Basket Option Pricing Research Articles

New control variates for pricing basket options

Abstract Accepted by: Giorgio Consigli Accurate pricing of basket options, which are financial derivatives on multiple underlying assets, is a challenging and practically important task for financial institutions. We propose several new control variates for accurate, fast and efficient pricing of basket options. The first approach to deriving new control variates is the use of Hermite polynomial approximation of appropriate function of the underlying asset prices, which leads to a Black–Scholes-like analytic solution. This approach is new in the option pricing context and opens up new possibilities in derivative pricing. Further control variates are analytically derived using Jensen’s inequality in one case, and distributional properties of multivariate Wiener processes in other cases. All the newly proposed control variates are shown to lead to excellent variance reduction in numerical experiments based on realistic data. The proposed methods are novel, computationally simple and have a strong potential to replace more conventional methods, such as the geometric lower bound in simulation-based pricing of basket options and similar products used in financial risk management.

Pricing of geometric basket Options with Default risk under Stochastic Volatility and Jump diffusion Model

Pricing of geometric basket Options with Default risk under Stochastic Volatility and Jump diffusion Model

Option Pricing and Portfolio Optimization under a Multi-Asset Jump-Diffusion Model with Systemic Risk

We explore a multi-asset jump-diffusion pricing model, combining a systemic risk asset with several conditionally independent ordinary assets. Our approach allows for analyzing and modeling a portfolio that integrates high-activity security, such as an exchange trading fund (ETF) tracking a major market index (e.g., S&P500), along with several low-activity securities infrequently traded on financial markets. The model retains tractability even as the number of securities increases. The proposed framework allows for constructing models with common and asset-specific jumps with normally or exponentially distributed sizes. One of the main features of the model is the possibility of estimating parameters for each asset price process individually. We present the conditional maximum likelihood estimation (MLE) method for fitting asset price processes to empirical data. For the case with common jumps only, we derive a closed-form solution to the conditional MLE method for ordinary assets that works even if the data are incomplete and asynchronous. Alternatively, to find risk-neutral parameters, the least-square method calibrates the model to option values. The number of parameters grows linearly in the number of assets compared to the quadratic growth through the correlation matrix, which is typical for many other multi-asset models. We delve into the properties of the proposed model, its parameter estimation using the MLE method and least-squares technique, the evaluation of VaR and CVaR metrics, the identification of optimal portfolios, and the pricing of European-style basket options. We propose a Laplace-transform-based approach to computing Value at Risk (VaR) and conditional VaR (also known as the expected shortfall) of portfolio returns. Additionally, European-style basket options written on the extreme and average stock prices or returns can be evaluated semi-analytically. For numerical demonstration, we examine a combination of the SPDR S&P 500 ETF (as a systemic risk asset) with eight ordinary assets representing diverse industries. Using historical assets and options prices, we estimate the real-world and risk-neutral parameters of the model with common jumps, construct several optimal portfolios, and evaluate various basket options with the eight assets. The results affirm the robustness and efficiency of the estimation and evaluation methodologies. Computational results are compared with Monte Carlo estimates.

Efficient pricing and calibration of high-dimensional basket options

This paper studies equity basket options – i.e. multi-dimensional derivatives whose payoffs depend on the value of a weighted sum of the underlying stocks – and develops a new and innovative approach to ensure consistency between options on individual stocks and the index comprising them. Specifically, we show how to resolve a well-known problem that when individual constituent distributions of an equity index are inferred from the single-stock option markets and combined in a multi-dimensional local/stochastic volatility model, the resulting basket option prices will not generate a skew matching that of the options on the equity index corresponding to the basket.

Multivariate Hawkes-based Models in Limit Order Book: European, Spread and Basket Option Pricing

Multivariate Hawkes-based Models in Limit Order Book: European, Spread and Basket Option Pricing

An augmented Lagrangian method with constraint generation for shape-constrained convex regression problems

Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a unified framework for computing the least squares estimator of a multivariate shape-constrained convex regression function in $${\mathbb {R}}^d$$ . We prove that the least squares estimator is computable via solving an essentially constrained convex quadratic programming (QP) problem with $$(d+1)n$$ variables, $$n(n-1)$$ linear inequality constraints and n possibly non-polyhedral inequality constraints, where n is the number of data points. To efficiently solve the generally very large-scale convex QP, we design a proximal augmented Lagrangian method (proxALM) whose subproblems are solved by the semismooth Newton method. To further accelerate the computation when n is huge, we design a practical implementation of the constraint generation method such that each reduced problem is efficiently solved by our proposed proxALM. Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that our proposed proxALM outperforms the state-of-the-art algorithms, and the proposed acceleration technique further shortens the computation time by a large margin.

