CARA效用函数下美式期权的定价

考虑当期权持有者的效用为CARA效用函数U(x)=-e<'-λx>时的关式期权定价问题.运用最优停止理论得到其在有限离散时间金融市场模型下的最佳实施期,并给出相应美式期权的定价公式.     

双指数跳扩散过程的最优停止问题

美式期权定价问题是金融数学的热点问题,一般要用最优停止理论。本文给出了双指数跳扩散过程的最优停止问题的解析解。     

具随机折现的博弈期权定价问题

对具随机折现的博弈期权定价问题进行了研究,在满足一个可积性条件的情况下,借用过份函数等工具给出了期权价格的表达式和买卖双方的最优停止策略.对于不满足可积性条件的情况,推广了相关文献的结果,并给出了τ＊存在的条件.最后给出了一个例子.     

On the regularity of American options with regime-switching uncertainty

We study the regularity of the stochastic representation of the solution of a class of initial–boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.     

Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

     

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.     

Existence and explicit determination of optimal stopping times

We consider large classes of continuous time optimal stopping problems for which we establish the existence and form of the optimal stopping times. These optimal times are then used to find approximate optimal solutions for a class of discrete time problems.     

A Mellin transform approach to pricing barrier options under stochastic elasticity of variance

This article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi-closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black–Scholes price for a barrier option with respect to the underlying price and computing the one-dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte-Carlo simulation and find the SEV effect on the Black–Scholes barrier option prices.     

A note on optimal stopping of diffusions with a two-sided optimal rule

We consider a class of optimal stopping problems of diffusions with a two-sided optimal rule. We propose an approach for finding and characterizing the solution. We establish that the optimal stopping rule can be associated with the unique fixed point of an auxiliary function. The results are illustrated with an explicit example.     

Pricing of perpetual American and Bermudan options by binomial tree method

In this paper, we consider the binomial tree method for pricing perpetual American and perpetual Bermudan options. The closed form solutions of these discrete models are solved. Explicit formulas for the optimal exercise boundary of the perpetual American option is obtained. A nonlinear equation that is satisfied by the optimal exercise boundaries of the perpetual Bermudan option is found.      

Optimal Timing to Initiate Medical Treatment for a Disease Evolving as a Semi-Markov Process

In this paper, we consider the problem of the optimal timing to initiate a medical treatment. In the absence of treatment, we model the disease evolution as a semi-Markov process. The optimal time to initiate the treatment is a stopping time, which maximizes the total expected reward for the patient. We propose a stochastic dynamic programming formulation to find this stopping time. Under some plausible conditions, we show that the maximum total expected reward at the start of a health state will be smaller when the patient is in a more severe state. We then prove that the optimal policy for initializing the treatment is determined by a time threshold for each given health state. That is, in each health state, the treatment should be planned to start, when the patient’s duration time in the health state reaches (or exceeds, in the case of a late observation of the patient’s health status) a certain threshold level. We also present numerical examples to illustrate our model and to provide managerial insights.     

An inverse problem of determining the implied volatility in option pricing

In the Black-Scholes world there is the important quantity of volatility which cannot be observed directly but has a major impact on the option value. In practice, traders usually work with what is known as implied volatility which is implied by option prices observed in the market. In this paper, we use an optimal control framework to discuss an inverse problem of determining the implied volatility when the average option premium, namely the average value of option premium corresponding with a fixed strike price and all possible maturities from the current time to a chosen future time, is known. The issue is converted into a terminal control problem by Green function method. The existence and uniqueness of the minimum of the control functional are addressed by the optimal control method, and the necessary condition which must be satisfied by the minimum is also given. The results obtained in the paper may be useful for those who engage in risk management or volatility trading.     

光滑树图期权定价模型的叉熵分析法

为了克服CRR模型收敛的波动性,以及强调历史信息的预测作用的情况,提出了一个新奇的光滑收敛的树图模型.新模型基于历史信息,运用最小叉熵原理
来推导树图的关键参数p,u,d, 然后使用倒推法推断期权的价格.显然,新模型所得的期权的价格隐含着历史信息.由于最小叉熵原理是一个凸规划问题,能求得唯一的最优解,所以,新模型也适用于不完全金融市场期权定价.最后,数值算例表明,相比于CRR模型,新模型收敛光滑平稳且有更高的计算精度;对上涨(下跌)的二元期权、欧式期权,新模型都能光滑收敛于B-S公式.     

THE ROLE OF STOCHASTIC MONOTONICITY IN THE DECISION TO CONSERVE OR HARVEST OLD-GROWTH FOREST

The problem of when, if ever, a stand of old-growth forest should be harvested is formulated as an optimal stopping problem, and a decision rule to maximize the expected present value of amenity services plus timber benefits is found analytically. This solution can be thought of as providing the “correct” way in which cost-benefit analysis should be carried out. Future values of amenity services provided by the standing forest and or timber are considered to be uncertain and are modeled by Geometric Poisson Jump (GPJ) processes. This specification avoids the ambiguity which arises with Geometric Brownian Motion (GBM) models, as to which form of stochastic integral (Itô or Stratonovich) should be employed, but more importantly allows for monotonic (yet stochastic) processes. It is shown that monotonicity (or lack of it) in the value of amenity services relative to timber values plays an important part in the solution. If amenity values never go down (or never go up) relative to timber values, then the certain-equivalence cost-benefit procedure provides the optimal solution, and there is no option value. It is only to the extent that the relative valuations can change direction that the certainty-equivalence procedure becomes sub-optimal and option value arises.     

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.     

Valuing continuous-installment options

Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.     

Stochastic Optimization Algorithms for Pricing American Put Options Under Regime-Switching Models

This work provides a Markov-modulated stochastic approximation based approach for pricing American put options under a regime-switching geometric Brownian motion market model. The solutions of pricing American options may be characterized by certain threshold values. Here, a class of Markov-modulated stochastic approximation (SA) algorithms is developed to determine the optimal threshold levels. For option pricing in a finite horizon, a SA procedure is carried out for a fixed time T. As T varies, the optimal threshold values obtained via SA trace out a curve, called the threshold frontier. Numerical experiments are reported to demonstrate the effectiveness of the approach. Our approach provides us with a viable computational tool and has advantage in terms of the reduced computational complexity compared with the variational or quasivariational inequality methods for optimal stopping.Communicated by C. T. LeondesThis research was supported in part by the National Science Foundation under Grant DMS-0304928, and in part by the National Natural Science Foundation of China under Grant 60574069.     

Two extensions to barrier option valuation

We first present a brief but essentially complete survey of the literature on barrier option pricing. We then present two extensions of European up-and-out call option valuation. The first allows for an initial protection period during which the option cannot be knocked out. The second considers an option which is only knocked out if a second asset touches an upper barrier. Closed form solutions, detailed derivations, and the economic rationale for both types of options are provided.     

Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times.