Research On The Non-parametric Geometric Lévy Model And Hedging Strategies For Foreign Exchange Options

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.