Models of Development of Financial Assets

Important relations in modern finance and the corresponding econometric calculations (e.g., calibrations of models in given financial environment, simulations for various financial scenarios, and the like) necessitate to model developments of financial assets (prices, volumes, returns) in continuous time (see also random processes in continuous time in Sects. 2.4 and 2.5 ). Such models mostly require relatively complex theoretical background denoted generally as stochastic calculus, which is the classical calculus of derivations and integrals modified for random variables (including special instruments such as stochastic integrals, martingales, Ito’s lemma, Wiener process, risk neutral probabilities, and others). In this chapter, we outline basic principles of this methodology, but only within the scope that enables to cope with simple practical applications for financial times series (in any case, there are references with technical and theoretical details, e.g., Campbell et al. (1997), Cipra (2010), Duffie (1988), Elliot and Kopp (2004), Enders (1995), Franke et al. (2004), Franses and van Dijk (2000), Gourieroux and Jasiak (2001), Hendry (1995), Hull (1993), Karatzas and Shreve (1988), Kwok (1998), Lim (2011), Malliaris and Brock (1982), McNeil et al. (2005), Mills (1993), Musiela and Rutkowski (2004), Neftci (2000), Poon (2005), Rachev et al. (2007), Ruppert (2004), Steele (2001), Taylor (1986), Tsay (2002), Wang (2003), and Wilmott (2000)).