The regularity of the free boundary for strike reset option

A strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder. The pricing model of the option can be formulated as a parabolic variational inequality and the optimal reset strategy is the free boundary. The smoothness of the free boundary in some cases was showed in our article published in JDE. We would prove its smoothness in the other case in this paper by a generalized comparison principle for the variational inequality.     

THE REGULARITY OF THE FREE BOUNDARY FOR STRIKE RESET OPTION

A strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder. The pricing model of the option can be formulated as a parabolic variational inequality and the optimal reset strategy is the free boundary. The smoothness of the free boundary in some cases was showed in our article published in JDE. We would prove its smoothness in the other case in this paper by a generalized comparison principle for the variational inequality.     

The British Put Option

Abstract

We present a new put option where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ‘best prediction’ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British put option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. The practical implications of this protection feature are most remarkable as not only can the option holder exercise at or above the strike price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive higher returns at a lesser price. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British put option that leads to the conclusions above and shows that with the contract drift properly selected the British put option becomes a very attractive alternative to the classic American put.     

A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids

This paper deals with a mathematical model describing the cell cycle dynamics and chemotactic driven cell movement in a multicellular tumor spheroid. Tumor cells consist of two types of cells: proliferating cells and quiescent cells, which have different chemotactic responses to an extracellular nutrient supply. The model is a free boundary problem for a nonlinear system of reaction-diffusion-advection equations, where the free boundary is the outer boundary of the spheroid. The free boundary condition is quite novel due to different velocity of two types of cells. The global existence and uniqueness of solutions to the model is proved. The proof is based on a fixed point argument, together with the Lp-theory for parabolic equations with the third boundary condition.     

国内外利率为随机的双币种重置型期权定价

双币种重置期权的特征是指在终端期T时的收益依赖于预先设定的t<,0>时刻标的资产的价格与执行价K>0(事先给定)的大小关系重新设置期权的执行价从而给出其定价,这种期权是投资于外国资产的一种合约,其风险不仅依赖外国资产价格的变化,还受外国货币的汇率以及国内外两种利率波动的影响,所以在实际应用方面十分广泛.本文首先就标的资...     

股价服从指数O-U过程的多点重置期权定价

本文在风险中性定价原则下,得到了股价服从指数O-U(Ornstein-Uhlenbeck)过程的n个重置日期m个执行价格的重置期权定价,又在利率服从扩展Vasicek模型下,得到了n个重置日期m个执行价格的重置期权定价.     

Sharp distribution free lower bounds for spread options and the corresponding optimal subreplicating portfolios

We derive in closed form distribution free lower bounds and optimal subreplicating strategies for spread options in a one-period static arbitrage setting. In the case of a continuum of strikes, we complement the optimal lower bound for spread options obtained in [Rapuch, G., Roncalli, T., 2002. Pricing multiasset options and credit derivatives with copula, Credit Lyonnais, Working Papers] by describing its corresponding subreplicating strategy. This result is explored numerically in a Black-Scholes and in a CEV setting. In the case of discrete strikes, we solve in closed form the optimization problem in which, for each asset S₁ and S₂, forward prices and the price of one option are used as constraints on the marginal distributions of each asset. We provide a partial solution in the case where the marginal distributions are constrained by two strikes per asset. Numerical results on real NYMEX (New York Mercantile Exchange) crack spread option data show that the one discrete lower bound can be far and also very close to the traded price. In addition, the one strike closed form solution is very close to the two strike.     

A GARCH option pricing model with α-stable innovations

We develop an option pricing model which is based on a GARCH asset return process with α-stable innovations with truncated tails. The approach utilizes a canonic martingale measure as pricing measure which provides the possibility of a model calibration to market prices. The GARCH-stable option pricing model allows the explanation of some well-known anomalies in empirical data as volatility clustering and heavy tailedness of the return distribution. Finally, the results of Monte Carlo simulations concerning the option price and the implied volatility with respect to different strike and maturity levels are presented.     

一类永久博弈期权的定价

博弈期权是由Kifer引进的,本质上是美式期权的一种,它使买卖双方都有权在到期日前的任何时刻中止合约来维护自己的权益。在股票波动率非常数时,对一类特殊类型的博弈期权进行了研究,通过解一个自由边界问题,得到了其价格的闭式解。     

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

Some properties of two-phase quadrature domains

In this paper, we investigate general properties of the two-phase quadrature domain, which was recently introduced by Emamizadeh, Prajapat and Shahgholian. The concept, which is a generalization of the well-known one-phase domain, introduces substantial difficulties with interesting features even richer than those of the one-phase counterpart.For given positive constants λ^± and two bounded and compactly supported measures μ^±, we investigate the uniqueness of the solution of the following free boundary problem:

(1)     

基于退休金保险的期权定价

本文引入一种基于退休年金的欧式看涨期权 ,它赋予合约持有者在退休年龄或其它年龄以某一约定的价格 (执行价格 )购买一份退休年金受益的机会 .通过建立相关的精算模型对一些特定情形的定价进行了阐述 ,并与传统的退休金合约进行了比较     

Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.     

On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.     

Option pricing under a normal mixture distribution derived from the Markov tree model

We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ⁺, and σ⁻ required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expires. Comparing against the Black–Scholes model, we find that the MT model’s prices are closer to market prices.     

Numerical pricing of options using high-order compact finite difference schemes

We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.     

Bifurcation of infinite Prandtl number rotating convection

We consider infinite Prandtl number convection with rotation which is the basic model in geophysical fluid dynamics. For the rotation free case, the rigorous analysis has been provided by Park (2005, 2007, revised for publication) [5], [6] and [25] under various boundary conditions. By thoroughly investigating we prove in this paper that the solutions bifurcate from the trivial solution u=0 to an attractor Σ_R which consists of only one cycle of steady state solutions and is homeomorphic to S¹. We also see how intensively the rotation inhibits the onset of convective motion. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation which was developed by Ma and Wang (2005); see [15].     

基于教育基金保险的期权定价

本文基于文献[1]引入一种基于教育年金保险的欧式看涨期权,它赋予合约持有人在约定时间以约定价格购买一份连续支付一定年限的教育年金的权利,本文运用保险精算和期权定价的二叉树方法对其进行的定价,并说明这种合约方便于一些低收入家庭进行教育投资.