非完全市场上奇异期权定价研究
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摘要
在本文中,我们研究当金融市场是随机波动率的非完全市场时,奇异期权(包括亚式期权,障碍期权和回望期权)的定价理论和风险管理理论。
利用Dannis Yang提出的随机控制应用于期权定价的理论,我们导出在非完全市场上,奇异期权的定价也是与投资者持有头寸相关联的。在指数效用函数的假定下,我们得到决定奇异期权均衡价值的偏微分方程,随机控制理论同时也给出最优的交易策略。众所周知,前人基于无套利定价理论在随机波动率的市场上给出期权的价格中,含有一个未定的参数(λ)-marke price ofrisk,不同的市场假设下,λ有不同的表现形式,如Heston著名的工作就是在这个参数是波动率的线性函数的假设下做出的。基于我们随机控制动态推导的方法,我们发现λ是与投资者持有头寸和投资品种相关的。在投资者只持有欧式期权和只持有亚式期权的两种情况下,λ即使对相同的投资者也是不同的。遵循Dennis Yang的定义,我们称这个参数为personal price of risk on Asian option(或者Lookback option)。
我们也讨论了当市场上包含所有金融期权产品,欧式期权和路径依赖的奇异期权,这些期权的定价是相互影响的。如,这时的欧式期权也会与回望期权的smdx(股票实现的最大值)有关,而不再是路径独立的。在研究这些期权的投资组合时,我们依然能够得到λ的显示表达式,它是与投资组合的各种期权的头寸相关,我们称之为"personal price of risk on Portfolio"。
在得到各种奇异期权在非完全市场的定价方程后,我们发展了解这类定价方程的有效数值方法。在给出奇异期权理论性质的同时,我们给出了期权价格的数值解,验证了理论分析的正确性。
最后,我们分析了最近市场上流行的累积期权(Knock Out Discount Accu-mulator),给出了在完全市场上基于风险中性的鞅测度框架下的期权定价公式,并分析了该类期权的一些套期保值参数。
In this thesis, we study the exotic option Pricing theory and risk management theory under incomplete market, including, Asian options, Barrier options, Lookback options.
Following Dennis Yang, we developed how to price exotic options with stochastic volatility in incomplete market. Under the exponential utility preference, we show that the fair price of exotic options is related to investor's position holding. As well-known, based on the arbitrage pricing theory the PDE satisfied by the option price is derived with a undetermined parameter-market price of risk. Under different assumption, the market price of risk is also different. Based on the dynamic derivation methodology, we get an explicit expression for market price of risk, and we found that market price of risk is related to the position and the type of option contained in the portfolio. Fol-lowing Dennis Yang, we call it personal price of risk on Asian option(or Lookback option,and so on)
We also discussed the portfolio optimization problem under stochastic volatility framework when the market consists all type of options, i.e. vanilla options and exotic options, we found that all options' price is mutual dependence, i.e. the fair price of vanilla option is dependent on smax and never path-independence when the portfolio contain European option and Lookback options. Furthermore, we give the numerical result of option price, and verified the theoretical properties of options.
Finally, we analysised the KODA option(Knock Out Discount Accumulator) which is popular in Hong Kong market recently. Under the risk-neutral martingale frame-work, we give an explicit formula for KODA option, and caculus some Hedging para-meters, like Delta and Gamma.
