This paper deals with pricing formulae for a European call option and an exchange option in the case where underlying asset price processes are represented by stochastic delay differential equations with jumps (hereafter “SDDEJ”). We introduce a new model in which Poisson jumps are added in stochastic delay differential equations to capture behaviors of an underlying asset process more precisely. We derive explicit pricing formulae for the European call option and the exchange option by proving a Lemma on the conditional expectation. Finally, we show that our “SDDEJ” model is meaningful through some numerical experiments and discussions.

For a Brownian bridge from 0 to y, we prove that the mean of the first exit time from the interval $\left( -h,h \right),h>0$ , behaves as ${\mathrm{O}}(h^2)$ when $h \downarrow 0$ . Similar behaviour is also seen to hold for the three-dimensional Bessel bridge. For the Brownian bridge and three-dimensional Bessel bridge, this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to provide a detailed proof of an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

Decision making in organic farming is related to risk and uncertainty, and options must be evaluated in the decision-making process. This paper presents the methodology of an integrated deterministic simulation system (KARSIM 1.0) application for decision-making support on organic farms in northeastern Slovenia. An emphasis to modify the net present value (NPVt) criterion by incorporating the real options approach was made. Its application is shown in organic spelt (Triticum aestivum ssp. spelta McKey) production and processing using two real options approaches, the Black–Scholes and binomial models. The NPVt indicates that the decision to process spelt for animal fodder is financially unfeasible, while the real options approach differentiates the results by organic spelt grain and flour production for human nutrition. It may be concluded that the real options approach can be useful when assessing projects with uncertainty, sunk costs and irreversibility, and it can provide for examining agricultural investment decisions.

We present a new valuation formula for a generic, multi-period binary option in a multi-asset Black–Scholes economy. The payoff of this so-called M-binary is the most general possible, subject to the condition that a simple analytic expression exists for the present value. Portfolios of M-binaries can be used to statically replicate many European exotics for which there exist closed-form Black–Scholes prices.

The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.