• Received : 2010.10.09
  • Accepted : 2010.11.29
  • Published : 2010.12.25


Often in practice, the implied volatility of an option is calculated to find the option price tomorrow or the prices of, nearby' options. To show that one does not need to adhere to the Black- Scholes formula in this scheme, Figlewski has provided a new pricing formula and has shown that his, alternating passive model' performs as well as the Black-Scholes formula [8]. The Figlewski model was modified by Henderson et al. so that the formula would have no static arbitrage [10]. In this paper, we show how to construct a huge class of such static no arbitrage pricing functions, making use of distortions, coherent risk measures and the pricing theory in incomplete markets by Carr et al. [4]. Through this construction, we provide a more elaborate static no arbitrage pricing formula than Black-Sholes in the above scheme. Moreover, using our pricing formula, we find a volatility curve which fits with striking accuracy the synthetic data used by Henderson et al. [10].


  1. P. Artzner, F. Delbaen, J. Eber and D. Heath, Coherent measures of risk , Mathematical Finance, 9 (1999), 203-228.
  2. Y. Z. Bergman, B. D. Grundy and Z. Wiener, General Properties of Option Prices, Journal of Finance, 51 (1996), 1573-1610.
  3. F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, May-June 81 (1973), 637-659.
  4. P. Carr, H. Geman and D. Madan, Pricing and Hedging in Incomplete Markets, Journal of Financial Economics, 62 (2001), 131-167.
  5. G. Choquet, Theory of Capacities, Ann. Inst. Fourier(Grenoble), 5 (1953), 131-295.
  6. F. Delbaen, Coherent risk measures on general probability spaces, in Advances in Finance and Stochastics- Essays in Honour of Dieter Sondermann, K. Sandmann and P. J. Schonbucher, eds. New York: Springer, 2002.
  7. D. Denneberg, Non-Additive Measure and Integral, Dordercht, The Netherlands: Kluwer Academic Publishers, 1994.
  8. S. Figlewski, Assessing the Incremental Value of Option Pricing Theory relative to an "Informationally Passive" Benchmark, Journal of Derivatives, Fall (2002), 81-96.
  9. S. Figlewski and T. Green, Market Risk and Model Risk for a Financial Institution Writing Options, Journal of Finance, 53(4) (1999), 1465-1499.
  10. V. Henderson, D. Hobson and T. Kluge, Is There an Informationally Passive Benchmark for Option Pricing Incorporating Maturity?, Quantitative Finance, 7(1) (2007), 75-86.
  11. R. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.
  12. D. Schmeidler, Integral Representation without additivity, Proceedings of the American Mathematical Society, 97(2) (1986), 255-261.
  13. S. Wang, A Class of Distortion Operators for Pricing Financial and Insurance Risks, The Journal of Risk and Insurance, 67(1) (2003), 15-36.