Showing papers in "Journal of Computational Finance in 2020"

Neural networks for option pricing and hedging: a literature review

Abstract: Neural networks have been used as a nonparametric method for option pricing and hedging since the early 1990s. Far over a hundred papers have been published on this topic. This note intends to provide a comprehensive review. Papers are compared in terms of input features, output variables, benchmark models, performance measures, data partition methods, and underlying assets. Furthermore, related work and regularisation techniques are discussed.

Pricing path-dependent Bermudan options using Wiener chaos expansion: an embarrassingly parallel approach

Abstract: In this work, we propose a new policy iteration algorithm for pricing Bermudan options when the payoff process cannot be written as a function of a lifted Markov process. Our approach is based on a modification of the well-known Longstaff Schwartz algorithm, in which we basically replace the standard least square regression by a Wiener chaos expansion. Not only does it allow us to deal with a non Markovian setting, but it also breaks the bottleneck induced by the least square regression as the coefficients of the chaos expansion are given by scalar products on the L^2 space and can therefore be approximated by independent Monte Carlo computations. This key feature enables us to provide an embarrassingly parallel algorithm.

On extensions of the Barone-Adesi and Whaley method to price American-type options

Abstract: This paper provides an efficient and accurate hybrid method to price American standard options in certain jump-diffusion models and American barrier-type options under the Black-Scholes framework. Our method generalizes the quadratic approximation scheme of Barone-Adesi and Whaley and several of its extensions. Using perturbative arguments, we decompose the early exercise pricing problem into subproblems of different orders and solve these subproblems successively. The solutions obtained are combined to recover approximations to the original pricing problem of multiple orders, with the zeroth-order version matching the general Barone-Adesi-Whaley ansatz. We test the accuracy and efficiency of the approximations via numerical simulations. The results show a clear dominance of higher-order approximations over their respective zeroth-order versions and reveal that significantly more pricing accuracy can be obtained by relying on approximations of the first few orders. In addition, they suggest that increasing the order of any approximation by one generally refines the pricing precision; however, this happens at the expense of greater computational costs.

Monte Carlo pathwise sensitivities for barrier options

Abstract: The Monte Carlo pathwise sensitivities approach is well established for smooth payoff functions. In this work, we present a new Monte Carlo algorithm that is able to calculate the pathwise sensitivities for discontinuous payoff functions. Our main tool is to combine the one-step survival idea of Glasserman and Staum with the stable differentiation approach of Alm, Harrach, Harrach and Keller. As an application we use the derived results for a two-dimensional calibration of a CoCo-Bond, which we model with different types of discretely monitored barrier options.

High-order approximations to call option prices in the Heston model

Abstract: In the present paper, a decomposition formula for the call price due to Alòs is transformed into a Taylor-type formula containing an infinite series with stochastic terms. The new decomposition may be considered as an alternative to the decomposition of the call price found in a recent paper by Als, Gatheral and Rodoičić. We use the new decomposition to obtain various approximations to the call price in the Heston model with sharper estimates of the error term than in previously known approximations. One of the formulas obtained in the present paper has five significant terms and an error estimate of the form O(v³(|P| + v)), where v and P are the volatility-of-volatility and the correlation in the Heston model, respectively. Another approximation formula contains seven more terms and the error estimate is of the form O(v⁴(1 + |P|v)). For the uncorrelated Heston model (P = 0), we obtain a formula with four significant terms and an error estimate O(v⁶) Numerical experiments show that the new approximations to the call price perform especially well in the high-volatility mode.

Gaussian process regression for derivative portfolio modeling and application to credit valuation adjustment computations

Abstract: Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all trades between each counterparty under both market and credit risk. We present a multi-Gaussian process regression approach, which is well suited for the over-the-counter derivative portfolio valuation involved in credit valuation adjustment (CVA) computation. Our approach avoids nested simulation or simulation and regression of cashflows by learning a Gaussian metamodel for the mark-to-market cube of a derivative portfolio. We model the joint posterior of the derivatives as a Gaussian process over function space, imposing the spatial covariance structure on the risk factors. Monte Carlo simulation is then used to simulate the dynamics of the risk factors. The uncertainty in portfolio valuation arising from the Gaussian process approximation is quantified numerically. Numerical experiments demonstrate the accuracy and convergence properties of our approach for CVA computations, including a counterparty portfolio of interest rate swaps.

