随机波动率模型中的金融衍生品定价问题
|
摘要
在过去的三十多年中,著名的Black-Scholes-Merton模型被认为是研究期权定价问题的重要且有效的工具.然而,最近有大量证据表明,Black-Scholes-Merton模型中常数波动率的假设并不准确.在金融市场中,风险资产的波动率更适合于用随机波动率模型来描述.本文研究随机波动率模型和金融衍生品定价问题,包括以下四个方面：
在第二章中,我们研究具多种风险原生资产的随机波动率模型及欧式期权定价问题.假设波动率是紧集上快扩散过程的函数,我们得到了类似于Black-Scholes方程的多资产欧式期权定价方程,通过奇异摄动分析方法和多尺度技巧,建立了该定价方程的渐近展式解和一致有效的误差估计
在第三章中,我们考虑带有随机利率的随机波动率模型.除了假设随机波动率过程是紧集上快扩散过程的函数之外,还假设随机利率过程为慢扩散过程,即满足CIR模型.在适当的条件下,我们建立了一类利率衍生品的定价偏微分方程,并得到衍生品一阶近似价格的精确表达式以及一致有效的误差估计.
在第四章中,我们考虑金融市场中随机波动率模型的杠杆效应.从数学角度描述了杠杆效应这一金融现象,给出了具有时间结构的杠杆作用(量)及杠杆效应存在的充分条件.
在第五章中,我们考虑非光滑收益函数条件下的随机波动率模型.通过奇异摄动分析方法,利用磨光逼近的技巧,我们求解带有小参数的定价方程,得到期权价格一致渐近展开式和一致有效的误差估计.
In the last30years, the famous Black-Scholes-Merton formula has been proved to be an efficient and important tool for pricing option. However, there are some evidences that the constant volatility assumption in Black-Scholes model is inaccurate, even Black himself noticed that the assumption of constant volatility is inaccurate, as the implied volatility in financial market fluctuates frequently around some mean value. It is more suitable to use stochastic volatility model to describe the complicated behaviors of derivatives. In this thesis, we consider stochastic volatility models and derivatives pricing problem. The main results we obtained are as follows:
Part I:Stochastic Volatility Model in Pricing Multi-Asset European Options.
In Chapter Two, we consider a stochastic volatility model for pricing multi-asset European options, which are widely used in the real world. Under the assumption that the volatilities are driven by different OU processes, we establish the option pricing equation. Using the singular perturbation method for multi-parameter and multiscale skill, we derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates.
The option pricing equation is Using the singular perturbation and multiscale method, we derive a uniform asymptotic expansion for option prices, as well as the uniform error estimates.
Theorem0.1Under the assumptions (A2.1),(A2.2) and (A2.3), we obtain the option price Pεδ in the following form, where the leading term P0(t,x1,x2) called main term of option price is the classical Black-Scholes price when we replacing the stochastic volatility f(y1) and g(y2) by their average and . The second order term P01(t,x1,x2) and P10(t,x1,x2) are called corrected terms of option price due to stochastic volatility and they are combinations of derivatives of P0(t,x1,x2) with respect to stock price x1,x2.
Actually, we can derive higher order corrected terms to Black-Scholes price and the full asymptotic expansion of the option price. Moreover we obtain the following uniform error estimates.
Theorem0.2Under the assumptions (A2.1),(A2.2) and (A2.3), we derive the uniform error estimate between the real option price Pε,δ and the approximate option price P0+(?)P10+(?)P01where P0(t,x1,x2), P10(t,x1,x2) and P01(t,x1,x2)have been calculated explicitly.
Part Ⅱ: Stochastic Model with Stochastic Interest Rate and Derivatives Pricing.
In Chapter Three, we consider the stochastic model with stochastic interest rate. It follows from the fast mean reverting stochastic volatility model and CIR model of short rate that we obtain the pricing equation of interest rate derivative. By the singular perturbation method and boundary layer theory, we derive the first order approximate price of derivative and the uniform error estimates.
First, we derive the pricing equation of interest rate derivatives, where h(xt) is the payoff function.
Second, following the singular perturbation method and multiscale skill, we get the explicit expression of first order approximate price, where the main term Po(t, x, z) is the classical Black-Scholes price with constant volatil-ity(?>) The corrected term due to stochastic volatility is Another corrected term due to stochastic interest rate is
At last, we give the uniform error estimate.
Theorem0.3Under the assumptions (A3.1),(A3.2),(A3.3) and (A3.4), we get the error estimate between real option price and approximate price The above error estimate is uniform in time stock price and interest rate.
Part III:Leverage Effect of Stochastic Volatility Model.
In Chapter Four, we consider the leverage effect of stochastic volatility model in financial market. We describe apparition of leverage effect and define quantitative leverage with time structure. Moreover we explicitly state the conditions for the ex-istence of leverage effect. We respectively study leverage effects for three classes of models:the general stochastic volatility model, the asset pricing model with jumps and the stochastic volatility model with Poisson jumps. The leverage of the first model depends on the correlation parameters. The leverage effect in the second model is actually amplification. The third model has the amplified leverage effect.
Considering general stochastic volatility model dSt=μStdt+σ(Yt)StdWt, dYt=α(Yt)dt+β(Yt)dBt, we now state the definition of leverage and condition of leverage effect in this model. Definition0.4We define the leverage quantity over the derivatives'available time [0,t] as the correlation ρRY between asset's return Rt and stochastic volatility σ(Yt). There exits leverage effect in the formulated model for asset price if L(t)<0, for any t∈[0,t].
Theorem0.5There exists leverage effect in the general stochastic volatility model if the following condition hold true for any u, s∈[0,t]. where
The second model considered is asset pricing model with jump, dR(t)=μdt+σdW(t)+d(Q(t)-βλt). The jump process Q(t) has a special leverage effect on the asset return R(t) in the above model.
