外汇期权定价的非参数几何Lévy模型与对冲策略研究
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摘要
外汇市场是全球交易量最大的金融市场,2010年12月的日均交易量达4万亿美元。布雷顿森林体系崩溃后,汇率波动加剧,国际经济活动面临巨大的汇率风险。在此背景下,外汇衍生产品应运而生并快速发展,其中外汇期权作为低成本的汇率风险管理工具更是受到投资者欢迎,产品结构不断完善。金融衍生产品创新过程中,产品定价与对冲策略设计是其中的重要环节。本文在非参数框架下,以Lévy过程为标的驱动过程,对外汇期权的定价与对冲策略进行了研究,涉及的问题包括Lévy过程的非参数估计、模型校正、外汇期权定价方法、隐含波动率特征和平方对冲策略等。
Lévy过程的样本路径具有间断点,可以刻画金融资产价格运动中的跳跃行为,因而在金融建模中得到广泛应用。现有的Lévy模型均对跳跃行为作了分布假设。资产价格运动中的跳跃行为由几个分布参数刻画,对于复杂的金融系统来说,具有一定的局限性。本文提出的非参数几何Lévy模型,不需要对跳跃行为作分布假设,突破了这一局限。但非参数框架下的几何Lévy模型更难估计,模型更难校正,期权定价方法更复杂。本文围绕定价与校正这两个主题展开研究。
首先采用非参数估计方法,从汇率的离散样本中估计出Lévy过程的特征三元组,其中Lévy测度由一个离散化的Lévy密度和跳跃强度表示。然后以此为先验知识,结合外汇期权市场价格数据中隐含的汇率运动信息,确定一个正则化的平方定价误差和,作为最优化问题的目标函数。采用大规模边界约束BFGS(LBFGSB)算法对此最优化问题进行寻优,获得模型参数。
其次,借鉴期权定价的傅立叶分析方法,将一个经修正的外汇期权时间价值函数的傅立叶变换,表示为驱动Lévy过程特征函数的函数。然后通过离散傅立叶逆变换获得修正的外汇期权时间价值函数的数值解,从而得到外汇期权价格。
最后研究了几何Lévy模型的平方对冲策略。平方对冲策略是对冲误差期望值最小化的结果,取期望的测度可为鞅测度或历史测度。鞅测度下,标的过程可分解为一个鞅部分和一个漂移部分;历史测度下,未定权益可表示为F(o|¨)llmer-Schweizer分解。利用这两种方法获得的平方对冲策略表达式还不能直接用于计算。本文对平方对冲策略在鞅测度和历史测度下的更直接表示进行了探讨,并分析了两者之间的关系。
全文中,还多处分析了Lévy模型的隐含波动率表面的形状。结果表明,通过参数选择,可使Lévy模型的隐含波动率表面比Black-Scholes模型更平坦。作为补充,讨论了隐含Lévy波动率模型的外汇期权定价方法和模型估计方法。
本文的创新之处在于提出一个非参数框架下的非参数几何Lévy模型,构造出相应的外汇期权定价方法和模型校正方法,并讨论了此模型下的平方对冲策略计算问题。非参数框架下,跳跃-扩散类模型不需要假定跳跃行为的分布,跳跃行为由一个离散化的Lévy密度表示,故现有跳跃-扩散类模型可推广为非参数几何Lévy模型。利用傅立叶变换推导出非参数几何Lévy模型下外汇期权的价格计算公式,并分析了数值实现方法的收敛速度和误差控制。模型校正问题涉及病态问题的良态化。本文利用相对熵作为正则函数,结合正则化方法和大规模边界约束BFGS算法,在MATLAB平台上实现非参数几何Lévy模型的校正。
Foreign exchange market is the largest financial market in the world, with an estimated $4 trillion average daily turnover. After the breakdown of the Bretton Woods System in 1970’s, exchange rate fluctuates widely and international economic activities carry a huge of risk. Under this background, foreign exchange derivatives were introduced and developed rapidly. Foreign exchange option, as a cheap tool of risk managing, receives a heated welcome from investors, and its product structure become completed more and more. While creating a new financial derivative, product pricing and devising a hedging strategy are two vital steps. Under a non-parametric framework, this dissertation studies these two problems of foreign exchange options with a Lévy process as the underlying driving process. Specifically speaking, it involves non-parametric estimation of a Lévy process, model calibration, method of option pricing, features of implied volatility surface, quadratic hedging strategy and so on.
Lévy processes can capture the jump behavior in the price movement of financial assets because their sample paths may be discontinuous. So they are widely used in financial modeling. All of established Lévy models make a distributional hypothesis about the jump behavior of Lévy process. The jumps in the price movement of asset are characterized by a few of distributional parameters. But by doing so in a complex financial setting, there exist limitations more or less. To break the back of this problem, this dissertation proposes a non-parametric geometric Lévy model with no distributional hypothesis about jump behavior of Lévy process. But it is difficult to estimate or calibrate this model, and pricing an option is a hard task in the model too. In this dissertation, Studies are expanded along these two subjects.
