Early-exercise Options Research Articles

Functional central limit theorems for rough volatility

The non-Markovian nature of rough volatility makes Monte Carlo methods challenging, and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker’s approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent H∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H \\in (0,1)$\\end{document}. Some of the most relevant consequences of this ‘rough Donsker (rDonsker) theorem’ are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme and find remarkable agreement (for a large range of values of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}). Our rDonsker theorem further provides a weak convergence proof for the hybrid scheme itself and allows constructing binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.

A simple and efficient numerical method for pricing discretely monitored early-exercise options

We present a simple, fast, and accurate method for pricing a variety of discretely monitored options in the Black-Scholes framework, including autocallable structured products, single and double barrier options, and Bermudan options. The method is based on a quadrature technique, and it employs only elementary calculations and a fixed one-dimensional uniform grid. The convergence rate is O(1/N4) and the complexity is O(MNlogN), where N is the number of grid points and M is the number of observation dates. Besides Black-Scholes, our method is also applicable to more general frameworks such as Merton’s jump diffusion model.

Solving high-dimensional optimal stopping problems using deep learning

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

A general continuous time Markov chain approximation for multi-asset option pricing with systems of correlated diffusions

Continuous time Markov Chain (CTMC) approximation techniques have received increasing attention in the option pricing literature, due to their ability to solve complex pricing problems, although existing approaches are mostly limited to one or two dimensions. This paper develops a general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes. This is accomplished with a general decorrelation procedure, which reduces the system of correlated diffusions to an uncorrelated system. This enables simple and efficient approximation of the driving processes by univariate CTMC approximations. Weak convergence of the approximation is demonstrated, with second order convergence in space. Numerical experiments demonstrate the accuracy and efficiency of the method for various European and early-exercise options in two and three dimensions.

A General Continuous Time Markov Chain Approximation for Multi-Asset Option Pricing With Systems of Correlated Diffusions

Continuous time Markov Chain (CTMC) approximation techniques have received increasing attention in the option pricing literature, due to their ability to solve complex pricing problems, although existing approaches are mostly limited to one or two dimensions. This paper develops a general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes. This is accomplished with a general de-correlation procedure, which reduces the system of correlated diffusions to an uncorrelated system. This enables simple and efficient approximation of the driving processes by uni-variate CTMC approximations. Weak convergence of the approximation is demonstrated, with second order convergence in space. Numerical experiments demonstrate the accuracy and efficiency of the method for various European and early-exercise options in two and three dimensions.

Pricing European-type, early-exercise and discrete barrier options using an algorithm for the convolution of Legendre series

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: http://www.chebfun.org/) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.

An SFP–FCC method for pricing and hedging early-exercise options under Lévy processes

This paper extends the singular Fourier–Padé (SFP) method proposed by Chan [Singular Fourier–Padé series expansion of European option prices. Quant. Finance, 2018, 18, 1149–1171] for pricing/hedging early-exercise options–Bermudan, American and discrete-monitored barrier options–under a Lévy process. The current SFP method is incorporated with the Filon–Clenshaw–Curtis (FCC) rules invented by Domínguez et al. [Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals. IMA J. Numer. Anal., 2011, 31, 1253–1280], and we call the new method SFP–FCC. The main purpose of using the SFP–FCC method is to require a small number of terms to yield fast error convergence and to formulate option pricing and option Greek curves rather than individual prices/Greek values. We also numerically show that the SFP–FCC method can retain a global spectral convergence rate in option pricing and hedging when the risk-free probability density function is piecewise smooth. Moreover, the computational complexity of the method is with N, a (small) number of complex Fourier series terms, , a number of Chebyshev series terms and L, the number of early-exercise/monitoring dates. Finally, we compare the accuracy and computational time of our method with those of existing techniques in numerical experiments.

Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise boundary for early-exercise options. Our approach is based on the Spitzer identities for general Lévy processes and on the Wiener–Hopf method. Our direct calculation of the price of α-quantile options combines for the first time the Dassios–Port–Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Lévy processes.

