IJPAM: Volume 109, No. 3 (2016)
AND SCHOLES APPROACH FOR THE LOG-RETURNS
AND VOLATILITY SMILE FITTING
Department of Computer Science
University of Verona
Strada le Grazie, 15-37134, Verona, ITALY
Abstract. In the present paper we perform a comparison between the standard Black and Scholes model and the Merton jump-diffusion one, from the point of view of the study of the leptokurtic feature of log-returns and also concerning the volatility smile fitting. Provided results are obtained by calibrating on market data and by mean of numerical simulations which clearly show how the jump-diffusion model outperforms the classical geometric Brownian motion approach.
Received: August 3, 2016
Revised: September 16, 2016
Published: October 1, 2016
AMS Subject Classification: 60H15, 60H35, 91B60, 91G20, 91G60
Key Words and Phrases: Black and Scholes model, Merton model, stochastic differential equations, log-returns, volatility smile
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- Black, F., and Scholes, M., The pricing of options and corporate liabilities, The Journal of Political Economy, 81, No. 3 (1973), 637-654, 10.1086/260062.
- Cordoni, F., and Di Persio, L.,Transition density for cir process by lie symmetries and application to zcb pricing(2013), International Journal of Pure and Applied Mathematics, 88 (2), pp. 239-246.
- Cordoni, F., and Di Persio, L.,Small noise expansion for the Lévy perturbed Vasicek model(2015), International Journal of Pure and Applied Mathematics, 98 (2), pp. 291-301.
- Cordoni, F., and Di Persio, L.,Invariant measure for the Vasicek interest rate model in the Heath-Jarrow-Morton-Musiela framework(2015), Infinite Dimensional Analysis, Quantum Probability and Related Topics, 18 (3), art. no. 1550022
- Di Persio, L., Pellegrini, G., Bonollo, M.,Polynomial chaos expansion approach to interest rate models,(2015) Journal of Probability and Statistics, No. 369053,
- Di Persio, L. and Frigo, M.,Gibbs sampling approach to regime switching analysis of financial time series(2016), Journal of Computational and Applied Mathematics, 300, pp. 43-55.
- Di Persio, L. and Frigo, M.,Maximum likelihood approach to markov switching models(2015), WSEAS Transactions on Business and Economics, 12, pp. 239-242.
- Cont, R., and Tankov, P., Financial Modelling With Jump Processes, Chapman and Hall/CRC, UK (2004).
- Hull, J.C, Options, Futures, And Other Derivatives, Pearson, USA (2012).
- Hanson, F.B., and Zhu, Z., Comparison of market parameters for jump-diffusion distributions using multinomial maximum likelihood estimation, 43rd IEEE Conference on Decision and Control, 4, 2004, 3919-3924, 10.1109/CDC.2004.1429353.
- Kim, C.J., Nelson, C.R., State-space Models with Regime Switching: Classical and Gibbs-sampling Approaches with Applications (1999), MIT Press, Cambridge.
- Marinelli, C., Di Persio, L. and Ziglio, G.,Approximation and convergence of solutions to semilinear stochastic evolution equations with jumps(2013), Journal of Functional Analysis, 264 (12), pp. 2784-2816.
- Merton, R.C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.
- Shreve, S.E., Stochastic Calculus for Finance II: Continuous-Time Models, Springer, USA (2004).
- Historical Quotes for Standard and Poor’s 500 Index and Nasdaq 100 Index, Option Chain for Russell 2000 Index. Data taken from https://finance.yahoo.com, 2005-2015.
- Karatzas, I., and Shreve, S.E., Brownian Motion and Stochastic Calculus, Springer-Verlag, 1988.
- Rogers, L.C.G., and Williams, D., Diffusions, Markov Processes and Martingales, Cambridge Mathamatical Library, 2000.
DOI: 10.12732/ijpam.v109i3.19 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Pages: 719 - 736