Singular perturbations for boundary value problems arising from exotic options

SIAM Journal on Applied Mathematics

Volumn 64, Issue 4, 2004, Pages 1268-1293

  Ilhan, Aytac ^a   Jonsson, Mattias ^b   Sircar, Ronnie ^a  

^a PRINCETON UNIVERSITY   (United States)

^b UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Asymptotic approximations; Option pricing; Stochastic volatility

Indexed keywords

ASYMPTOTIC APPROXIMATIONS; OPTION PRICING; STOCHASTIC VOLATILITY; VOLATILITY DIFFUSION MODEL;

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; BOUNDARY VALUE PROBLEMS; COMPUTATIONAL COMPLEXITY; INTEGRAL EQUATIONS; LINEAR SYSTEMS; MARKOV PROCESSES; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; OPTIMAL CONTROL SYSTEMS;

PERTURBATION TECHNIQUES;

EID: 4544264572     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139902420043     Document Type: Article

References (29)

1. S. ALIZADEH, M. BRANDT, AND F. DIEBOLD, Range-baaed estimation of stochastic volatility modela, J. Finance, 57 (2002), pp. 1047-1091.
2. G. BAKSHI, C. CAO, AND Z. CHEN, Empirical performance of alternative option pricing models, J. Finance, 52 (1997), pp, 2003-2049.
3. H. CARSLAW AND J. JAEGER, Conduction of Heat in Solids, 2nd ed., Oxford Clarendon Press, Oxford, UK, 1959.
4. M. CHERNOV, R. GALLANT, E. GHYSELS, AND G. TAUCHEN, Alternative models for stock price dynamics, J. Econometrics, 116 (2003), pp. 225-257.
5. J. CONLON AND M. SULLIVAN, Convergence to Black-Scholes for ergodic volatility models, preprint, Department of Mathematics, University of Michigan, Ann Arbor, MI, 2003.
6. D. DARIUS, A. ILHAN, J. MULVEY, K. SIMSEK, AND K. R. SIRCAR, Trend-following hedge funds and multi-period asset allocation, Quant. Finance, 2 (2002), pp. 354-361.
7. D. DAVYDOV AND V. LINETSKY, Pricing and hedging path- dependent options under the CEV process, Management Sci., 47 (2001), pp. 949-965.
8. D. DUFFIE, Dynamic Asset Pricing Theory, 3rd ed., Princeton University Press, Princeton, NJ, 2001.
9. D. DUFFIE, J. PAN, AND K. SINGLETON, Transform analysis and option pricing for affine jump-diffusions, Econometrica, 68 (2000), pp. 1343-1377.
10. R. ENGLE AND A. PATTON, What good is a volatility model?, Quant. Finance, 1 (2001), pp. 237-245.
11. H. FÖLLMER AND P. LEUKERT, Efficient hedging: Cost versus shortfall risk, Finance Stoch., 4 (2000), pp. 117-146.
12. J.-P. FOUQUE AND C.-H. HAN, Pricing Asian options with stochastic volatility, Quant. Finance, 3 (2003), pp. 353-362.
13. J.-P. FOUQUE, G. PAPANIOOLAOU, AND K. R. SIRCAR, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, London, 2000.
14. J.-P. FOUQUE, G. PAPANICOLAOU, K. R. SIRCAR, AND K. SOLNA, Maturity cycles in implied volatility, Finance Stooh., to appear.
15. J.-P. FOUQUE, G. PAPANICOLAOU, R. SIRCAR, AND K. SOLNA, Multiscale stochastic volatility asymptotics, Multiscale Model. Simul., 2 (2003), pp. 22-42.
16. J.-P. FOUQUE, G. PAPANICOLAOU, K. R. SIRCAR, AND K. SOLNA, Short time-scale in SOP 500 volatility, J. Computational Finance, 6 (2003), pp. 1-23.
17. J.-P. FOUQUE, G. PAPANICOLAOU, R. SIRCAR, AND K. SOLNA, Singular perturbations in option pricing, SIAM J. Appl. Math., 63 (2003), pp. 1648-1665.
18. M. GOLDMAN, H. SOZIN, AND M. GATTO, Path dependent options: Buy at the low, sell at the high, J. Finance, 34 (1979), pp. 1111-1128.
19. V. HENDERSON AND D. HOBSON, Passport options with stochastic volatility, Applied Math. Finance, 8 (2001), pp. 97-118.
20. S. HESTON, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), pp. 327-343.
21. S. HOWISON AND D. LAMPER, Trading volume in models of financial derivatives, Applied Math. Finance, 8 (2001), pp. 119-135.
22. S. HOWISON, A. RAFAILIDIS, AND H. RASMUSSEN, On the Pricing and Hedging of Volatility Derivatives, preprint, Mathematical Institute, University of Oxford, Oxford, UK, 2003.
23. J. HULL AND A. WHITE, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), pp. 281-300.
24. T. HYER, A. LIPTON-LIFSCHITZ, AND D. PUGACHEVSKY, Passport to success, Risk, 10 (1997), pp. 127-131.
25. M. JONSSON AND K. R. SIRCAR, Optimal investment problems and volatility homogenization approximations, in Modern Methods in Scientific Computing and Applications, NATO Sci. Ser. II Math. Phys. Chem. 75, A. Bourlioux, M. Gander, and G. Sabidussi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002, pp. 255-281.
26. M. JONSSON AND K. R. SIRCAR, Partial hedging in a stochastic volatility environment, Math. Finance, 12 (2002). pp. 375-409.
27. V. LIKETSKY, Lookback Options and Diffusion Hitting Times: A Spectral Expansion Approach, technical report, IEMS Department, Northwestern University, Evanston, IL, 2002.
28. S. E. SHREVE AND J. VECER, Options on a traded account: Vacation calls, vacation puts, and passport options. Finance Stooh., 4 (2000), pp. 225-274.
29. P. WILMOTT, S. HOWISON, AND J. DEWYNNE, Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, Cambridge, UK, 1996.