Basket Options Research Articles

High-order ADI finite difference schemes for parabolic equations in the combination technique with application in finance

In this article we combine high-order (HO) finite difference discretisations with alternating direction implicit (ADI) schemes for parabolic partial differential equations with mixed derivatives in a sparse grid setting. In each implicit leg of the ADI schemes, we propose a high-order-compact (HOC) discretisation, such that only tridiagonal systems have to be solved. With the help of HO spatial discretisations and ADI schemes solutions with second order accuracy in time and fourth order accuracy in space can be computed. In order to reduce the number of involved grid points we use the combination technique to construct the so called sparse grid solution. The theoretical findings are illustrated by numerical examples with European basket options.

A general control variate method for multi-dimensional SDEs: An application to multi-asset options under local stochastic volatility with jumps models in finance

This paper presents a new control variate method for general multi-dimensional stochastic differential equations (SDEs) including jumps in order to reduce the variance of Monte Carlo method. Our control variate method is based on an asymptotic expansion technique, and does not require an explicit characteristic function of SDEs. This is an extension of previous researches using asymptotic expansions to obtain the control variates for such general models. Moreover, in our control variate method, the regression estimators can be chosen for each number of jump times with a stratified sampling, and improve the efficiency of the variance reduction. This paper also provides the asymptotic bias and variance of our method in terms of its terminal time and a small noise parameter used in an asymptotic expansion method.For an application of our method, we evaluate multi-asset options whose payoffs are expressed as linear functions of the underlying asset price processes in general local stochastic volatility with jumps models, and show a calculation scheme of control variates for Greeks.In numerical experiments, we apply the new control variate method to pricing basket, spread, and average options and Delta of basket options under a ZABR-type local stochastic volatility model with jumps, and confirm that our method works very well.

An Environment for Rapid Derivatives Design and Experimentation

In the highly competitive world of modern finance, new derivatives are continually required to take advantage of changes in financial markets, and to hedge businesses against new risks. The research described in this paper aims to accelerate the development and pricing of new derivatives in two different ways. First, new derivatives can be specified mathematically within a general framework, enabling new mathematical formulae to be specified rather than just new parameter settings. This Generic Pricing Engine (GPE) is expressively powerful enough to specify a wide range of standard pricing engines. Second, the associated price simulation using the Monte Carlo method is accelerated using GPU or multicore hardware. The parallel implementation (in OpenCL) is automatically derived from the mathematical description of the derivative. As a test, for a Basket Option Pricing Engine (BOPE) generated using the GPE, on the largest problem size, an NVidia GPU runs the generated pricing engine at 45 times the speed of a sequential, specific hand-coded implementation of the same BOPE. Thus, a user can more rapidly devise, simulate, and experiment with new derivatives without actual programming

Accelerated trinomial trees applied to American basket options and American options under the Bates model

The accelerated trinomial tree (ATT) is a derivatives pricing lattice method that circumvents the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs), in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps, each of which contains an inner lattice/trinomial tree with N nonuniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, is constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be interpreted as using risk-neutral extended probabilities: extended in the sense that values can lie outside the range [0, 1] on the substep scale but aggregate to probabilities within the range [0,1] on the superstep scale. Hence, it is only strictly at the end of each superstep that a practically meaningful solution may be extracted from the tree. We demonstrate that ATTs with L supersteps are more efficient or have comparable efficiency to competing implicit methods that use L time steps in pricing Black–Scholes American put options and two-dimensional American basket options. Crucially, this performance is achieved using an algorithm that requires only a modest modification of a standard trinomial tree. This is in contrast to implicit FDMs, which may be relatively complex in their implementation. We also extend ATTs to the pricing of American options under the Heston model and the Bates model in order to demonstrate the general applicability of the approach.

A moment matching approach to log-normal portfolio optimization

We consider the problem where a manager aims to minimize the probability of his portfolio return falling below a threshold while keeping the expected return no worse than a target, under the assumption that stock returns are Log-Normally distributed. This assumption, common in the finance literature for daily and weekly returns, creates computational difficulties because the distribution of the portfolio return is difficult to estimate precisely. We approximate it with a single Log-Normal random variable using the Fenton–Wilkinson method and investigate an iterative, data-driven approximation to the problem. We propose a two-stage solution approach, where the first stage requires solving a classic mean-variance optimization model and the second step involves solving an unconstrained nonlinear problem with a smooth objective function. We suggest an iterative calibration method to improve the accuracy of the method and test its performance against a Generalized Pareto Distribution approximation. We also extend our results to the design of basket options.

Pricing and hedging basket options with exact moment matching

Theoretical models applied to option pricing should take into account the empirical characteristics of financial time series. In this paper, we show how to price basket options when the underlying asset prices follow a displaced log-normal process with jumps, capable of accommodating negative skewness and excess kurtosis. Our technique involves Hermite polynomial expansion that can match exactly the first m moments of the model-implied basket return. This method is shown to provide superior results for basket options not only with respect to pricing but also for hedging.

