Speculative option valuation and the fractional diffusion equation

In financial markets not only returns, but also waiting times between consecutive trades are random variables and it is possible to apply continuoustime random walks (CTRWs) as phenomenological models of high-frequency prices. Based on these considerations, in this extended abstract, some results are outlined which can be useful for speculative option valuation. 1 The basic mapping onto continuous-time random walks High-frequency financial data can be phenomenologically mapped onto continuous-time random walks (CTRWs), also called point or renewal processes with reward [1]. Let denote the price of an asset or the value of an index at time . In finance, returns rather than prices are more convenient variables. Following Parkinson [2], let us introduce the variable , that is the logarithm of the price. For small price variations, "! $#% & , the return '( ) +* and the logarithmic return '-,/.10 % 324 "! 5* 6 87 virtually coincide. As also waiting times 9: ; < =! #> between two consecutive trades are stochastic variables, the time series ?@ BA is characterised by CD EGF593 , the joint probability density function of log-returns EH I > =! #J and of waiting times 9: K % "! #L . The joint density satisfies the normalization condition M MON EPN 93CD EQF59R S UT . An important property SCALAS, GORENFLO, MAINARDI, MEERSCHAERT of CTRWs is that log-returns and waiting times are independent and identically distributed random variables. However, there can be a dependence between the two random variables. One can define the two marginal densities in the usual way: VW EX Y <MON 93CD EQF59R and