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.

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.

Optimal portfolio and consumption for a Markovian regime-switching jump-diffusion process

We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases. doi:10.1017/S1446181121000122

Pricing Vulnerable Options in the Bifractional Brownian Environment with Jumps

In this paper, we study the valuation of European vulnerable options where the underlying asset price and the firm value of the counterparty both follow the bifractional Brownian motion with jumps, respectively. We assume that default event occurs when the firm value of the counterparty is less than the default boundary. By using the actuarial approach, analytic formulae for pricing the European vulnerable options are derived. The proposed pricing model contains many existing models such as Black–Scholes model (1973), Merton jump-diffusion model (1976), Klein model (1996), and Tian et al. model (2014).

Detecting Jump Risk and Jump-Diffusion Model for Bitcoin Options Pricing and Hedging

In this paper, we conduct a fast calibration in the jump-diffusion model to capture the Bitcoin price dynamics, as well as the behavior of some components affecting the price itself, such as the risk of pitfalls and its ambiguous effect on the evolution of Bitcoin’s price. In addition, in our study of the Bitcoin option pricing, we find that the inclusion of jumps in returns and volatilities are significant in the historical time series of Bitcoin prices. The benefits of incorporating these jumps flow over into option pricing, as well as adequately capture the volatility smile in option prices. To the best of our knowledge, this is the first work to analyze the phenomenon of price jump risk and to interpret Bitcoin option valuation as “exceptionally ambiguous”. Crucially, using hedging options for the Bitcoin market, we also prove some important properties: Bitcoin options follow a convex, but not strictly convex function. This property provides adequate risk assessment for convex risk measure.