Proposition0.6In the asset pricing model, the jump process Q(t) have amplified effect on R(t) at any time. Namely,(?)(t)=λt(β2+η)≤0for any t>0,有A>0. We call (?)(t) the magnification of amplified leverage which implied that the jumps' frequency and sizes reinforce the return or loss of asset.
The third model is the stochastic model with Poisson jumps. dS(t)=μS(t)dt+σ(Y(t))S(t)dW(t)+S(t-)d(Q(t)-βλt) dY(t)=αa(Yt)dt+β(Yt)dB(t). We find that the leverage of this model consists of two part: one part is the amplified effect (?)(t) of jump process Q(t), the other is related to the leverage effect L(t).
Definition0.7We define the leverage quantity L(t) between asset return R{t) and stochastic volatility process σ(Y{t)) at time t as L(t)=(?)(t)Cov(R(t),σ(Y(t))), where (?)(t)=cov(R(t), Q(t)) is called magnification of amplified leverage. There exits amplified leverage effect in model if L(t)<0for any t>0.
Theorem0.8(The leverage condition) There exits leverage effect in the stochastic model with Poisson jumps at time t,if the following inequality hold true for any u, s∈[0, t], where
Part IV:Stochastic Volatility Model Under Non-Smoothness Conditions
In Chapter Five, we consider a stochastic volatility model for pricing European options. By the perturbation analysis and smoothing approximation method, we solve a backward PDE with one small parameter and derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates, under non-smoothness conditions on the payoff function.
Theorem0.9Under the assumptions (A5.1),(A5.2),and (A3.5), we get the error estimate between real option price P(t,x,y) and first order approximate price where P0(t,x) is the Black-Scholes price with content volatility (?), namely Pi(t,x) is the combination of derivatives of P0(t,x), namely The error estimate obtained above is uniform in time t∈[0,T], stock price x∈R and volatility driving process y∈S.
在第二章中,我们研究具多种风险原生资产的随机波动率模型及欧式期权定价问题.假设波动率是紧集上快扩散过程的函数,我们得到了类似于Black-Scholes方程的多资产欧式期权定价方程,通过奇异摄动分析方法和多尺度技巧,建立了该定价方程的渐近展式解和一致有效的误差估计
在第三章中,我们考虑带有随机利率的随机波动率模型.除了假设随机波动率过程是紧集上快扩散过程的函数之外,还假设随机利率过程为慢扩散过程,即满足CIR模型.在适当的条件下,我们建立了一类利率衍生品的定价偏微分方程,并得到衍生品一阶近似价格的精确表达式以及一致有效的误差估计.
在第四章中,我们考虑金融市场中随机波动率模型的杠杆效应.从数学角度描述了杠杆效应这一金融现象,给出了具有时间结构的杠杆作用(量)及杠杆效应存在的充分条件.
在第五章中,我们考虑非光滑收益函数条件下的随机波动率模型.通过奇异摄动分析方法,利用磨光逼近的技巧,我们求解带有小参数的定价方程,得到期权价格一致渐近展开式和一致有效的误差估计.
In the last30years, the famous Black-Scholes-Merton formula has been proved to be an efficient and important tool for pricing option. However, there are some evidences that the constant volatility assumption in Black-Scholes model is inaccurate, even Black himself noticed that the assumption of constant volatility is inaccurate, as the implied volatility in financial market fluctuates frequently around some mean value. It is more suitable to use stochastic volatility model to describe the complicated behaviors of derivatives. In this thesis, we consider stochastic volatility models and derivatives pricing problem. The main results we obtained are as follows:
Part I:Stochastic Volatility Model in Pricing Multi-Asset European Options.
In Chapter Two, we consider a stochastic volatility model for pricing multi-asset European options, which are widely used in the real world. Under the assumption that the volatilities are driven by different OU processes, we establish the option pricing equation. Using the singular perturbation method for multi-parameter and multiscale skill, we derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates.
The option pricing equation is Using the singular perturbation and multiscale method, we derive a uniform asymptotic expansion for option prices, as well as the uniform error estimates.
Theorem0.1Under the assumptions (A2.1),(A2.2) and (A2.3), we obtain the option price Pεδ in the following form, where the leading term P0(t,x1,x2) called main term of option price is the classical Black-Scholes price when we replacing the stochastic volatility f(y1) and g(y2) by their average
Actually, we can derive higher order corrected terms to Black-Scholes price and the full asymptotic expansion of the option price. Moreover we obtain the following uniform error estimates.
Theorem0.2Under the assumptions (A2.1),(A2.2) and (A2.3), we derive the uniform error estimate between the real option price Pε,δ and the approximate option price P0+(?)P10+(?)P01where P0(t,x1,x2), P10(t,x1,x2) and P01(t,x1,x2)have been calculated explicitly.
Part Ⅱ: Stochastic Model with Stochastic Interest Rate and Derivatives Pricing.
In Chapter Three, we consider the stochastic model with stochastic interest rate. It follows from the fast mean reverting stochastic volatility model and CIR model of short rate that we obtain the pricing equation of interest rate derivative. By the singular perturbation method and boundary layer theory, we derive the first order approximate price of derivative and the uniform error estimates.
First, we derive the pricing equation of interest rate derivatives, where h(xt) is the payoff function.
Second, following the singular perturbation method and multiscale skill, we get the explicit expression of first order approximate price, where the main term Po(t, x, z) is the classical Black-Scholes price with constant volatil-ity(?>) The corrected term due to stochastic volatility is Another corrected term due to stochastic interest rate is
At last, we give the uniform error estimate.
Theorem0.3Under the assumptions (A3.1),(A3.2),(A3.3) and (A3.4), we get the error estimate between real option price and approximate price The above error estimate is uniform in time stock price and interest rate.
Part III:Leverage Effect of Stochastic Volatility Model.