In Search of the Perfect Hedge Underlying

This report attempts to answer the question: What underlying portfolio should one use to hedge an active fund? We introduce a framework which allows us to conduct analysis on simulated realistic active portfolios in order to build intuition as to how hedge mismatch error affects the level of protection from a given hedge. We show that for typical market conditions, hedge effectiveness improves dramatically when using a hedge portfolio that more accurately reflects the underlying portfolio. This has clear consequences for using generic index options to hedge highly active portfolios. We also showcase several active return and tracking error decompositions that allow one to precisely quantify and thus manage the sources of risk and rewards within a given portfolio. We then discuss a mixed integer quadratic programming formulation that enables us to search across a large investment universe in order to find the subset of stocks that most closely replicates a given portfolio’s performance, whilst simultaneously complying with realistic market constraints. Motivated by these three elements, we introduce several alternative hedging methods for the fund manager to implement a better hedge for their active portfolio. In this report, we focus specifically on the use of long-only and long/short custom basket options as a means of creating an appropriate portfolio hedge.

Single Stock and Custom Basket Options in Fund Management

This report provides an overview of the utility of single stock and custom basket options in fund management. It is shown that managers of active equity funds can limit possible negative return contributions of their over - and underweight positions via single stock options and thus help to hedge against underperformance of the fund relative to the benchmark.We highlight the fact that non-index equity portfolios are imperfectly hedged by index derivatives. In this context we describe an analysis that helps to quantify the risk associated with a portfolio-underlying mismatch, and discuss how this risk can be addressed via an option or future on a custom basket of stocks.We also discuss challenges in pricing single stock options and provide an overview of a single stock volatility skew calibration system developed at Peregrine Securities that addresses an important need for accurate volatility estimates. Using these single stock volatility skews, we are able to implement a simple procedure for the calculation of implied volatility for options on custom baskets.

Reconstructing the Joint Probability Distribution From Basket Prices: A Mildly Ill-Posed Problem

In this thesis, we study basket options which are contracts on N assets. It is known that the risk neutral joint probability density of the assets is uniquely defined by the prices of basket options with positive weights. However, the proof of this fact requires a step involving the inversion of a Laplace transform. It is known that inversion of a Laplace transform is a severely ill-posed problem. The purpose of this thesis is to understand whether inversion of the basket transform is ill-posed or not.In our dissertation, we propose three methods to determine and prove the order of ill-posedness for the case when the basket contains only one asset and then map the problem to Radon transform to address two or higher dimensional case. Our proofs show that that problem is mildly ill-posed of order 1/2 (N -1) 2 given N dimensions.We also conduct numerical experiments under the Black-Scholes setting. First, using the reconstruction method from Cohen and Pagliacci, our results show that this method functions well in one-dimension but it becomes numerically intensive in our two-dimensional implementation, and we were unable to achieve convergence. Alternatively, we have looked at using an inverse Radon transform method to reconstruct the joint density numerically. As this requires knowledge of basket prices with negative weights, we have tried extending the basket price in this region as an odd function. For comparison, we compared this against the reconstruction from approximate basket prices calculated directly in the negative weight region. It is fascinating that these two approaches give similar results. This may indicate that the errors are due to the two-moment matching approximation rather than the odd extension. This intriguing possibility would certainly merit further investigation if more time were available.We believe that our mathematical particularly, but also our numerical results indicate there is cause for optimism that a stable numerical algorithm to reconstruct the joint density function from basket prices will be found in the future.

The CLV Framework - A Fresh Look at Efficient Pricing with Smile

It is a market practice to price exotic derivatives, like callable basket options, with the local volatility model (B. Dupire, 1994, E. Derman & I. Kani, 1994) which can, contrary to stochastic volatility frameworks, handle multi-dimensionality easily. On the other hand a well-known limitation of the nonparametric local volatility model is the necessity of a short-stepping simulation, which, in high dimensions, is computationally expensive. In this article we propose a new local volatility framework called the Collocating Local Volatility (CLV) model which allows for large Monte Carlo steps and therefore it is computationally efficient. The CLV model is by its construction guaranteed to be almost perfectly calibrated to implied volatility smiles/skews at a given set of expiries. Additionally, the framework allows to control forward volatilities without affecting the fit to plain vanillas. The model requires only a fraction of a second for complete calibration to simple vanilla products and, finally, it allows for calculation of the Greeks without re-simulating of the Monte Carlo paths. We will apply the CLV model for pricing FX barrier options and also show that under this new framework we can easily price Bermudan options on 100-dimensional baskets of correlated underlyings within just a few seconds. Illustrative examples will be also presented.

Dealing a Discrete Dividend with Basket Option Pricing Models

Dealing a Discrete Dividend with Basket Option Pricing Models

The Practice of Local Correlation: An Empirical Study of Multi-Currency Option Pricing

We introduce local correlation models for pricing FX basket options. We review the general methodology and analyze three main specifications used in the industry. Numerical examples are showcased, and prices obtained using local correlation compared with corresponding “rule-of-thumb” methods.

