# American option pricing under GARCH with non-normal innovations

- 5 Downloads

## Abstract

As it is well known from the time-series literature, GARCH processes with non-normal shocks provide better descriptions of stock returns than GARCH processes with normal shocks. However, in the derivatives literature, American option pricing algorithms under GARCH are typically designed to deal with normal shocks. We thus develop here an approach capable of pricing American options with non-normal shocks. The approach uses an equilibrium pricing model with shocks characterized by a Johnson \(S_{u}\) distribution and a simple algorithm inspired from the quadrature approaches recently proposed in the option pricing literature. Numerical experiments calibrated to stock index return data show that this method provides accurate option prices under GARCH for non-normal and normal cases.

## Keywords

American options GARCH Johnson distribution Quadrature## JEL Classification

C63 G13## Notes

### Acknowledgements

This research was supported by the Social Sciences and Humanities Research Council of Canada. I would like to thank Lynn Mailloux for helpful comments

## References

- Andricopoulos A, Widdicks M, Newton D, Duck P (2007) Extending quadrature methods to value multi-asset and complex path dependent options. J Financ Econ 83:471–499CrossRefGoogle Scholar
- Andricopoulos A, Widdicks M, Duck P, Newton D (2003) Universal option variation using quadrature. J Financ Econ 67:447–471CrossRefGoogle Scholar
- Ben-Ameur H, Breton M, Martinez JM (2009) Dynamic programming approach for valuing options in the GARCH Model. Manag Sci 55:252–266CrossRefzbMATHGoogle Scholar
- Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–659MathSciNetCrossRefzbMATHGoogle Scholar
- Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econ 31:307–327MathSciNetCrossRefzbMATHGoogle Scholar
- Bollerslev T (1987) A Conditionally heteroskedastic time-series model for speculative prices and rates of return. Rev Econ Stat 69:542–547CrossRefGoogle Scholar
- Broadie M, Detemple J (1996) American option valuation: new bounds, approximations, and a comparison of existing methods. Rev Financ Stud 9:1211–1250CrossRefGoogle Scholar
- Cakici N, Topyan K (2000) The GARCH option pricing model: a lattice approach. J Comput Finance 3:71–85CrossRefGoogle Scholar
- Chen D, Härkönen H, Newton D (2014) Advancing the universality of quadrature methods to any underlying process for option pricing. J Financ Econ 114:600–612CrossRefGoogle Scholar
- Christoffersen P, Elkamhi R, Feunou B, Jacobs K (2010) Option valuation with conditional heteroskedasticity and non-normality. Rev Financ Stud 23:2139–2183CrossRefGoogle Scholar
- Christoffersen P, Jacobs K, Ornthanalai C (2013) GARCH option valuation: theory and evidence. J Deriv 21:8–41CrossRefGoogle Scholar
- Chung SL, Ko K, Shackleton M, Yeh CY (2010) Efficient quadrature and node positioning for exotic option valuation. J Futures Mark 30:1026–1057CrossRefGoogle Scholar
- Cosma A, Galluccio S, Pederzoli P, Scaillet O (2016) Early exercise decision in american options with dividends, stochastic volatility and jumps, Swiss Finance Institute Research paper no. 16-73. https://ssrn.com/abstract=2883202
- Duan JC (1995) The GARCH option pricing model. Math Finance 5:13–32MathSciNetCrossRefzbMATHGoogle Scholar
- Duan JC (1999) Conditionally fat-tailed distributions and the volatility smile in options, working paper, Hong Kong University of Science and TechnologyGoogle Scholar
- Duan JC, Simonato JG (2001) American option pricing under GARCH by a Markov chain approximation. J Econ Dyn Control 25:1689–1718MathSciNetCrossRefzbMATHGoogle Scholar
- Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50:987–1008MathSciNetCrossRefzbMATHGoogle Scholar
- Glasserman P (2004) Monte Carlo Methods in Financial Engineering. Springer, New YorkzbMATHGoogle Scholar
- Hansen B (1994) Autoregressive conditional density estimation. Int Econ Rev 35:705–730CrossRefzbMATHGoogle Scholar
- Hsieh D (1989) Modeling heteroscedasticity in daily foreign-exchange rates. J Bus Econ Stat 7:307–317Google Scholar
- Johnson N (1949) Systems of frequency curves generated by methods of translation. Biometrika 36:149–176MathSciNetCrossRefzbMATHGoogle Scholar
- Judd K (1998) Numerical methods in economics. MIT Press, CambridgezbMATHGoogle Scholar
- Lyuu Y, Wu C (2005) On accurate and provably efficient GARCH option pricing algorithms. Quant Finance 5:181–198MathSciNetCrossRefzbMATHGoogle Scholar
- Ritchken P, Trevor R (1999) Pricing options under generalized GARCH and stochastic volatility processes. J Finance 54:377–402CrossRefGoogle Scholar
- Simonato JG (2011) Computing American option prices in the lognormal jump-diffusion framework with a Markov chain. Finance Res Lett 8:220–226CrossRefGoogle Scholar
- Simonato JG (2012) GARCH processes with skewed and leptokurtic innovations: revisiting the Johnson \(S_{u}\) case. Finance Res Lett 9:213–219CrossRefGoogle Scholar
- Simonato JG (2016) A simplified quadrature approach for computing Bermudan option prices. Int Rev Finance 16:647–658CrossRefGoogle Scholar
- Simonato JG, Stentoft L (2015) Which pricing approach for options under GARCH with non-normal innovations? CREATES research paper 2015-32Google Scholar
- Sullivan M (2000) Valuing American put options using Gaussian quadrature. Rev Financ Stud 13:75–94CrossRefGoogle Scholar
- Stentoft L (2008) American option pricing using GARCH models and the Normal Inverse Gaussian distribution. J Financ Econom 6:540–582CrossRefGoogle Scholar
- Su H, Chen D, Newton D (2017) Option pricing via QUAD: from Black–Scholes–Merton to Heston with jumps. J Deriv 24:9–27CrossRefGoogle Scholar