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An easy computable upper bound for the price of an arithmetic Asian option. (English) Zbl 0964.91021

In an arbitrage-free complete market, the price of an arithmetic Asian option with \(n\) averaging dates is of the form \( E[ ( {1\over n} \sum_{i=1}^n S_{t_i} - K)^+ ] \). This is bounded from above by the price of a portfolio of European call options, namely \( {1\over n} \sum_{i=1}^n E[ ( S_{t_i} - k_i)^+ ] \), for any choice of \(k_i\) summing to \(nK\). By using results from [M. J. Goovaerts and J. Dhaene, Insur. Math. Econ. 24, 281-290 (1999; Zbl 0942.60008)] on comonotone random variables, this paper provides explicit formulae for the optimal choice of \(k_i\) in terms of the marginal distributions \(F_i\) of \(S_{t_i}\). In comparison to prices computed by simulation, the resulting bounds in the Black-Scholes model seem to be rather sharp.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60E15 Inequalities; stochastic orderings
62P05 Applications of statistics to actuarial sciences and financial mathematics

Citations:

Zbl 0942.60008
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References:

[1] Dennenberg, D., 1994. Non-additive Measure and Integral. Kluwer Academic Publishers, Boston, MA.
[2] Dhaene, J., Wang, S., Young, V., Goovaerts, M., 1997. Comonotonicity and Maximal Stop-loss Premiums, submitted for publication. Research Report 9730, Department of Applied Economics, K.U. Leuven. · Zbl 1187.91099
[3] Goovaerts, M.J., Dhaene, J., 1999. Supermodular ordering and stochastic annuities. Insurance Mathematics and Economics, 24, 281-290. · Zbl 0942.60008
[4] Goovaerts, M.J., Dhaene, J., De Schepper, A., 1999. Stochastic bounds for present value functions. Research Report 9914. Department of Applied Economics, K.U. Leuven. · Zbl 1092.91527
[5] Harrison, J.; Kreps, D., Martingales and arbitrage in multiperiod securities markets, Journal of economic theory, 20, 381-408, (1979) · Zbl 0431.90019
[6] Harrison, J.; Pliska, R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic processes and their applications, 11, 215-260, (1981) · Zbl 0482.60097
[7] Jacques, M., On the hedging portfolio of Asian options, ASTIN bulletin, 26, 165-183, (1996)
[8] Kemna, A.G.Z.; Vorst, A.C.F., A pricing method for options based on average asset values, Journal of banking and finance, 14, 113-129, (1990) · Zbl 0638.90013
[9] Rogers, L.C.G.; Shi, Z., The value of an Asian option, Journal of applied probability, 32, 1077-1088, (1995) · Zbl 0839.90013
[10] Vazquez-Abad, F.J., Dufresne, D., 1998. Accelerated simulation for pricing Asian options. Research Paper No. 62. Centre for Actuarial Studies, The University of Melbourne.
[11] Wang, S., Dhaene, J., 1998. Comonotonicity, Correlation Order and Premium Principles. Insurance Mathematics and Economics 22, 235-242. · Zbl 0909.62110
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