欧式期权定价的随机波动率模型
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摘要
在本文中,我们考虑带有多个原生资产的欧式期权定价问题的随机波动率模型。本文假设随机波动率是OU扩散过程的遍历函数。利用奇异摄动分析方法和带有多个参数的边界层理论,我们得到欧式期权价格的一致渐近展开式和该近似解的一致有效的误差估计。
In the last 20 years the famous Black-Scholes model has been proved to be an efficient and important tool for pricing option.However, recently,there are some evidences that the constant volatility assumption in Black-Scholes model 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,where the volatilityσ(t,ω)=f(Y) is usually supposed to be a function of an OU process Y(t) which is ergodic.
There are many papers devoted to the study of the stochastic volatility model.It was Hull and White who first studied the vanilla option with stochastic volatility f~2(Y)=Y in[13].Recently,Fouque [18],[19]found a drawback in Hull-White's model that it can produce a negative volatility,and alternatively presented a more suitable assumption that the stochastic volatility is fast mean-reverting,on a time scale smaller than the typical maturities.By the singular pertur- bation analysis,the first order approximation to the option price and its accuracy were obtained.Moreover,it was shown in[20]that the process driving stochastic volatility should be supposed to be a more generalized diffusion on a compact set.While Wong and Chan[21] studied the stochastic volatility with many factors in dealing with the lookback option pricing.
However,in respect to the above mentioned investigations,the stochastic volatility models for pricing option are considered under only one kind of assets.In this paper,we consider the multi-asset European call option with stochastic volatilities.As in[19],[20]we assume the stochastic volatilities to be fast mean-reverting and the driving diffusion process are on circles.Multi-asset option models are more natural in real world.To avoid risk,investors would like to choose portfolio which consist of any combination of stocks,bonds and so on.For instance,a currency basket option provides a cheaper method for multinational corporations to receive or sell a basket of several currencies for one specified currency,since the volatility of a basket of underlying assets is relatively low.Unfortunately,Jiang pointed in[22]that it is impossible to consider the basket option pricing model as a 1-dimensional problem through any mathematical transformation.For simplicity,we consider the European call option under two kinds of assets.For example,the quanto option with two kinds of assets,the stock and the exchange rate or the better-of option with stock A and stock B.
Our main goal in this paper is to provide asymptotic and approximate solutions of the option pricing equation by the technique of singular perturbation and multi-scale method.Notice that it is much more difficult to deal with the singular problem with two different small parameters.The leading term in the asymptotic expansion is the solution of classical Black-Scholes equation with a constant volatility. This constant volatilty is actually the average of the stochastic volatilities with respect to the distribution of driving processes.The first order terms of the asymptotic expansion consist of derivatives of classical Black-Scholes solutions above.As in[19],the first order terms are also called corrected terms because they reveal the correction to the shape of the implied volatility skewness.However,the second or higher order terms in the asymptotic expansions have singularities as time approach to maturity.Thus we introduce a new time scaling to derive the boundary layers which ensure us to work out a uniform asymptotic approximation and the corresponding error estimates.
The rest of this paper is arranged as follows.In Section 2,we review the stochastic volatility model for European option under only one kind of asset.We state the main results and main theorem.In section 3,we introduce the stochastic volatility model under two kinds of underlying assets and seek for the uniform asymptotic expansions of the prices of derivatives,we present and prove the main theorem on the accuracy of uniform asymptotic expansions to the option price.
In the last 20 years the famous Black-Scholes model has been proved to be an efficient and important tool for pricing option.However, recently,there are some evidences that the constant volatility assumption in Black-Scholes model 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,where the volatilityσ(t,ω)=f(Y) is usually supposed to be a function of an OU process Y(t) which is ergodic.
There are many papers devoted to the study of the stochastic volatility model.It was Hull and White who first studied the vanilla option with stochastic volatility f~2(Y)=Y in[13].Recently,Fouque [18],[19]found a drawback in Hull-White's model that it can produce a negative volatility,and alternatively presented a more suitable assumption that the stochastic volatility is fast mean-reverting,on a time scale smaller than the typical maturities.By the singular pertur- bation analysis,the first order approximation to the option price and its accuracy were obtained.Moreover,it was shown in[20]that the process driving stochastic volatility should be supposed to be a more generalized diffusion on a compact set.While Wong and Chan[21] studied the stochastic volatility with many factors in dealing with the lookback option pricing.
However,in respect to the above mentioned investigations,the stochastic volatility models for pricing option are considered under only one kind of assets.In this paper,we consider the multi-asset European call option with stochastic volatilities.As in[19],[20]we assume the stochastic volatilities to be fast mean-reverting and the driving diffusion process are on circles.Multi-asset option models are more natural in real world.To avoid risk,investors would like to choose portfolio which consist of any combination of stocks,bonds and so on.For instance,a currency basket option provides a cheaper method for multinational corporations to receive or sell a basket of several currencies for one specified currency,since the volatility of a basket of underlying assets is relatively low.Unfortunately,Jiang pointed in[22]that it is impossible to consider the basket option pricing model as a 1-dimensional problem through any mathematical transformation.For simplicity,we consider the European call option under two kinds of assets.For example,the quanto option with two kinds of assets,the stock and the exchange rate or the better-of option with stock A and stock B.