Firstly, to estimate a Lévy process’characteristic triplet from a discrete sample of exchange rate, whose Lévy measure is expressed in terms of a discrete Lévy density and a jump intensity parameter, a non-parametric estimation method is employed. Then as prior knowledge, the triplet estimated from historical data of exchange rate, is integrated the information of exchange movement implied in the dataset of market price of foreign exchange option into the objective function of a optimization problem, which is a sum of squared pricing errors. The optimization problem is solved by a large scale bound constrained BFGS (LBFGSB) algorithm. The resulting solution is also the model parameters.
Secondly, borrowing the Fourier analysis method of option pricing, the characteristic function of a modified time value function of foreign exchange option, is represented as a function of the driving Lévy process’characteristic function. The modified time value function is numerically solved by the inverse Discrete Fourier Transform, so the foreign exchange option price can be obtained from it.
Finally, the quadratic hedging strategy of geometric Lévy model is studied. A quadratic hedging strategy is the result of minimizing the expectation of hedging error. The measure with respect to which the expectation is taken may be a martingale measure or a historical measure. Under a martingale measure, the underlying process can be decomposed into a martingale part and a drift term, while under a historical measure, a contingent claim can represented as a F(o|¨)llmer-Schweizer decomposition. The formula of hedging ratio directly based on these two types of decompositions is not enough implicit for the purpose of computation. Two more implicit representations of quadratic hedging strategy under martingale measure and historical measure are investigated in this dissertation. The relations between the two representations are also analyzed.
The shape of implied volatility surface is analyzed in several chapters of this dissertation. The results indicate that, the implied volatility surface of Lévy model can get flatter than that of Black-Scholes model by choosing suitable parameters. As a complement, this dissertation also discusses the methods of foreign exchange option pacing and model estimation in an implied Lévy volatility model.
The main contributions of this dissertation lie in proposing a non-parametric geometric Lévy model under a non-parametric framework, developing relevant methods of foreign exchange option pricing and model calibration, and discussing the problem of computing the quadratic hedging strategy in this model. Under the non-parametric framework, the distributional hypothesis about jump behavior in a jump-diffusion model is not needed anymore, and the jumps are characterized by a discrete Lévy density. So all the jump-diffusion models can be generalized into the non-parametric geometric Lévy model. With the Fast Fourier Transform technique, the formula of computing foreign exchange option price in this model is derived, convergence rate and error control of this numerical methods are investigated. The problem of Model calibration involves well-posing an ill-posed problem. In this dissertation, the non-parametric geometric Lévy model is calibrated on the MALAB platform, using a LBFGSB algorithm and a regularization method whose penalized term is a relative entropy function together.
Lévy过程的样本路径具有间断点,可以刻画金融资产价格运动中的跳跃行为,因而在金融建模中得到广泛应用。现有的Lévy模型均对跳跃行为作了分布假设。资产价格运动中的跳跃行为由几个分布参数刻画,对于复杂的金融系统来说,具有一定的局限性。本文提出的非参数几何Lévy模型,不需要对跳跃行为作分布假设,突破了这一局限。但非参数框架下的几何Lévy模型更难估计,模型更难校正,期权定价方法更复杂。本文围绕定价与校正这两个主题展开研究。
首先采用非参数估计方法,从汇率的离散样本中估计出Lévy过程的特征三元组,其中Lévy测度由一个离散化的Lévy密度和跳跃强度表示。然后以此为先验知识,结合外汇期权市场价格数据中隐含的汇率运动信息,确定一个正则化的平方定价误差和,作为最优化问题的目标函数。采用大规模边界约束BFGS(LBFGSB)算法对此最优化问题进行寻优,获得模型参数。
其次,借鉴期权定价的傅立叶分析方法,将一个经修正的外汇期权时间价值函数的傅立叶变换,表示为驱动Lévy过程特征函数的函数。然后通过离散傅立叶逆变换获得修正的外汇期权时间价值函数的数值解,从而得到外汇期权价格。
最后研究了几何Lévy模型的平方对冲策略。平方对冲策略是对冲误差期望值最小化的结果,取期望的测度可为鞅测度或历史测度。鞅测度下,标的过程可分解为一个鞅部分和一个漂移部分;历史测度下,未定权益可表示为F(o|¨)llmer-Schweizer分解。利用这两种方法获得的平方对冲策略表达式还不能直接用于计算。本文对平方对冲策略在鞅测度和历史测度下的更直接表示进行了探讨,并分析了两者之间的关系。
全文中,还多处分析了Lévy模型的隐含波动率表面的形状。结果表明,通过参数选择,可使Lévy模型的隐含波动率表面比Black-Scholes模型更平坦。作为补充,讨论了隐含Lévy波动率模型的外汇期权定价方法和模型估计方法。
本文的创新之处在于提出一个非参数框架下的非参数几何Lévy模型,构造出相应的外汇期权定价方法和模型校正方法,并讨论了此模型下的平方对冲策略计算问题。非参数框架下,跳跃-扩散类模型不需要假定跳跃行为的分布,跳跃行为由一个离散化的Lévy密度表示,故现有跳跃-扩散类模型可推广为非参数几何Lévy模型。利用傅立叶变换推导出非参数几何Lévy模型下外汇期权的价格计算公式,并分析了数值实现方法的收敛速度和误差控制。模型校正问题涉及病态问题的良态化。本文利用相对熵作为正则函数,结合正则化方法和大规模边界约束BFGS算法,在MATLAB平台上实现非参数几何Lévy模型的校正。