In Chapter Four, we consider the leverage effect of stochastic volatility model in financial market. We describe apparition of leverage effect and define quantitative leverage with time structure. Moreover we explicitly state the conditions for the ex-istence of leverage effect. We respectively study leverage effects for three classes of models:the general stochastic volatility model, the asset pricing model with jumps and the stochastic volatility model with Poisson jumps. The leverage of the first model depends on the correlation parameters. The leverage effect in the second model is actually amplification. The third model has the amplified leverage effect.
Considering general stochastic volatility model dSt=μStdt+σ(Yt)StdWt, dYt=α(Yt)dt+β(Yt)dBt, we now state the definition of leverage and condition of leverage effect in this model. Definition0.4We define the leverage quantity over the derivatives'available time [0,t] as the correlation ρRY between asset's return Rt and stochastic volatility σ(Yt). There exits leverage effect in the formulated model for asset price if L(t)<0, for any t∈[0,t].
Theorem0.5There exists leverage effect in the general stochastic volatility model if the following condition hold true for any u, s∈[0,t]. where
The second model considered is asset pricing model with jump, dR(t)=μdt+σdW(t)+d(Q(t)-βλt). The jump process Q(t) has a special leverage effect on the asset return R(t) in the above model.
Proposition0.6In the asset pricing model, the jump process Q(t) have amplified effect on R(t) at any time. Namely,(?)(t)=λt(β2+η)≤0for any t>0,有A>0. We call (?)(t) the magnification of amplified leverage which implied that the jumps' frequency and sizes reinforce the return or loss of asset.
The third model is the stochastic model with Poisson jumps. dS(t)=μS(t)dt+σ(Y(t))S(t)dW(t)+S(t-)d(Q(t)-βλt) dY(t)=αa(Yt)dt+β(Yt)dB(t). We find that the leverage of this model consists of two part: one part is the amplified effect (?)(t) of jump process Q(t), the other is related to the leverage effect L(t).
Definition0.7We define the leverage quantity L(t) between asset return R{t) and stochastic volatility process σ(Y{t)) at time t as L(t)=(?)(t)Cov(R(t),σ(Y(t))), where (?)(t)=cov(R(t), Q(t)) is called magnification of amplified leverage. There exits amplified leverage effect in model if L(t)<0for any t>0.
Theorem0.8(The leverage condition) There exits leverage effect in the stochastic model with Poisson jumps at time t,if the following inequality hold true for any u, s∈[0, t], where
Part IV:Stochastic Volatility Model Under Non-Smoothness Conditions
In Chapter Five, we consider a stochastic volatility model for pricing European options. By the perturbation analysis and smoothing approximation method, we solve a backward PDE with one small parameter and derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates, under non-smoothness conditions on the payoff function.
Theorem0.9Under the assumptions (A5.1),(A5.2),and (A3.5), we get the error estimate between real option price P(t,x,y) and first order approximate price where P0(t,x) is the Black-Scholes price with content volatility (?), namely Pi(t,x) is the combination of derivatives of P0(t,x), namely The error estimate obtained above is uniform in time t∈[0,T], stock price x∈R and volatility driving process y∈S.
引文
[1]姜礼尚,徐承龙,任学敏.金融衍生产品定价的数学模型与案例分析[M].北京：高等教育出版社,2008.[2]姜礼尚.期权定价的数学模型和方法[M].北京：高等教育出版社,2008.[3]伍卓群,李勇.常微分方程[M].北京：高等教育出版社,2004.[4]AGGARWL R, ZHAO X. The leverage-value relationship puzzle:An industry effects resolution[J]. Journal of Economics and Business59(2007), no.4,286-297.[5]AHN S, DENIS D J and DENIS D K. Leverage and investment in diversified firms[J]. Journal of Financial Economics79(2006), no.2,317-337.[6]ANDERSEN T G. Return volatility and trading volume:An information flow inter-pretation of stochastic volatility[J]. Journal of Finance51(1996), no.1,169-204.[7]ANDERSEN T G, BOLLERSLEV T and DOBREV D. No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and i.i.d. noise:theory and testable distributional implications[J]. J. Econometrics138(2007), no.1,125-180.[8]ANE T, GEMAN H. Order flow, transaction clock, and normality of asset returns[J]. The Journal of Finance55(2000), no.5,2259-2284.[9]BACHELIER L. Theorie de la speculation[M]. Gauthier-Villars,1900.[10]BARNDORFF-NIELSEN O E. Superposition of ornstein-uhlenbeck, type processes[J]. Theory of Probability and its Applications45, no.2.[11]BARNDORFF-NIELSEN O E. Processes of normal inverse gaussian type[J]. Finance and stochastics2(1998), no.1,41-68.
[12] BELOMESTNY D, SCHOENMAKERS J. A jump-difusion Libor model and its robustcalibration[J]. Quant. Finance11(2011), no.4,529–546.
[13] BLACK F. Stuedies of stock price volatility changes[C]. Proceedings of the1976Meet-ings of the Business and Economics Statistics Section, American Statistical Association(1976),177–81.
[14] BLACK F, KARASINSKI P. Bond and option pricing when short rates are lognor-mal[J]. Financial Analysts Journal47(1991), no.4,52–59.
[15] BLACK F, SCHOLES M. The pricing of options and corporate liabilities[J]. The jour-nal of political economy81(1973), no.3,637–654.
[16] BONESS A J. Elements of a theory of stock-option value[J]. The Journal of PoliticalEconomy72(1964), no.2,163–175.
[17] BOUCHAUD J P, MATACZ A and POTTERS M. Leverage efect in financial markets:The retarded volatility model[J]. Physical Review Letters87(2001), no.22,228701.
[18] BOUCHAUD J P, POTTERS M. More stylized facts of financial markets: leverage ef-fect and downside correlations[J]. Physica A: Statistical Mechanics and its Applications299(2001), no.1,60–70.
