Pascucci, Andrea and Foschi, Paolo (2006): Path dependent volatility.

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Abstract
We propose a general class of nonconstant volatility models with dependence on the past. The framework includes pathdependent volatility models such as that by Hobson&Rogers and also path dependent contracts such as options of Asian style. A key feature of the model is that market completeness is preserved. Some empirical analysis, based on the comparison with the performance of standard local volatility and Heston models, shows the effectiveness of the path dependent volatility.
Item Type:  MPRA Paper 

Institution:  Università di Bologna 
Original Title:  Path dependent volatility 
Language:  English 
Keywords:  option pricing; stochastic volatility; path dependent option 
Subjects:  G  Financial Economics > G1  General Financial Markets C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods 
Item ID:  973 
Depositing User:  Andrea Pascucci 
Date Deposited:  30 Nov 2006 
Last Modified:  29 Sep 2019 04:45 
References:  [1] C. Alexander and L. Nogueira, Hedging options with scaleinvariant models, tech. report, ICMA Centre, University of REading, June 2006. [2] E. Barucci, S. Polidoro, and V. Vespri, Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci., 11 (2001), pp. 475–497. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), pp. 637–654. [4] V. Blaka Hallulli and T. Vargiolu, Financial models with dependence on the past: a survey, Applied and Industrial Mathematics in Italy, M. Primicerio, R. Spigler, V. Valente, editors, Series on Advances in Mathematics for Applied Sciences, World Scientific 2005, 69 (2005). [5] P. Carr and D. Madan, Option pricing and the fast fourier transform, Journal of Computational Finance, 2 (1999), pp. 61–73. [6] C. Chiarella and K. Kwon, A complete Markovian stochastic volatility model in the HJM framework, AsiaPacific Financial Markets, 7 (2000), pp. 293–304. [7] R. Cont, Model uncertainty and its impact on the pricing of derivative instruments, Math. Finance, 16 (2006), pp. 519–547. [8] M. Di Francesco and A. Pascucci, On the complete model with stochastic volatility by Hobson and Rogers, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), pp. 3327– 3338. [9] , On a class of degenerate parabolic equations of Kolmogorov type, AMRX Appl. Math. Res. Express, (2005), pp. 77–116. [10] B. Dumas, J. Fleming, and R. E. Whaley, Implied volatility functions: empirical tests, J. Finance, 53 (1998), pp. 2059–2106. [11] B. Dupire, Pricing and hedging with smiles, in Mathematics of derivative securities (Cambridge, 1995), vol. 15 of Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997, pp. 103– 111. [12] G. Fig`aTalamanca and M. L. Guerra, Complete models with stochastic volatility: further implications, Working Paper, Universit`a della Tuscia, Facolt`a di Economia, 5 (2000). [13] P. Foschi and A. Pascucci, Calibration of the Hobson&Rogers model: empirical tests., Preprint AMS Acta, University of Bologna, (2005). [14] M. Hahn, W. Putsch¨ogl, and J. Sass, Portfolio optimization with nonconstant volatility and partial information, preprint, (2006). [15] S. Heston, A closedform solution for options with stochastic volatility with applications to bond and currency options., Review of Financial Studies, 6 (1993), pp. 327–343. [16] D. G. Hobson and L. C. G. Rogers, Complete models with stochastic volatility, Math. Finance, 8 (1998), pp. 27–48. [17] L. H¨ormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), pp. 147–171. [18] F. Hubalek, J. Teichmann, and R. Tompkins, Flexible complete models with stochastic volatility generalising HobsonRogers, working paper, (2004). [19] R. C. Merton, Theory of rational option pricing, Bell J. Econom. and Management Sci., 4 (1973), pp. 141–183. [20] A. Platania and L. C. G. Rogers, Putting the Hobson&Rogers model to the test, working paper, (2006). [21] S. Polidoro, Uniqueness and representation theorems for solutions of KolmogorovFokker Planck equations, Rend. Mat. Appl. (7), 15 (1995), pp. 535–560. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/973 