调和稳定Lévy过程驱动的双重跳跃模型及期权应用
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摘要
为准确刻画证券价格波动过程中的跳跃特征,捕获收益率尖峰厚尾、有偏等非高斯特征以及波动率集聚、异方差性等效应,建立了证券价格与相应波动率均存在跳跃的随机波动率模型,其中跳跃分布为两类纯跳跃Lévy分布(调和稳定分布和速降调和稳定分布)。最终,构建得到调和稳定Lévy过程驱动的双重跳跃随机波动模型。利用恒生和标普500股指数据进行实证,结果表明:与仿射跳扩散相比,纯跳跃Lévy分布能捕获随机信息的尖峰厚尾特征,拟合能力更优越,股指收益尾部分布存在速降特征。在此基础上的期权定价结果表明,速降调和稳定过程驱动的双重跳跃随机波动模型更有效。
In order to characterize the jump feature of securities price and the associated volatility process,which involves the leptokurtosis,heavy tailed and skewed non-Gaussian phenomena in stock returns,as well to simultaneously address the volatility clustering and heteroskedasticity effect,this paper introduces two pure jump Lévy processes(classical tempered stable and rapidly decreasing tempered stable processes)into the double jump stochastic volatility model in which both the returns and volatility processes jumps.Thus we establish double jump stochastic volatility models driven by tempered stable Lévy processes.The empirical studies of hang Seng and SP 500 index show that,compared with affine jump diffusion processes,pure jump Lévy distribution models can capture the above-mentioned characteristics more accurately and the tail distribution of stock index returns express rapidly decreasing phenomenon.Option pricing results demonstrate that the double jump stochastic volatility model driven by rapidly decreasing tempered stable process is more effective.
In order to characterize the jump feature of securities price and the associated volatility process,which involves the leptokurtosis,heavy tailed and skewed non-Gaussian phenomena in stock returns,as well to simultaneously address the volatility clustering and heteroskedasticity effect,this paper introduces two pure jump Lévy processes(classical tempered stable and rapidly decreasing tempered stable processes)into the double jump stochastic volatility model in which both the returns and volatility processes jumps.Thus we establish double jump stochastic volatility models driven by tempered stable Lévy processes.The empirical studies of hang Seng and SP 500 index show that,compared with affine jump diffusion processes,pure jump Lévy distribution models can capture the above-mentioned characteristics more accurately and the tail distribution of stock index returns express rapidly decreasing phenomenon.Option pricing results demonstrate that the double jump stochastic volatility model driven by rapidly decreasing tempered stable process is more effective.
引文
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[16]Zaevski T S,Kim Y S,Fabozzi F J.Option pricing under stochastic volatility and tempered stable Lévy jumps[J].International Review of Financial Analysis,2014,31(1):101-108.
[17]吴恒煜,朱福敏,胡根华,等.基于参数学习的GARCH动态无穷活动率Levy过程的欧式期权定价[J].系统工程理论与实践,2014,34(10):2465-2482.
[18]Heston S L.A closed form solution for options with stochastic volatility with applications to bond and currency options[J].The Review of Financial Studies,1993,6(2):327-343.
[19]Bates D S.Jumps and stochastic volatility:Exchange rate processes implicit in deutsche mark options[J].The Review of Financial Studies,1996,9(1):69-107.
[20]Eraker B,Johannes M J,Polson N G.The impact of jumps in returns and volatility[J].Journal of Finance,2003,53(3):1269-1300.
[21]Bollerslev T,Kretschmer U,Pigorsch C,et al.A discrete-time model for daily S&P returns and realized variations:Jumps and leverage effects[J].Journal of Econometrics,2009,150(2):151-166.
[22]周伟,何健敏,余德建.随机跳变广义双指数分布下的双重跳跃扩散模型及应用[J].系统工程理论与实践,2013,33(11):2746-2756.
[23]Rosinski J.Tempering stable processes[J].Stochastic Processes and Their Applications,2007,117(6):677-707.