[19] BRACE A, MUSIELA M, et al., The market model of interest rate dynamics[J]. Math-ematical finance7(1997), no.2,127–155.
[20] BRIGO D MERCURIO F. A deterministic-shift extension of analytically-tractable andtime-homogeneous short-rate models[J]. Finance Stoch.5(2001), no.3,369–387.
[21] BRIGO D MERCURIO F. Interest rate models: theory and practice: with smile, infla-tion, and credit[M]. Springer Verlag,2006.
[22] BUETOW G W Jr, SOCHACKI J. Term-structure models using binomial trees[M].CFA Institute,2001.
[23] CAIRNS A J G. Interest rate models: an introduction, vol.10[M]. Princeton Univ Pr,2004.
[24] CARR P, GEMAN H, MADAN D B, YOR M. The fine structure of asset returns: Anempirical investigation[J]. Journal of business75(2002), no.2,305–332.
[25] CARVERHILL A. When is the short rate Markovian?[J]. Math. Finance4(1994), no.4,305–312.
[26] CHEN L. Interest rate dynamics, derivatives pricing, and risk management[M].Springer,1996.
[27] CHEN L, LEONARD N. Stochastic mean and stochastic volatility: A three-factor modelof the term structure of interest rates and its applications in derivatives pricing andrisk management[M]. Blackwell publishers,1996.
[28] CHEN L, ZHAO X. Mechanical mean reversion of leverage ratios[J], Economics Letters95(2007), no.2,223–229.
[29] CHERNOV M, GALLANT A R, CHYSELS E and TAUCHEN G. Alternative modelsfor stock price dynamics[J]. J. Econometrics116(2003), no.1-2,225–257.
[30] CHRISTITE A A. The stochastic behavior of common stock variances: Value, leverageand interest rate efects[J]. Journal of financial Economics10(1982), no.4,407–432.
[31] CLARK P K. A subordinated stochastic process model with finite variance for specula-tive prices[J]. Econometrica41(1973), no.1,135–155.
[32] COX J C, INGERSOLL Jr, JONATHAN E and ROSS S A. A theory of the termstructure of interest rates[J]. Econometrica53(1985), no.2,385–407.
[33] DUFFIE D, KAN R. A yield-factor model of interest rates[J]. Mathematical finance6(1996), no.4,379–406.
[34] EBERLEIN E,O¨ZKAN F. The L′evy LIBOR model[J]. Finance Stoch.9(2005), no.3,327–348.
[35] EISLER Z, KERTESZ J. Multifractal model of asset returns with leverage efect[J].Physica A: Statistical Mechanics and its Applications343(2004),603–622.
[36] ELLIOTT R J, TSOI A H, LUI S H. Short rate analysis and marked point processes[J].Math. Methods Oper. Res.50(1999), no.1,149–160.
[37] ENGLE R F, NG V K. Measuring and testing the impact of news on volatility[J].Journal of Finance48(1993), no.5,1749–1778.
[38] ENGLE R F, PATTON A J. What good is a volatility model?[J].1(2001),237–245.
[39] FLORESCU I, PA sA ricA C G. A study about the existence of the leverage efect instochastic volatility models[J]. Physica A: Statistical Mechanics and its Applications388(2009), no.4,419–432.
[40] FOUQUE J P, HAN C H. A martingale control variate method for option pricing withstochastic volatility[J]. ESAIM Probab. Stat.11(2007),40–54.
[41] FOUQUE J P, PAPANICOLAOU G and SIRCAR K. Derivatives in financial marketswith stochastic volatility[M]. Cambridge University Press, Cambridge,2000.
[42] FOUQUE J P, PAPANICOLAOU G, SIRCAR and SOLNA K. Singular perturbationsin option pricing[J]. SIAM J. Appl. Math.63(2003), no.5,1648–1665.
[43] FOUQUE J P, HAN C H. Pricing Asian options with stochastic volatility[J]. Quant.Finance3(2003), no.5,353–362.
[44] FOUQUE J P. Variance reduction for Monte Carlo methods to evaluate option pricesunder multi-factor stochastic volatility models[J]. Quant. Finance4(2004), no.5,597–606.
[45] FOUQUE J P, ZHOU X. Perturbed Gaussian copula, Econometrics and risk manage-ment[J]. Adv. Econom., vol.22, Emerald/JAI, Bingley,2008, pp.103–121.
[46] FRANCIS B, HASAN I and SHARMA Z. Leverage and growth: efect of stock op-tions[J]. Journal of Economics and Business63(2011), no.6,558–581.
[47] GAPEEV P V. On arbitrage and Markovian short rates in fractional bond markets[J].Statist. Probab. Lett.70(2004), no.3,211–222.
[48] GLASSERMAN P, KOU S G. The term structure of simple forward rates with jumprisk[J]. Mathematical finance13(2003), no.3,383–410.
[49] GLASSERMAN P, ZHAO X. Arbitrage-free discretization of lognormal forward Liborand swap rate models[J]. Finance Stoch.4(2000), no.1,35–68.
[50] HARVEY A C, SHEPHARD N. Estimation of an asymmetric stochastic volatility modelfor asset returns[J]. Journal of Business&Economic Statistics14(1996), no.4,429–434.
[51] HEATH D, JARROW Rand MORTON A. Bond pricing and the term structure ofinterest rates: A new methodology for contingent claims valuation[J]. Econometrica:Journal of the Econometric Society60(1992), no.1,77–105.
[52] HENS T, STEUDE S C. The leverage efect without leverage[J]. Finance ResearchLetters6(2009), no.2,83–94.
[53] HESTON S L. A closed-form solution for options with stochastic volatility with appli-cations to bond and currency options[J]. Review of financial studies6(1993), no.2,327–343.
[54] HO T S Y, LEE S B. Term structure movements and pricing interest rate contingentclaims[J]. Journal of Finance41(1986), no.5,1011–1029.
