Pricing Continuously Monitored Barrier Options under the SABR Model:A Closed‐Form Approximation

The stochastic alpha beta rho(SABR)model introduced by Hagan et al.(2002)is widely used in both fixed income and the foreign exchange(FX)markets.Continuously monitored barrier option contracts are among the most popular derivative contracts in the FX markets.In this paper,we develop closed-form formulas to approximate various types of barrier option prices(down-and-out/in,up-and-out/in)under the SABR model.We first derive an approximate formula for the survival density.The barrier option price is the one-dimensional integral of its payoff function and the survival density,which can be easily implemented and quickly evaluated.The approximation error of the survival density is also analyzed.To the best of our knowledge,it is the first time that analytical(approximate)formulas for the survival density and the barrier option prices for the SABR model are derived.Numerical experiments demonstrate the validity and efficiency of these formulas.

Dopamine improves apple replant disease resistance by regulating physiological resilience and rhizosphere microbial community structure

Apple replant disease(ARD)is a complex agricultural problem caused by multiple stressors that can lead to increased reactive oxygen species(ROS)levels and limited nutrient utilization in plants.However,existing countermeasures cannot effectively address this challenge.Here,we used Malus hupehensis as a test organism to investigate whether the pleiotropic molecule dopamine can alleviate ARD using pot experiments.Exogenous application of 100μmol L-1 dopamine significantly promoted the growth of apple seedlings in the replanted soil,with a relative growth rate increase of 17.44%.Our results revealed two major pathways by which dopamine regulates ARD resistance in apple trees.First,dopamine effectively reduces the level of ROS and activates the expression of genes related to nitrogen(N)transport and metabolism.Among those genes,MdNLP5,MdNRT1.1,MdNLP2,MdNRT2.5,MdNLP3,MdNRT2.4,MdNADH-GAGOT,and MdFd-GAGOT were strongly regulated by dopamine.These regulatory effects promoted the uptake and utilization of soil N by the plants.Second,dopamine improved the physical and chemical properties,enhanced microbial community diversity,and promoted mutual cooperation between microbial communities in the soil.Furthermore,dopamine altered the microbial structure of rhizosphere soil(upregulating Clostridiales,Gaiellales,Sordariales and Mortierellales;downregulating Micrococcales,Longimicrobiales,Hypocreales and Cystobasidiales).Notably,dopamine significantly upregulated the abundances of Gaiella and Mortierella,both of which were positively correlated with soil urease activity,soil available N content,plant growth and N uptake.Dopamine also significantly downregulated the abundance of the plant pathogen Gibberella(by 11.71-fold)in replant soil.Our results provide insights into the mechanisms by which dopamine promotes ARD resistance,and can promote the sustainable development of the apple industry.