Our main goal in this paper is to provide asymptotic and approximate solutions of the option pricing equation by the technique of singular perturbation and multi-scale method.Notice that it is much more difficult to deal with the singular problem with two different small parameters.The leading term in the asymptotic expansion is the solution of classical Black-Scholes equation with a constant volatility. This constant volatilty is actually the average of the stochastic volatilities with respect to the distribution of driving processes.The first order terms of the asymptotic expansion consist of derivatives of classical Black-Scholes solutions above.As in[19],the first order terms are also called corrected terms because they reveal the correction to the shape of the implied volatility skewness.However,the second or higher order terms in the asymptotic expansions have singularities as time approach to maturity.Thus we introduce a new time scaling to derive the boundary layers which ensure us to work out a uniform asymptotic approximation and the corresponding error estimates.
The rest of this paper is arranged as follows.In Section 2,we review the stochastic volatility model for European option under only one kind of asset.We state the main results and main theorem.In section 3,we introduce the stochastic volatility model under two kinds of underlying assets and seek for the uniform asymptotic expansions of the prices of derivatives,we present and prove the main theorem on the accuracy of uniform asymptotic expansions to the option price.
引文
[1]CLARK P K.A subordinated stochastic process model with fixed variance for speculative prices [J].Econometrica,1973,41(1):135-156.
[2]BOCHNER S.Diffusion equation and stochastic processes [J].Proceedings of the National Academy of Science of the United States of America,1949,35(7):368-370.
[3]MADAN D B,SENETA E.The VG model for share market returns [J].Journal of Business,1990,63(4):511-524.
[4]EBERLEIN E,KELLER U.Hyperbolic distributions in finance [J].Bernoulli,1995,1(3):281-299.
[5]BARNDORFF-NIELSEN O E.Processes of normal inverse Gaussian type [J].Finance and Stochastics,1998,2(1):41-68.
[6]CARR P,GEMAN H,MADAN D B,YOR M.The fine structure of asset returns:an empirical investigation [J].Journal of Business 2002,75(2):305-32.
[7]TAYLOR S J.Modelling Financial Time Series [M].2nd ed.New Jersey:World Scientific,1986.
[8]TAYLOR S J.Conjectured models for trends in financial prices,tests and forecasts [J].Journal of the Royal Statistical Society Series A,1980,143(3):338-362.
[9]TAYLOR S J.Financial returns modeled by the product of two stochastic processes.A study of daily sugar prices 1961-79,in:O.D.,Anderson,(Ed.).Time Series Analysis:Theory and Practice 1,North-Holland Amsterdam 1982,203-226.
[10]JOHNSON H.Option pricing when the variance rate is changing,working paper.Los Angeles:University of California,1979.
[11]SHANNO D.Option pricing when the variance is changing [J].Journal of Financial and Quantitative Analysis,1987,22(2):143-151.
[12]WIGGINS J B.Option values under stochastic volatilities [J].Journal of Financial Economics 1987,19(2):351-372.
[13]HULL J,WHITE A.The pricing of options on assets with stochastic volatilities [J].Journal of Finance,1987,42(2):281-300.
[14]IKEDA N,WATANABE S.Stochastic Differential Equations and Diffusion Processes [M].2nd ed.Amsterdam:North-Holland,1989.
[15]HESTION S L.A closed-form solution for options with stochastic volatility,with applications to bond and currency options [J].Review of Financial Studies,1993,6(2):327-343.
[16]BLACK F,SCHOLES M.The pricing of options and corporate liabilities [J].Journal of Political Economy,1973,81(3):637-654.
[17]MERTONR C.Rational theory of option pricing [J].Bell Journal of Economics and Management Science,1973,4(1):141-183.
[18]FOUQUE J P,PAPANICOLAOU G,SIRCAR R K.Derivatives in Financial Markets with Stochastic Volatility [M].Cambridge:Cambridge University Press,2000.
[19]FOUQUE J P,PAPANICOLAOU G,SIRCAR R K,SOLNA K.Singular perturbations in option pricing [J].SIAM J.Appl.Math.,2003,63(5):1648-1665.
[20]KHASMINSKII R Z,YIN G.Uniform asymptotic expansions for pricing European options [J].Appl.Math.Optim,2005,52(3):279-296.
[21]WONG H Y,CHAN C M.Lookback options and dynamic fund protection under multiscale stochastic volatility [J].Insur.Math.Econ,2007,40(3):357-385.
[22]JIANG L S.Mathematical Modeling and Methods of Option Pricing [M].Beijing:Higher Education Press,2003.
[23]SOHNSON R S.Singular Perturbation Theory,Mathematical and Analytical Techniques wtih Applications to Engineering [M].Boston:Springer,2005.
[24]FREY R.Derivative asset analysis in models with level-dependent and stochastic volatility [C].Discussion Paper Serie B,401,University of Bonn,Germany,1997
[25]GHYSELS E,HARVEY A C,RENAULT E.Stochastic volatility.In Rao C R,Maddala G S.(eds.),Statistical Methods in Finance,Amsterdam:North-Holland,1996:119-191.
