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標題: 考慮隨機波動度之SABR-LMM模型對利率衍生性金融商品之評價
Pricing Interest Rate Derivatives Products in SABR(Stochastic Alpha Beta Rho Model)-LMM Model
作者: 游承翰
Yu, Cheng-Han
關鍵字: Volatility Smile (Skew);波動度微笑曲線;LIBOR Market Model;SABR Model;SABR-LMM Model;Stochastic Volatility LMM;Constant Maturity Swap (CMS);LIBOR市場模型;SABR模型;SABR-LMM模型;隨機波動度LIBOR市場模型;固定期限利率交換
出版社: 財務金融系所
引用: Andersen, L. and Andreasen, J. (2000) Volatility Skews and Extensions of the LIBOR Market Model, Applied Mathematical Finance, Vol. 7, March, pp. 1-32. Andersen, L. and Brotherton-Ratcliffe, R. (2005) Extended LIBRO Market Model with Stochastic Volatility, Journal of Computational Finance, Vol. 9, No. 1, Fall. Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, pp. 637-59. Brace, A., Gatarek, D. et Musiela, M. (1997) The Market Model of Interest Rate Dynamics, Mathematical Finance, 7(2), pp. 127-154. Brigo, D. and Mercurio, F. (2006) Interest Rate Models: Theory and Practice, New York: Springer-Verlag. Cox, J., Infersoll, J. and Ross, S. (1985) A Theory of the Term Structure of Interest Rate, Econometrica, Vol. 53, No. 2, March, pp. 385-407. Dupire, B. (1994) Pricing with a Smile, Risk, January, pp. 18-20. Dupire, B. (1997) Pricing and Hedging with Smiles, In Mathematics of Derivative Securities, Ch. 8, 103-111. Cambridge University Press. Derman, E. and Kani, I. (1998) Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility, International Journal of Theoretical Applied Finance, 1, pp. 61-110. Hagan, P.S., Kumar, D., Lesniewski, A.S. and Woodward, D. E. (2002) Managing smile risk, Wilmott, September, pp. 84-108. Hagan, P. S. and Lesniewski,A.S. (2008) Libor Market Model with SABR Style Stochastic Volatility, Working paper. Henry-Labordère (2007) Unifying the BGM and SABR Models: A Short Ride in Hyperbolic Geometry, Working paper available at Henry-Labordère (2007) Combining the SABR and LMM Model, Risk, October, pp. 102-107. Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies, 6, pp. 327-343. Ho, T. and Lee, S. (1986) Term Structure Movement and Pricing Interest Rate Contingent Claims, Journal of Finance, Vol. 41, No. 5, December, pp.1011-1029. Hull, J. and White, A. (1987) The pricing of options on assets with stochastic volatilities, Journal of Finance, Vol.42, pp.281-300. Hull, J. and White, A. (1990) Pricing Interest Rate Derivative Securities, Review of Finance Studies, Vol. 3, No. 4, pp. 573-592. Jäckel, P., and Rebonato, R. (2000) Linking Caplet and Swaption Volatilities in a BGM/J Framework: Approximate Solutions. Joshi, M., and Rebonato, R. (2003) A stochastic-volatility, displaced-diffusion extension of the LIBOR market model. Quantitative Finance, 3, pp. 458-469. Mercurio, F. and Pallavicini, A. (2006) Swaption Skews and Convexity Adjustments, Working paper available at Mercurio, F. and Morini, M. (2007a) A Note on Correlation in Stochastic Volatility Term Structure Models, Working paper available at Mercurio, F. and Morini, M.(2007b) No-arbitrage Dynamics for a Tractable SABR Term Structure LIBOR Model, Available at Mercurio, F. and Morini, M. (2009) Joining the SABR and LIBOR Models Together, Risk March, pp. 80-85. Piterbarg,V.V. (2005) A Stochastic Volatility model with Time Dependent Skew, Applied Mathematical Finance, 12(2), June ,pp. 147-185. Rebonato, R. (1998) Interest Rate Option Models, Second Edition, Wiley. Rebonato, R. (2007) A Time-homogeous, SABR-consistent Extension of the LMM: Calibration and Numerical Results, Risk November, pp. 92-97. Rebonato, R., McKay, K. and White,R. (2009) The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives, John Wiley & Sons. Vasicek, O.(1977) An Equilibrium Characterization of the Term Structure, Journal of financial Economics, 5, pp. 177-188. West, G.(2005) Calibration of the SABR Model in Illiquid Markets, Applied Mathenatical Finance, Vol. 12, No. 4, December, pp. 371-385. Wu, L. and Zhang, F. (2006) LIBOR Market Model with Stochastic Volatility, Journal of Industrial and Management Optimization, 2, May, pp. 199 - 227. Zhu, J. (2007) An Extended LIBOR Market Model with Nested Stochastic Volatility Dynamics, Working paper available at
波動度微笑曲線為1987年10月美國股市大崩盤之後首次顯現於選擇權市場中的現象,同時也顯示出Black and Scholes (1973)所提出的選擇權評價公式於波動度為常數函數的設定並不符合市場上的實際狀況, 因此針對模型加入隨機波動度的假設將更能反映出資產在市場上的真實動態。
本研究使用Mercurio and Morini (2009)所提出的SABR-LMM模型評價利率衍生性金融商品,該模型為結合了Hagan et al. (2002)的SABR模型以及Brace, Gaterek and Musiela (1997)的LIBOR市場模型所發展出來隨機波動度LIBOR市場模型。由於Mercurio and Morini (2009)沿用了Hagan et al. (2002) SABR模型的隱含波動度封閉公式,因此在參數校準的工作效率上優於傳統的蒙地卡羅模擬。

The appearance of volatility smile has been documented since at the least the time of the USA stock market crash of 1987 .This situation has pointed out the failure of constant volatility assumption in Black and Scholes model (1973).The followed researches have also found the behavior of implied volatility is more volatile than constant or deterministic function in the options market. Therefore, joint the stochastic volatility process into traditional models in order to reflect the true dynamics of volatility in real market has been considered.
This research used SABR-LMM model proposed by Mercurio and Morini (2009) for pricing interest derivative products. This is one kind of stochastic volatility LIBOR market model consistent with SABR dynamics and develops approximations that allow for use of the SABR implied volatility formula with modified inputs.
The SABR model is proposed by Hagan et al. (2002) and is now the market standard for dealing with volatility smiles. And the LIBOR market model can generate the joint evolution of forward rates when pricing complicate term structure products.
In the empirical study, we found that SABR-LMM model is capable to calibrate the whole swaption volatility surface and capture the volatility smile accurately. Therefore, this model can reinforcement the weakness of traditional LIBOR market model for dealing with the volatility skew effect. Besides, the procedure of calibration to SABR-LMM model is more efficient than simulation due to the SABR formula. And it's more elasticity on operation compare to LIBOR market model.
Finally, we simulated the forward rates and stochastic volatility dynamics under SABR-LMM model by using Monte Carlo method, and apply the result to CMS relative products pricing.
其他識別: U0005-2807201017475900
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