# 臺灣博碩士論文加值系統

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 在這篇論文中，我們推導一個對股價報酬的變異數交換做定價公式，並且與Carr and Wu(2004)此篇論文中的定價公式在相同的架構之下做比較。此外，我們也在Duan, Ritchken and Sun (2004)所提出假設股價報酬跟變異數都產生內容的NGARCH(1,1) 不連續模型之下，推導出對變異數交換的定價公式。我們也發現，跳躍現象的發生在變異數跟報酬均不連續的情況之下，對變異數交換的價格有明顯的影響。
 In this paper we developed a model for valuing variance swaps with jumps in the returns of underlying asset.We compare our simulation results with those of Carr and Wu (2004) model under the same framework.We find that our model value of variance swap contracts are very close to those of Carr and Wu model.We then applied Duan, Ritchken and Sun (2004) GARCH jump framework which analyzes which jumps could happen in both asset return and volatilityto develop a more general model for valuing variance swaps.From the simulation results, we find that both jumps in return and volatility will significantly affect the values of variance swaps.
 Contents1 Introduction 12 Variance swaps 32.1 How to price variance when the underlying asset price is continuous . 32.2 Hedging for Variance Swaps . . . . . . . . . . . . . . . . . . . . . . . 63 Pricing variance swaps when underlying stock return processes arediscontinued 63.1 The mixed processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Hedging for Variance Swaps under Mixed-Jump Process . . . . . . . . 103.5 The Comparison of our Model and Carr-Wu (2004) model . . . . . . 104 Pricing Variance Swaps When the Underlying Stocks Return ProcessesFollow GARCH Jump Model 124.1 A NGARCH(1,1) Jump Model . . . . . . . . . . . . . . . . . . . . . . 124.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Conclusion 18References 20A The proof of equations (26)-(27) 22B The proof of equation (40) 23C The proof of equation (41)-(42) 24
 Black, F., M. Scholes, 1973,”The pricing of options and corporate liabilities”, Journalof Political Economy 81, pp.637-654.Brenner, M., E. Ou, and J. Zhang, 2001,”Hedging Volatility Risk”, NYU workingpaper.Brenner, M., and D. Galai, 1989,”New financial instruments for hedging changesin volatility”, Financial Analyst’s Journal July-August, pp.61-65.Carr, P., and Wu, Liuren, 2004, ”Variance Risk Premia”.Carr, P. , and K. Lewis, 2002, ”Corridor variance swaps”, NYU working paper.Carr, P.,and D. Madan, 1998, ”Towards a theory of volatility trading”, Volatility, RiskPublications, R. Jarrow, ed., pp.417-427.Reprinted in Option Pricing, Interest Rates,and Risk Management, Musiella, Jouini, Cvitanic, ed., Cambridge University Press,2001, pp. 458-476. Available at http://www.math.nyu.edu/research/carrp/papersDemeterfi, K., E. Derman, M. Kamal, and J. Zhou, 1999, ”A guide to varianceswaps”, Journal of Derivatives, 6, 4, pp. 9-32.Duan, J. (1995) ”The GARCH Option Pricing Model”, Mathematical Finance, 5,13-32.Duan, J., Ritchken, P., Sun, Z, 2004, ”Jump Starting GARCH: Pricing and HedgingOption With Jumps in Riturns and Volatilities”, working paperNavas. J. F., 2003, ” Calculation of Volatility in a Jump-Diffusion Model”, Journalof Derivates, 11, pp. 66-72 Heston, S., and S. Nandi, 2000,”Derivatives on volatility:Some simple solutions based on observables”, Federal Reserve Bank of AtlantaWorking Paper.Hull, J., 2002, Options, Futures and Other Derivatives, fifth edition, Prentice Hall.Jarrow, R., and S.Turnbull, 2000, Derivatives Securities, second edition, South-Western.Merton, R. 1976, ”Option pricing when underlying returns are discontinues”, Journalof Financial Economics 4, pp. 124-44.Musiela, M., and M. Rutkowski, 1997, Martingale Methods in Financial Modelling,Springer.Neftci, S., 2004, Principles of Financial Engineering, first edition, Academic Press.Neuberger, A., 1994, ”The log contract: A new instrument to hedge volatility.”,Journal of Portfolio Management Winter, pp. 74-80.Ross, S., 1996, Stochastic Processes, second edition, John Wiley Sons.Runggaldier, W., 2003, ”Jump-diffusion models”, Handbook of Heavy Tailed Distributionsin Finance (S.T. Rachev, ed.),Elesevier/North-Holland.Sulima, C., 2001, ”Volatility and variance swaps”, Capital Market News, Federal ReserveBank of Chicago.
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