Pricing Methods for Alpha-Quantile and Perpetual Early Exercise Options Based on Spitzer Identities

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise barrier for early-exercise options. Our approach is based on the Spitzer identities for general Levy processes and on the Wiener-Hopf method. Our direct calculation of the price of alpha-quantile options combines for the first time the Dassios-Port-Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Levy processes.

Hedging and pricing early-exercise options with complex fourier series expansion

We introduce a new numerical method called the complex Fourier series (CFS) method proposed by Chan (2017) to price options with an early-exercise feature—American, Bermudan and discretely monitored barrier options—under exponential Lévy asset dynamics. This new method allows us to quickly and accurately compute the values of early-exercise options and their Greeks. We also provide an error analysis to demonstrate that, in many cases, we can achieve an exponential convergence rate in the pricing method as long as we choose the correct truncated computational interval. Our numerical analysis indicates that the CFS method is computationally more comparable or favourable than the methods currently available. Finally, the superiority of the CFS method is illustrated with real financial data by considering Standard & Poor’s depositary receipts (SPDR) exchange-traded fund (ETF) on the S&P 500® index options, which are American options traded from November 2017 to February 2018 and from 30 January 2019 to 21 June 2019.

Rolling Adjoints: Fast Greeks along Monte Carlo scenarios for early-exercise options

In this paper we extend the Stochastic Grid Bundling Method (SGBM), a regress-later Monte Carlo scheme for pricing early-exercise options, with an adjoint method to compute in a highly efficient manner the option sensitivities (the “Greeks”) along the Monte Carlo paths, with reasonable accuracy. The path-wise SGBM Greeks computation is based on the conventional path-wise sensitivity analysis, however, for a regress-later technique. The resulting sensitivities at the end of the monitoring period are implicitly rolled over into the sensitivities of the regression coefficients of the previous monitoring date. For this reason, we name the method Rolling Adjoints, which facilitates Smoking Adjoints [M. Giles, P. Glasserman, Smoking adjoints: fast Monte Carlo Greeks, Risk 19 (1) (2006) 88–92] to compute conditional sensitivities along the paths for options with early-exercise features.

Hedging and Pricing Early-Exercise Options With Complex Fourier Series Expansion

Hedging and Pricing Early-Exercise Options With Complex Fourier Series Expansion

Functional Central Limit Theorems for Rough Volatility

We extend Donsker’s approximation of Brownian motion to fractional Brownian motion with Hurst exponent H∈(0,1) and to Volterra-like processes. Some of the most relevant consequences of our ‘rough Donsker (rDonsker) Theorem’ are convergence results for discrete approximations of a large class of rough models. This justifies the validity of simple and easy-to-implement Monte-Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark Hybrid scheme and find remarkable agreement (for a large range of values of H). This rDonsker Theorem further provides a weak convergence proof for the Hybrid scheme itself, and allows to construct binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan.

An American Call IS Worth More Than a European Call: The Value of American Exercise When the Market Is Not Perfectly Liquid

An easily proved theoretical principle in option pricing is that an American call option should never be exercised early, except possibly just before an ex-dividend date. This result depends on the ability to liquidate an option position, European or American, for its theoretical value (the European call price) in the market. But this is normally impossible in a real world option market, where the best bid for an in-the-money short maturity option is generally below intrinsic value even for an option on a liquid stock. An American option can always be exercised to recover intrinsic value while a European contract must be liquidated at the best available bid in the market. If an investor has to exit an American call position before maturity, exercise can often return more than selling at the market's best bid. This paper derives the value of liquidity-based early exercise of both calls and puts in closed-form, as a function of the bid-ask spread. An illustrative example using a small amount of market data shows that the value of Americanness can be substantial, even for the most actively traded options whose spreads are very tight. For somewhat less active options the premium for liquidity-based early exercise can easily exceed the theoretical value of the early exercise option for an American put or a call on a typical dividend-paying stock.