The closed-form approximation to price basket options under stochastic interest rate

The paper presents closed-form approximation formulas for pricing basket options. We assume that the underlying asset prices follow geometric Brownian motions and the interest rate follows the one-factor Hull-White model. Under a given forward measure, we obtain the analytical lower and upper bounds of basket options. By finding a simple random variable to replace the sum of the lognormal random variables, we combine the conditioning and the moment matching approaches, and derive approximation formulas to price basket options. Numerical results illustrate that our results fall in the sharp lower and upper bounds of basket options and are consistent with Monte Carlo simulation results.

The Laplace transform of the integrated Volterra Wishart process

AbstractWe establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore, we link the characteristic exponents to a system of non‐standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear‐quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

American Basket Option Pricing Formulas in Uncertain Environment

American basket option is a contract containing multiple underlying assets, and its payoff is correlated with average prices or weighted average prices of these assets on or before the expiration date. The type of option entitles a holder the right to trade at the strike price within a specified date, and this right can be waived. Therefore, there is a certain price to be paid for acquiring this right, which produces the problem of option pricing. A lot of literature shows that basket option price is usually cheaper than option portfolios on individual underlying assets. Based on this advantage, basket option becomes popular among investors. Consequently, this paper predominantly explores four types of American basket option pricing in uncertain financial environment. Specifically they are American arithmetic basket call option, American arithmetic basket put option, American geometric basket call option and American geometric basket put option. Assuming that these stocks prices follow corresponding uncertain differential equations, we derive corresponding option pricing formulas. Some numerical examples are taken to illustrate the feasibility of pricing formulas. Simultaneously, this paper discusses the relationship between option price and some parameters.

Pricing of Basket Options by Conditioning and Moment Matching

The price of basket options can be represented as an exact analytical part and an approximate part by using a conditional variable. The first part is calculated by conditioning on the price process of the underlying asset and the second part is calculated by moment matching approach. In order to calculate the second part, we find a new single random variable, which has an analytically known distribution, to approximate the sum of log-normal random variables and to obtain a closed form pricing formula. Our method can be viewed as a combination of conditioning and moment matching methods. Numerical studies demonstrated that our formula is more accurate in handling both homogeneous and heterogeneous lognormal random variable cases. TOPICS:Derivatives, options Key Findings • The price of basket options can be represented as an exact analytical part and an approximate part by using a conditional variable. In order to calculate the approximate part, we find a new single random variable, which has an analytically known distribution, to approximate the sum of log-normal random variables and to obtain a closed form pricing formula. • Numerical studies demonstrate that our formula is more accurate in handling both homogeneous and heterogeneous lognormal random variable cases.

American-type basket option pricing: a simple two-dimensional partial differential equation

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.

A General Accurate Approximation for Pricing and Hedging Basket Options with Exact Moment Matching

This article provides a new methodology for pricing and hedging basket options. The authors approximate the basket by using the shifted log-normal distribution with the polynomial expansion, which can match exactly any required m moments of the basket, to give quasi-analytical formulas for the prices and hedging parameters of basket options. Numerical simulations show that the methodology provides superior results for basket option prices and hedging parameters. This methodology works well not only for regular baskets but also for negative-weight baskets and negative-value baskets. Compared with the best available methods, the authors’ methodology appears to perform better.

Spread and basket option pricing in a Markov‐modulated Lévy framework with synchronous jumps

AbstractThis paper considers the evaluation of spread and basket options when the underlying asset prices are driven by Markov‐modulated Lévy processes with synchronous jumps. In particular, the asset prices may jump whenever there is a change of phase of the underlying Markov process. We further allow for dependence between the different price dynamics. In this general regime‐switching framework, we provide lower and upper bounds to the exact option prices based upon ideas from the literature without regime switching. These bounds are obtained via univariate Fourier inversion under the assumption that the joint characteristic functions of the Markov‐modulated Lévy processes are known. We study these obtained spread and basket option price approximations in different regime‐switching models. Several numerical experiments are included and these show that, especially, the lower bounds have a very high precision.