[55] HULL J. Fundamentals of Futures and Options Markets and Derivagem[M]. PearsonPrentice Hall,2008.
[56] HULL J. Options, futures and other derivatives[M]. Pearson Prentice Hall,2009.
[57] HULL J, WHITE A. The pricing of options on assets with stochastic volatilities[J].Journal of finance42(1987), no.2,281–300.
[58] HULL J, WHITE A. Pricing interest-rate-derivative securities[J]. Review of financialstudies3(1990), no.4,573–592.
[59] JACQUIER E, POLSON N G and ROSSI P E. Bayesian analysis of stochastic volatilitymodels with fat-tails and correlated errors[J]. Journal of Econometrics122(2004), no.1,185–212.
[60] JOHNSON H SHANNO D. Option pricing when the variance is changing[J]. Journalof Financial and Quantitative Analysis22(1987), no.02,143–151.
[61] KARATZAS I, SHREVE S E. Brownian motion and stochastic calculus[M]. SpringerVerlag,1991.
[62] KELLER R M, STEINER T. Yield curve shapes and the asymptotic short rate distri-bution in afne one-factor models[J]. Finance Stoch.12(2008), no.2,149–172.
[63] KHASMINSKII R Z, YIN G. On averaging principles: an asymptotic expansion ap-proach[J]. SIAM J. Math. Anal.35(2004), no.6,1534–1560.
[64] KHASMINSKII R Z, YIN G. Limit behavior of two-time-scale difusions revisited[J].J. Diferential Equations212(2005), no.1,85–113.
[65] KHASMINSKII R Z, YIN G. Uniform asymptotic expansions for pricing Europeanoptions[J]. Appl. Math. Optim.52(2005), no.3,279–296.
[66] LANDE′N C. Bond pricing in a hidden Markov model of the short rate[J]. FinanceStoch.4(2000), no.4,371–389.
[67] LANG L, OFEK E and STULZ R M. Leverage, investment, and firm growth[J]. Journalof financial Economics40(1996), no.1,3–29.
[68] LEE S, KOH Y and KANG K H. Moderating efect of capital intensity on the relation-ship between leverage and financial distress in the us restaurant industry[J]. Interna-tional Journal of Hospitality Management30(2010),429–438.
[69] LONGSTAFF F A, SCHWARTZ E S. Interest rate volatility and the term structure: Atwo-factor general equilibrium model[J]. Journal of finance47(1992), no.4,1259–1282.
[70] LUOMA G A, SPILLER JR E A. Financial accounting return on investment andfinancial leverage[J]. Journal of accounting education20(2002), no.2,131–138.
[71] MADAN D B, SENTA E. The variance gamma (vg) model for share market returns[J].Journal of business (1990),511–524.
[72] MALLIAVIN P, MANCINO M E and RECCHIONI M C. A non-parametric calibrationof the HJM geometry: an application of It o calculus to financial statistics[J]. Jpn. J.Math.2(2007), no.1,55–77.
[73] MANOS R, MURINDE V and GREEN C J. Leverage and business groups: Evidencefrom indian firms[J]. Journal of Economics and Business59(2007), no.5,443–465.
[74] MARINELLI C. Well-posedness and invariant measures for HJM models with deter-ministic volatility and L′evy noise[J]. Quant. Finance10(2010), no.1,39–47.
[75] MERTON R C. Theory of rational option pricing[J]. Bell J. Econom. and ManagementSci.4(1973),141–183.
[76] MERTON R C. An intertemporal capital asset pricing model[J]. Econometrica41(1973), no.5,867–887.
[77] MEYER R, YU J. Bugs for a bayesian analysis of stochastic volatility models[J]. Theeconometrics journal3(2000), no.2,198–215.
[78] MILSTEIN G N, PLATEN E and SCHURZ H. Balanced implicit methods for stifstochastic systems[J]. SIAM J. Numer. Anal.35(1998), no.3,1010–1019.
[79] MUSIELA M, RUTKOWSKI. Martingale methods in financial modelling[J]. second ed.,Stochastic Modelling and Applied Probability, vol.36, Springer-Verlag, Berlin,2005.
[80] NELSON D B. Conditional heteroskedasticity in asset returns: a new approach[J].Econometrica59(1991), no.2,347–370.
[81] NELSON D B. Arch models as difusion approximations[J]. Journal of Econometrics45(1990), no.1-2,7–38.
[82] NING C, XU D and WIRJANTO T S. Modeling the leverage efect with copulas andrealized volatility[J]. Finance Research Letters5(2008), no.4,221–227.
[83] OKSENDAL B. Stochastic diferential equations, sixth ed.[M]. Universitext, Springer-Verlag, Berlin,2003, An introduction with applications.
[84] PE′REZ A, RUIZ E and VEIGA H. A note on the properties of power-transformed re-turns in long-memory stochastic volatility models with leverage efect[J]. ComputationalStatistics&Data Analysis53(2009), no.10,3593–3600.
[85] PRAETZ P D. The distribution of share price changes[J]. Journal of business45(1972),no.1,49–55.
[86] PRESS S J. A compound events model for security prices[J]. Journal of Business40(1967), no.3,317–335.
[87] REBONATO R. Modern pricing of interest-rate derivatives: The libor market modeland beyond[M]. Princeton Univ Pr,2002.
[88] RENDLEMAN R, BARTTER B. The pricing of options on debt securities[J]. Journalof Financial and Quantitative Analysis15(1980), no.1,11–24.
[89] RUIZ E, VEIGA H. Modelling long-memory volatilities with leverage efect: A-LMSVversus FIEGARCH [J]. Comput. Statist. Data Anal.52(2008), no.6,2846–2862.
[90] SAMUELSON P A. Rational theory of warrant pricing[J]. Industrial management re-view6(1965),13–31.
[91] SCOTT L O. Option pricing when the variance changes randomly: Theory, estimation,and an application[J]. Journal of Financial and Quantitative analysis22(1987), no.4,419–438.