利用Dannis Yang提出的随机控制应用于期权定价的理论,我们导出在非完全市场上,奇异期权的定价也是与投资者持有头寸相关联的。在指数效用函数的假定下,我们得到决定奇异期权均衡价值的偏微分方程,随机控制理论同时也给出最优的交易策略。众所周知,前人基于无套利定价理论在随机波动率的市场上给出期权的价格中,含有一个未定的参数(λ)-marke price ofrisk,不同的市场假设下,λ有不同的表现形式,如Heston著名的工作就是在这个参数是波动率的线性函数的假设下做出的。基于我们随机控制动态推导的方法,我们发现λ是与投资者持有头寸和投资品种相关的。在投资者只持有欧式期权和只持有亚式期权的两种情况下,λ即使对相同的投资者也是不同的。遵循Dennis Yang的定义,我们称这个参数为personal price of risk on Asian option(或者Lookback option)。
我们也讨论了当市场上包含所有金融期权产品,欧式期权和路径依赖的奇异期权,这些期权的定价是相互影响的。如,这时的欧式期权也会与回望期权的smdx(股票实现的最大值)有关,而不再是路径独立的。在研究这些期权的投资组合时,我们依然能够得到λ的显示表达式,它是与投资组合的各种期权的头寸相关,我们称之为"personal price of risk on Portfolio"。
在得到各种奇异期权在非完全市场的定价方程后,我们发展了解这类定价方程的有效数值方法。在给出奇异期权理论性质的同时,我们给出了期权价格的数值解,验证了理论分析的正确性。
最后,我们分析了最近市场上流行的累积期权(Knock Out Discount Accu-mulator),给出了在完全市场上基于风险中性的鞅测度框架下的期权定价公式,并分析了该类期权的一些套期保值参数。
In this thesis, we study the exotic option Pricing theory and risk management theory under incomplete market, including, Asian options, Barrier options, Lookback options.
Following Dennis Yang, we developed how to price exotic options with stochastic volatility in incomplete market. Under the exponential utility preference, we show that the fair price of exotic options is related to investor's position holding. As well-known, based on the arbitrage pricing theory the PDE satisfied by the option price is derived with a undetermined parameter-market price of risk. Under different assumption, the market price of risk is also different. Based on the dynamic derivation methodology, we get an explicit expression for market price of risk, and we found that market price of risk is related to the position and the type of option contained in the portfolio. Fol-lowing Dennis Yang, we call it personal price of risk on Asian option(or Lookback option,and so on)
We also discussed the portfolio optimization problem under stochastic volatility framework when the market consists all type of options, i.e. vanilla options and exotic options, we found that all options' price is mutual dependence, i.e. the fair price of vanilla option is dependent on smax and never path-independence when the portfolio contain European option and Lookback options. Furthermore, we give the numerical result of option price, and verified the theoretical properties of options.
Finally, we analysised the KODA option(Knock Out Discount Accumulator) which is popular in Hong Kong market recently. Under the risk-neutral martingale frame-work, we give an explicit formula for KODA option, and caculus some Hedging para-meters, like Delta and Gamma.
引文
[1]Angus J A note on pricing derivatives with continuous geometric averaging. Journal of Fu-tures Market 19,845-858,1999
[2]F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities. Journal of Political Economy. Vol.81, No.3, pp.637-654, May/June 1973.
[3]Robert C.Merton. Continuous Time Finance, Basil Blackwell, Cambridge, MA,1992.
[4]Robert C. Merton. Optimum consumption and protfolio rules in a continuous-time model. J. Economic Theory,3:373-413,1971.
[5]Vicky Henderson. Valuation of claims on non-traded assets using utility maximization. Math-ematical Finance,12(4):351-373,2002.
