We establish a local martingale M associate with f(X,Y) under some restrictions on f, where Y is a process of bounded variation (on compact intervals) and either X is a jump diffusion (a special case being a Lévy process) or X is some general (càdlàg metric-space valued) Markov process. In the latter case, f is restricted to the form f(x,y)=∑k=1Kξk(x)ηk(y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes an L2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both in L2 and almost sure is also considered and sufficient conditions for functions for which this happens are identified.

A new approach to jump diffusion is introduced, where the jump is treated as a vertical shift of the price (or volatility) function. This method is simpler than the previous methods and it is applied to the portfolio model with a stochastic volatility. Moreover, closed-form solutions for the optimal portfolio are obtained. The optimal closed-form solutions are derived when the value function is not smooth, without relying on the method of viscosity solutions.

In this paper, we study carbon emission trading whose market is gaining in popularity as a policy instrument for global climate change. The mathematical model is presented for pricing options on CO2 emission allowance futures with jump diffusion processes, and a so-called fitted finite volume method is proposed to solve the pricing model for the spatial discretization, in which the Crank-Nicolson is employed for time stepping. In addition, the stability and the convergence of the fully discrete scheme are given, and some numerical results, which are compared with the closed form solution and the Monte Carlo simulation solution, are provided to demonstrate the rates of convergence and the robustness of the numerical method.

Investing in a new perennial crop variety involves an irreversible commitment of capital and generates an uncertain return stream. As a result, the decision to adopt a new variety includes a significant real option value. Waiting for returns to rise above this real option causes a delay in adoption because of economic hysteresis. This study tests for hysteresis in the adoption of wine grape varieties using a sample of district-level data from the state of California. The empirical results show a significant hysteretic effect in wine grape investment, which might be reduced by activities that smooth earnings over time.

An asset manager invests the savings of some investors in a portfolio of defaultable bonds. The manager pays the investors coupons at a constant rate and receives a management fee proportional to the value of the portfolio. He/she also has the right to walk out of the contract at any time with the net terminal value of the portfolio after payment of the investors' initial funds, and is not responsible for any deficit. To control the principal losses, investors may buy from the manager a limited protection which terminates the agreement as soon as the value of the portfolio drops below a predetermined threshold. We assume that the value of the portfolio is a jump diffusion process and find an optimal termination rule of the manager with and without protection. We also derive the indifference price of a limited protection. We illustrate the solution method on a numerical example. The motivation comes from the collateralized debt obligations.

This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.

We describe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0,1] that jumps to the boundary before diffusing back into the interior. We show that the behavior of the time-reversed process depends on whether the boundaries are accessible to the diffusive motion of the forward process. If a boundary point is inaccessible to the forward diffusion then time reversal leads to a jump diffusion that jumps immediately into the interior whenever it arrives at that point. If, instead, a boundary point is accessible then the jumps off of that point are governed by a weighted local time of the time-reversed process.

We study defaultable bond prices in the Black–Cox model with jumps in the asset value. The jump-size distribution is arbitrary, and following Longstaff and Schwartz (1995) and Zhou (2001) we assume that, if default occurs, the recovery at maturity depends on the ‘severity of default’. Under this general setting, the vehicle for our analysis is an integral equation. With the aid of this, we prove some properties of the bond price which are consistent numerically and empirically with earlier works. In particular, the limiting credit spread as time to maturity tends to 0 is nonzero. As a byproduct, we show that the integral equation implies an infinite-series expansion for the bond price.

We consider a dynamical system in $\mathbb{R}$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tailnature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.

We provide bounds for perpetual American option prices in a jump diffusion model in terms of American option prices in the standard Black–Scholes model. We also investigate the dependence of the bounds on different parameters of the model.