In this paper, we consider a Heston local–stochastic volatility (HLSV) model to study an optimal investment strategy problem, and analyze the optimal strategy when the volatility component of the model obeys a slow varying process and a fast varying process, respectively. For the optimal investment objective with a constant absolute risk aversion (CARA) utility function, the analytical solution under the HLSV model cannot be obtained due to the complicated nonlinearity of the partial differential equation. In this paper we employ a dual method, Legendre transformation, and an asymptotic expansion technique to derive an asymptotic solution. We also apply a Monte Carlo method to compute the optimal strategy, which can be compared with the asymptotic solution. Finally, numerical examples are provided to support our theoretical results.

The problem of generalizing the compound option pricing model to incorporate more empirical features becomes an urgent and necessary event. In this study, a new N-fold compound option pricing method is designed for the economic uncertainty and technical uncertainty. The economic uncertainty is modelled by a fractional jump-diffusion model, which incorporates the long-term dependence of financial markets, the kurtosis of returns and the unpredictable shocks of real world. The technical uncertainty is modelled as a simplified version Poisson-type jump process, which describes the catastrophic impact of the technical risk in multi-stage projects. The main contribution of this paper is that we firstly develop the N-fold compound option pricing model with the fractional Brownian motion and the technical risk variable. Further, the analytic solutions of pricing compound options are achieved and verified by the recursive formula of option price. Numerical examples are provided to support the theoretical results of this model. Moreover, from the sensitivity analysis, some results are presented to illustrate that the N-fold compound option price without considering the phase-specific characteristics of technical risk is improperly estimated. Thereby, it is essential for investors to take into account the technical risk when making decisions.

In this paper, we present a new pricing method on N -fold compound option by adopting the theory of fuzzy sets into a fractional stochastic financial model. Considering the characteristics of correlations and kurtosis of returns in the long run, we employ the fractional Brownian motion to model the dynamic price of underlying assets. Then, a trapezoidal fuzzy stochastic process is employed to depict the fuzziness of underlying asset price. Involving the decision-maker’s subjective judgment, the mean value with the possibility–necessity weight and pessimistic–optimistic index is expressed. Further, the formulas of N -fold compound option price are derived by martingale method. Moreover, the valuation and properties of the formulas are analyzed under some reasonable assumptions. In the end, some numerical examples are provided to support our theoretical results and illustrate the mean of N -fold compound option pricing in fuzzy and fractional environments.

The wealth substitution rate, which describes the substitution relationship between agents’ investment in wealth, is introduced into the collision kernel of the Boltzmann equation to study wealth distribution. Using the continuous trading limit, the Fokker–Planck equation is derived and the steady-state solution is obtained. The results show that the inequality of wealth distribution decreases as the wealth substitution rate increases under certain assumptions. The wealth distribution has a bimodal shape if the wealth substitution rate does not equal one.

In this paper, we develop an extended constant elasticity of variance (CEV) model with stochastic volatility to study an optimal investment strategy problem. This extended CEV model remedies the shortcoming of classical CEV model. In CEV model, the volatility term is an power function of stock price, which only covers firm-specific risks. Consequently, we consider the coefficient of volatility with the mean reverting process to make the volatility involve the market risks to improve the classical CEV model. For the optimal investment objective with a constant absolute risk aversion (CARA) utility function, the analytical solution under the extended CEV model cannot be obtained due to the complicated nonlinearity of the partial differential equation. In this paper we successfully employ a dual method, Legendre transformation, and an asymptotic expansion technique to approach an asymptotic solution. The numerical examples indicate the optimal strategy is an increasing function of the expectation of stock returns and correlations between the two market risks. Besides, it is a decreasing function of interest rate and risk aversion coefficient. In addition, by statistical analysis, we find that the power parameter, the expectation of stock returns and interest rate are all significant factors affecting the investment strategy.

In common stock loan, lenders face the risk that their loans will not be repaid if the stock price falls below loan, which limits the issuance and circulation of stock loans. The empirical test suggests that the log-return series of stock price in the US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the model of the margin call stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of the margin call stock loan can be described by a space-fractional partial differential equation with a time-varying free boundary condition. We transform the free boundary problem to a linear complementarity problem, and the fully-implicit finite difference method that we used is unconditionally stable in both the integer and fractional order. The numerical experiments are carried out to demonstrate differences of the margin call stock loan model under the FMLS and the standard normal distribution. Last, we analyze the impact of key parameters in our model on the margin call stock loan evaluation and give some reasonable explanation.