An asymptotic expansion for local-stochastic volatility with jump models

This paper develops an asymptotic expansion method for general stochastic differential equations with jumps and their functions. By applying the method, we derive an explicit approximation formula for pricing options on functions of multiple assets under local-stochastic volatility with jump models. Moreover, we present numerical examples for pricing basket options based on the parameters calibrated to the actual market data, which confirms the validity of our method in practice.

A parsimonious model for generating arbitrage-free scenario trees

Simulation models of economic, financial and business risk factors are widely used to assess risks and support decision-making. Extensive literature on scenario generation methods aims at describing some underlying stochastic processes with the least number of scenarios to overcome the ‘curse of dimensionality’. There is, however, an important requirement that is usually overlooked when one departs from the application domain of security pricing: the no-arbitrage condition. We formulate a moment matching model to generate multi-factor scenario trees for stochastic optimization satisfying no-arbitrage restrictions with a minimal number of scenarios and without any distributional assumptions. The resulting global optimization problem is quite general. However, it is non-convex and can grow significantly with the number of risk factors, and we develop convex lower bounding techniques for its solution exploiting the special structure of the problem. Applications to some standard problems from the literature show that this is a robust approach for tree generation. We use it to price a European basket option in complete and incomplete markets.

Stochastic covariance and dimension reduction in the pricing of basket options

This paper presents a tailor-made method for dimension reduction aimed at approximating the price of basket options in the context of stochastic volatility and stochastic correlation. The methodology is built on a modification to the Principal Component Stochastic Volatility (PCSV) model, a stochastic covariance model that accounts for most stylized facts in prices. The method to reduce dimension is first derived theoretically. Afterwards the results are applied to a multivariate lognormal context as a special case of the PCSV model. Finally empirical results for the application of the method to the general PCSV model are illustrated.

A General Closed Form Approximation Pricing Formula for Basket and Multi-Asset Spread Options

The aims of this paper are twofold. Firstly, we present an approximating formula for pricing basket and multi-asset spread options, which genuinely extends Caldana and Fusai’s (2013) two-asset spread options formula. Secondly, under the lognormal setting, we show that our formula becomes a Black and Scholes type formula, extending Bjerksund and Stensland’s (2011). Numerical experiments and comparison with Monte Carlo simulations and other methods available in the literature are discussed. The main contribution of this paper is to provide practitioners with a pricing formula, which can be used for pricing basket and multi-asset spread options, even under a non-Gaussian framework.

Pricing Basket Options by Polynomial Approximations

We propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model. The method is based on Taylor and Chebyshev expansions and involves mixed exponential-power moments of a Gaussian distribution. Our numerical results show that both approaches are comparable in accuracy to a standard Monte Carlo method, with a lesser computational effort.

Pricing American Options Under High-Dimensional Models with Recursive Adaptive Sparse Expectations

We introduce a novel numerical framework for pricing American options in high dimensions. Such settings naturally arise for derivatives with multiple underlying assets, like basket options. They are equally important for single-asset options because high-dimensional models are best capable of capturing observed price dynamics. Yet, higher-dimensional settings come at the cost of a loss of tractability due to the accompanying exponential growth of computational complexity. Our scheme manages to alleviate the problem of dimension scaling through the use of adaptive sparse grids. We approximate the value function with a low number of points and recursively apply fast approximations of the expectation operator from an exercise period to the previous one. The algorithm copes with discretely spaced, possibly nonuniform, time grids. This makes it particularly fast for options with a limited number of exercise periods, like Bermudan options, and options for which the optimal exercise schedule is known ex ante. As compared to Monte Carlo simulations, our scheme processes an entire cross section of options in a single execution. It thereby offers an immediate solution to the estimation of hedging coefficients through finite differences and is ideal when multiple related options need to be analyzed. The algorithm is also capable of dealing with discrete dividends with no performance deterioration, thus improving on the documented inefficiency of exercise policies under continuous dividend yield approximations. We benchmark our algorithm under both the canonical model of Black and Scholes and the stochastic volatility model of Heston in the presence of discrete dividends. We illustrate the massive improvement of complexity scaling over dense grids with a basket option study including up to eight underlying assets. We show how the high degree of parallelism of our scheme makes it suitable for deployment on massively parallel computing units to scale to higher dimensions or further speed up the solution process.

Computing lower bounds on basket option prices by discretizing semi-infinite linear programming

The problem of finding static-arbitrage bounds on basket option prices has received a growing attention in the literature. In this paper, we focus on the lower bound case and propose a novel efficient solution procedure that is based on the separation problem. The computational burden of the proposed method is polynomial in the input data size. We also discuss the case of possibly negative weight vectors which can be applied to spread options.

General closed-form basket option pricing bounds

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate.