Financial derivatives and Lie symmetries

References

• Andriopoulos, K., Dimas, S., Leach, P.G.L. & Tsoubelis, D. 2009. On the systematic approach to the classification of differential equations by group theoretical methods. Journal of Computational and Applied Mathematics 230: 224–232.10.1016/j.cam.2008.11.002
   Web of Science ®Google Scholar
• Bernstein, P.L. 1996. Against the Gods: The Remarkable Story of Risk. New York, NY, John Wiley and Sons.
   Google Scholar
• Black, F. & Scholes, M. 1972. The valuation of option contracts and a test of market efficiency. Journal of Finance 27: 399–417.10.2307/2978484
   Web of Science ®Google Scholar
• Black, F. & Scholes, M. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654.10.1086/260062
   Web of Science ®Google Scholar
• Brown, R. 1828. A brief account of microscopical observations made on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Philosophical Magazine 4: 161–173. (Also to be found in German as an article in Annalen der Physik und Chemie 14: 294–313 of the same year.)
   Google Scholar
• Burghes, D.N. & Graham, A. 1980. Introduction to Control Theory including Optimal Control. New York, NY, Ellis Horwood.
   Google Scholar
• Cairns, A. 2004. Interest Rate Models: An Introduction. Princeton, NJ, Princeton University Press.
   Google Scholar
• Caister, N.C., O'hara, J.G. & Govender, K.S. 2010. Solving the Asian option PDE using Lie symmetry methods. International Journal of Theoretical and Applied Finance 13: 1265–1277.10.1142/S0219024910006194
   Google Scholar
• Cox, J.C., Ingersoll, J.E. & Ross, S.A. 1985. An intertemporal general equilibrium model of asset prices. Econometrica 53: 363–384.10.2307/1911241
   Web of Science ®Google Scholar
• Dimas, S. & Tsoubelis, D. 2005. SYM: A new symmetry-finding package for Mathematica. In Ibragimov, N.H., Sophocleous, C. & Damianou, P.A. (Eds.) Group Analysis of Differential Equations. Nicosia, University of Cyprus. pp. 64–70.
   Google Scholar
• Dimas, S. & Tsoubelis, D. 2006. A new Mathematica-based program for solving overdetermined systems of PDEs. Paper presented at the 8th International Mathematica Symposium (IMS 2006), Palais des Papes, Avignon, France, 19–23 June.
   Google Scholar
• Dimas, S. 2008. Partial differential equations: Algebraic computing and nonlinear systems. Unpublished Ph.D. thesis, University of Patras, Greece.
   Google Scholar
• Dimas, S., Andriopoulos, K., Tsoubelis, D. & Leach, P.G.L. 2009. Complete specification of some partial differential equations that arise in Financial Mathematics. Journal of Nonlinear Mathematical Physics 16 s01: 73–92.10.1142/S1402925109000339
   Web of Science ®Google Scholar
• Gazizov, R. & Ibragimov, N.H. 1998. Lie symmetry analysis of differential equations in finance. Nonlinear Dynamics 17: 387–407.10.1023/A:1008304132308
   Web of Science ®Google Scholar
• Henderson, V. & Hobson, D.G. 2002. Real options with constant relative risk aversion. Journal of Economics Dynamics and Control 27: 329–355.10.1016/S0165-1889(01)00052-5
   Web of Science ®Google Scholar
• Henderson, V. 2002. Valuation of claims on nontraded assets using utility maximisation. Mathematical Finance 12: 351–373.10.1111/j.1467-9965.2002.tb00129.x
   Web of Science ®Google Scholar
• Hull, J. & White, A. 1990. Pricing interest-rate-derivative securities. The Review of Financial Studies 3: 573–592.10.1093/rfs/3.4.573
   Web of Science ®Google Scholar
• Ibragimov, N.H. & Wafo, S.C. 1997. Solution of the Cauchy problem for the Black-Scholes equation using its symmetries. Paper presented at the International Conference for Modern Group Analysis VII: Lie Groups and Contemporary Symmetry Analysis, Sophus Lie Conference Center, Nordfjordeid, Norway, 30 June–5 July.
   Google Scholar
• Itô, K. 1944. Stochastic integral. Proceedings of the Imperial Academy 20: 519–524.10.3792/pia/1195572786
   Google Scholar
• Jouini, E., Cvitanic, J. & Musiela, M. (Eds.) 2001. Option Pricing, Interest Rates and Risk Management. Cambridge, Cambridge University Press.