[6]JC Cox,JE Ingersoll Jr, SA Ross A Theory of the Term Structure of Interest Rate. Economet-rica, Vol 53,385-407 1985
[7]Zhu J., Modular Pricing of Options Wroking paper, Eberhar-Karls-University, Tubingen
[8]J Ericsson, J Reneby A framework for valuing corporate securities Applied Mathematical Finance 1998
[9]Rich,D. The mathematical foundations of barrier option pricing theory. Advances in Futures Options Research 7,267-312.1996
[10]Heynen,R.C.,Kat, H.M. Crossing barriers Risk 7,46-49,1994. Correction(1995), Risk 8
[11]Gao, B. Huang,J.,Subrahmanyam,M.G., The valuation of American barrier options using the decomposition technique. Journal of Economic Dynamics and Control,2000
[12]Goldman,M.B., Sosin, H.B., Gatto,M.A. Path-dependent options. J.Finance 34,1111-1127,1979a
[13]Goldman,M.B., Sosin, H.B., Shepp,L. On contingent claims that insure ex-post optimal stock market timing J.Finance 34,401-413,1979b
[14]Kunitomo,N., Ikeda,M. Pricing options with Curved Boundaries. Mathematical Finance 2,275-298.1992
[15]M Broadie, J Detemple American capped call options on dividend-paying assets Review of Financial Studies,1995
[16]Heynen,R.C.,Kat, H.M. Partial barrier options. Journal of Financial Engineering 3,253-274 1994
[17]Heynen,R.C.,Kat, H.M. Lookback options with discrete and partial monitorning of the under-lying price. Applied Mathematical Finance 2,273-284,1995
[18]Boyle, P.P., Lau, S.H. Bumping up against the barrier with the binomial method Journal of Derivatives2,6-14 1994
[19]Merton, R.C. On the pricing of corporate debts:the risky structure of interest rate. Journal of Finance 29,449-469
[20]MB Garman The duration of option portfolios Journal of Financial Economics,1985
[21]J. Hull and A. White. The pricing of options on assets with stochastic volatilities. Journal of Finance,42(2):281-300,1987.
[22]Alberto Bressan. Viscosity solutions of Hamilton-Jacobi equations and optimal control prob-lems:an illustrated tutorial,2001.
[23]Vorst T. Prices and Hedge ratios of average exchange rate options Int.Rev.Financial Anall 179-193,1992
[24]Kwok Y K and Wong H Y. Currency-translated foreign equity options with path dependent features and their multi-asset extensions. Int.J.Theor.Appl.Finance3 257-278,2000
[25]Fouque J P and Han C H Pricing Asian options with stochastic volatility Quantitative Finance 3288-295,2003
[26]Hoi Ying Wong and Ying Lok Cheung Geometric Asian options:valuation and calibration with stochastic volatility Quantitative Finance 4 301-314,2004
[27]Peter K. Clark A subordinated stochastic process model with finite variance for speculative prices. Econometrica,41(1):135-155,1973
[28]Fred Espen Benth and Kenneth Hvistendahl Karlsen. A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets. Stochastics:An International Journal of Probability and Stochastic Processes,77,109-137,2005
[29]M.R. Grasselli and T.R. Hurd. Indifference pricing and hedging in stochastic volatility models. Preprint,2004
[30]Sasha Stoikov. Pricing options from the point of view of a trader. Preprint,2005
[31]Vicky Henderson and David Hobson. Utility indifferent pricing-an overview. Princeton Uni-versity Press.2004.
[32]LeBaron B Stochastic volatility as a simple generator of apparent financial power laws and long memory Quant. Finance 1621-31
[33]Linetsky V Spectral expansion for Asian(average price) options Preprint Northwestern Uni-versity,2002
[34]Vecer J Unified Pricing of Asian options Risk June 113-6,2002
[35]Vecer J and Xu M Pricing Asian options in a semimartingale model Preprint
[36]Steven Heston. A closed-form solution for options with stochastic volatility withapplications to bond and currency options. Review of Financial Studies,6(2):327-343,1993.
[37]M Avellaneda, A Paras, Managing the volatility risk of portfolios of derivative securities:the Lagrangian uncertain volatility model. Applied Mathematical Finance,1996.
[38]Steven E.shreve Stochastic Calculus for Finance Springer-Verkag New York,LLC
[39]Anh, V. and Inoue. Financial Markets with Memory Ⅰ:Dynamic Models. Stochastic Analysis and Applications, Vol.23 Issue 2, p275-300,26p,2005.
[40]Black, Fischer, and Myron S. Scholes. The Pricing of Options and Corporate Liabilities. Jour-nal of Political Economy 81 (May/June 1973):637-54.