In this paper, we study a dynamic auction for allocating a single indivisible project while different participants have different bid values for the project. When the price rises continuously, the bidders can retreat the auction and obtain the compensation by the difference between the price at retreating time and the previous bid price. The final successful bidder achieves the project and pays compensations to others. We show that the auction of bidders with constant relative risk aversion (CRRA) has a unique equilibrium. While the relative risk aversion coefficient approaches to zero, the equilibrium with CRRA bidders would approach to the equilibrium with risk-neutral bidders.

Power exchange option is an exotic option which combines power option and exchange option. In this paper, we consider the pricing of the power exchange option under exchange rate volatility risk and issuing company bankruptcy risk. Meanwhile, considering the major events between the two countries, we add the Poisson jump process to the option model in order to reflect the impact of sudden factors on the price of transnational derivatives in the international market. According to the no-arbitrage principle, a mathematical model for pricing such problems is established, and explicit solutions are obtained. The numerical examples show that the model established in this paper is effective.

Traditional derivative pricing theories usually focus on the risk-neutral price or the equilibrium price. However, in highly competitive financial markets, we observed two prices which are called bid and ask prices; then the unique risk-neutral price fails to hold. In this paper, within the framework of conic finance, we provide a useful approach to evaluate the ask and bid prices of geometric Asian options and obtain the explicit formulas for the ask and bid prices. Finally, numerical examples show that the higher the market liquidity parameter γ, the wider the spread and hence the less the liquidity.

Let $R_{r_{0}}, R_{r_{1}}: \mathbb{S}^{1}\longrightarrow \mathbb{S} ^{1}$ be rotations on the unit circle $\mathbb{S}^{1}$ and define $f: \varSigma _{2}\times \mathbb{S}^{1}\longrightarrow \varSigma _{2}\times \mathbb{S}^{1}$ as $$ f(x, t)=\bigl(\sigma (x), R_{r_{x_{1}}}(t)\bigr), $$ for $x=x_{1}x_{2}\cdots \in \varSigma _{2}:=\{0, 1\}^{\mathbb{N}}$ , $t\in \mathbb{S}^{1}$ , where $\sigma: \varSigma _{2}\longrightarrow \varSigma _{2}$ is the shift, and $r_{0}$ and $r_{1}$ are rotational angles. It is first proved that the system $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ exhibits maximal distributional chaos for any $r_{0}, r_{1}\in \mathbb{R}$ (no assumption of $r_{0}, r_{1}\in \mathbb{R}\setminus \mathbb{Q}$ ), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91–99, 2014). It is also obtained that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is cofinitely sensitive and $(\hat{\mathscr{M}} ^{1}, \hat{\mathscr{M}}^{1})$ -sensitive and that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is densely chaotic if and only if $r_{1}-r_{0} \in \mathbb{R}\setminus \mathbb{Q}$ .

In this paper, we study the valuation of swing options on electricity in a model where the underlying spot price is set to be the product of a deterministic seasonal pattern and Ornstein-Uhlenbeck process with Markov-modulated parameters. Under this setting, the difficulties of pricing swing options come from the various constraints embedded in contracts, e.g., the total number of rights constraint, the refraction time constraint, the local volume constraint, and the global volume constraint. Here we propose a framework for the valuation of the swing option on the condition that all the above constraints are nontrivial. To be specific, we formulate the pricing problem as an optimal stochastic control problem, which can be solved by the trinomial forest dynamic programming approach. Besides, empirical analysis is carried out on the model. We collect historical data in Nord Pool electricity market, extract the seasonal pattern, calibrate the Ornstein-Uhlenbeck process parameters in each regime, and also get market price of risk. Finally, on the basis of calibration results, a specific numerical example concerning all typical constraints is presented to demonstrate the valuation procedure.

In this paper, the TODIM method is used to solve the multi-attribute decision-making problem with unknown attribute weight in venture capital, and the decision information is given in the form of single-valued neutrosophic numbers. In order to consider the objectivity and subjectivity of decision-making problems reasonably, the optimal weight is obtained by combining subjective weights and objective weights. Subjective weights are given directly by decision makers. Objective weights are obtained by establishing a weight optimization model with known decision information, then this method will compare with entropy weight method. These simulation results also validate the effectiveness and reasonableness of this proposed method.

