European Basket Options Research Articles

BOUNDS ON MULTI-ASSET DERIVATIVES VIA NEURAL NETWORKS

Using neural networks, we compute bounds on the prices of multi-asset derivatives given information on prices of related payoffs. As a main example, we focus on European basket options and include information on the prices of other similar options, such as spread options and/or basket options on subindices. We show that, in most cases, adding further constraints gives rise to bounds that are considerably tighter. Our approach follows the literature on constrained optimal transport and, in particular, builds on the work of Eckstein & Kupper (2018) [Computation of optimal transport and related hedging problems via penalization and neural networks, Appl. Math. Optimiz. 1–29].

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.

Optimal Hedging of Basket Barrier Options with Additive Models and Its Application to Equity Value Separation Problem

At the heart of optimal hedging with additive models in Yamada (Recent advances in financial engineering: proceedings of the KIER-TMU international workshop on financial engineering, World Scientific, pp 225–245, 2010; Proceedings of the 2011 American control conference, pp 3856–3861, 2011; Asia-Pac Financ Mark 19(2):149–179, 2012) is to replicate the payoff of European basket options using separate options as close as possible. In this paper, we extend their technique for the case of path-dependent barrier options, where the mean square error of the payoffs between the basket barrier option and the sum of options on the individual assets is minimized over any smooth payoff functions. To this end, we propose to represent the underlying assets using the Brownian bride decomposition and show that computations involving conditional expectations of basket barrier options boil down to those of unconditional expectations. This procedure enables us to provide an algorithm to compute the necessary and sufficient condition for the optimal hedging problem based on the Monte Carlo method. Then, we consider to apply our methodology to the Black–Cox type first passage time structural model, where a defaultable company possesses/runs multiple assets/projects and the default may occur the first time the asset value hits a certain lower threshold before the maturity. We formulate the equity value separation problem using additive models, in which individual equity values are introduced so that their sum approximates the total equity value as close as possible. It is also shown that any portion of total equity value may be assigned as an initial value of each individual equity when using the optimal smooth functions. Finally, we examine the contributions of individual equity values to default or survival by applying a certain normalization for conditional expectations via numerical experiments to illustrate our proposed methodology.

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 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.

Lower Bound Approximation to Basket Option Values for Local Volatility Jump-Diffusion Models

In this paper we derive an easily computed approximation of Rogers and Shi’s lower bound for a local volatility jump-diffusion model and then use it to approximate European basket option values. If the local volatility function is time independent then there is a closed-form expression for the approximation. Numerical tests show that the lower bound approximation is fast and accurate in comparison with the Monte Carlo method, the partial-exact approximation method and the asymptotic expansion method.

Generating Multi-Asset Arbitrage-Free Scenario Trees with Global Optimization

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 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 assets, and scenarios 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 also use it to price a European basket option in complete and incomplete markets.

European basket option pricing by maximizing over a subset of lower bounds

In this paper we derive a lower bound for the price of a European basket option under the Black and Scholes framework. The bound is based on the conditioning method of [L.C.G. Rogers and Z. Shi, The value of an Asian option, J. of Appl. Prob. 32 (1995), 1077–1088] and [M. Curran, Valuing Asian and portfolio options by conditioning on the geometric mean, Management Science 40(12) (1994), 1705–1711]. We adapt the method of [K.W. Chen and Y.D. Lyuu, Accurate pricing formulas for Asian options, Applied Mathematics and Computation 188 (2007), 1711–1724] applied for Asian options to the case of the European basket options. Using an improved Monte Carlo method based on the antithetic variates, we benchmark our results. We also calculate the lower bound for various parameters involved, compare these results with those obtained by the improved Monte Carlo simulations and conclude that the proposed bound is indeed optimal.

Computing arbitrage upper bounds on basket options in the presence of bid–ask spreads

We study the problem of computing the sharpest static-arbitrage upper bound on the price of a European basket option, given the bid–ask prices of vanilla call options in the underlying securities. We show that this semi-infinite problem can be recast as a linear program whose size is linear in the input data size. These developments advance previous related results, and enhance the practical value of static-arbitrage bounds as a pricing technique by taking into account the presence of bid–ask spreads. We illustrate our results by computing upper bounds on the price of a DJX basket option. The MATLAB code used to compute these bounds is available online at www.andrew.cmu.edu/user/jfp/arbitragebounds.html.