[24]Yu C L,Li H T,Wells M T.MCMC estimation of Lévy jump models using stock and option prices[J].Mathematical Finance,2011,21(3):383-422.
[2]张高勋,田益祥,李秋敏.多元非线性期权定价模型及实证分析[J].系统管理学报,2014,23(2):200-207.
[3]Andreas K,Carol A.Stochastic volatility jumpdiffusions for european equity index dynamics[J].European Financial Management,2013,19(3):470-496.
[4]Todorov V.Econometric analysis of jump-driven stochastic volatility models[J].Journal of Econometrics,2011,160(1):12-21.
[5]Kaeck A,Alexander C.Stochastic volatility jumpdiffusions for european equity index dynamics[J].European Financial Management,2013,19(3):470-496.
[6]Merton R C.Option pricing when underlying stock returns are discontinuous[J].Journal of Financial Economics,1976,3(1):125-144.
[7]Duffie D,Pan J,Singleton K.Transform analysis and asset pricing for affine jump-diffusions[J].Econometrica,2000,68(6):1343-1376.
[8]Kou S G,Wang Hui.Option pricing under a double exponential jump diffusion model[J].ManagementScience,2004,50(9):1178-1192.
[9]Barndorff-Nielsen O E.Processes of normal inverse Gaussian type[J].Finance and Stochastics,1997,2(1):41-68.
[10]Madan D,Carr P,Chang E.The variance gamma process and option pricing[J].European Finance Review,1998,2(1):79-105.
[11]Carr P,Geman H,Madan D B,et al.The fine structure of asset returns:An empirical investigation[J].The Journal of Business,2002,75(2):205-332.
[12]Li H,Wells M T,Yu C L.A bayesian analysis of return dynamics with Lévy jumps[J].The Review of Financial Studies,2008,21(5):2345-2378.
[13]Kim Y S,Rachev S,Bianchi M,et al.Tempered stable and tempered infinitely divisible GARCH models[J].Journal of Banking and Finance,2010,34(9):2096-2109.
[14]Rachev S,Kim Y S,Bianchi M,et al.Financial models with Lévy processes and volatility clustering[M].New York:Wiley,2010.
[15]Kim Y S,Rachev S,Bianchi M,et al.Time series analysis for financial market meltdowns[J].Journal of Banking and Finance,2011,35(8):1879-1891.
[16]Zaevski T S,Kim Y S,Fabozzi F J.Option pricing under stochastic volatility and tempered stable Lévy jumps[J].International Review of Financial Analysis,2014,31(1):101-108.
[17]吴恒煜,朱福敏,胡根华,等.基于参数学习的GARCH动态无穷活动率Levy过程的欧式期权定价[J].系统工程理论与实践,2014,34(10):2465-2482.
[18]Heston S L.A closed form solution for options with stochastic volatility with applications to bond and currency options[J].The Review of Financial Studies,1993,6(2):327-343.
[19]Bates D S.Jumps and stochastic volatility:Exchange rate processes implicit in deutsche mark options[J].The Review of Financial Studies,1996,9(1):69-107.
[20]Eraker B,Johannes M J,Polson N G.The impact of jumps in returns and volatility[J].Journal of Finance,2003,53(3):1269-1300.
[21]Bollerslev T,Kretschmer U,Pigorsch C,et al.A discrete-time model for daily S&P returns and realized variations:Jumps and leverage effects[J].Journal of Econometrics,2009,150(2):151-166.
[22]周伟,何健敏,余德建.随机跳变广义双指数分布下的双重跳跃扩散模型及应用[J].系统工程理论与实践,2013,33(11):2746-2756.
[23]Rosinski J.Tempering stable processes[J].Stochastic Processes and Their Applications,2007,117(6):677-707.
[24]Yu C L,Li H T,Wells M T.MCMC estimation of Lévy jump models using stock and option prices[J].Mathematical Finance,2011,21(3):383-422.