Fuzzy information in venture capital can be well expressed by neutrosophic numbers, and TODIM method is an effective tool for multi-attribute decision-making. The distance measure is an essential step in TODIM method. The keystone of this paper is to define several new distance measures, in particular the improved interval neutrosophic Euclidean distance, and these measures are applied in the TODIM method for multi-attribute decision-making. Firstly, the normalized generalized interval neutrosophic Hausdorff distance is defined and proved to be valid in this paper. Secondly, we define a weighted parameter interval neutrosophic distance and discuss whether different weight parameters affect the decision result based on TODIM method. Thirdly, considering the preference perspective of decision-makers in behavioral economics, we define the improved interval neutrosophic Euclidean distance with the known parameter of risk preference. Finally, an application example is given to compare the effects of different parameters on the result and discuss the feasibility of these two distance measures in TODIM method.

In this paper, we take financial crisis into consideration for American call options and put options pricing problems by using a jump diffusion model. Under no‐arbitrage pricing principle, we obtain a PDE (partial differential equation), which is different from the PDE derived from the classical Black‐Scholes model, it adds a postcrash market index to the primary equation. Then, we introduce the penalty method for solving the nonlinear PDE. Numerical results suggest that the option value will be affected by the crash.

In this paper, we study the valuation of swing options on electricity markets with local volume and refraction time constraints, under the setting that the dynamic of the underlying spot price is a 2-state regime-switching mean-reverting process. We derive the corresponding optimal multiple stopping problem, reduce it to a sequence of optimal single stopping problems, and further find that those value functions satisfy HJB variational inequalities subject to suitable conditions. Then after a prior estimation for the value functions, the viscosity solutions approach is adopted to get existence and uniqueness results in viscosity sense.

The empirical test suggests that the log-return series of stock price in US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of stock loan can be described by a space-fractional partial differential equation with time-varying free boundary condition. Firstly, a penalty term is introduced to change the original problem to be defined on a fixed domain, and then a fully-implicit difference scheme has been developed. Secondly, based on the fully-implicit scheme, we prove that the stock loan value generated by the penalty method cannot fall below the value obtained when the stock loan is exercised early. Thirdly, the numerical experiments are carried out to demonstrate differences of stock loan model under the FMLS and the standard normal distribution. Optimal redemption strategy of stock loan has been achieved. Furthermore the impact of key parameters in our model on the stock loan evaluation are analyzed, and some reasonable explanation are given.

We introduce in this paper a new technique, a semiexplicit linearized Crank-Nicolson finite difference method, for solving the generalized Rosenau-Kawahara equation. We first prove the second-order convergence in L∞-norm of the difference scheme by an induction argument and the discrete energy method, and then we obtain the prior estimate in L∞-norm of the numerical solutions. Moreover, the existence, uniqueness, and satiability of the numerical solution are also shown. Finally, numerical examples show that the new scheme is more efficient in terms of not only accuracy but also CPU time in implementation.

This paper presents a dynamic model of capital financing, taking into consideration unexpected major events occurring within continuous time model. We are considering a special jump-diffusion model first described by Samuelson (1973) while using traditional geometric Brownian motion. This paper seeks to accurately show the innovative project valuation when unexpected major events occur and get the analytical results of the project option value. Furthermore, we analyzed the impact of multistaged financing; results indicated that both sources of uncertainty positively impact the project option value; particularly, the option price when considering unexpected major events occurrence is larger than the option price without unexpected major events. Based on a comparative-static analysis, new propositions for optimal amount of investment and optimal level of project are derived from simulations.

Market crashes often appear in daily trading activities and such instantaneous occurring events would affect the stock prices greatly. In an unstable market, the volatility of financial assets changes sharply, which leads to the fact that classical option pricing models with constant volatility coefficient, even stochastic volatility term, are not accurate. To overcome this problem, in this paper we put forward a dynamic elasticity of variance (DEV) model by extending the classical constant elasticity of variance (CEV) model. Further, the partial differential equation (PDE) for the prices of European call option is derived by using risk neutral pricing principle and the numerical solution of the PDE is calculated by the Crank-Nicolson scheme. In addition, Kalman filtering method is employed to estimate the volatility term of our model. Our main finding is that the prices of European call option under our model are more accurate than those calculated by Black-Scholes model and CEV model in financial crashes.

Using variational minimizing methods, we prove the existence of a connecting orbit between the center of mass and infinity of Newtonian-like N-body problems with Newtonian-type weak force potentials.