   Google Scholar
• Korniotis, M. 2009. A multifactor quadratic stochastic volatility model with applications in finance and insurance. Unpublished Ph.D. thesis, University of California, Irvine, US.
   Google Scholar
• Kushner, H.J. 1971. Introduction to Stochastic Control. New York, NY, Holt.
   Google Scholar
• Lie, S. 1881. Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch for Math 6: 328–368. (Translation by Ibragimov, N.H.: LIE, S. 1994. On the integration of a class of linear differential equations by means of definite integrals. In Ibragimov, N.H., CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 2. Boca Raton, FL, CRC Press. pp. 473–508.)
   Google Scholar
• Lie, S. 1884. Klassifikation und Integration von gewohnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten IV. Archiv for Matematik og Naturvidenskab 9: 431–448. (Reprinted in Lie's Ges. Abhandl. 5, paper XVI, 1924, 432–446.)
   Google Scholar
• Lie, S. 1967. Differentialgleichungen. New York, NY, Chelsea.
   Google Scholar
• Lie, S. 1970a. Theorie der Transformationsgruppen, vol 1. New York, NY, Chelsea.
   Google Scholar
• Lie, S. 1970b. Theorie der Transformationsgruppen, vol 2. New York, NY, Chelsea.
   Google Scholar
• Lie, S. 1970c. Theorie der Transformationsgruppen, vol 3. New York, NY, Chelsea.
   Google Scholar
• Lie, S. 1971. Vorlesungen über Continuerliche Gruppen mit geometrischen und anderen Anwendungen. New York, NY, Chelsea.
   Google Scholar
• Lie, S. 1977. Ber ührungstransformation. New York, NY, Chelsea.
   Google Scholar
• Merton, R.C. 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4: 141–183.
   Google Scholar
• Merton, R.C. 1974. On the pricing of corporate data: The risk structure of interest rates. Journal of Finance 29: 449–470.
   Web of Science ®Google Scholar
• Morozov, V.V. 1958. Classification of six-dimensional nilpotent Lie algebras. Izvestia Vysshikh Uchebn Zavendeniĭ Matematika 5: 161–171.
   Google Scholar
• Mubarakzyanov, G.M. 1963a. Classification of real structures of five-dimensional Lie algebras. Izvestia Vysshikh Uchebn Zavendeniĭ Matematika 34: 99–106.
   Google Scholar
• Mubarakzyanov, G.M. 1963b. Classification of solvable six-dimensional Lie algebras with one nilpotent base element. Izvestia Vysshikh Uchebn Zavendeniĭ Matematika 35: 104–116.
   Google Scholar
• Mubarakzyanov, G.M. 1963c. On solvable Lie algebras. Izvestia Vysshikh Uchebn Zavendeniĭ Matematika 32: 114–123.
   Google Scholar
• O'hara, J.G., Sophocleous, C. & Leach, P.G.L. 2013a. Application of Lie point symmetries to the resolution of certain problems in financial mathematics with a terminal condition. Journal of Engineering Mathematics 82: 67–75.10.1007/s10665-012-9595-4
   Web of Science ®Google Scholar
• O'hara, J.G., Sophocleous, C. & Leach, P.G.L. 2013b. Symmetry analysis of a model for the exercise of a barrier option. Communications in Nonlinear Science and Numerical Simulation 18: 2367–2373.10.1016/j.cnsns.2012.12.027
   Web of Science ®Google Scholar
• Schwartz, E.S. 1997. The stochastic behaviour of commodity prices: Implications for valuation and hedging. The Journal of Finance 52: 923–973.10.1111/j.1540-6261.1997.tb02721.x
   Web of Science ®Google Scholar
• Sinkala, W., Leach, P.G.L. & O'hara J.G. 2008a. Invariance properties of a general bond-pricing equation. Journal of Differential Equations 244: 2820–2835.10.1016/j.jde.2008.02.044
   Web of Science ®Google Scholar
• Sinkala, W., Leach, P.G.L. & O'hara J.G. 2008b. Zero-coupon bond prices in the Vasicek and CIR models: Their computation as group-invariant solutions. Mathematical Methods in the Applied Sciences 31: 665–678.10.1002/mma.935
   Web of Science ®Google Scholar
• Vasicek, O. 1977. An equilibrium characterisation of the term structure. Journal of Financial Economics 5: 177–188.10.1016/0304-405X(77)90016-2
   Web of Science ®Google Scholar