Computational Approaches to Economic Problems pp 127-141 | Cite as
Neural Networks for Contingent Claim Pricing via the Galerkin Method
Chapter
Abstract
We use Neural Networks as a Semi-NonParametric technique to approximate, by means of the Galerkin method, contingent claim prices defined by a no-arbitrage Partial Differential Equation. The Neural Networks’ weights are determined as to satisfy the no-arbitrage Partial Differential Equation. A general solution procedure is developed for European Contingent Claims. The main feature of the Neural Network is that its weights are time varying, they change as the time to expiration of the claim changes. The method has been evaluated for option pricing in the standard Black and Scholes framework.
Keywords
Asset Price Galerkin Method Option Price Trial Function Implied Volatility
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
- Bansal, R. and S. Viswanathan, 1993, ‘No arbitrage and arbitrage pricing: A new approach’, Journal of Finance XLVIII(4), 1231–1262.CrossRefGoogle Scholar
- Black, E. and M. Scholes, 1973, ‘The pricing of options and corporate liabilities’, Journal of Political Economy 81, 637–654.CrossRefGoogle Scholar
- Courant, R. and D. Hilbert, 1989, Methods of Mathematical Physics, New York: Wiley.CrossRefGoogle Scholar
- Duffle, D., 1992, Dynamic Asset Pricing Theory, Princeton: Princeton University Press.Google Scholar
- Dupire, B., 1993, ‘Pricing and hedging with smiles’, in Proceedings of the AFFI Conference, La-Baule.Google Scholar
- Fletcher, C., 1984, Computational Galerkin Methods, New York: Springer-Verlag.CrossRefGoogle Scholar
- Fletcher, C., 1991, Computational Techniques for Fluid Dynamics, Vols. I—II, New York: Springer-Verlag.CrossRefGoogle Scholar
- Gallant, A. and G. Tauchen, 1989, ‘Semi-nonparametric estimation of conditionally constraint heterogenous processes: Asset pricing implications’, Econometrica 57, 1091–1120.CrossRefGoogle Scholar
- Gallant, A. and H. White (1990), ‘Connectionist non-parametric regression: Multi-layer feedforward networks can learn arbitrary mappings’, Neural Networks 3, 535–550.CrossRefGoogle Scholar
- Grenander, U. (1981), Abstract Inference, New York: WileyGoogle Scholar
- Harrison, M.J. and D.M. Kreps, 1979, ‘Martingales and arbitrages in Multiperiod securities markets’, Journal of Economic Theory 20, 381–408.CrossRefGoogle Scholar
- Harrison, M.J. and S. Pliska, 1981, ‘Martingales and stochastic integrals in the theory of continuous trading’, Stochastic Processes and Applications 11, 215–260.CrossRefGoogle Scholar
- Hutchinson, J.M., A.W. Lo, and T. Poggio, 1994, ‘A nonparametric approach to pricing and hedging derivatives securities via learning networks’, Journal of Finance 49 (3), 851–889.CrossRefGoogle Scholar
- Kantorovich, L.V. and V.I. Krilov, 1958, Approximate Methods of Higher Analysis, New York: Noordhoff, Gröningen and Interscience.Google Scholar
- Madan, D.B. and F. Milne, 1994, ‘Contingent claims valued and hedged by pricing and investing in a basis’, Mathematical Finance 4 (3), 223–245.CrossRefGoogle Scholar
- Meade, A. and A. Fernandez, 1994a, ‘The numerical solution of linear ordinary differential equations by feedforward neural networks’, Mathematical and Computer Modelling 19, 1–25.CrossRefGoogle Scholar
- Meade, A. and A. Fernandez, 1994b, ‘Solution of nonlinear ordinary differential equations by feedforward neural networks’, Mathematical and Computer Modelling, to appear.Google Scholar
- Mikhlin, S.G., 1971, The Numerical Performance of Variational Methods, Wolters-Noordhoff.Google Scholar
- Panton, R.Z. and H.B. Sallee, 1975, Computers and Fluids, Vol. 3, pp. 257–269, New York: Springer-Verlag.Google Scholar
- Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, 1992, Numerical Recipies in C, Cambridge: Cambridge University Press.Google Scholar
- Rumelhart, D.E., G.E. Hintos, R.J. Williams, 1986, ‘Learning internal representation by error propagation’, in Parallel Distributed Processing: Exploration in the Microstructure of Cognition. Vol. 1: Foundations, D. Rumelhart and J. McClelland (Eds), Cambridge, MA: MIT Press.Google Scholar
Copyright information
© Springer Science+Business Media Dordrecht 1997