The Effect of Correlation and Transaction Costs on the Pricing of Basket Options

In this article, we examine the problem of evaluating the option price of a European call option written on N underlying assets when there are proportional transaction costs in the market. Since the portfolio under consideration consists of multiple risky assets, which makes numerical methods formidable, we use perturbation analyses. The article extends the model for option pricing on uncorrelated assets, which was proposed by Atkinson and Alexandropoulos (2006; Pricing a European basket option in the presence of proportional transaction cost, Applied Mathematical Finance, 13(3), pp. 191–214). We determine optimal hedging strategies as well as option prices on both correlated and uncorrelated assets. The option valuation problem is obtained by comparing the maximized utility of wealth with and without option liability. The two stochastic control problems, which arise from the transaction costs, are transformed to free boundary and partial differential equation problems. Once the problems have been formulated, we establish optimal trading strategies for each of the portfolios. In addition, the optimal hedging strategies can be found by comparing the trading strategies of the two portfolios. We provide a general procedure for solving N risky assets, which shows that for ‘small’ correlations the N asset problem can be replaced by N (N − 1)/2 two-dimensional problems and give numerical examples for the two risky assets portfolios.

Static-arbitrage lower bounds on the prices of basket options via linear programming

We show that the problem of computing sharp upper and lower static-arbitrage bounds on the price of a European basket option, given the prices of other similar options, can be cast as a linear program (LP). The LP formulations readily yield super-replicating (subreplicating) strategies for the upper (lower) bound problem. The dual counterparts of the LP formulations in turn yield underlying asset price distributions that replicate the given option prices, and the bound on the new basket option’s price. In the special case when the given option prices are those of vanilla options on the underlying assets, we show that the LP formulations admit further simplifications. In particular, for the upper bound problem we derive closed-form formulas for the basket’s price bound, and for the corresponding superreplicating strategy. In addition, our LP approach admits efficient modeling of additional features such as basket options with negative weights, bid/ask spreads, transaction costs, and diversification constraints. We provide numerical experiments to illustrate some of our results.

T-Copula generation for control variates

The standard method for generating multi- t vectors is simple and convenient but it has the disadvantage that the generated multi-normal and multi- t vectors are not similar. For t-copula models this destroys much of the variance reduction when using the result of the multi-normal model as external control variate. Therefore we develop a new generation method for multi- t vectors. It is based on the polar method and numerical inversion, and generates multi-normal and multi- t vectors that are very similar. Numerical experiments with simple functions of the weighted sum of t-copula vectors and with pricing European basket options with a t-copula model confirm that the obtained variance reduction factors of the new method are high; 2–100 times higher than when using the standard generation method.

A Parallel Implementation on GPUs of ADI Finite Difference Methods for Parabolic PDEs with Applications in Finance

We study a parallel implementation on a Graphics Processing Unit (GPU) of Alternating Direction Implicit (ADI) time-discretization methods for solving time-dependent parabolic Partial Differential Equations (PDEs) in three spatial dimensions with mixed spatial derivatives in a variety of applications in computational finance. Finite differences on uniform grids are used for the spatial discretization of the PDEs. As examples, we apply the GPU-based parallel methods to price European rainbow and European basket options, each written on three assets. Numerical results showing the efficiency of the parallel methods are provided.

Pricing a European Basket Option in the Presence of Proportional Transaction Costs

A crucial assumption in the Black–Scholes theory of options pricing is the no transaction costs assumption. However, following such a strategy in the presence of transaction costs would lead to immediate ruin. This paper presents a stochastic control approach to the pricing and hedging of a European basket option, dependent on primitive assets whose prices are modelled as lognormal diffusions, in the presence of costs proportional to the size of the transaction. Under certain assumptions on the individual preferences, it is able to reduce the dimensionality of the resulting control problem. This facilitates considerably the study of the value function and the characterisation of the optimal trading policy. For solution of the problem a perturbation analysis scheme is utilized to derive a non‐trivial, asymptotically optimal result. The findings reveal that this result can be expressed by means of a small correction to the corresponding solution of the frictionless Black–Scholes type problem, resembling a multi‐dimensional ‘bandwidth’ around the vanilla case, which, moreover, is readily tractable.