股票价格遵循几何分式Brown运动的期权定价

讨论了股票价格过程遵循几何分式B row n运动的欧式期权定价.由于该过程存在套利机会使得传统的期权定价方法(如资本资产定价模型(CAPM),套利定价模型(APT),动态均衡定价理论(DEPT))不可能对该期权定价.利用保险精算定价法,在对市场无其它任何假设条件下,获得了欧式期权的定价公式.并讨论了在有效期内股票支付已知红利和红利率的推广公式.     

Pricing of Ratchet equity-indexed annuities under stochastic interest rates

We consider the valuation of simple and compound Ratchet equity-indexed annuities (EIAs) in the presence of stochastic interest rates. We assume that the equity index follows a geometric Brownian motion and the short rate follows the extended Vasicek model. Under a given forward measure, we obtain an explicit multivariate normal characterization for multiple log-returns on the equity index. Using such a characterization, closed-form price formulas are derived for both simple and compound Ratchet EIAs. An efficient Monte Carlo simulation scheme is also established to overcome the computational difficulties resulting from the evaluation of high-dimensional multivariate normal cumulative distribution functions (CDFs) embedded in the price formulas as well as the consideration of additional complex contract features. Finally, numerical results are provided to illustrate the computational efficiency of our simulation scheme and the effects of various model and contract parameters on pricing.     

An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit

In this paper we explore an identity in distribution of hitting times of a finite variation process (integrated geometric Brownian motion) and a diffusion process (geometric Brownian motion with affine drift), both of which arise from various applications in financial mathematics. We develop semi-analytical solutions to fair charges of variable annuity guaranteed minimum withdrawal benefit from both a policyholder’s perspective and an insurer’s perspective. The pricing framework from the policyholder’s perspective was known previously in the literature only by numerical methods, whereas the insurer’s pricing method was used in the industry but only with Monte Carlo simulations. While comparing their similarities and differences, we prove under the assumption of no friction cost the two pricing approaches are equivalent. In the presence of friction cost, the semi-analytic solutions in this paper lead to a fast and accurate algorithm for determining rider charges and other management fees.     

标的资产价格服从分数布朗运动的几种新型期权定价

在等价鞅测度下,研究标的资产价格服从分数布朗运动的几种新型股票期权定价公式——n次幂期权、(幂型)上封顶及下保底型欧式看涨期权.并与基于标准布朗运动的期权定价公式进行比较分析,进一步论证布朗运动只是分数布朗运动的一种特例,可基于分数布朗运动对原有的期权定价模型进行推广.     

Closed-form valuations of basket options using a multivariate normal inverse Gaussian model

This paper uses a multivariate normal inverse Gaussian model to develop closed-form pricing formulas for both geometric and arithmetic basket options. For geometric basket options, an exact analytical solution is possible; for arithmetic basket options, the formula is an approximation. The model is based on a jump-driven financial process, which is known empirically to be more realistic than a geometric Brownian motion. By comparing our results to Monte Carlo experiments, we confirm the internal consistency of our formulas. The “Greeks” can be derived from the closed-form formulas in a straightforward manner.     

A continuous-time dynamic pricing model knowing the competitor’s pricing strategy

In this paper we consider a dynamic pricing model for a firm knowing that a competitor adopts a static pricing strategy. We establish a continuous time model to analyze the effect of dynamic pricing on the improvement in expected revenue in the duopoly. We assume that customers arrive to purchase tickets in accordance with a geometric Brownian motion. We derive an explicit closed-form expression for an optimal pricing policy to maximize the expected revenue. It is shown that when the competitor adopts a static pricing policy, dynamic pricing is not always effective in terms of maximizing expected revenue compared to a fixed pricing strategy. Moreover, we show that the size of the reduction in the expected revenue depends on the competitor’s pricing strategy. Numerical results are presented to illustrate the dynamic pricing policy.     