[26]FOUQUE J P,PAPANICOLAOU G,SIRCAR R K.Mean-reverting stochastic volatility [J].International Journal of Theoretical and Applied Finance,2000,3:101-142.
[27]KHASMINSKII R Z,YIN G.Limit behavior of two-time-scale diffusions revisited [J].J.Differential Equations,2005,212(1):85-113.
[28](?)KSENDAL B.Stochastic Differential Equations:An Introduction with Applications [M].6th ed,New York:Corrected Printing Springer-Verlag,2005.
[29]KLEBANER F C.Introduction to Stochastic Calculus with Ap- plication [M].London:Imperical College Press,1998.
[2]BOCHNER S.Diffusion equation and stochastic processes [J].Proceedings of the National Academy of Science of the United States of America,1949,35(7):368-370.
[3]MADAN D B,SENETA E.The VG model for share market returns [J].Journal of Business,1990,63(4):511-524.
[4]EBERLEIN E,KELLER U.Hyperbolic distributions in finance [J].Bernoulli,1995,1(3):281-299.
[5]BARNDORFF-NIELSEN O E.Processes of normal inverse Gaussian type [J].Finance and Stochastics,1998,2(1):41-68.
[6]CARR P,GEMAN H,MADAN D B,YOR M.The fine structure of asset returns:an empirical investigation [J].Journal of Business 2002,75(2):305-32.
[7]TAYLOR S J.Modelling Financial Time Series [M].2nd ed.New Jersey:World Scientific,1986.
[8]TAYLOR S J.Conjectured models for trends in financial prices,tests and forecasts [J].Journal of the Royal Statistical Society Series A,1980,143(3):338-362.
[9]TAYLOR S J.Financial returns modeled by the product of two stochastic processes.A study of daily sugar prices 1961-79,in:O.D.,Anderson,(Ed.).Time Series Analysis:Theory and Practice 1,North-Holland Amsterdam 1982,203-226.
[10]JOHNSON H.Option pricing when the variance rate is changing,working paper.Los Angeles:University of California,1979.
[11]SHANNO D.Option pricing when the variance is changing [J].Journal of Financial and Quantitative Analysis,1987,22(2):143-151.
[12]WIGGINS J B.Option values under stochastic volatilities [J].Journal of Financial Economics 1987,19(2):351-372.
[13]HULL J,WHITE A.The pricing of options on assets with stochastic volatilities [J].Journal of Finance,1987,42(2):281-300.
[14]IKEDA N,WATANABE S.Stochastic Differential Equations and Diffusion Processes [M].2nd ed.Amsterdam:North-Holland,1989.
[15]HESTION S L.A closed-form solution for options with stochastic volatility,with applications to bond and currency options [J].Review of Financial Studies,1993,6(2):327-343.
[16]BLACK F,SCHOLES M.The pricing of options and corporate liabilities [J].Journal of Political Economy,1973,81(3):637-654.
[17]MERTONR C.Rational theory of option pricing [J].Bell Journal of Economics and Management Science,1973,4(1):141-183.
[18]FOUQUE J P,PAPANICOLAOU G,SIRCAR R K.Derivatives in Financial Markets with Stochastic Volatility [M].Cambridge:Cambridge University Press,2000.
[19]FOUQUE J P,PAPANICOLAOU G,SIRCAR R K,SOLNA K.Singular perturbations in option pricing [J].SIAM J.Appl.Math.,2003,63(5):1648-1665.
[20]KHASMINSKII R Z,YIN G.Uniform asymptotic expansions for pricing European options [J].Appl.Math.Optim,2005,52(3):279-296.
[21]WONG H Y,CHAN C M.Lookback options and dynamic fund protection under multiscale stochastic volatility [J].Insur.Math.Econ,2007,40(3):357-385.
[22]JIANG L S.Mathematical Modeling and Methods of Option Pricing [M].Beijing:Higher Education Press,2003.
[23]SOHNSON R S.Singular Perturbation Theory,Mathematical and Analytical Techniques wtih Applications to Engineering [M].Boston:Springer,2005.
[24]FREY R.Derivative asset analysis in models with level-dependent and stochastic volatility [C].Discussion Paper Serie B,401,University of Bonn,Germany,1997
[25]GHYSELS E,HARVEY A C,RENAULT E.Stochastic volatility.In Rao C R,Maddala G S.(eds.),Statistical Methods in Finance,Amsterdam:North-Holland,1996:119-191.
[26]FOUQUE J P,PAPANICOLAOU G,SIRCAR R K.Mean-reverting stochastic volatility [J].International Journal of Theoretical and Applied Finance,2000,3:101-142.
[27]KHASMINSKII R Z,YIN G.Limit behavior of two-time-scale diffusions revisited [J].J.Differential Equations,2005,212(1):85-113.
[28](?)KSENDAL B.Stochastic Differential Equations:An Introduction with Applications [M].6th ed,New York:Corrected Printing Springer-Verlag,2005.
[29]KLEBANER F C.Introduction to Stochastic Calculus with Ap- plication [M].London:Imperical College Press,1998.