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On a Heath-Jarrow-Morton approach for stock options. (English) Zbl 1390.91302
Summary: This paper aims at transferring the philosophy behind Heath-Jarrow-Morton to the modelling of call options with all strikes and maturities. Contrary to the approach by R. Carmona and S. Nadtochiy [ibid. 13, No. 1, 1–48 (2009; Zbl 1199.91202)] and related to their recent contribution [ibid. 16, No. 1, 63–104 (2012; Zbl 1259.91047)], the key parameterisation of our approach involves time-inhomogeneous Lévy processes instead of local volatility models. We provide necessary and sufficient conditions for absence of arbitrage. Moreover, we discuss the construction of arbitrage-free models. Specifically, we prove their existence and uniqueness given basic building blocks.

91G20 Derivative securities (option pricing, hedging, etc.)
60G51 Processes with independent increments; Lévy processes
91G30 Interest rates, asset pricing, etc. (stochastic models)
Full Text: DOI
[1] Albert, A.: Regression and the Moore-Penrose Pseudoinverse. Academic Press, New York (1972) · Zbl 0253.62030
[2] Barndorff-Nielsen, O.; Shephard, N., Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc., Ser. B, Stat. Methodol., 63, 167-241, (2001) · Zbl 0983.60028
[3] Belomestny, D.; Reiß, M., Spectral calibration of exponential Lévy models, Finance Stoch., 10, 449-474, (2006) · Zbl 1126.91022
[4] Bennani, N.: The forward loss model: a dynamic term structure approach for the pricing of portfolios of credit derivatives. Technical report (2005). Available at http://www.defaultrisk.com/pp_crdrv_95.htm · Zbl 1224.91156
[5] Bühler, H., Consistent variance curve models, Finance Stoch., 10, 178-203, (2006) · Zbl 1101.91031
[6] Carmona, R.; Carmona, R. (ed.); etal., HJM: a unified approach to dynamic models for fixed income, credit and equity markets, 2005, Berlin
[7] Carmona, R.; Nadtochiy, S., Local volatility dynamic models, Finance Stoch., 13, 1-48, (2009) · Zbl 1199.91202
[8] Carmona, R.; Nadtochiy, S., Tangent models as a mathematical framework for dynamic calibration, Int. J. Theor. Appl. Finance, 14, 107-135, (2011) · Zbl 1208.91169
[9] Carmona, R.; Nadtochiy, S., Tangent Lévy market models, Finance Stoch., 16, 63-104, (2012) · Zbl 1259.91047
[10] Carr, P.; Geman, H.; Madan, D.; Yor, M., Stochastic volatility for Lévy processes, Math. Finance, 13, 345-382, (2003) · Zbl 1092.91022
[11] Cont, R.; Durrleman, V.; Fonseca, I., Stochastic models of implied volatility surfaces, Econ. Notes, 31, 361-377, (2002)
[12] Cont, R., Tankov, P.: Financial Modelling with Jump Processes. CRC Press, Boca Raton (2004) · Zbl 1052.91043
[13] Davis, M.; Hobson, D., The range of traded option prices, Math. Finance, 17, 1-14, (2007) · Zbl 1278.91158
[14] Ethier, S., Kurtz, T.: Markov Processes. Characterization and Convergence. Wiley, New York (1986)
[15] Filipović, D., Time-inhomogeneous affine processes, Stoch. Process. Appl., 115, 639-659, (2005) · Zbl 1079.60068
[16] Filipović, D.; Tappe, S.; Teichmann, J., Term structure models driven by Wiener processes and Poisson measures: existence and positivity, SIAM J. Financ. Math., 1, 523-554, (2010) · Zbl 1207.91068
[17] Filipović, D.; Tappe, S.; Teichmann, J., Invariant manifolds with boundary for jump-diffusions, Electron. J. Probab., 51, 1-28, (2014) · Zbl 1301.60072
[18] Heath, D.; Jarrow, R.; Morton, A., Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 60, 77-105, (1992) · Zbl 0751.90009
[19] Jacod, J.; Protter, P., Risk neutral compatibility with option prices, Finance Stoch., 14, 285-315, (2010) · Zbl 1224.91156
[20] Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003) · Zbl 1018.60002
[21] Kallsen, J., \(σ\)-localization and \(σ\)-martingales, Theory Probab. Appl., 48, 152-163, (2004) · Zbl 1069.60042
[22] Kallsen, J.; Kabanov, Y. (ed.); Liptser, R. (ed.); Stoyanov, J. (ed.), A didactic note on affine stochastic volatility models, 343-368, (2006), Berlin · Zbl 1104.60024
[23] Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004) · Zbl 1041.60005
[24] Schönbucher, P.: Portfolio losses and the term structure of loss transition rates: a new methodology for the pricing of portfolio credit derivatives. Technical report (2005). Available at http://www.econbiz.de/archiv1/2008/50106_portfolio_credit_derivatives.pdf
[25] Schweizer, M.; Wissel, J., Arbitrage-free market models for option prices: the multi-strike case, Finance Stoch., 12, 469-505, (2008) · Zbl 1199.91218
[26] Schweizer, M.; Wissel, J., Term structures of implied volatilities: absence of arbitrage and existence results, Math. Finance, 18, 77-114, (2008) · Zbl 1138.91481
[27] Sidenius, J.; Piterbarg, V.; Andersen, L., A new framework for dynamic credit portfolio loss modelling, Int. J. Theor. Appl. Finance, 11, 163-197, (2008) · Zbl 1211.91246
[28] Wissel, J.: Arbitrage-free market models for liquid options. Ph.D. thesis, ETH Zürich (2008). Available at doi:10.3929/ethz-a-005559619 · Zbl 1069.60042
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