On information costs, short sales and the pricing of extendible options, steps and Parisian options

This paper provides a simple framework for the valuation of exotic derivatives within shadow costs of incomplete information and short sales. The specific features of the OTC markets with comparison to the organized markets require an additional investment to obtain information about the financial products, to process data, to elaborate models, etc. The shadow cost includes two components. The first component is the product of pure information cost due to imperfect knowledge. The second component represents the additional cost caused by the short-selling constraint. Information costs are linked to Merton’s (J Fianance 42:483–510, 1987) model of capital market equilibrium with incomplete information, CAPMI. This model is extended by Wu et al. (Rev Quant Finance Account 7:119–136, 1996) who propose incomplete-information capital market equilibrium with heterogeneous expectations and short sale restrictions, GCAPM. This model is used in our paper to provide for the first time in the literature analytic solutions for derivatives in the presence of both shadow costs of incomplete information and short sales. Our methodology incorporates shadow costs of incomplete information and short sales in the options and their underlying securities. We provide formulas using the standard Black and Scholes method or the martingale method. Since shadow costs are important in the presence of illiquidity, the formulas are useful for the valuation of OTC derivatives.