Developing a Risk-Based Approach for American Basket Option Pricing

Options are one of the important financial contracts for reducing the risk of investors. Many active practitioners in the financial markets really believe that mispricing or incorrect valuation of these securities would be the main reason of collapse of some financial institutions. The complexity of option pricing/valuation, especially in the case of American basket options, as high dimensional options, has motivated many researchers to develop numerical and simulation-based models. In this paper, a new simulation-based approach for pricing/valuation of American basket option with risk consideration is proposed. Having the prices obtained through Longstaff–Schwartz methodology, which is based on Approximate Dynamic Programming as a risk-neutral approach, we propose a new approach for pricing the American basket option according to the worst-case (pessimistic/risk-averse) and the best-case (optimistic/risk-taking) scenarios. Furthermore, for scenarios generation, we use a Monte Carlo simulation technique using a t-student copula-GARCH method and Extreme Value Theory to handle the nonlinearity of dependencies between variables. To verify the computational efficiency and the accuracy of the proposed methodology, we compare the results of prices obtained through the proposed models with those achieved through the Monte Carlo simulation and the method developed by Ju for European basket options. Moreover, the developed models are tested using out-sample scenarios to explore what would happen if investors bought the option with the obtained prices through three different strategies: risk-averse, risk-neutral, and risk-taking approaches.

A hybrid Monte Carlo acceleration method of pricing basket options based on splitting

Pricing basket options has always been one of the key problems in financial engineering because of high dimensionality and low convergence rate. This paper proposes a hybrid Monte Carlo variance reduction method for pricing basket options. First, by splitting the payoff of the basket option into two parts, we can price basket options by value the two parts respectively. The first part has a closed-form expectation formula, the second part can be considered as a small probability event. To reduce variance for simulating the second part, the conditional Monte Carlo (CMC) method combined with the importance sampling(IS) method is adapted. Because these two methods are all effective to deal with small probability events. For IS method, it is a challenge to compute the optimal parameters with as little computational cost as possible. Therefore, an efficient prediction–correction(PC) iteration algorithm based on moment estimation is proposed to determine the optimal parameters in the importance sampling method. Some theoretical analyses for the existence and uniqueness of the optimal parameters in the IS method and the convergence of the PC method are also given. Numerical results show that the hybrid variance reduction method has great variance reduction effect and PC iteration algorithm can save a lot of computing costs comparing with the traditional Newton’s iteration method.

A Simple Closed-Form Formula for Pricing Basket Options

Options involving more than one underlying, often called <i>basket options</i>, are quite common, but valuation presents a mathematical problem. Although returns for a single stock can be assumed to follow a lognormal diffusion, leading to a variety of possible standard pricing models, sums, averages, or differences among lognormals are not lognormal. A number of approaches to approximating the returns distribution for an underlying basket have been proposed. Among the more successful is Levy’s technique of fitting an approximating lognormal for the basket by calculating and matching the first two moments of the basket’s return distribution. A problem arises, however, when the component assets have heterogeneous volatilities. Kan proposes a new technique that breaks the valuation formula into several pieces, each of which can be well approximated by moment matching. The accuracy of the Kan model over a wide range of parameter values is shown to be better than that of alternative models, and computation speed is nearly as good as the Levy model, which is much less accurate in general. <b>TOPICS:</b>Options, statistical methods

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black–Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 35.

Pricing of basket options in subdiffusive fractional Black–Scholes model

In this paper we generalize the classical multidimensional Black-Scholes model to the subdiffusive case. In the studied model the prices of the underlying assets follow subdiffusive multidimensional geometric Brownian motion. We derive the corresponding fractional Fokker–Plank equation, which describes the probability density function of the asset price. We show that the considered market is arbitrage-free and incomplete. Using the criterion of minimal relative entropy we choose the optimal martingale measure which extends the martingale measure from used in the standard Black–Scholes model. Finally, we derive the subdiffusive Black–Scholes formula for the fair price of basket options and use the approximation methods to compare the classical and subdiffusive prices.

A simple efficient approximation to price basket stock options with volatility smile

This paper develops a new approach to obtain the price and risk sensitivities of basket options which have a volatility smile. Using this approach, the Black–Scholes model and the Stochastic Volatility Inspired model have been used to obtain an approximate analytical pricing formula for basket options with a volatility smile. It is found that our approximate formula is quite accurate by comparing it with Monte Carlo simulations. It is also proved the option value of our approach is consistent with the option value generated by Levy’s and Gentle’s approaches for typical ranges of volatility. Further, we give a theoretical proof that the option values from Levy’s and Gentle’s works are the upper bound and the lower bound, respectively, for our option value. The calibration procedure and a practical example are provided. The main advantage of our approach is that it provides accurate and easily implemented basket option prices with volatility smile and hedge parameters and avoids the need to use time-consuming numerical procedures such as Monte Carlo simulation.

The pricing of basket options: A weak convergence approach

We consider a limit price of basket options in a large portfolio where the dynamics of basket assets is described as a CEV jump diffusion system. The explicit representation of the limiting price is established using weak convergence of empirical measure valued processes generated by the system. As an application, the closed-form formula of the limit price is derived when the price dynamics of basket assets follows a mixed-double exponential jump-diffusion system.