The existence of a weak solution and the uniqueness in law are assumed for the equation, the coefficients $b$ and $\sigma$ being generally $C\left({ℝ}^{+}\right)$-progressive processes. Any weak solution $X$ is called a $(b,\sigma )$-stock price and Girsanov Theorem jointly with the DDS Theorem on time changed martingales are applied to establish the probability distribution ${\mu }_{\sigma }$ of $X$ in $C\left({ℝ}^{+}\right)$ in the special case of a diffusion volatility $\sigma (X,t)=\tilde{\sigma }\left(X\left(t\right)\right).$ A martingale option pricing method is presented.

This paper continues the research started in [J. Štěpán and P. Dostál: The $\mathrm{d}X\left(t\right)=Xb\left(X\right)\mathrm{d}t+X\sigma \left(X\right)\mathrm{d}W$ equation and financial mathematics I. Kybernetika 39 (2003)]. Considering a stock price $X\left(t\right)$ born by the above semilinear SDE with $\sigma (x,t)=\tilde{\sigma }\left(x\left(t\right)\right),$ we suggest two methods how to compute the price of a general option $g\left(X\right(T\left)\right)$. The first, a more universal one, is based on a Monte Carlo procedure while the second one provides explicit formulas. We in this case need an information on the two dimensional distributions of $ℒ\left(Y\right(s),\tau (s\left)\right)$ for $s\ge 0,$ where $Y$ is the exponential...

This paper discusses option valuation logic and four selected methods for the valuation of real options in the light of their modeling choices. Two of the selected methods the Datar–Mathews method and the Fuzzy Pay-off Method represent later developments in real option valuation and the Black & Scholes formula and the Binomial model for option pricing the more established methods used in real option valuation. The goal of this paper is to understand the big picture of real option valuation models...