[92] SHREVE S E. Stochastic calculus models for finance ii: Continuous time models[M].Springer-Verlag, New York, NY,2004.
[93] SPRENKLE C M. Warrant prices as indicators of expectations and preferences[J]. YaleEconomic Essays1(1961), no.2,178–231.
[94] STEIN E M and STEIN J C.Stock price distributions with stochastic volatility: ananalytic approach[J]. Review of financial Studies4(1991), no.4,727–752.
[95] TAKAMIZAWA H and SHOJI I. Modeling the term structure of interest rates withgeneral short-rate models[J]. Finance Stoch.7(2003), no.3,323–335.
[96] TAYLOR S. Modelling financial time series[M].1986.
[97] VASICEK O. An equilibrium characterization of the term structure[J]. Journal of fi-nancial economics5(1977), no.2,177–188.
[98] WIGGINS J B. Option values under stochastic volatility: Theory and empirical esti-mates[J]. Journal of financial economics19(1987), no.2,351–372.
[99] WONG H Y, CHAN C M. Lookback options and dynamic fund protection under mul-tiscale stochastic volatility[J]. Insurance Math. Econom.40(2007), no.3.
[100] WU Z, YIN J and WANG C. Elliptic and parabolic equations[J]. World Scientific PubCo Inc,2006.
[101] YONG J M. Remarks on some short rate term structure models[J]. J. Ind. Manag.Optim.2(2006), no.2,119–134.
[102] YU J. On leverage in a stochastic volatility model[J]. Journal of Econometrics127(2005), no.2,165–178.
[103] ZEGHAL A B, MNIF M. Portfolio optimization with stochastic volatilities: a backwardapproach[J]. Stoch. Anal. Appl.29(2011), no.5,723–748.
[104] ZHU S P. A simple approximation formula for calculating the optimal exercise boundaryof American puts[J]. J. Appl. Math. Comput.37(2011), no.1-2,611–623.
[105] ZHU S P, CHEN W T. Pricing perpetual American options under a stochastic-volatilitymodel with fast mean reversion[J]. Appl. Math. Lett.24(2011), no.10,1663–1669.
[106] ZHU S P, LIAN G H. A closed-form exact solution for pricing variance swaps withstochastic volatility[J]. Math. Finance21(2011), no.2,233–256.
[12] BELOMESTNY D, SCHOENMAKERS J. A jump-difusion Libor model and its robustcalibration[J]. Quant. Finance11(2011), no.4,529–546.
[13] BLACK F. Stuedies of stock price volatility changes[C]. Proceedings of the1976Meet-ings of the Business and Economics Statistics Section, American Statistical Association(1976),177–81.
[14] BLACK F, KARASINSKI P. Bond and option pricing when short rates are lognor-mal[J]. Financial Analysts Journal47(1991), no.4,52–59.
[15] BLACK F, SCHOLES M. The pricing of options and corporate liabilities[J]. The jour-nal of political economy81(1973), no.3,637–654.
[16] BONESS A J. Elements of a theory of stock-option value[J]. The Journal of PoliticalEconomy72(1964), no.2,163–175.
[17] BOUCHAUD J P, MATACZ A and POTTERS M. Leverage efect in financial markets:The retarded volatility model[J]. Physical Review Letters87(2001), no.22,228701.
[18] BOUCHAUD J P, POTTERS M. More stylized facts of financial markets: leverage ef-fect and downside correlations[J]. Physica A: Statistical Mechanics and its Applications299(2001), no.1,60–70.
[19] BRACE A, MUSIELA M, et al., The market model of interest rate dynamics[J]. Math-ematical finance7(1997), no.2,127–155.
[20] BRIGO D MERCURIO F. A deterministic-shift extension of analytically-tractable andtime-homogeneous short-rate models[J]. Finance Stoch.5(2001), no.3,369–387.
[21] BRIGO D MERCURIO F. Interest rate models: theory and practice: with smile, infla-tion, and credit[M]. Springer Verlag,2006.
[22] BUETOW G W Jr, SOCHACKI J. Term-structure models using binomial trees[M].CFA Institute,2001.
[23] CAIRNS A J G. Interest rate models: an introduction, vol.10[M]. Princeton Univ Pr,2004.
[24] CARR P, GEMAN H, MADAN D B, YOR M. The fine structure of asset returns: Anempirical investigation[J]. Journal of business75(2002), no.2,305–332.
[25] CARVERHILL A. When is the short rate Markovian?[J]. Math. Finance4(1994), no.4,305–312.
[26] CHEN L. Interest rate dynamics, derivatives pricing, and risk management[M].Springer,1996.
[27] CHEN L, LEONARD N. Stochastic mean and stochastic volatility: A three-factor modelof the term structure of interest rates and its applications in derivatives pricing andrisk management[M]. Blackwell publishers,1996.
[28] CHEN L, ZHAO X. Mechanical mean reversion of leverage ratios[J], Economics Letters95(2007), no.2,223–229.
[29] CHERNOV M, GALLANT A R, CHYSELS E and TAUCHEN G. Alternative modelsfor stock price dynamics[J]. J. Econometrics116(2003), no.1-2,225–257.
[30] CHRISTITE A A. The stochastic behavior of common stock variances: Value, leverageand interest rate efects[J]. Journal of financial Economics10(1982), no.4,407–432.
[31] CLARK P K. A subordinated stochastic process model with finite variance for specula-tive prices[J]. Econometrica41(1973), no.1,135–155.
[32] COX J C, INGERSOLL Jr, JONATHAN E and ROSS S A. A theory of the termstructure of interest rates[J]. Econometrica53(1985), no.2,385–407.
[33] DUFFIE D, KAN R. A yield-factor model of interest rates[J]. Mathematical finance6(1996), no.4,379–406.
[34] EBERLEIN E,O¨ZKAN F. The L′evy LIBOR model[J]. Finance Stoch.9(2005), no.3,327–348.
