dunod, 1978.
[159] Jaillet, J., Lamberton, D., and Lapeyre, B. Variational inequalities and the pricing of merican options. Acta Applicandae Mathematicae 21:263–289 (1990).
[160] Crandall, M.G., Ishii, H., and Lions, P.L. User's guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27(1):1–67 (1992).
[161] Lions, P.L. Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, part 2: Viscosity solutions and uniqueness. Communications in Partial Differential Equations 8(11):1229–1276 (1983).
[162] Barles, G., Daher, C.H., and Souganidis, P. Convergence of numerical schemes for parabolic equations arising in finance theory. Mathematical Models and Methods in Applied Science 5:125–143 (1995).
[163] Wilmott, P. Derivatives: The Theory and Practice of Financial Engineering. Hoboken, NJ: Wiley, 1998.
[164] Clarke, N., and Parrot, K. Multigrid American option pricing with stochastic volatility. Applied Mathematical Finance 6:177–195 (1999).
[165] Forsyth, P.A., and Vetzal, K. Quadratic convergence for valuing american options using a penalty method. SIAM Journal on Scientific Computation 23:2096–2123 (2002).
[166] Parés, C., Castro, M., and Macías, J. On the convergence of the Bermúdez-Moreno algorithm with constant parameters. Numerische Mathematik 92:113–128 (2002).
[167] Morton, K.W. Numerical Solution of Convection-Diffusion Problems. Boca Raton, FL: Chapman & Hall, 1996.
[168] Pironneau, O., and Hetch, F. Mesh adaptation for the Black and Scholes equations. Journal of Numerical Mathematics, 8(1):25–35 (2000).
[169] Figlewski, S., and Gao, B. The adaptive mesh model: A new approach to efficient option pricing. Working Paper, Stern School of Business, New York University, 1997.
[170] Zvan, R., Forsyth, P.A., and Vetzal, K.R. PDE methods for pricing barrier options. Journal of Economic Dynamics and Control, 24 (2000).
[171] Pooley, D.M., Forsyth, P.A., Vetzal, K.R., and Simpson, R.B. Unstructured meshing for two asset barrier options. Applied Mathematical Finance 7:33–60 (2000).
[172] Winkler, G., Apel, T., and Wystup, U. Valuation of options in heston's stochastic volatility model using finite element methods. In Foreign Exchange Risk. London: Risk Publications, 2001.
[173] Topper, J. Finite element modeling of exotic options. Discussion paper 216, Department of Economics, University of Hannover, December 1998.
[174] D'Halluin, Y., Forsyth, P., Vetzal, K., and Labahn, G. A numerical PDE approach for pricing callable bonds. Applied Mathematical Finance, 8:49–77 (2001).
[175] Zvan, R., Forsyth, P.A., and Vetzal, K.R. A finite volume approach for contingent claims valuation. IMA Journal of Numerical Analysis 21:703–721 (2001).
[176] Zvan, R., Forsyth, P.A., and Vetzal, K.R. Convergence of lattice and PDE methods valuing path dependent options with interpolation. Review of Derivatives Research 5:273–314, 2002.
[177] Zvan, R., Forsyth, P.A., and Vetzal, K.R. Robust numerical methods for PDE models of Asian options. Journal of Computational Finance 1:39–78 (1998).
[178] Zvan, R., Forsyth, P.A., and Vetzal, K.R. A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Applied Mathematical Finance 6:87–106 (1999).
[179] Selmin, V., and Formaggia, L. Unified construction of finite element and finite volume discretisation for compressible flows. International Journal for Numerical Methods in Engineering 39:1–32, (1996).
[180] Ciarlet, P.G. The Finite Element Method for Elliptic Problems, vol. 4 of Studies in Mathematics and its Applications. Amsterdam: North-Holland, 1978.
[181] Zienkiewicz, O.C., Taylor, R.L., and Zhu, J.Z. The Finite Element Method: Its Basis and Fundamentals. Amsterdam: 6th ed., Elsevier Butterworth-Heinemann, 2005.
[182] Suli, E. Stability and convergence of the Lagrange-Galerkin method with nonexact integration. In J. R. Whiteman, ed. The Proceedings of the Conference on the Mathematics of Finite Elements and Applications, MAFELAP VI. Academic Press: London, 1998: 435–442.
[183] Bause, M., and Knabner, P. Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems. SIAM Journal of Numerical Analysis 39:1954–1984 (2002).
[184] Ewing, R.E., and Russel, T.F. Multistep Galerkin methods along characteristics for convection-diffusion problems. In R. Vichtneveski and R.S. Stepleman, ed. Advances in Computer Methods for Partial Differential Equations IV. IMACS Publications, 1981: 28–36.
[185] Boukir, K., Maday, Y., Metivet, B., and Razafindrakoto, E. A high-order characteristics/finite element method for incompressible Navier-Stoke equations. International Journal on Numerical Methods in Fluids 25:1421–1454 (1997).
[186] Priestley, A. Exact projections and the Lagrange-Galerkin method: A realistic alternative to quadrature. Journal of Computational Physics, 112:316–333 (1994).
[187] Morton, K.W., Priestley, A., and Suli, E. Stability of the Lagrange-Galerkin method with nonexact integration. Mathematical Modeling and Numerical Analysis 22:625–653, 1988.
[188] Broadie, M., and Glasserman, P. (1973). Pricing American-style securities using simulation. Journal of Economic Dynamics and Control 21(8–9): 1323–1352 (1997).
[189] Longstaff, F., and Schwartz, E. Valuing American options by simulation: A simple least-squares approach. Review Financial Studies 14:113–148 (2001).
[190] Fries, Christian P. Foresight bias and suboptimality correction in Monte-Carlo pricing of options with early exercise: Classification, calculation and removal. http://www.christian-fries.de/finmath/foresightbias.
[191] Tilley, J.A. Valuing American options in a path simulation model. Transactions of the Society of Actuaries 45:83–104 (1993).
[192] Rogers, C. Monte Carlo valuation of American options. Mathematical Finance 12:271–286 (2002).
[193] Andersen, L., and Broadie, M. “A primal-dual simulation algorithm for pricing multidimensional American options. Management Science 50(9): 1222–1234 (2004).
[194] Andersen, L.B.G. A simple approach to the pricing of Bermudan swaptions in the multifactor LIBOR market model. (March 5, 1999). http://ssm.com/abstract=155208.
[195] Windcliff, H., Forsyth, P., and Vetzal, K. Asymptotic boundary conditions for the Black-Scholes equation. Working Paper, University of Waterloo, October 2001.
[196] Topper, J. 2005. Financial Engineering with Finite Elements. Hoboken, NJ: Wiley-Finance.
[197] Bermúdez, A., Nogueiras, M.R., and Vázquez, C. 2006 Numerical analysis of convection-diffusion-reaction problems with higher order characteristic/finite elements. Part II: Fully discretized scheme and quadrature formulas. To appear in Siam Journal on Numerical Analysis.