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Pricing Model for Convertible Bonds: A Mixed Fractional Brownian Motion with Jumps

Published online by Cambridge University Press:  07 September 2015

Jie Miao  and
Xu Yang
Jie Miao*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China Department of Mathematics, Changji College, Changji, Xinjiang 831100, China
Xu Yang
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
*
*Corresponding author. Email addresses: [email protected] (J. Miao), [email protected] (X. Yang)
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Abstract

A mathematical model to price convertible bonds involving mixed fractional Brownian motion with jumps is presented. We obtain a general pricing formula using the risk neutral pricing principle and quasi-conditional expectation. The sensitivity of the price to changing various parameters is discussed. Theoretical prices from our jump mixed fractional Brownian motion model are compared with the prices predicted by traditional models. An empirical study shows that our new model is more acceptable.

Keywords

60J7560G2291G80Mixed fractional Brownian motionPoisson jumpconvertible bondempirical study
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 3 , August 2015 , pp. 222 - 237
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Ingersoll, J.E. Jr., A contingent-claims valuation of convertible securities, J. Financial Economics 4, 289–321 (1977).Google Scholar
[2]Brennan, M.J. and Schwartz, E.S., Convertible bonds: Valuation and optimal strategies for call and conversion, J. Finance 32, 1699–1715 (1977).Google Scholar
[3]Brennan, M.J. and Schwartz, E.S., Analyzing convertible bonds, J. Financial and Quantitative Analysis 15, 907–929 (1980).Google Scholar
[4]Nyborg, K.G., The use and pricing of convertible bonds, Applied Mathematical Finance 3, 167–190 (1996).CrossRefGoogle Scholar
[5]Mantegna, R.N. and Stanley, H.E., Turbulence and financial markets, Nature 383, 587–588 (1996).Google Scholar
[6]Cajueiro, D.O. and Tabak, B.M., Long-range dependence and multifractality in the term structure of LIBOR interest rates, Physica A 373, 603–614 (2007).Google Scholar
[7]Kang, S. and Yoon, S.M., Long memory features in the high frequency data of the Korean stock market, Physica A 387, 5189–5196 (2008).Google Scholar
[8]Wang, Y., Wei, Y. and Wu, C., Cross-correlations between Chinese A-share and B-share markets, Physica A 389, 5468–5478 (2010).Google Scholar
[9]Ding, Z., Granger, C.W.J. and Engle, R.F., A long memory property of stock market returns and a new model, J. Empirical Finance 1, 83–106 (1993).Google Scholar
[10]Liu, Y., Gopikrishnan, P., Cizeau, P., Meyer, M., Peng, C. and Stanley, H.E., The statistical properties of the volatility of price fluctuations, Phys. Rev. E 60, 1390–1400 (1999).CrossRefGoogle ScholarPubMed
[11]Necula, C., Option pricing in a fractional Brownian motion environment, available at SSRN 1286833 (2002).Google Scholar
[12]Lv, L., Ren, F. and Qiu, W., The application of fractional derivatives in stochastic models driven by fractional Brownian motion, Physica A 389, 4809–4818 (2010).Google Scholar
[13]Wang, X., Scaling and long range dependence in option pricing, IV: pricing European options with transaction costs under the multifractional Black-Scholes model, Physica A 389, 789–796 (2010).Google Scholar
[14]Wang, X., Wu, M., Zhou, Z. and Jing, W., Pricing European option with transaction costs under the fractional long memory stochastic volatility model, Physica A 391, 1469–1480 (2012).Google Scholar
[15]Shen, M., Convertible bond pricing in fractional brownian motion environment, J. Anhui University of Technology and Science 25, 72–74 (2010).Google Scholar
[16]Lin, S., Stochastic analysis of fractional Brownian motion, Stochastics and Stochastic Reports 55, 121–140 (1995).Google Scholar
[17]Mishura, Y., Stochastic Calculus for Fractional Brownian Motions and Related Processes, Springer Verlag, Berlin (2008).Google Scholar
[18]Li, L., Hu, J., Chen, Y. and Zhang, Y., PCA based Hurst exponent estimator for fbm signals under disturbances, IEEE Trans. Signal Processing 57, 2840–2846 (2009).CrossRefGoogle Scholar
[19]Cheridito, P., Mixed fractional Brownian motion, Bernoulli 7, 913–934 (2001).CrossRefGoogle Scholar
[20]Zili, M., On the mixed fractional Brownian motion, J. Applied Math. and Stochastic Analysis 2006, 1–9 (2006).Google Scholar
[21]Chernov, M., Gallant, A.R., Ghysels, E. and Tauchen, G., Alternative models for stock price dynamics. J. Econometrics 116, 225–257 (2003).Google Scholar
[22]Eraker, B., Do stock prices and volatility jump? Reconciling evidence from spot and option prices. J. Finance 59, 1367–1403 (2004).Google Scholar
[23]Pan, J., The jump-risk premia implicit in options: evidence from an integrated time-series study. J. Financial Economics 63, 3–50 (2002).CrossRefGoogle Scholar
[24]Merton, R., Option pricing when underlying stock returns are discontinuous, J. Financial Economics 3, 125–144 (1976).Google Scholar
[25]Davis, M. and Lischka, F.R., Convertible bonds with market risk and credit risk, Working Paper, Tokyo-Mitsubishi International (1999).Google Scholar
[26]Gapeev, P.V. and Kühn, C., Perpetual convertible bonds in jump diffusion models, Statistics and Decision 23, 15–31 (2005).Google Scholar
[27]Hua, H. and Cheng, X., Pricing convertible bond based on the jump-diffusion process, Application of Statistics and Management 28, 347–351 (2009).Google Scholar
[28]Bender, C., Sottinen, T. and Valkeila, E., Pricing by hedging and no-arbitrage beyond semimartin-gales, Finance Stochastics 12, 441–468 (2008).Google Scholar
[29]Xiao, W., Zhang, W., Zhang, X. and Wang, Y., Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm, Physica A 391, 6418–6431 (2012).Google Scholar
[30]Shokrollahi, F. and Kilicman, A., Pricing currency option in a mixed fractional Brownian motion with jumps environment, Mathematical Problems in Engineering 2014, 1–13 (2014).Google Scholar
[31]He, L. and Qian, W., A Monte Carlo simulation to the performance of the R/S and V/S method statistical revisit and real world application, Physica A 391, 3770–3782 (2012).CrossRefGoogle Scholar
[32]Kendall, M. and Stuart, A., The Advanced Theory of Statistics, Macmillan, New York (1977).Google Scholar