[41]Dennis Yang. Quantitative Strategies for Derivatives Trading. Published by:ATMIF LLC, New Jersey, USA.2006
[42]Dennis Yang, Derivatives Pricing and Trading in Incomplete Markets:A Tutorial on Con-cepts. Available at http://atmif.com/papers,2006.
[43]Bruce P.Ayati.Glemn F.Webb and Alexander R.A.Anderson. Computational methods and re-sults for structured mltiscale models of tumor invasion Multiscal Modeling & Simulation:A SIAM Interdisciplinary Journal,5(1):1-20,2006
[44]Daniel J.Duffy. Finite difference methods in financial engineering:A partial Differential Equation Approach. John Wiley and Sons Ltd.,2006
[45]Daniel J.Duffy Financial Instrument Pricing Using C++ J ohn Wiley and Sons Ltd.,2005
[46]Hardy Hulley and Grant Lotter. Numerical Methods in Finance,2004 University of the Wit-watersrand.
[47]K.W.Morton and D.F.Mayers Numerical Solution of Partial Differential Equations. Cam-bridge University Press,1996
[48]William H. Press, Saul A. Teukolsky, William T.Vetterling, and Brain PFlannery. Numerical recipes in C:The Art of Scientific Computing. Cambridge University Press,2 edition,1992.
[49]Ahn,D.G., Figlewski.S., Gao,B. Pricing discrete barrier options with an adaptive mesh. Jour-nal of Derivatives,633-44,1999
[50]Andricopoulos,A., Widdicks, M.,Duck,P.,Newton, D. Universal option valuation using quadrature methods. Journal of Financial Economics 67,447-471,2003
[51]Dennis Yang, A Simple Jump to Default Model. Available at http://atmif.com/papers,2006.
[52]Dennis Yang, Beyond Black-Scholes Option Theory. Available at http://atmif.com/papers, 2007.
[53]Dennis Yang, A Note on Risk Neutral Pricing in Incomplete Markets. Available at http://atmif.com/papers,2008.
[54]Dupire, B. Pricing With A Smile, RISK,7 (January),18-20,1994.
[55]Thaleia Zariphopoulou. A solution approach to valuation with unhedgalbe risks. Finance and stochastics,5:61-82,2001
[56]Stein, E. M. and J. C. Stein. Stock Price Distributions with Stochastic Volatility:An analytic Approach. Review of Financial Studies,4,727-752,1991.
[57]Jun Liu. Portfolio Selection in Stochastic Environments. The review of Financial Studies v20. 2007.
[58]Sasha Stoikov. Pricing options from the point of view of a trader, preprint,2005.
[59]Robert Tompkins. Stock index future markets:Stochastic volatility models and smiles. Jour-nal of Futures Markets,21(1):43-78,2001.
[60]G Bakshi, C Cao,Z Chen Empirical performance of alternative option pricing models Journal ofFinance,1997
[61]R Schbel,J zhu Stochastic volatility with an Ornstein-Uhlenbeck process:an extension Review of Finance,3(1):23-46,1999
[62]Paul Wilmott. Paul Willmott on Quantitative Finance. John Wiley and Sons Ltd.2000
[63]Alan L. Lewis. Option Valuation under Stochastic Volatility. Financial Press, Newport Beach, CA,2000.
[64]Robert Kohn and Oana Papazoglu. On the equiavlence of the static and dynamic asset alloca-tion problems. Quantitative Finance,6(2):173-183,2006.
[65]Ioannis Karatzas. Lectures on the Mathematics of Finance. American Mathematical Society, Providence, RI,1997.
[66]Carl M. Bender and Steven A. Orszag. Advanced Mathematical Methods for Scientists adn Engineers, Springer,1999.
[67]Martin B. Haugh and Andrew W. Lo. Asset allocation and derivatives. Quantitative Finance, 1:45-72,2001.
