×

Moment explosions in stochastic volatility models. (English) Zbl 1142.65004

The authors consider a class of stochastic volatility models that are presented in terms of vector stochastic differential equations with correlated Brownian motions and power volatility coefficient. It is shown that the models based on log-normal or displaced-diffusion volatility specifications can exhibit moment explosions with fairly typical values of parameters but the CEV(constant elasticity variance)-based models are free of such problems. The behaviour of the Heston model, where the power coefficient is 1/2, is studied and a description of circumstances under which this model produces stable moments is provided. It is demonstrated that under the conditions of absence of moment stability the reasonably parameterised models can produce infinite prices for Eurodollar futures and for some swaps with floating legs.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
91B24 Microeconomic theory (price theory and economic markets)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andersen L., Andreasen J. (2002): Volatile volatilities. RISK 15(12): 163–168
[2] Andersen L., Brotherton-Ratcliffe R. (2005): Extended LIBOR market models with stochastic volatility. J. Comput. Financ. 9, 1–40
[3] Andersen, L., Piterbarg, V.: Moment explosions in stochastic volatility models: Supplement. Supplemental Note, Bank of America and Barclays Capital (2006), http://dx.doi.org/DOI 10.1007/s00780-006-0011-7
[4] Beckenbach E.F, Bellman R. (1965): Inequalities. Springer, Berlin Heidelberg New York
[5] Chen, R.R., Scott, L.: Stochastic volatility and jumps in interest rates: An empirical analysis. Working Paper, Rutgers University and Morgan Stanley (2002), http://www.rci.rutgers.edu/ chen/papers.html
[6] Collin-Dufresne P., Goldstein R. (2002): Do bonds span fixed income markets: Theory and evidence for unspanned stochastic volatility. J. Financ. 57, 1685–1730 · doi:10.1111/1540-6261.00475
[7] Cox A., Hobson D. (2005): Local martingales, bubbles and option prices. Financ. Stoch. 9, 477–492 · Zbl 1092.91023 · doi:10.1007/s00780-005-0162-y
[8] Cox J., Ingersoll J., Ross S. (1985): A theory of the term structure of interest rates. Econometrica 3, 385–408 · Zbl 1274.91447 · doi:10.2307/1911242
[9] Cox J., Ross S. (1976): The valuation of options for alternative stochastic processes. J. Financ. Econ. 7, 229–263 · Zbl 1131.91333 · doi:10.1016/0304-405X(79)90015-1
[10] Dragulescu A., Yakovenko V. (2002): Probability distribution of returns in the Heston model with stochastic volatility. Quant. Financ. 2, 443–453
[11] Duffie D. (2001): Dynamic Asset Pricing Theory. Princeton University Press, New Jersey · Zbl 1140.91041
[12] Hagan P., Kumar D., Lesniewski A., Woodward D. (2002): Managing smile risk. Wilmott Mag. 1, 84–108
[13] Han, B.: Stochastic volatilities and correlations of bond yields. Working Paper, Ohio State University (2003), http://fisher.osu.edu/an_184
[14] Heston S. (1993): A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327
[15] Hogan, M., Weintraub, K.: The lognormal interest rate model and Eurodollar futures. http://www.uni-bonn.de/www/Mathematik/Faculty/pdf/Sandmann7.pdf
[16] Hu, Y.: Exponential integrability of diffusion processes. In: Hill, T.P., Houdré, C. (eds.) Advances in stochastic inequalities. Contemporary Mathematics, vol. 234, pp 75–84. American Mathematical Society, New York (1999) · Zbl 0936.60075
[17] Hull J., White A. (1987): The pricing of options on assets with stochastic volatilities. J. Financ. 42, 281–300 · Zbl 1126.91369 · doi:10.2307/2328253
[18] Jamshidian F. (1997): Libor and swap market models and measures. Financ. Stoch. 1, 293–330 · Zbl 0888.60038 · doi:10.1007/s007800050026
[19] Jarrow, R., Li, H., Zhao, F.: Interest rate caps smile too! But can the Libor market models capture it? J. Financ., http://www.afajof.org/afa/forthcoming/2495.pdf
[20] Joshi M., Rebonato R. (2003): A stochastic volatility displaced diffusion extension of the Libor market model. Quant. Financ. 6, 458–469 · doi:10.1088/1469-7688/3/6/305
[21] Karatzas I., Shreve S. (1991): Brownian Motion and Stochastic Calculus. Springer, New York · Zbl 0734.60060
[22] Karlin S., Taylor H. (1981): A Second Course in Stochastic Processes. Academic, Boston · Zbl 0469.60001
[23] Lee R. (2004): The moment formula for implied volatility at extreme strikes. Math. Financ. 14, 469–480 · Zbl 1134.91443 · doi:10.1111/j.0960-1627.2004.00200.x
[24] Lewis A. (2000): Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach · Zbl 0937.91060
[25] Li H., Zhao F. (2006): Unspanned stochastic volatility: Evidence from hedging interest rate caps. J. Financ. 61, 341–378 · doi:10.1111/j.1540-6261.2006.00838.x
[26] Musiela M. (1985): Divergence, convergence and moments of some integral functionals of diffusions. Z. Wahrsch. verw. Gebiete 70, 49–65 · Zbl 0548.60075 · doi:10.1007/BF00532237
[27] Piterbarg V. (2005): Stochastic volatility model with time-dependent skew. Appl. Math. Financ. 12, 147–185 · Zbl 1148.91021 · doi:10.1080/1350486042000297225
[28] Revuz D., Yor M. (1991): Continuous Martingales and Brownian Motion. Springer, Berlin Heidelberg New York · Zbl 0731.60002
[29] Sandmann K., Sondermann D. (1997): A note on the stability of lognormal interest rate models and the pricing of Eurodollar futures. Math. Financ. 7, 119–125 · Zbl 0884.90049 · doi:10.1111/1467-9965.00027
[30] Sin C. (1998): Complications with stochastic volatility models. Adv. Appl. Probab. 30, 256–268 · Zbl 0907.90026 · doi:10.1239/aap/1035228003
[31] Wong B., Heyde C. (2004): On the martingale property of stochastic exponentials. J. Appl. Probab. 41, 654–664 · Zbl 1066.60064 · doi:10.1239/jap/1091543416
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.