Yes We Can (Price Derivatives on Survivor Indices)

AbstractWe propose a simulation approach to value derivatives when the underlying dynamics are estimated using the survivor indices directly. Our results show that survivor forward and swap premiums increase with maturity and with the market price of risk. Our results also confirm that taking the optionality into consideration is important from a pricing perspective, for both U.S. women and men. We compare our results to what is obtained using an alternative modeling approach in which a Lee–Carter model is used to indirectly model the survivor index. Compared to this method, our estimated premiums and prices are higher for all longevity products. Moreover, comparing American‐style with European‐style options we find that, although the early exercise option has value when using survivor indices directly, the relative value of the early exercise option is significantly less than when the Lee–Carter model is used to indirectly model the survivor index. It follows that the assumed mortality dynamics have important implications for the term structure of forward and swap premiums and for the effect that changes in the market price of risk has on them.

Rolling Adjoints : Fast Greeks Along Monte Carlo Scenarios for Early-Exercise Options

In this paper we extend the stochastic grid bundling method (SGBM), a regress-later based Monte Carlo scheme for pricing early-exercise options, with an adjoint method to compute in a highly efficient manner sensitivities along the paths, with reasonable accuracy. With the ISDA standard initial margin model being adopted by the financial markets, computing sensitivities along scenarios is required to compute quantities like the margin valuation adjustment.

EFFICIENT COMPUTATION OF EXPOSURE PROFILES FOR COUNTERPARTY CREDIT RISK

Three computational techniques for approximation of counterparty exposure for financial derivatives are presented. The exposure can be used to quantify so-called Credit Valuation Adjustment (CVA) and Potential Future Exposure (PFE), which are of utmost importance for modern risk management in the financial industry, especially since the recent credit crisis. The three techniques all involve a Monte Carlo path discretization and simulation of the underlying entities. Along the generated paths, the corresponding values and distributions are computed during the entire lifetime of the option. Option values are computed by either the finite difference method for the corresponding partial differential equations, or the simulation-based Stochastic Grid Bundling Method (SGBM), or by the COS method, based on Fourier-cosine expansions. In this research, numerical results are presented for early-exercise options. The underlying asset dynamics are given by either the Black–Scholes or the Heston stochastic volatility model.

GPU Acceleration of the Stochastic Grid Bundling Method for Early-Exercise Options

In this work, a parallel GPU version of the Monte Carlo Stochastic Grid Bundling Method (SGBM) for pricing multi-dimensional early-exercise options is presented. To extend the method's applicability, the problem dimensions and the number of bundles will be increased drastically. This makes SGBM very expensive in terms of computational costs on conventional hardware systems based on CPUs. A parallelization strategy of the method is developed and the GPGPU paradigm is used to reduce the execution time. An improved technique for bundling asset paths, which is more efficient on parallel hardware is introduced. Thanks to the performance of the GPU version of SGBM, a general approach for computing the early-exercise policy is proposed. Comparisons between sequential and GPU parallel versions are presented.

Efficient Computation of Exposure Profiles for Counterparty Credit Risk

Three computational techniques for approximation of counterparty exposure for financial derivatives are presented. The exposure can be used to quantify so-called Credit Valuation Adjustment (CVA) and Potential Future Exposure (PFE), which are of utmost importance for modern risk management in the financial industry, especially since the recent credit crisis. The three techniques all involve a Monte Carlo path discretization and simulation of the underlying entities. Along the generated paths, the corresponding values and distributions are computed during the entire lifetime of the option. Option values are computed by either the finite difference method for the corresponding partial differential equations, or the simulation based Stochastic Grid Bundling Method (SGBM), or by the COS method, based on Fourier-cosine expansions. In this research, numerical results are presented for early-exercise options. The underlying asset dynamics are given by either the Black-Scholes or the Heston stochastic volatility model.

A bias-reduction technique for Monte Carlo pricing of early-exercise options

We present a new method for reducing the bias present in Monte-Carlo estimators of the price of American-style contingent claims. At each exercise opportunity (in a time discretization), we assume there is an unbiased estimator of the claim value at the next exercise opportunity. We approximate the distribution of this statistic using the central limit theorem, and use this to derive an asymptotic expression for the bias. This expression is easily estimated in the context of a simulation, which allows for the straightforward computation of bias-reduced estimators of the claim value. We conclude by presenting a well-studied multivariate pricing example to show that this method offers significant improvements over the vanilla stochastic mesh technique, and that it is much more computationally efficient approach to reducing bias than nonparametric bootstrapping.