[68]Michael Tehranchi. Explicit solutions of some utility maximization problems in imcomplete markets. Stochastic Processes and Their Applicaitons,114:109-125,2004.
[69]Bernt Oksendal. Stochastic Differential Equations:an introduction with applications. Springer, New York,6th edition,2003.
[70]Freilling G. A survey of nonsymmetric Riccati equations. Linear Algebra and its applications, 351-352,243-270,2002.
[71]Bjork T. Arbitrage Theory in Continuous Time. Oxford University Press, New York,1998.
[72]Ilhan, A., M. Jonsson and R. Sircar. Portfolio Optimization with Derivatives and Indifference Pricing forthcoming in volume on Indifference Pricing. Princeton Unicersity Press,2004.
[73]Sircar, R., and T. Zariphopoulou. Bounds and Asymptotic Approximations for Utility Prices when Volatility is Random to appear in SIAM J. Control and optimizations,43,1328-1353, 2004.
[74]Musiela, M. and T. Zariphopoulou. The Spot and Forward Dynamic Utilities and the Associ-ated Pricing Systems:Case Study of the Binomial Model, Volume on Indifference Pricing, R. Carmona (ed.), Princeon University Press, to appear.2005.
[75]Rouge, R., and N. El Karoui. Pricing via Utility Maximazition and Entropy. Mathematical Finance,10(2),259-276,2000.
[76]Davis, M.H.A. Option Valuation and Hedging with Basis Risk Systems Theory. Modelling, Analysis and Control, T.E. Djaferis and I.C. Schick (eds.), Kluver, Amsterdam.1999.
[77]Hagan, P.S., Kumar, D., Lesniewski, A.S and D.E. Woodward. Managing Smile Risk. WILMOTT Magazine,84-108,September,2002.
[78]Follmer, H. and D. Sondermann. Hedging of Non-Redundant Contingent Claims. Contribu-tions to mathematical economics in honor of Gerard Debreu. North Holland, Amsterdam, 205-223,1986.
[79]Follmer, H. and P. Leukert. Quantile Hedging. Finance and Stochastics.3,251-273,1999.
[80]Follmer, H. and P. Leukert. Efficient Hedging:Cost Versus Shortfall Risk. Finance and Sto-chasitcs.4,117-146,2000.
[81]Hodges, S.D., and A. Neuberger. Optimal Replication of Contingent Claims under Transac-tion Costs. Rev. Fut. Markets,8,222-239,1989.
[82]F. Delbaen, P. Grandits, T. Rheinlander, D. Samperi, M.Schweizer and C. Stricker, Exponen-tial hedging and entropic penalties, Mathematical Finance 12,99-123,2002.
[83]Rubinstein,M.,and H.E.Leland Replicating Options with Positions in Stock and Cash, Finan-cial Analysists Journal,37,63-72
[84]Davis, M.H.A. Complete-market models of stochastic volatility. Proc. R. Soc. Lond. A, 460:11-26,2004.
[85]Hobson, D.G. and Rogers, L.C.G. Complete models with stochasitc volatility. Mathematical Finance,8(1):27-48,1998.
[86]Frittelli M. The minimal entropy martingale measure and the valuation problem in incomplete markets. Mathmatics Finance,10(1):39-52,2000.
[87]A. J. Birkbeck. Asymptotic Skew under Stochastic Volatility. Working Paper.
[2]F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities. Journal of Political Economy. Vol.81, No.3, pp.637-654, May/June 1973.
[3]Robert C.Merton. Continuous Time Finance, Basil Blackwell, Cambridge, MA,1992.
[4]Robert C. Merton. Optimum consumption and protfolio rules in a continuous-time model. J. Economic Theory,3:373-413,1971.
[5]Vicky Henderson. Valuation of claims on non-traded assets using utility maximization. Math-ematical Finance,12(4):351-373,2002.
