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Matched Asymptotic Expansions in Financial Engineering

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Abstract

Modern financial practice depends heavily on mathematics and a correspondingly large theory has grown up to meet this demand. This paper focuses on the use of matched asymptotic expansions in option pricing; it presents illustrations of the approach in ‘plain vanilla’ option valuation, in valuation using a fast mean-reverting-stochastic volatility model, and in a model for illiquid markets. A tentative framework for matched asymptotic expansions applied directly to stochastic processes of diffusion type is also proposed.

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References

  1. M.J. Lighthill, Newer Uses of Mathematics. London: Penguin books (1978).

    MATH  Google Scholar 

  2. M.J. Lighthill, Introduction to Fourier Analysis and Generalised Functions. Cambridge: Cambridge University Press (1958).

    MATH  Google Scholar 

  3. F. Black and M. Scholes, The pricing of options and corporate liabilities. J. Political Economy 81 (1973) 637-654

    Article  Google Scholar 

  4. R.C. Merton, Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4 (1973) 141-183

    Article  MathSciNet  Google Scholar 

  5. P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives. Cambridge: Cambridge University Press (1995).

    MATH  Google Scholar 

  6. M. Widdicks, P. Duck, A. Andricopoulos and D.P. Newton, The Black-Scholes equation revisited: asymptotic expansions and singular perturbations. Math. Finance 15 (2005) 373-391

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Almgren and N. Chriss, Optimal execution of portfolio transactions. J. Risk 3 (2001) 5-39

    Google Scholar 

  8. D. Bakstein and S.D. Howison, An arbitrage-free liquidity model with observable parameters for derivatives, working paper, Mathematical Institute, Oxford University (2004).

  9. R. Frey and A. Stremme, Market volatility and feedback effects from dynamic hedging. University of Bonn, Discussion Paper (1995).

  10. V. Putyatin and J. Dewynne, Market liquidity and its effects on option valuation and hedging. Phil. Trans. R. Soc. London 357 (1999) 2093-2108

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. P. Schönbucher and P. Wilmott, The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61 (2000) 232-272

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Sircar and G. Papanicolaou, General Black–Scholes models accounting for increased market volatility from hedging strategies. Appl. Math. Finance 5 (1998) 45-82

    Article  MATH  Google Scholar 

  13. M. Mitton, Derivative pricing in an illiquid market. Transfer Thesis, OCIAM, Mathematical Institute, Oxford University (2003).

  14. J.R. King and C.P. Please, Diffusion of dopant in crystalline silicon: an asymptotic analysis. I.M.A. J. Appl. Maths 37 (1986) 185-197

    Article  MATH  MathSciNet  Google Scholar 

  15. J.P. Fouque, G. Papanicolaou and K.R. Sircar, Derivatives in Financial Markets with Stochastic Volatility. Cambridge: Cambridge University Press (2000).

    MATH  Google Scholar 

  16. S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6 (1993) 237-343

    Article  Google Scholar 

  17. S.D. Howison, A. Rafailidis and H. Rasmussen, On the pricing and hedging of volatility derivatives. Appl. Math. Finance 11 (2004) 317-346

    Article  MATH  Google Scholar 

  18. H.O. Rasmussen and P. Wilmott, Asymptotic analysis of stochastic volatility models. In: P. Wilmott and H.O. Rasmussen (eds.) New Directions in Mathematical Finance. New York: Wiley (2002) pp.

    Google Scholar 

  19. J.P. Fouque, G. Papanicolaou, K.R. Sircar and K. Solna, Singular perturbations in option pricing. SIAM J. Appl. Math. 63 (2003) 1648-1665

    Article  MATH  MathSciNet  Google Scholar 

  20. J.G. Conlon and M.G. Sullivan, Convergence to Black–Scholes for ergodic volatility models. (2005) Preprint.

  21. S.D. Howison, Practical Applied Mathematics. Cambridge: Cambridge University Press (2005).

    MATH  Google Scholar 

  22. S.D. Howison and J.R. King, Ray methods for free boundary problems. Quart. Appl. Math. (2005) In press.

  23. S.D. Howison and M. Steinberg, A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 1: barrier options. (2005) Preprint.

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  1. OCIAM and Nomura Centre for Quantitative Finance, Mathematical Institute, Oxford University, 24–29 St Giles, Oxford, OX1 3LB, UK

    Sam Howison

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  1. Sam Howison
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Correspondence to Sam Howison.

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Howison, S. Matched Asymptotic Expansions in Financial Engineering. J Eng Math 53, 385–406 (2005). https://doi.org/10.1007/s10665-005-7716-z

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  • DOI: https://doi.org/10.1007/s10665-005-7716-z

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