On the asymptotic free boundary for the American put option problem

In practical work with American put options, it is important to be able to know when to exercise the option, and when not to do so. In computer simulation based on the standard theory of geometric Brownian motion for simulating stock price movements, this problem is fairly easy to handle for options with a short lifespan, by analyzing binomial trees. It is considerably more challenging to make the decision for American put options with long lifespan. In order to provide a satisfactory analysis, we look at the corresponding free boundary problem, and show that the free boundary—which is the curve that separates the two decisions, to exercise or not to—has an asymptotic expansion, where the coefficient of the main term is expressed as an integral in terms of the free boundary. This raises the perspective that one could use numerical simulation to approximate the integral and thus get an effective way to make correct decisions for long life options.     

A Note on the Valuation of American Option

American options give holder a right to exercise it at any time at will, the holder should to make the exercise policy in such a way that the expected payoff from the option will be maximized. In this note we prove that it is equivalent to a fact which makes the option value and option delta continuous.     

Optimal stopping, free boundary, and American option in a jump-diffusion model

This paper considers the American put option valuation in a jump-diffusion model and relates this optimal-stopping problem to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black-Scholes (BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary and also give one estimate near maturity, generalizing a recent result of Barleset al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American put option price and its critical stock price.     

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。     

求解美式回望期权的有限元方法

期权作为一种金融衍生产品,在欧美国家一直很受欢迎.由于其规避风险的特性,期权也吸引了中国投资者的兴趣.基于市场的需求,2015年初,上海证券交易所推出了中国首批期权产品,期权定价问题的研究热潮正席卷全球.本文研究的美式回望期权,是一种路径相关的期权,其支付函数不仅依赖于标的资产的现值,也依赖其历史最值.分析回望期权的特点,不难发现:1)这类期权空间变量的变化范围为二维无界不规则区域,难以应用数值方法直接求解;2)最佳实施边界未知,使得该问题变得高度非线性.本文的主要工作就是解决这两个困难,得到回望期权和最佳实施边界的数值逼近结果.现有的处理问题1)的有效方法是采用标准变量替换、计价单位变换以及Landau变换将定价模型化为一个[0,1]区间上的非线性抛物问题,本文也将沿用这些技巧处理问题1).进一步,采用有限元方法离散简化后的定价模型,并论证了数值解的非负性,提出了利用Newton法求解离散化的非线性系统.最后,通过数值模拟,验证了本文所提算法的高效性和准确性.     

Electricity swing options: Behavioral models and pricing

Electricity swing options are supply contracts for power, which give the owner the right to change the required delivery on short time notice. It gives more flexibility than fixed base load or peak load contracts. The name “option” is a bit misleading, since it gives the owner multiple exercise rights at many different time horizons with exercise amounts on a continuous scale. We look at the problem to determine a rational ask price for such a contract from the viewpoint of the contract seller. The pricing of these contracts differs drastically from the pricing of financial options. First, peculiar properties arise from the non-storability of the underlying (the energy) and therefore the impossibility to hedge with the underlying, hedging is only possible with some future contracts. Second, the behavior of the owner plays an important role. Based on some behavioral model for the option holder, we develop a game-theoretic model, which allows to identify the equilibrium price. Besides some theoretical results, we present some numerical results which clarify the dependence of the asked price on the amount of flexibility offered in the swing option.     

American put option with regime‐switching volatility (finite time horizon)—Variational inequality approach

We study the fair price of American put option with regime‐switching volatility. Assuming that volatility σ(t) takes two different values σ₁ and σ₂, applying Δ hedging technique we obtain a system of evolutionary variational inequalities, which possesses two free boundaries (optimal exercise boundaries). The following are the main results of this paper.

• 1. Two free boundaries are monotonic and infinitely differentiable.
• 2. The optimal exercise boundary of American put option with regime‐switching volatility in the bearish (or bullish) market is smaller (or higher) than the one of standard American put option. And the price of American put option with regime‐switching volatility in the bearish (or bullish) market is higher (or smaller) than the one of standard American put option.
• 3. The solution of problem (1) is unique.

These results are original in the option pricing with regime‐switching volatility, the proof is technical. Copyright © 2008 John Wiley & Sons, Ltd.     

The American Foreign Exchange Option in Time-Dependent One-Dimensional Diffusion Model for Exchange Rate

The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent case is the continuity in time argument of the value function of the American put option and the regularity properties of the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for the American puts with different strikes, maturities and volatilities. Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value function of the American foreign exchange put option as the sum of the European put option value function and the early exercise premium. The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise boundary which can be studied by numerical methods.     

