Topics in Numerical Methods for Finance

Consistent and Realistic Dynamics of the Implied Volatility Surface under Local Volatility Models

In this paper, we discuss the reconstruction of a local volatility surface from American option prices. First of all, the American option prices are calculated by an accurate and fast finite difference scheme. Then the local volatility is obtained by minimizing the distance between theoretical prices and market option prices, which yields an optimization problem. The Bicubic spline regularization technique is used to overcome the ill-posedness of the reconstruction problem. We solve the nonlinear optimization problem by using a gradient-based optimization algorithm. Finally, we test our model with numerical examples and real market American put option data. The results show the good performance of our method.

Machine Learning for Quantitative Finance: Fast Derivative Pricing, Hedging and Fitting

A Note on Relative Efficiency of Some Numerical Methods for Pricing of American Options Under Levy Processes

In this paper, we study the problem of hedging a basket credit derivatives, in particular, we are interested in basket default swaps. For the pricing of credit derivatives, we consider a factor Copula approach. Single-name credit default swaps will be chosen as the hedging instruments. The hedging mechanism is tested using simulated data with a given measure. Numerical results reveal the efficiency of our proposed hedging method.

The option pricing model with constant volatility in illiquid markets has been expanded by introducing two uncertain volatility models in this paper volatility. To conquer some insufficient existed in some literature, for example the time step should be small enough to satisfy the stability, the implicit difference equation has been established and numerical solution of the modified model with uncertain volatility has been discussed. Numerical results show that the method is nice and the accurate results can be gained with less computation.

Influenced by factors which are mainly internal share structure, insider control, bond market and security market in financing, food listed companies have financing preference to some extent. In order to solve the food listed company’s financing preference caused in the financing process. This study is to analyze the company’s financial status and its current marketing environment from the perspective of the influences of its financing environment on its financing preference. This study put forward the countermeasures for the current preference on financing in food listed companies from four aspects, including regulating dividend distributions, developing corporation bond market, optimizing corporation governance structure and perfecting the law and market systems, so as to adjust and reorganize the share structure of food listed companies and to establish incentive system and alleviate the financing preference of food listed companies.

In this paper we present an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing American and European option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov–Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the solutions of the fully discretized system. Numerical results are presented to validate the theoretical results.

The Impact of CDS Trading on the Bond Market: Evidence from Asia

Forecasting the Distribution of Option Returns

We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial approximation is close to the true price of the option, the bounds are also guaranteed to be close. We also explicitly characterize the worst-case performance of the pricing bounds. The computation of the lower bound is straightforward and relies on simulating the suboptimal exercise strategy implied by the approximate option price. The upper bound is also computed using Monte Carlo simulation. This is made feasible by the representation of the American option price as a solution of a properly defined dual minimization problem, which is the main theoretical result of this paper. Our algorithm proves to be accurate on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. These numerical results suggest that our pricing method can be successfully applied to problems of practical interest.

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black–Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed form formula for integration. Moreover, these two methods solve the backward dynamic programing problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of the binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite that this model is only bidimensional, the whole history of the process impacts on the price, and how to handle all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.

We consider the delay control problem for multiuser opportunistic scheduling policies in wireless systems. First, the user selection behaviors of opportunistic scheduling policies are examined to highlight the delay and resource starvation issues. We then propose and analyze two schemes for delay control and starvation prevention that are both effective and flexible for use with existing scheduling policies. The first scheme, the Round-Robin Mixing (RRM) scheme, provides a simple and accurate mechanism for controlling the starvation probability in an i.i.d. environment. Based on the foundation of the RRM scheme and a small set of carefully chosen system approximations, we develop another improved scheme, the Opportunistic Bernoulli Mixing (OBM) scheme, that can support different user channel conditions and resource allocation requirements. The OBM scheme performs much closer in achievable system rate to the original non delay-controlled policy due to its flexible opportunistic delay adjustments while enabling very accurate control on user starvation probabilities. Numerical results illustrate excellent delay/performance tradeoff for the OBM scheme with negligible approximation errors for even moderately small systems.

The authors consider the problem of controlling the packet delay and packet loss in a fast packet switch supporting integrated services. Finite buffering capacity for each output port is assumed. Two classes of services are considered: real-time service and nonreal-time service. The delay and loss control strategy is to assign priorities and to allocate an appropriate amount of buffering capacity to different services. Real-time packets are given a higher service priority at the output port. The finite buffer at each output port is partitioned into two buffer pools, one for real-time packets and the other for nonreal-time packets, in accordance with an allocation rule. Numerical results indicate that the proposed delay and loss control strategy is quite effective. >

This paper proposes for the first time an approximate computing based energy-efficient hardware accelerator using Ramanujan Sums for edge detection applications. We exploit the inherent error resilience in the edge detection algorithm to propose an approximation technique by combining two very efficient approximation methods, viz., precision scaling and loop skipping that reduces the energy consumption of the edge detection system. We propose a gradient descent based novel heuristic to automatically configure the two approximation knobs to result in the least energy consumption for a specified application-level quality, that reduces the energy consumption of the total accelerator by almost 30% for negligible application-level quality degradation. The energy savings increase to a range of as much as 70%-80% for 5-20% quality degradation.