Foreign exchange market is the largest financial market in the world, with an estimated $4 trillion average daily turnover. After the breakdown of the Bretton Woods System in 1970’s, exchange rate fluctuates widely and international economic activities carry a huge of risk. Under this background, foreign exchange derivatives were introduced and developed rapidly. Foreign exchange option, as a cheap tool of risk managing, receives a heated welcome from investors, and its product structure become completed more and more. While creating a new financial derivative, product pricing and devising a hedging strategy are two vital steps. Under a non-parametric framework, this dissertation studies these two problems of foreign exchange options with a Lévy process as the underlying driving process. Specifically speaking, it involves non-parametric estimation of a Lévy process, model calibration, method of option pricing, features of implied volatility surface, quadratic hedging strategy and so on.
Lévy processes can capture the jump behavior in the price movement of financial assets because their sample paths may be discontinuous. So they are widely used in financial modeling. All of established Lévy models make a distributional hypothesis about the jump behavior of Lévy process. The jumps in the price movement of asset are characterized by a few of distributional parameters. But by doing so in a complex financial setting, there exist limitations more or less. To break the back of this problem, this dissertation proposes a non-parametric geometric Lévy model with no distributional hypothesis about jump behavior of Lévy process. But it is difficult to estimate or calibrate this model, and pricing an option is a hard task in the model too. In this dissertation, Studies are expanded along these two subjects.
Firstly, to estimate a Lévy process’characteristic triplet from a discrete sample of exchange rate, whose Lévy measure is expressed in terms of a discrete Lévy density and a jump intensity parameter, a non-parametric estimation method is employed. Then as prior knowledge, the triplet estimated from historical data of exchange rate, is integrated the information of exchange movement implied in the dataset of market price of foreign exchange option into the objective function of a optimization problem, which is a sum of squared pricing errors. The optimization problem is solved by a large scale bound constrained BFGS (LBFGSB) algorithm. The resulting solution is also the model parameters.
Secondly, borrowing the Fourier analysis method of option pricing, the characteristic function of a modified time value function of foreign exchange option, is represented as a function of the driving Lévy process’characteristic function. The modified time value function is numerically solved by the inverse Discrete Fourier Transform, so the foreign exchange option price can be obtained from it.
Finally, the quadratic hedging strategy of geometric Lévy model is studied. A quadratic hedging strategy is the result of minimizing the expectation of hedging error. The measure with respect to which the expectation is taken may be a martingale measure or a historical measure. Under a martingale measure, the underlying process can be decomposed into a martingale part and a drift term, while under a historical measure, a contingent claim can represented as a F(o|¨)llmer-Schweizer decomposition. The formula of hedging ratio directly based on these two types of decompositions is not enough implicit for the purpose of computation. Two more implicit representations of quadratic hedging strategy under martingale measure and historical measure are investigated in this dissertation. The relations between the two representations are also analyzed.
The shape of implied volatility surface is analyzed in several chapters of this dissertation. The results indicate that, the implied volatility surface of Lévy model can get flatter than that of Black-Scholes model by choosing suitable parameters. As a complement, this dissertation also discusses the methods of foreign exchange option pacing and model estimation in an implied Lévy volatility model.
The main contributions of this dissertation lie in proposing a non-parametric geometric Lévy model under a non-parametric framework, developing relevant methods of foreign exchange option pricing and model calibration, and discussing the problem of computing the quadratic hedging strategy in this model. Under the non-parametric framework, the distributional hypothesis about jump behavior in a jump-diffusion model is not needed anymore, and the jumps are characterized by a discrete Lévy density. So all the jump-diffusion models can be generalized into the non-parametric geometric Lévy model. With the Fast Fourier Transform technique, the formula of computing foreign exchange option price in this model is derived, convergence rate and error control of this numerical methods are investigated. The problem of Model calibration involves well-posing an ill-posed problem. In this dissertation, the non-parametric geometric Lévy model is calibrated on the MALAB platform, using a LBFGSB algorithm and a regularization method whose penalized term is a relative entropy function together.
引文
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