[35] EISLER Z, KERTESZ J. Multifractal model of asset returns with leverage efect[J].Physica A: Statistical Mechanics and its Applications343(2004),603–622.
[36] ELLIOTT R J, TSOI A H, LUI S H. Short rate analysis and marked point processes[J].Math. Methods Oper. Res.50(1999), no.1,149–160.
[37] ENGLE R F, NG V K. Measuring and testing the impact of news on volatility[J].Journal of Finance48(1993), no.5,1749–1778.
[38] ENGLE R F, PATTON A J. What good is a volatility model?[J].1(2001),237–245.
[39] FLORESCU I, PA sA ricA C G. A study about the existence of the leverage efect instochastic volatility models[J]. Physica A: Statistical Mechanics and its Applications388(2009), no.4,419–432.
[40] FOUQUE J P, HAN C H. A martingale control variate method for option pricing withstochastic volatility[J]. ESAIM Probab. Stat.11(2007),40–54.
[41] FOUQUE J P, PAPANICOLAOU G and SIRCAR K. Derivatives in financial marketswith stochastic volatility[M]. Cambridge University Press, Cambridge,2000.
[42] FOUQUE J P, PAPANICOLAOU G, SIRCAR and SOLNA K. Singular perturbationsin option pricing[J]. SIAM J. Appl. Math.63(2003), no.5,1648–1665.
[43] FOUQUE J P, HAN C H. Pricing Asian options with stochastic volatility[J]. Quant.Finance3(2003), no.5,353–362.
[44] FOUQUE J P. Variance reduction for Monte Carlo methods to evaluate option pricesunder multi-factor stochastic volatility models[J]. Quant. Finance4(2004), no.5,597–606.
[45] FOUQUE J P, ZHOU X. Perturbed Gaussian copula, Econometrics and risk manage-ment[J]. Adv. Econom., vol.22, Emerald/JAI, Bingley,2008, pp.103–121.
[46] FRANCIS B, HASAN I and SHARMA Z. Leverage and growth: efect of stock op-tions[J]. Journal of Economics and Business63(2011), no.6,558–581.
[47] GAPEEV P V. On arbitrage and Markovian short rates in fractional bond markets[J].Statist. Probab. Lett.70(2004), no.3,211–222.
[48] GLASSERMAN P, KOU S G. The term structure of simple forward rates with jumprisk[J]. Mathematical finance13(2003), no.3,383–410.
[49] GLASSERMAN P, ZHAO X. Arbitrage-free discretization of lognormal forward Liborand swap rate models[J]. Finance Stoch.4(2000), no.1,35–68.
[50] HARVEY A C, SHEPHARD N. Estimation of an asymmetric stochastic volatility modelfor asset returns[J]. Journal of Business&Economic Statistics14(1996), no.4,429–434.
[51] HEATH D, JARROW Rand MORTON A. Bond pricing and the term structure ofinterest rates: A new methodology for contingent claims valuation[J]. Econometrica:Journal of the Econometric Society60(1992), no.1,77–105.
[52] HENS T, STEUDE S C. The leverage efect without leverage[J]. Finance ResearchLetters6(2009), no.2,83–94.
[53] HESTON S L. A closed-form solution for options with stochastic volatility with appli-cations to bond and currency options[J]. Review of financial studies6(1993), no.2,327–343.
[54] HO T S Y, LEE S B. Term structure movements and pricing interest rate contingentclaims[J]. Journal of Finance41(1986), no.5,1011–1029.
[55] HULL J. Fundamentals of Futures and Options Markets and Derivagem[M]. PearsonPrentice Hall,2008.
[56] HULL J. Options, futures and other derivatives[M]. Pearson Prentice Hall,2009.
[57] HULL J, WHITE A. The pricing of options on assets with stochastic volatilities[J].Journal of finance42(1987), no.2,281–300.
[58] HULL J, WHITE A. Pricing interest-rate-derivative securities[J]. Review of financialstudies3(1990), no.4,573–592.
[59] JACQUIER E, POLSON N G and ROSSI P E. Bayesian analysis of stochastic volatilitymodels with fat-tails and correlated errors[J]. Journal of Econometrics122(2004), no.1,185–212.
[60] JOHNSON H SHANNO D. Option pricing when the variance is changing[J]. Journalof Financial and Quantitative Analysis22(1987), no.02,143–151.
[61] KARATZAS I, SHREVE S E. Brownian motion and stochastic calculus[M]. SpringerVerlag,1991.
[62] KELLER R M, STEINER T. Yield curve shapes and the asymptotic short rate distri-bution in afne one-factor models[J]. Finance Stoch.12(2008), no.2,149–172.
[63] KHASMINSKII R Z, YIN G. On averaging principles: an asymptotic expansion ap-proach[J]. SIAM J. Math. Anal.35(2004), no.6,1534–1560.
[64] KHASMINSKII R Z, YIN G. Limit behavior of two-time-scale difusions revisited[J].J. Diferential Equations212(2005), no.1,85–113.
[65] KHASMINSKII R Z, YIN G. Uniform asymptotic expansions for pricing Europeanoptions[J]. Appl. Math. Optim.52(2005), no.3,279–296.
[66] LANDE′N C. Bond pricing in a hidden Markov model of the short rate[J]. FinanceStoch.4(2000), no.4,371–389.
[67] LANG L, OFEK E and STULZ R M. Leverage, investment, and firm growth[J]. Journalof financial Economics40(1996), no.1,3–29.
[68] LEE S, KOH Y and KANG K H. Moderating efect of capital intensity on the relation-ship between leverage and financial distress in the us restaurant industry[J]. Interna-tional Journal of Hospitality Management30(2010),429–438.
[69] LONGSTAFF F A, SCHWARTZ E S. Interest rate volatility and the term structure: Atwo-factor general equilibrium model[J]. Journal of finance47(1992), no.4,1259–1282.