[6]JC Cox,JE Ingersoll Jr, SA Ross A Theory of the Term Structure of Interest Rate. Economet-rica, Vol 53,385-407 1985
[7]Zhu J., Modular Pricing of Options Wroking paper, Eberhar-Karls-University, Tubingen
[8]J Ericsson, J Reneby A framework for valuing corporate securities Applied Mathematical Finance 1998
[9]Rich,D. The mathematical foundations of barrier option pricing theory. Advances in Futures Options Research 7,267-312.1996
[10]Heynen,R.C.,Kat, H.M. Crossing barriers Risk 7,46-49,1994. Correction(1995), Risk 8
[11]Gao, B. Huang,J.,Subrahmanyam,M.G., The valuation of American barrier options using the decomposition technique. Journal of Economic Dynamics and Control,2000
[12]Goldman,M.B., Sosin, H.B., Gatto,M.A. Path-dependent options. J.Finance 34,1111-1127,1979a
[13]Goldman,M.B., Sosin, H.B., Shepp,L. On contingent claims that insure ex-post optimal stock market timing J.Finance 34,401-413,1979b
[14]Kunitomo,N., Ikeda,M. Pricing options with Curved Boundaries. Mathematical Finance 2,275-298.1992
[15]M Broadie, J Detemple American capped call options on dividend-paying assets Review of Financial Studies,1995
[16]Heynen,R.C.,Kat, H.M. Partial barrier options. Journal of Financial Engineering 3,253-274 1994
[17]Heynen,R.C.,Kat, H.M. Lookback options with discrete and partial monitorning of the under-lying price. Applied Mathematical Finance 2,273-284,1995
[18]Boyle, P.P., Lau, S.H. Bumping up against the barrier with the binomial method Journal of Derivatives2,6-14 1994
[19]Merton, R.C. On the pricing of corporate debts:the risky structure of interest rate. Journal of Finance 29,449-469
[20]MB Garman The duration of option portfolios Journal of Financial Economics,1985
[21]J. Hull and A. White. The pricing of options on assets with stochastic volatilities. Journal of Finance,42(2):281-300,1987.
[22]Alberto Bressan. Viscosity solutions of Hamilton-Jacobi equations and optimal control prob-lems:an illustrated tutorial,2001.
[23]Vorst T. Prices and Hedge ratios of average exchange rate options Int.Rev.Financial Anall 179-193,1992
[24]Kwok Y K and Wong H Y. Currency-translated foreign equity options with path dependent features and their multi-asset extensions. Int.J.Theor.Appl.Finance3 257-278,2000
[25]Fouque J P and Han C H Pricing Asian options with stochastic volatility Quantitative Finance 3288-295,2003
[26]Hoi Ying Wong and Ying Lok Cheung Geometric Asian options:valuation and calibration with stochastic volatility Quantitative Finance 4 301-314,2004
[27]Peter K. Clark A subordinated stochastic process model with finite variance for speculative prices. Econometrica,41(1):135-155,1973
[28]Fred Espen Benth and Kenneth Hvistendahl Karlsen. A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets. Stochastics:An International Journal of Probability and Stochastic Processes,77,109-137,2005
[29]M.R. Grasselli and T.R. Hurd. Indifference pricing and hedging in stochastic volatility models. Preprint,2004
[30]Sasha Stoikov. Pricing options from the point of view of a trader. Preprint,2005
[31]Vicky Henderson and David Hobson. Utility indifferent pricing-an overview. Princeton Uni-versity Press.2004.
[32]LeBaron B Stochastic volatility as a simple generator of apparent financial power laws and long memory Quant. Finance 1621-31
[33]Linetsky V Spectral expansion for Asian(average price) options Preprint Northwestern Uni-versity,2002
[34]Vecer J Unified Pricing of Asian options Risk June 113-6,2002
[35]Vecer J and Xu M Pricing Asian options in a semimartingale model Preprint
[36]Steven Heston. A closed-form solution for options with stochastic volatility withapplications to bond and currency options. Review of Financial Studies,6(2):327-343,1993.