Option pricing in exponential Lévy models with transaction costs

Abstract: We present an approach for pricing European call options in the presence of proportional transaction costs, when the stock price follows a general exponential Levy process. The model is a generalization of the celebrated 1993 work of Davis, Panas and Zariphopoulou, in which the value of the option is defined as the utility indifference price. This approach requires the solution of two stochastic singular control problems in a finite horizon that satisfy the same Hamilton–Jacobi–Bellman equation with different terminal conditions. We introduce a general formulation for these portfolio selection problems, and then focus on a special case in which the probability of default is ignored. We solve the optimization problems numerically using the Markov chain approximation method and show results for diffusion, Merton and variance Gamma processes. Option prices are computed for both the writer and the buyer.

Numerical simulation and applications of the convection–diffusion–reaction equation with the radial basis function in a finite-difference mode

Abstract: This paper develops two local mesh-free methods for designing stencil weights and spatial discretization, respectively, for parabolic partial differential equations (PDEs) of convection–diffusion–reaction type. These are known as the radial-basis-function generated finite-difference method and the Hermite finite-difference method. The convergence and stability of these schemes are investigated numerically using some examples in two and three dimensions with regularly and irregularly shaped domains. Then we consider the numerical pricing of European and American options under the Heston stochastic volatility model. The European option leads to the solution of a two-dimensional parabolic PDE, and the price of the American option is given by a linear complementarity problem with a two-dimensional parabolic PDE of convection–diffusion–reaction type. Then we use the operator splitting method to perform time-stepping after space discretization. The resulting linear systems of equations are well conditioned and sparse, and by numerical experiments we show that our numerical technique is fast and stable with respect to the change in the shape parameter of the radial basis function. Finally, numerical results are provided to illustrate the quality of approximation and to show how well our approach converges with the results presented in the literature.

Second-order Monte Carlo sensitivities in linear or constant time

Abstract: We consider the problem of efficiently computing the full matrix of second-order sensitivities of a Monte Carlo price when the number of inputs is large. Specifically, we analyze and compare methods with run times of at most O(NT), where N is the dimension of the input and T is the time required to compute the price. Since none of the alternatives from previous literature appears to be satisfactory in all settings, we propose two original methods: the first method is based on differentiation in a distributional sense, while the second method leverages a functional relation between first- and second-order derivatives. The former shows excellent generality and computational times to achieve a given target accuracy. The latter is by far the most effective in at least one relevant example and has theoretical interest in that it is the first practical estimator of the full Hessian whose complexity, as a multiple of that of the only-price implementation, does not grow with the dimension of the problem.

A shrinking horizon optimal liquidation framework with lower partial moments criteria

Abstract: A novel quasi-multi-period model for optimal position liquidation in the presence of both temporary and permanent market impact is proposed. Two main features distinguish the proposed approach from alternatives. First, a shrinking horizon framework is implemented to update intraday parameters by incorporating new incoming information while maintaining standard non-anticipativity constraints. The method is data-driven, numerically tractable, and reactive to the market. Second, lower partial moments, a downside risk measure, is used which, unlike symmetric measures such as variance, captures traders’ increased risk aversion to losses. The performance of the proposed strategies is tested using historical, high-frequency New York Stock Exchange (NYSE) data. All proposed strategies outperform classic strategies such as a Time Weighted Average Price (TWAP) strategy as well as more unrealistic strategies such as an Ex-post Volume Weighted Average Price (VWAP) strategy that violates non-anticipativity on days with unfavorable market conditions, thus, strongly supporting the use of lower partial moments as a risk measure. Additionally, results validate the use of a shrinking horizon framework as an adaptive, tractable alternative to dynamic programming for trading.

Extremal risk management: expected shortfall value verification using the bootstrap method

Abstract: In this paper, we refer to the axiomatic theory of risk and investigate the problem of formal verification of the expected shortfall (ES) model based on a sample ES. Recognizing the infeasibility of parametric methods, we explore the bootstrap technique, which, unlike the current value-at-risk model-based (VaR model-based) Basel III testing framework, permits the creation of more powerful sample ES-based procedures. Our contribution to the debate on the possibilities of sample ES-based testing is twofold. First, we introduce a bootstrap test based on the idea of ES prediction corrected variables. In this way, we obtain a procedure that makes no distributional assumptions about the underlying returns process, and whose p-value computation does not assume any asymptotic convergence. Second, we provide a unifying framework for ES value verification, in which we compare alternative sample ES-based approaches: the residual-based procedures versus the ES prediction corrected tests as well as the VaR model-dependent approach versus the fixed failure rate tests. By examining its statistical properties and practical applicability, we find evidence that the proposed bootstrap procedure, based on ES prediction corrected variables, is superior to other methods. This provides important guidance for developing international standards of market risk management.

