Path dependent volatility

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

Abstract

We propose a general class of non-constant volatility models with dependence on the past. The framework includes path-dependent 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
• 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

References

[1] C. Alexander and L. Nogueira, Hedging options with scale-invariant 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, Asia-Pacific 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`a-Talamanca 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 non-constant volatility and partial information, preprint, (2006). [15] S. Heston, A closed-form 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 Hobson-Rogers, 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 Kolmogorov-Fokker- Planck equations, Rend. Mat. Appl. (7), 15 (1995), pp. 535–560.