Explore the vast range of titles available.

This paper aims to extend the analytical tractability of the Black-Scholes model to alternative models with arbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixed-exponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and double-exponential jump diffusion models, because the mixed-exponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavy-tailed distributions. The mixed-exponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for path-dependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a high-order integro-differential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day. This paper was accepted by Michael Fu, stochastic models and simulation.

In recent years, there has been a bloom in the stock investors due to availability of various platforms that have provided an opportunity even for small scale investors to earn profits from the market. However, due to very high uncertainty, bad investments can lead to large financial losses and hence need for tools that can predict stock behaviour, arises. The main objective of this article is to provide a comparative empirical analysis of stochastic models with artificial neural networks in the prediction of stock indices across different markets. We consider three types of models, namely the time series models: autoregressive integrated moving average and autoregressive fractionally integrated moving average; jump diffusion models: Merton jump diffusion and Kou jump diffusion; the artificial neural network models: feed-forward network and the long short term memory. These models are used to forecast 10, 20 and 30 days ahead prices of major stock indices across different markets which include both developed and emerging economies. It is shown that the long short-term memory performs better than other considered models on most of the considered indices over all the time horizons. The results also indicate the forecasts provided by the LSTM model are significant from both statistical point of view and can possibly be used for profitable investments.

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. In the first part of this paper, fitting of PH distributions to human lifetimes is considered. The class of generalized Coxian distributions is given special attention. In part, some new software is developed. In the second part, pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

Generally, the semiclosed-form option pricing formula for complex financial models depends on unobservable factors such as stochastic volatility and jump intensity. A popular practice is to use an estimate of these latent factors to compute the option price. However, in many situations this plug-and-play approximation does not yield the appropriate price. This article examines this bias and quantifies its impacts. We decompose the bias into terms that are related to the bias on the unobservable factors and to the precision of their point estimators. The approximated price is found to be highly biased when only the history of the stock price is used to recover the latent states. This bias is corrected when option prices are added to the sample used to recover the states’ best estimate. We also show numerically that such a bias is propagated on calibrated parameters, leading to erroneous values. Les formules de tarification d’options pour les modèles financiers complexes dépendent généralement de facteurs non observables tels que la volatilité stochastique et l’intensité des sauts. Une pratique courante consiste à utiliser un estimateur ponctuel de ces facteurs latents pour calculer le prix d’options. Malheureusement, cette approche simple et intuitive produit souvent des prix erronés. Les auteurs examinent ce biais et en quantifient les impacts. Ils le décomposent en termes liés au biais des facteurs non observables et à la précision de leurs estimateurs ponctuels. Le prix approximatif s’avère fortement biaisé lorsque seul l’historique du cours de l’action est utilisé pour récupérer les variables latentes, mais l’ajout des prix des options à l’échantillon utilisé corrige ce problème en procurant la meilleure estimation des facteurs latents. Les auteurs montrent aussi numériquement qu’un tel biais se propage sur les paramètres calibrés.

In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Itô-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control u under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of u in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions.

This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results, a coupling method turns out to be useful.

Exact simulation of SDEs is a very important and challenging problem. In this paper we discuss exact simulation problems for jump-diffusion processes. Motivated by statistical applications, our main contribution is to propose an algorithm that performs exact simulation of a class of jump-diffusion bridges. We also present and discuss the existing methods for forward simulation and propose an extension of one of them to account for unbounded jump rate. Finally, the exact algorithms are compared to competing non-exact ones in some simulated examples.

We consider a Markov process X, which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of Z outside any neighborhood of the origin, we obtain a small-time second-order polynomial expansion for the tail distribution and the transition density of the process X. Our method of proof combines a recent regularizing technique for deriving the analog small-time expansions for a Lévy process with some new tail and density estimates for jump-diffusion processes with small jumps based on the theory of Malliavin calculus, flow of diffeomorphisms for SDEs, and time-reversibility. As an application, the leading term for out-of-the-money option prices in short maturity under a local jump-diffusion model is also derived.

We study de Finetti's optimal dividend problem, also known as the optimal harvesting problem, in the dual model. In this model, the firm value is affected both by continuous fluctuations and by upward directed jumps. We use a fixed point method to show that the solution of the optimal dividend problem with jumps can be obtained as the limit of a sequence of stochastic control problems for a diffusion. In each problem, the optimal dividend strategy is of barrier type, and the rate of convergence of the barrier and the corresponding value function is exponential.

In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving parabolic partial integro-differential equations (PIDEs), which describe the jump-diffusion option pricing model in finance. It is shown that the variable step-size IMEX BDF2 method is stable for abstract PIDEs under suitable time step restrictions. Based on the time regularity analysis of abstract PIDEs, the consistency error and the global error bounds for the variable step-size IMEX BDF2 method are provided. After time semidiscretization, spatial differential operators are treated by using finite difference methods, and the jump integral is computed using the composite trapezoidal rule. A local mesh refinement strategy is also considered near the strike price because of the nonsmoothness of the payoff function. Numerical results illustrate the effectiveness of the proposed method for European and American options under jump-diffusion models.