[37]M Avellaneda, A Paras, Managing the volatility risk of portfolios of derivative securities:the Lagrangian uncertain volatility model. Applied Mathematical Finance,1996.
[38]Steven E.shreve Stochastic Calculus for Finance Springer-Verkag New York,LLC
[39]Anh, V. and Inoue. Financial Markets with Memory Ⅰ:Dynamic Models. Stochastic Analysis and Applications, Vol.23 Issue 2, p275-300,26p,2005.
[40]Black, Fischer, and Myron S. Scholes. The Pricing of Options and Corporate Liabilities. Jour-nal of Political Economy 81 (May/June 1973):637-54.
[41]Dennis Yang. Quantitative Strategies for Derivatives Trading. Published by:ATMIF LLC, New Jersey, USA.2006
[42]Dennis Yang, Derivatives Pricing and Trading in Incomplete Markets:A Tutorial on Con-cepts. Available at http://atmif.com/papers,2006.
[43]Bruce P.Ayati.Glemn F.Webb and Alexander R.A.Anderson. Computational methods and re-sults for structured mltiscale models of tumor invasion Multiscal Modeling & Simulation:A SIAM Interdisciplinary Journal,5(1):1-20,2006
[44]Daniel J.Duffy. Finite difference methods in financial engineering:A partial Differential Equation Approach. John Wiley and Sons Ltd.,2006
[45]Daniel J.Duffy Financial Instrument Pricing Using C++ J ohn Wiley and Sons Ltd.,2005
[46]Hardy Hulley and Grant Lotter. Numerical Methods in Finance,2004 University of the Wit-watersrand.
[47]K.W.Morton and D.F.Mayers Numerical Solution of Partial Differential Equations. Cam-bridge University Press,1996
[48]William H. Press, Saul A. Teukolsky, William T.Vetterling, and Brain PFlannery. Numerical recipes in C:The Art of Scientific Computing. Cambridge University Press,2 edition,1992.
[49]Ahn,D.G., Figlewski.S., Gao,B. Pricing discrete barrier options with an adaptive mesh. Jour-nal of Derivatives,633-44,1999
[50]Andricopoulos,A., Widdicks, M.,Duck,P.,Newton, D. Universal option valuation using quadrature methods. Journal of Financial Economics 67,447-471,2003
[51]Dennis Yang, A Simple Jump to Default Model. Available at http://atmif.com/papers,2006.
[52]Dennis Yang, Beyond Black-Scholes Option Theory. Available at http://atmif.com/papers, 2007.
[53]Dennis Yang, A Note on Risk Neutral Pricing in Incomplete Markets. Available at http://atmif.com/papers,2008.
[54]Dupire, B. Pricing With A Smile, RISK,7 (January),18-20,1994.
[55]Thaleia Zariphopoulou. A solution approach to valuation with unhedgalbe risks. Finance and stochastics,5:61-82,2001
[56]Stein, E. M. and J. C. Stein. Stock Price Distributions with Stochastic Volatility:An analytic Approach. Review of Financial Studies,4,727-752,1991.
[57]Jun Liu. Portfolio Selection in Stochastic Environments. The review of Financial Studies v20. 2007.
[58]Sasha Stoikov. Pricing options from the point of view of a trader, preprint,2005.
[59]Robert Tompkins. Stock index future markets:Stochastic volatility models and smiles. Jour-nal of Futures Markets,21(1):43-78,2001.
[60]G Bakshi, C Cao,Z Chen Empirical performance of alternative option pricing models Journal ofFinance,1997
[61]R Schbel,J zhu Stochastic volatility with an Ornstein-Uhlenbeck process:an extension Review of Finance,3(1):23-46,1999
[62]Paul Wilmott. Paul Willmott on Quantitative Finance. John Wiley and Sons Ltd.2000
[63]Alan L. Lewis. Option Valuation under Stochastic Volatility. Financial Press, Newport Beach, CA,2000.
