1. Pricing Asian Options in Affine Garch models Lorenzo Mercuri Dip. Metodi Quantitativi per le Scienze Economiche e Aziendali Milano-Bicocca 28-29-30th of January, 2009. X Workshop on Quantitative Finance

2. Outline Affine Garch models Geometric Asian options Arithmetic Asian options References

3. Affine Garch models Affine Garch Models yield a closed form formula for option prices by means of Fourier Transform Stock price dynamics where is an affine Garch process 1.The Heston and Nandi model Ref: Heston, Nandi (2000) Log-returns dynamics

4. Affine Garch models 2.The Christoffersen, Heston and Jacobs model Ref: Christoffersen, Heston and Jacobs (2006) Log-returns dynamic under real measure: 3.The Gamma model Ref: Bellini, Mercuri (2007) Log-returns dynamic under real measure:

5. Affine Garch models 4.The Tempered Stable model Ref: Mercuri (2008) Log-returns dynamic under real measure: Tempered Stable distribution Ref: Tweedie (1984), Hougaard (1986) Let be the positively skewed stable density function with We say that is a Tempered Stable distribution with if its density is given by: The characteristic function is given by:

6. Affine Garch models: Tempered Stable distribution Fig. 1. Behavior of the Tempered Stable density as function of b, from top to down and

7. Affine Garch models: Tempered Stable distribution Fig. 2. Convergence of the Tempered Stable density to the Gamma density (upper part) and to the Inverse Gaussian (lower part)

8. Affine Garch models: Change of measure Conditional Esscher Transform Ref. Siu et Al. (2004), Buhlmann et Al. (1996), Gerber and Shiu (1994). Definition Given a predictable stochastic process and an adapted process such that We denote with We define the stochastic process as To price a contingent claim, we need an equivalent martingale measure Q such that In order to obtain the martingale condition, we have to solve the following equation with respect to This equation is called Conditional Esscher Equation

9. Geometric Asian options: Option pricing formula We consider a geometric Asian call option with fixed strike where the underlying is observed at equally-spaced times. The pay-off is given by: Under Q measure, we price a geometric Asian call option with the formula: In Affine Garch Models, it’s possible to compute the moment generating function (m.g.f) of by a recursive procedure and then to price an option by the inversion of Fourier transform.

10. Geometric Asian options: Recursive procedure for m.g.f. We define: We write the m.g.f. of in exponential form: 1.Heston and Nandi model We get the following recursive relations: 2.The Christoffersen, Heston and Jacobs model

11. Geometric Asian options: Recursive procedure for m.g.f. 3.The Gamma model 4.The Tempered Stable model With terminal conditions

12. Geometric Asian options: Comparison between semianalytical price and Monte Carlo simulations

14. Arithmetic Asian options: Option pricing formula We consider an arithmetic Asian call option with fixed strike where the underlying is observed at equally-spaced times. The pay-off is given by: Under Q measure, we need to evaluate the following expected value: The distribution of the variable in unknown. Turnbull and Wakeman (1992) Approximate the true distribution with a more tractable distribution that matches the first fourth moments

15. Arithmetic Asian options: Recursive procedure for the moments We define: The nth-moment, given the information a time zero, is obtained by: In Affine Garch models, it is possible to compute recursively the quantity: Indeed

16. Arithmetic Asian options: Recursive procedure for the moments 1.The Heston and Nandi model 2.The Christoffersen Heston Jacobs model

17. Arithmetic Asian options: Recursive procedure for the moments 3.The Gamma model 4. The Tempered Stable model

18. Arithmetic Asian options: Approximation formula Rif. Lévy (1991) Turnbull and Wakeman (1992) Let the lognormal density function where the parameters match the mean and variance of the We can approximate the true distribution by using the fourth-order Edgeworth series Where With Therefore the approximate Asian option price is given by:

19. Arithmetic Asian options: Comparison between approximate formula and Monte Carlo simulations

21. References Bellini, F. Mercuri, L. (2007). “Option Pricing in the Garch Models” Working paper n.124. Buhlmann, H. Delbaen, F. Embrechts, P. Shiryaev, A.N. (1996) "No arbitrage, Change of Measure and Conditional Esscher Transform" CWI Quarterly 9(4) (1996) pp. 291- 317. Carr, P. Madan, D. B. (1999). “Option Valuation using the Fast Fourier Transform” Journal of Computational Finance 2 (4), 61-73. Christoffersen, P. Heston, S.L. Jacobs, C (2006). "Option valuation with conditional skewness" Journal of Econometrics, 131, 253-284. Fusai, G. Meucci, A. (2008). "Pricing discretely monitored Asian options under Levy processes," Journal of Banking & Finance, 32 (10), pp. 2076-2088. Fusai, G. Roncoroni, A. (2008). “Asian Options: An Average Problem, in Problem Solving in Quantitative Finance: A Case-Study Approach” Gerber, H.U. Shiu, E.S.W (1994) "Option pricing by Esscher transforms“ Transactions of the Society of Actuaries 46 (1994) pp. 99-191. Heston, S.L. Nandi, S. (2000). "A closed form option pricing model" Review of financial studies 13,3, pp. 585-562. Hougaard, P. (1986) "Survival models for heterogeneous populations derived from stable distributions" Biometrika 73, pp.387-396. Lévy, E., (1992). “Pricing European Average Rate Currency Options” Journal of International Money and Finance 11, 474-491. Mercuri, L. (2008). “Option Pricing in a Garch Model with Tempered Stable Innovations“ Finance Research Letters 5, pp.172-182. Siu, T.K. Tong, H. Yang, H. (2004)"On pricing derivatives under Garch models: a dynamic Gerber-Shiu approach" North American Actuarial Journal 8(3) pp. 17-31. Tweedie, M. C. K. (1984). “An Index wich Distinguishes between some important exponential families” Statistics: Applications and New Directions: Proc. Indian Statistical Institute Golden Jubilee International Conference (ed. J. Ghosh and J. Roy), pp.579-604.