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...     

A closed-form analytic correction to the Black–Scholes–Merton price for perpetual American options

This is a complementary study of a recent work by Yoon et al. (2013) [1] [J.-H. Yoon, J.-H. Kim, S.-Y. Choi, Multiscale analysis of a perpetual American option with the stochastic elasticity of variance, Appl. Math. Lett. 26 (7) (2013)] which excludes a certain level of the elasticity of variance. A second-order correction to the Black–Scholes option price and optimal exercise boundary for a perpetual American put option is made under the stochastic elasticity of variance of a risky asset. Contrary to the case of Yoon et al. (2013) [1], it is given by an explicit closed-form analytic expression so that one can access clearly the sensitivity of the option price and the optimal exercise boundary to changes in model parameters as well as the impact of the presence of a stochastic elasticity term on the option price and the optimal time to exercise.     

模糊环境下美式看跌期权的定价研究

应用模糊集理论将无风险利率和波动率进行模糊化,以梯形模糊数替代精确值,将美式期权的定价模型扩展到美式期权模糊定价模型.得到了模糊风险中性概率表达式,并在此概率测度下推导出多期二叉树模糊定价模型,以及二叉树上各节点以梯形模糊数表示的模糊期权价值,以数值模拟演示了美式看跌期权的模糊定价过程.最后分析了不同风险偏好投资者在不确定环境下的套利决策行为,结果表明风险偏好大的投资者具有较高的置信水平、较小的主观模糊期权价格以及较大的无风险套利区间.     

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 American options under the CEV model by Laplace-Carson transforms

This paper investigates American option pricing under the constant elasticity of variance (CEV) model. Taking the Laplace-Carson transform (LCT) to the corresponding free-boundary problem enables the determination of the optimal early exercise boundary to be separated from the valuation procedure. Specifically, a functional equation for the LCT of the early exercise boundary is obtained. By applying Gaussian quadrature formulas, an efficient method is developed to compute the early exercise boundary, American option price and Greeks under the CEV model.     

Infinite reload options: Pricing and analysis

Infinite reload options allow the user to exercise his reload right as often as he chooses during the lifetime of the contract. Each time a reload occurs, the owner receives new options where the strike price is set to the current stock price. We consider a modified version of the infinite reload option contract where the strike price of the new options received by the owner is increased by a certain percentage; we refer to this new contract as an increased reload option. The pricing problem for this modified contract is characterized as an impulse control problem resulting in a Hamilton–Jacobi–Bellman equation. We use fully implicit timestepping and prove that the discretized equations are monotone, stable and consistent, implying convergence to the viscosity solution. We also derive a globally convergent iterative method for solving the non-linear discrete equations. Numerical examples show that both the exercise policy and the option value are very sensitive to the percentage increase in the reload strike.     

具限制投资组合和交易费用的博弈未定权益的保值

博弈期权是由kifer(2000)提出的,但就其本质而言,仍是美式期权的一种,只是增加了卖方中止合约的权利.本文主要对连续市场模型中具交易费用和限制投资组合的博弈未定权益的保值问题进行了研究,给出了买卖双方的保值价格和一个无套利区间.     

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.     

美式看跌期权定价的一种混合数值方法

本文对美式看跌期权的定价提供了一种新的混合数值方法,即快速傅里叶变换法加龙格-库塔法.首先将美式看跌期权价格所满足的Black-Scholes微分方程定解问题转化为一个标准的抛物型初、边值问题,然后通过傅里叶变换,使之转换为一个不带股价变量的常微分方程初值问题,再利用龙格-库塔法对其进行数值求解.数值实验表明,本文算法是一种快速的高精度的算法.     

A moment-based analytic approximation of the risk-neutral density of American options

The price of a European option can be computed as the expected value of the payoff function under the risk-neutral measure. For American options and path-dependent options in general, this principle cannot be applied. In this paper, we derive a model-free analytical formula for the implied risk-neutral density based on the implied moments of the implicit European contract under which the expected value will be the price of the equivalent payoff with the American exercise condition. The risk-neutral density is semi-parametric as it is the result of applying the multivariate generalized Edgeworth expansion, where the moments of the American density are obtained by a reverse engineering application of the least-squares method. The theory of multivariate truncated moments is employed for approximating the option price, with important consequences for the hedging of variance, skewness and kurtosis swaps.     

A new predictor-corrector scheme for valuing American puts

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution [35], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.