Power Exchange Option with a Hybrid Credit Risk under Jump-Diffusion Model

In this paper, we study the valuation of power exchange options with a correlated hybrid credit risk when the underlying assets follow the jump-diffusion processes. The hybrid credit risk model is constructed using two credit risk models (the reduced-form model and the structural model), and the jump-diffusion processes are proposed based on the assumptions of Merton. We assume that the dynamics of underlying assets have correlated continuous terms as well as idiosyncratic and common jump terms. Under the proposed model, we derive the explicit pricing formula of the power exchange option using the measure change technique with multidimensional Girsanov’s theorem. Finally, the formula is presented as the normal cumulative functions and the infinite sums.

Valuation of Insurance Products with Shout Options in a Jump-Diffusion Model

The insurance product with shout options which permit the holders to modify the contract rules is one of the most popular products in European and American markets today. Therefore, it is of great significance to price more precisely. A new mathematical model consisting of a partial differential inequality and constraint conditions is derived for the price of insurance products in a jump-diffusion model. The numerical experiments are performed to analyze the impact of parameters on the insurance product with shout put options, especially for the jump times and the quantities of shout opportunities. The experiment results show that the value of the product is strongly affected by the quantities of shouting opportunities, especially for high values of the underlying asset, while it is only weakly affected for low values. Meanwhile, another meaningful discovery is that the valuation has changed little as the jump times are less than five, while it has shown a sharp increase once the jump times are more than five. Furthermore, the indicator results of course grid errors show that the values of shout put options in the jump-diffusion model are more accurate than those in a Brownian motion.

Robust equilibrium investment and reinsurance strategy with bounded memory and common shock dependence

In this paper, we consider the robust investment and reinsurance problem with bounded memory and risk co-shocks under a jump-diffusion risk model. The insurer is assumed to be ambiguity-averse and make the optimal decision under the mean-variance criterion. The insurance market is described by two-dimensional dependent claims while the risky asset is depicted by the jump-diffusion model. By introducing the performance in the past, we derive the wealth process depicted by a stochastic delay differential equation (SDDE). Applying the stochastic control theory under the game-theoretic framework, together with stochastic control theory with delay, the robust equilibrium investment-reinsurance strategy and the corresponding robust equilibrium value function are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment and reinsurance strategy.

Locally Scaled and Stochastic Volatility Metropolis– Hastings Algorithms

Markov chain Monte Carlo (MCMC) techniques are usually used to infer model parameters when closed-form inference is not feasible, with one of the simplest MCMC methods being the random walk Metropolis–Hastings (MH) algorithm. The MH algorithm suffers from random walk behaviour, which results in inefficient exploration of the target posterior distribution. This method has been improved upon, with algorithms such as Metropolis Adjusted Langevin Monte Carlo (MALA) and Hamiltonian Monte Carlo being examples of popular modifications to MH. In this work, we revisit the MH algorithm to reduce the autocorrelations in the generated samples without adding significant computational time. We present the: (1) Stochastic Volatility Metropolis–Hastings (SVMH) algorithm, which is based on using a random scaling matrix in the MH algorithm, and (2) Locally Scaled Metropolis–Hastings (LSMH) algorithm, in which the scaled matrix depends on the local geometry of the target distribution. For both these algorithms, the proposal distribution is still Gaussian centred at the current state. The empirical results show that these minor additions to the MH algorithm significantly improve the effective sample rates and predictive performance over the vanilla MH method. The SVMH algorithm produces similar effective sample sizes to the LSMH method, with SVMH outperforming LSMH on an execution time normalised effective sample size basis. The performance of the proposed methods is also compared to the MALA and the current state-of-art method being the No-U-Turn sampler (NUTS). The analysis is performed using a simulation study based on Neal’s funnel and multivariate Gaussian distributions and using real world data modeled using jump diffusion processes and Bayesian logistic regression. Although both MALA and NUTS outperform the proposed algorithms on an effective sample size basis, the SVMH algorithm has similar or better predictive performance when compared to MALA and NUTS across the various targets. In addition, the SVMH algorithm outperforms the other MCMC algorithms on a normalised effective sample size basis on the jump diffusion processes datasets. These results indicate the overall usefulness of the proposed algorithms.

An Algorithm for the Pricing and Timing of the Option to make a Two-Stage Investment with Credit Guarantees

We develop a jump-diffusion model for a guarantee-investment combination financing mode (G-I mode) that is recently popular in financial practice. We assume that a borrower has exclusively an option to invest in a project in two stages. The project’s cash flow follows a double exponential jump-diffusion process and it is increased by a growth factor once the second-stage investment is exercised. The first-stage investment cost is financed by a bank loan with the guarantee provided by an insurer, who promises to provide the second-stage investment cost as well as take the lender’s all default losses. In return for the guarantee and investment, the borrower pays a guarantee fee upon first investment and grants a fraction of equity upon second investment to the insurer. In sharp contrast to prior papers on guarantee, the guarantee costs are contracted prior to investment. We provide closed-form solutions and produce a numerical algorithm for the timing and pricing of the two investment options.