Abstract
We consider asset price processes Xt which are weak solutions of one-dimensional stochastic differential equations of the form (equation (2)) Such price models can be interpreted as non-lognormally-distributed generalizations of the geometric Brownian motion. We study properties of the Iα-divergence between the law of the solution Xt and the corresponding drift-less measure (the special case α=1 is the relative entropy). This will be applied to some context in statistical information theory as well as to arbitrage theory and contingent claim valuation. For instance, the seminal option pricing theorems of Black-Scholes and Merton appear as a special case.
Highlights
ÒÖ ÔÔÐà ÝÛÓÖ × Á« ÚÖÒÖÐØÚ ÒØÖÓÔÝ ×Ø Ø ×Ø Ð Ò ÓÖÑ Ø ÓÒ ÕÙ Ú Ð ÒØ Ñ Ö1 Ø Ò Ð Ñ ×ÙÖ ÓÔØ ÓÒ ÔÖ Ò Ð 1Ë ÓÐ ×1Å ÖØÓÒo.
Û Ø ×ÓÑ ÓÒ×Ø ÒØ× 1⁄2 3⁄4 ÁÊ Ò Ó 1⁄4 ̧ Ò ÖÓÛÒ Ò ÑÓØ ÓÒ Ï× Ë ÑÙ Ð×ÓÒ 3⁄4 ̧ Å ÖØÓÒ
ÓÖ ÖØ Ò3⁄4μo ́3⁄4μ ØÓ ÔÔÐÝ Ø Ö ×ÙÐØ× Ó1⁄2μ Ò ÓÖ Ö ØÓ Ù ×× ÖØ ÓÒ× ÙÔÓÒ Ø ×Ø Ø ×Ø Ð Ò ÓÖÑ Ø ÓÒ.
Summary
à ÝÛÓÖ × Á« ÚÖÒÖÐØÚ ÒØÖÓÔÝ ×Ø Ø ×Ø Ð Ò ÓÖÑ Ø ÓÒ ÕÙ Ú Ð ÒØ Ñ Ö1 Ø Ò Ð Ñ ×ÙÖ ÓÔØ ÓÒ ÔÖ Ò Ð 1Ë ÓÐ ×1Å ÖØÓÒo. Û Ø ×ÓÑ ÓÒ×Ø ÒØ× 1⁄2 3⁄4 ÁÊ Ò Ó 1⁄4 ̧ Ò ÖÓÛÒ Ò ÑÓØ ÓÒ Ï× Ë ÑÙ Ð×ÓÒ 3⁄4 ̧ Å ÖØÓÒ
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