Finding the nearest covariance matrix: the foreign exchange market case

Abstract: We consider the problem of finding a valid covariance matrix in the foreign exchange market given an initial nonpositively semidefinite (non-PSD) estimate of such a matrix. The common no-arbitrage assumption imposes additional linear constraints on such matrixes, inevitably making them singular. As a result, even the most advanced numerical techniques will predictably balk at a seemingly standard optimization task. The reason is that the problem is ill posed, while its PSD solution is not strictly feasible. In order to deal with this issue we describe a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which is both well posed and of a smaller scale. We show that, after solving the reduced problem, the solution to the initial problem can be recovered uniquely in one step. We run numerous numerical experiments to compare the performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that its solution can effectively be found by the application of virtually any numerical method.

An adaptive Monte Carlo approach for pricing Parisian options with general boundaries

Abstract: We propose a new, flexible framework using Monte Carlo methods to price Parisian options not only with constant boundaries but also with general curved boundaries. The proposed approach also enables a direct simulation of the Parisian time, namely the first time when a Parisian contract is triggered. Further, we employ an adaptive control variable method to improve the accuracy of the Monte Carlo simulation. Finally, we present numerical examples for the flat and curved barriers and show that our method produces better simulation results than the alternative procedures considered in this paper.

Pricing multiple barrier derivatives under stochastic volatility

Abstract: This work generalizes existing one- and two-dimensional pricing formulas with an equal number of barriers to a setting of n dimensions and up to two barriers in the presence of stochastic volatility This allows for the consideration of multidimensional single-barrier derivatives with, for example, a collateral triggered by a barrier default of the issuing company We introduce stochastic volatility to a multidimensional Black–Scholes framework via the common Cox–Ingersoll–Ross process and present semianalytical solutions for collateralized structured products with two barriers, one representing default and the other a market-related option Our model accommodates implied volatility skew along the lines of displaced diffusions We show that our semianalytical formulas are more efficient in terms of computational speed than Monte Carlo simulations, particularly for tail scenarios Moreover, our proposed analytical simplifications permit a twenty-fivefold gain in time savings compared with the results given by the main theorem These multidimensional structured products gained increasing popularity after the subprime and financial crises Therefore, we perform comprehensive sensitivity analyses with respect to stochastic volatility parameters and contribute to a better understanding of multidimensional barrier derivatives in a stochastic volatility framework

Introducing two mixing fractions to a lognormal local-stochastic volatility model

Abstract: A single parameter, termed the mixing fraction, is used to calibrate current local stochastic volatility (LSV) models to traded exotic prices as well as vanilla options. This single parameter has been multiplied by both the volatility-of-volatility parameter and the correlation between spot and volatility of the original stochastic volatility model. In this paper, we introduce two mixing fractions that can be controlled separately to apply impact to the volatility-of-volatility and the correlation in a lognormal LSV model. For the case study using USD/JPY market data, we observe significant improvement in calibration accuracy to one-touch exotic option prices with the introduction of a second mixing fraction, without losing accuracy in the replication of European options. Importantly, when the LSV model with enhanced calibration accuracy is used to price other one-touch options, the discrepancy with market prices is not improved noticeably for the out-of-sample exotics, which indicates that either the market prices of traded one-touch options are not self-consistent or the LSV model does not capture the underlying dynamics exactly.

A Libor market model including credit risk under the real-world measure

Abstract: We present a methodology to generate future scenarios of interest rates for different credit ratings under a real-world probability measure. More precisely, we explain how to perform simulations of the real-world forward rates for different rating classes by generalizing the multidimensional shifted lognormal London Interbank Offered Rate market model to account for credit ratings and a specification of the market prices of risk vector processes. The proposed methodology allows for the presence of negative interest rates, as currently observed in the markets, and guarantees the monotonicity of forward rates with respect to credit ratings.