Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

The pricing and hedging of financial derivatives have become one of the hot research issues in mathematical finance today. In the case of non-risk neutrality, this article uses the martingale method and probability measurement method to study the pricing method and hedging strategy of financial derivatives. This paper also further studies the hedging strategy of financial derivatives in the incomplete market based on the BSM model and converts the solution of this problem into solving a vector on the Hilbert space to its closure. The problem of space projection is to use projection theory to decompose financial derivatives under a given martingale measure. In the imperfect market, the vertical projection theory is used to obtain the approximate pricing method and hedging strategy of financial derivatives in which the underlying asset follows the martingale process; the projection theory is further expanded, and the pricing problem of financial derivatives under the mixed-asset portfolio is obtained. Approximate pricing of financial derivatives; in the discrete state, the hedging investment strategy of financial derivatives H in the imperfect market is found through the method of variance approximation.

The Condition of the Conditionality

The interdependence of options is common among compound options. Moreover, this interconnectedness is synonymous with probability theory-how a set of axioms are treated. The conditionality, where one option value is dependent on another option, has spilled over to option pricing, especially exchange options. However, it seems that no study has explored whether that simultaneous occurrence of two options is conditional or not. This study uses conditional approaches (Radon–Nikodým derivative and probability theory) to illustrate conditionality in an exchange option. Furthermore, hedging strategy is derived based on straddles. The results show that due to conditionality another exotic option, tri-conditional option (also known as triple option) is derived. The hedging of a triple option encompasses both dynamic and static techniques.

Why Are Banks Exposed to Monetary Policy?

We propose a model of banks’ exposure to movements in interest rates and their role in the transmission of monetary shocks. Since bank deposits provide liquidity, higher interest rates allow banks to earn larger spreads on deposits. Therefore, if risk aversion is higher than one, banks’ optimal dynamic hedging strategy is to take losses when interest rates rise. This risk exposure can be achieved by a traditional maturity-mismatched balance sheet and amplifies the effects of monetary shocks on the cost of liquidity. The model can match the level, time pattern, and cross-sectional pattern of banks’ maturity mismatch. (JEL E43, E44, E51, E52, G21, G32)

“Indirect development” increases reproductive plasticity and contributes to the success of scyphozoan jellyfish in the oceans

Ecologists and evolutionary biologists have been looking for the key(s) to the success of scyphomedusae through their long evolutionary history in multiple habitats. Their ability to generate young medusae (ephyrae) via two distinct reproductive strategies, strobilation or direct development from planula into ephyra without a polyp stage, has been a potential explanation. In addition to these reproductive modes, here we provide evidence of a third ephyral production which has been rarely observed and often confused with direct development from planula into ephyra. Planulae of Aurelia relicta Scorrano et al. 2017 and Cotylorhiza tuberculata (Macri 1778) settled and formed fully-grown polyps which transformed into ephyrae within several days. In distinction to monodisk strobilation, the basal polyp of indirect development was merely a non-tentaculate stalk that dissolved shortly after detachment of the ephyra. We provide a fully detailed description of this variant that increases reproductive plasticity within scyphozoan life cycles and is different than either true direct development or the monodisk strobilation. Our observations of this pattern in co-occurrence with mono- and polydisk strobilation in Aurelia spp. suggest that this reproductive mode may be crucial for the survival of some scyphozoan populations within the frame of a bet-hedging strategy and contribute to their long evolutionary success throughout the varied conditions of past and future oceans.

Staying in touch: how highly specialised moth pollinators track host plant phenology in unpredictable climates

Background For specialised pollinators, the synchrony of plant and pollinator life history is critical to the persistence of pollinator populations. This is even more critical in nursery pollination, where pollinators are obligately dependant on female host plant flowers for oviposition sites. Epicephala moths (Gracillariidae) form highly specialised nursery pollination mutualisms with Phyllanthaceae plants. Several hundred Phyllanthaceae are estimated to be exclusively pollinated by highly specific Epicephala moths, making these mutualisms an outstanding example of plant–insect coevolution. However, there have been no studies of how Epicephala moths synchronise their activity with host plant flowering or persist through periods when flowers are absent. Such knowledge is critical to understanding the ecology and evolutionary stability of these mutualisms. We surveyed multiple populations of both Breynia oblongifolia (Phyllanthaceae) and it’s Epicephala pollinators for over two years to determine their phenology and modelled the environmental factors that underpin their interactions. Results The abundance of flowers and fruits was highly variable and strongly linked to local rainfall and photoperiod. Unlike male flowers and fruits, female flowers were present throughout the entire year, including winter. Fruit abundance was a significant predictor of adult Epicephala activity, suggesting that eggs or early instar larvae diapause within dormant female flowers and emerge as fruits mature. Searches of overwintering female flowers confirmed that many contained pollen and diapausing pollinators. We also observed diapause in Epicephala prior to pupation, finding that 12% (9/78) of larvae emerging from fruits in the autumn entered an extended diapause for 38–48 weeks. The remaining autumn emerging larvae pupated directly without diapause, suggesting a possible bet-hedging strategy. Conclusions Epicephala appear to use diapause at multiple stages in their lifecycle to survive variable host plant phenology. Furthermore, moth abundance was predicted by the same environmental variables as male flowers, suggesting that moths track flowering through temperature. These adaptations may thereby mitigate against unpredictability in the timing of fruiting and flowering because of variable rainfall. It remains to be seen how widespread egg diapause and pre-pupal diapause may be within Epicephala moths, and, furthermore, to what degree these traits may have facilitated the evolution of these highly diverse mutualisms.

Optimal futures hedging strategies based on an improved kernel density estimation method

In this paper, we study the hedging effectiveness of crude oil futures on the basis of the lower partial moments (LPMs). An improved kernel density estimation method is proposed to estimate the optimal hedge ratio. We investigate crude oil price hedging by contributing to the literature in the following two-fold: first, unlike the existing studies which focus on univariate kernel density method, we use bivariate kernel density to calculate the estimated LPMs, wherein the two bandwidths of the bivariate kernel density are not limited to the same, which is our main innovation point. According to the criterion of minimizing the mean integrated square error, we derive the conditions that the optimal bandwidths satisfy. In the process of derivation, we make a distribution assumption “locally” in order to simplify calculation, but this type of “local” distribution assumption is far better than “global” distribution assumption used in parameter method theoretically and empirically. Second, in order to meet the requirement of bivariate kernel density for independent random variables, we adopt ARCH models to obtain the independent noises with related to the returns of crude oil spot and futures. Genetic algorithm is used to tune the parameters that maximize Quasi-likelihood. Empirical results reveal that, at first, the hedging strategy based on the improved kernel density estimation method is of highly efficiency, then it achieves better performance than the hedging strategy based on the traditional parametric method. We also compare the risk control effectiveness of static hedge ratio vs. time-varying hedge ratio, and find that static hedging has a better performance than time-varying hedging.