标的资产服从几何分数布朗运动的期权定价

本文在M ogens B ladt和T ina H av iid R ydberg无市场假设,仅利用价格过程的实际概率的期权保险精算定价模型的基础上,得出了标的资产服从几何分数布朗运动的欧式期权定价公式,并说明了几何布朗运动是本文的一种特殊情况.     

分数随机利率模型中的期权定价

本文研究分数随机利率模型中的期权定价问题.通过选取不同的资产作为计价单位及相应的测度交换,将经典模型中的测度变换方法推广到分数布朗运动市场环境,既丰富了分数期权定价的拟鞅方法,也得到了股票价格与利率分别服从几何分数布朗运动时的期权定价公式.     

Fractional Liu process with application to finance

As a fuzzy counterpart of Brownian motion, Liu process has attracted more and more attention in the recent literature. In this paper, the concept of fractional Liu process is proposed as an extension of Liu process. Furthermore, we obtain the expressions of the membership functions, expected values and variances of arithmetic and geometric fractional Liu processes for each fixed time. As an application, geometric fractional Liu process is assumed to characterize the stock price, which formulates a new fuzzy stock model. Based on this proposed model, European option pricing formulas are gained and two numerical examples are given with different parameters.     

Esscher变换下扩展的欧式交换期权定价

文章研究Esscher变换下标的资产价格服从几何布朗运动的扩展的几种欧式交换期权(包括广义交换期权,复合交换期权,障碍交换期权,红绿灯期权)定价问题.首先,给出了带漂移布朗运动的反射原理和性质;其次,借助Gerber和Shiu (1994)给出了多维独立平稳增量过程和二维带漂移布朗运动的Esscher变换定义及其性质;最后,应用Esscher变换的相关理论给出了标的资产价格服从几何布朗运动的扩展的多种欧式交换期权定价公式.特别,本文所得到的期权定价公式与以往文献中给出的结果是一致的.     

混合分数布朗运动下欧式期权模糊定价研究

本文采用混合分数布朗运动来刻画标的股票价格的动态变化,以此体现金融市场的长记忆性特征。在混合分数Black-Scholes模型的基础上,基于标的股票价格、无风险利率和波动率均是模糊数的假定下,构建了欧式期权模糊定价模型。其次,分析了金融市场长记忆性的度量指标Hurst指数H对欧式期权模糊定价模型的影响。最后,数值实验表明：考虑长记忆性特征得到的欧式期权模糊定价模型更符合实际。     

G框架下的欧式期权定价公式

用G几何布朗运动描述标的资产的价格变动,得到了欧式看涨期权定价的动态公式,并给出了动态复制策略的显示表达.     

Time series model for GLWB with surrender benefit and stochastic interest rate: Dynamic withdrawal approach

This study investigates the pricing problem of a variable annuity (VA) contract embedded with a guaranteed lifetime withdrawal benefit (GLWB) rider. VAs are annuities in which the value is linked to a bond and equity sub-account fund. The guaranteed lifetime withdrawal benefit rider regularly provides a series of payments to the policyholder for the term of the policy while he/she is alive, regardless of portfolio performance. At the time of the policyholder's death, the remaining fund value is given to his nominee. Therefore, proper fund modeling is critical in the pricing of VA products. Several writers in the literature used a GBM model in which variance is considered to be constant to represent the fund value in a variable annuity contract. However, on the other hand, the returns on financial assets are non-normally distributed in real life. A bit much Kurtosis, leverage effect, and Non-zero Skewness characterize the returns. The generalized autoregressive conditional heteroscedastic (GARCH) models are also used for presenting a discrete framework for the pricing of GLWB. Still, the interest rate was kept constant without including the surrender benefit and the static withdrawal approach, which keeps the model far from the real scenario. Thus, in this research, the generalized GARCH models are used with surrender benefit and dynamic withdrawal strategy to develop a time series model for the pricing of annuity that overcomes the constraints of previous models. A numerical illustration and sensitivity analysis are used to examine the suggested model.     