[64]Robert Kohn and Oana Papazoglu. On the equiavlence of the static and dynamic asset alloca-tion problems. Quantitative Finance,6(2):173-183,2006.
[65]Ioannis Karatzas. Lectures on the Mathematics of Finance. American Mathematical Society, Providence, RI,1997.
[66]Carl M. Bender and Steven A. Orszag. Advanced Mathematical Methods for Scientists adn Engineers, Springer,1999.
[67]Martin B. Haugh and Andrew W. Lo. Asset allocation and derivatives. Quantitative Finance, 1:45-72,2001.
[68]Michael Tehranchi. Explicit solutions of some utility maximization problems in imcomplete markets. Stochastic Processes and Their Applicaitons,114:109-125,2004.
[69]Bernt Oksendal. Stochastic Differential Equations:an introduction with applications. Springer, New York,6th edition,2003.
[70]Freilling G. A survey of nonsymmetric Riccati equations. Linear Algebra and its applications, 351-352,243-270,2002.
[71]Bjork T. Arbitrage Theory in Continuous Time. Oxford University Press, New York,1998.
[72]Ilhan, A., M. Jonsson and R. Sircar. Portfolio Optimization with Derivatives and Indifference Pricing forthcoming in volume on Indifference Pricing. Princeton Unicersity Press,2004.
[73]Sircar, R., and T. Zariphopoulou. Bounds and Asymptotic Approximations for Utility Prices when Volatility is Random to appear in SIAM J. Control and optimizations,43,1328-1353, 2004.
[74]Musiela, M. and T. Zariphopoulou. The Spot and Forward Dynamic Utilities and the Associ-ated Pricing Systems:Case Study of the Binomial Model, Volume on Indifference Pricing, R. Carmona (ed.), Princeon University Press, to appear.2005.
[75]Rouge, R., and N. El Karoui. Pricing via Utility Maximazition and Entropy. Mathematical Finance,10(2),259-276,2000.
[76]Davis, M.H.A. Option Valuation and Hedging with Basis Risk Systems Theory. Modelling, Analysis and Control, T.E. Djaferis and I.C. Schick (eds.), Kluver, Amsterdam.1999.
[77]Hagan, P.S., Kumar, D., Lesniewski, A.S and D.E. Woodward. Managing Smile Risk. WILMOTT Magazine,84-108,September,2002.
[78]Follmer, H. and D. Sondermann. Hedging of Non-Redundant Contingent Claims. Contribu-tions to mathematical economics in honor of Gerard Debreu. North Holland, Amsterdam, 205-223,1986.
[79]Follmer, H. and P. Leukert. Quantile Hedging. Finance and Stochastics.3,251-273,1999.
[80]Follmer, H. and P. Leukert. Efficient Hedging:Cost Versus Shortfall Risk. Finance and Sto-chasitcs.4,117-146,2000.
[81]Hodges, S.D., and A. Neuberger. Optimal Replication of Contingent Claims under Transac-tion Costs. Rev. Fut. Markets,8,222-239,1989.
[82]F. Delbaen, P. Grandits, T. Rheinlander, D. Samperi, M.Schweizer and C. Stricker, Exponen-tial hedging and entropic penalties, Mathematical Finance 12,99-123,2002.
[83]Rubinstein,M.,and H.E.Leland Replicating Options with Positions in Stock and Cash, Finan-cial Analysists Journal,37,63-72
[84]Davis, M.H.A. Complete-market models of stochastic volatility. Proc. R. Soc. Lond. A, 460:11-26,2004.
[85]Hobson, D.G. and Rogers, L.C.G. Complete models with stochasitc volatility. Mathematical Finance,8(1):27-48,1998.
[86]Frittelli M. The minimal entropy martingale measure and the valuation problem in incomplete markets. Mathmatics Finance,10(1):39-52,2000.
[87]A. J. Birkbeck. Asymptotic Skew under Stochastic Volatility. Working Paper.