[70] LUOMA G A, SPILLER JR E A. Financial accounting return on investment andfinancial leverage[J]. Journal of accounting education20(2002), no.2,131–138.
[71] MADAN D B, SENTA E. The variance gamma (vg) model for share market returns[J].Journal of business (1990),511–524.
[72] MALLIAVIN P, MANCINO M E and RECCHIONI M C. A non-parametric calibrationof the HJM geometry: an application of It o calculus to financial statistics[J]. Jpn. J.Math.2(2007), no.1,55–77.
[73] MANOS R, MURINDE V and GREEN C J. Leverage and business groups: Evidencefrom indian firms[J]. Journal of Economics and Business59(2007), no.5,443–465.
[74] MARINELLI C. Well-posedness and invariant measures for HJM models with deter-ministic volatility and L′evy noise[J]. Quant. Finance10(2010), no.1,39–47.
[75] MERTON R C. Theory of rational option pricing[J]. Bell J. Econom. and ManagementSci.4(1973),141–183.
[76] MERTON R C. An intertemporal capital asset pricing model[J]. Econometrica41(1973), no.5,867–887.
[77] MEYER R, YU J. Bugs for a bayesian analysis of stochastic volatility models[J]. Theeconometrics journal3(2000), no.2,198–215.
[78] MILSTEIN G N, PLATEN E and SCHURZ H. Balanced implicit methods for stifstochastic systems[J]. SIAM J. Numer. Anal.35(1998), no.3,1010–1019.
[79] MUSIELA M, RUTKOWSKI. Martingale methods in financial modelling[J]. second ed.,Stochastic Modelling and Applied Probability, vol.36, Springer-Verlag, Berlin,2005.
[80] NELSON D B. Conditional heteroskedasticity in asset returns: a new approach[J].Econometrica59(1991), no.2,347–370.
[81] NELSON D B. Arch models as difusion approximations[J]. Journal of Econometrics45(1990), no.1-2,7–38.
[82] NING C, XU D and WIRJANTO T S. Modeling the leverage efect with copulas andrealized volatility[J]. Finance Research Letters5(2008), no.4,221–227.
[83] OKSENDAL B. Stochastic diferential equations, sixth ed.[M]. Universitext, Springer-Verlag, Berlin,2003, An introduction with applications.
[84] PE′REZ A, RUIZ E and VEIGA H. A note on the properties of power-transformed re-turns in long-memory stochastic volatility models with leverage efect[J]. ComputationalStatistics&Data Analysis53(2009), no.10,3593–3600.
[85] PRAETZ P D. The distribution of share price changes[J]. Journal of business45(1972),no.1,49–55.
[86] PRESS S J. A compound events model for security prices[J]. Journal of Business40(1967), no.3,317–335.
[87] REBONATO R. Modern pricing of interest-rate derivatives: The libor market modeland beyond[M]. Princeton Univ Pr,2002.
[88] RENDLEMAN R, BARTTER B. The pricing of options on debt securities[J]. Journalof Financial and Quantitative Analysis15(1980), no.1,11–24.
[89] RUIZ E, VEIGA H. Modelling long-memory volatilities with leverage efect: A-LMSVversus FIEGARCH [J]. Comput. Statist. Data Anal.52(2008), no.6,2846–2862.
[90] SAMUELSON P A. Rational theory of warrant pricing[J]. Industrial management re-view6(1965),13–31.
[91] SCOTT L O. Option pricing when the variance changes randomly: Theory, estimation,and an application[J]. Journal of Financial and Quantitative analysis22(1987), no.4,419–438.
[92] SHREVE S E. Stochastic calculus models for finance ii: Continuous time models[M].Springer-Verlag, New York, NY,2004.
[93] SPRENKLE C M. Warrant prices as indicators of expectations and preferences[J]. YaleEconomic Essays1(1961), no.2,178–231.
[94] STEIN E M and STEIN J C.Stock price distributions with stochastic volatility: ananalytic approach[J]. Review of financial Studies4(1991), no.4,727–752.
[95] TAKAMIZAWA H and SHOJI I. Modeling the term structure of interest rates withgeneral short-rate models[J]. Finance Stoch.7(2003), no.3,323–335.
[96] TAYLOR S. Modelling financial time series[M].1986.
[97] VASICEK O. An equilibrium characterization of the term structure[J]. Journal of fi-nancial economics5(1977), no.2,177–188.
[98] WIGGINS J B. Option values under stochastic volatility: Theory and empirical esti-mates[J]. Journal of financial economics19(1987), no.2,351–372.
[99] WONG H Y, CHAN C M. Lookback options and dynamic fund protection under mul-tiscale stochastic volatility[J]. Insurance Math. Econom.40(2007), no.3.
[100] WU Z, YIN J and WANG C. Elliptic and parabolic equations[J]. World Scientific PubCo Inc,2006.
[101] YONG J M. Remarks on some short rate term structure models[J]. J. Ind. Manag.Optim.2(2006), no.2,119–134.
[102] YU J. On leverage in a stochastic volatility model[J]. Journal of Econometrics127(2005), no.2,165–178.
[103] ZEGHAL A B, MNIF M. Portfolio optimization with stochastic volatilities: a backwardapproach[J]. Stoch. Anal. Appl.29(2011), no.5,723–748.
[104] ZHU S P. A simple approximation formula for calculating the optimal exercise boundaryof American puts[J]. J. Appl. Math. Comput.37(2011), no.1-2,611–623.
[105] ZHU S P, CHEN W T. Pricing perpetual American options under a stochastic-volatilitymodel with fast mean reversion[J]. Appl. Math. Lett.24(2011), no.10,1663–1669.
[106] ZHU S P, LIAN G H. A closed-form exact solution for pricing variance swaps withstochastic volatility[J]. Math. Finance21(2011), no.2,233–256.