Hitting Time Problems of Sticky Brownian Motion and Their Applications in Optimal Stopping and Bond Pricing

This paper investigates the hitting time problems of sticky Brownian motion and their applications in optimal stopping and bond pricing. We study the Laplace transform of first hitting time over the constant and random jump boundary, respectively. The results about hitting the constant boundary serve for solving the optimal stopping problem of sticky Brownian motion. By introducing the sharpo ratio, we settle the bond pricing problem under sticky Brownian motion as well. An interesting result shows that the sticky point is in the continuation region and all the results we get are in closed form.

     

分期付费的投资连结保险定价研究

本文在投资基金价格服从几何布朗运动假设下,根据布朗运动的定义和Schwarz不等式得到了具有最小保证金和分期付费的投资连结保险保费的一个上限,并通过Monte Carlo模拟方法检验了它的合理性.这结论对于此保险的定价具有较大的理论与实践意义.     

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In this paper, we establish the option pricing model under sub-fractional Brownian motion, and consider the situation of the continuous dividend payments. Firstly, Wick-It\^{o} integral and partial differential method are used to get the option price of partial differential equation, and then through variable substitution into Cauchy problem, we can get the pricing formula of European call option with dividend-paying in sub-fractional Brownian motion environment. According to the pricing formula of European call option, the European put option pricing formula is obtained. Moreover, we study the parameter estimation in the model, and consider the unbiasedness and the strong convergence of the estimator.     

Critical Thresholds for Dynamic Routing in Queueing Networks

This paper studies dynamic routing in a parallel server queueing network with a single Poisson arrival process and two servers with exponential processing times of different rates. Each customer must be routed at the time of arrival to one of the two queues in the network. We establish that this system operating under a threshold policy can be well approximated by a one-dimensional reflected Brownian motion when the arrival rate to the network is close to the processing capacity of the two servers. As the heavy traffic limit is approached, thresholds which grow at a logarithmic rate are critical in determining the behavior of the limiting system. We provide necessary and sufficient conditions on the growth rate of the threshold for (i) approximation of the network by a reflected Brownian motion (ii) positive recurrence of the limiting Brownian diffusion and (iii) asymptotic optimality of the threshold policy.     

Valuation of the interest rate guarantee embedded in defined contribution pension plans

In this research, we derive the valuation formulae for a defined contribution pension plan associated with the minimum rate of return guarantees. Different from the previous studies, we work on the rate of return guarantee which is linked to the δ-year spot rate. The payoffs of interest rate guarantees can be viewed as a function of the exchange option. By employing Margrabe’s [Margrabe, W., 1978. The value of an option to exchange one asset for another. Journal of Finance 33, 177–186] option pricing approach, we derive general pricing formulae under the assumptions that the interest rate dynamics follow a single-factor HJM (1992) [Heath. D. et al., 1992. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60, 77–105] interest rate model and the asset prices follow a geometric Brownian motion. The volatility of the forward rates is assumed to be exponentially decaying. The formula is explicit for valuing maturity guarantee (type-I guarantee). For multi-period guarantee (type-II guarantee), the analytical formula only exists when the guaranteed rate is the one-year spot rate. The accuracy of the valuation formulae is illustrated with numerical analysis. We also investigate the effect of mortality and the sensitivity of key parameters on the value of the guarantee. We find that type-II guarantee is much more costly than the type-I guarantee, especially with a long duration policy. The closed form solution provides the advantage in valuing pension guarantees.     

Arbitrage in skew Brownian motion models

Empirical skewness of asset returns can be reproduced by stochastic processes other than the Brownian motion with drift. Some authors have proposed the skew Brownian motion for pricing as well as interest rate modelling. Although the asymmetric feature of random return involved in the stock price process is driven by a parsimonious one-dimensional model, we will show how this is intrinsically incompatible with a modern theory of arbitrage in continuous time. Application to investment performance and to the Black-Scholes pricing model clearly emphasize how this process can provide some kind of arbitrage.     

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.