Efficient L-stable method for parabolic problems with application to pricing American options under stochastic volatility

Fully discrete approximation methods are studied for parabolic boundary value problems evolving from nonsmooth (and noncompatible) initial data. Galerkin methods are used for the spatial discretization, and single step or backwards differentiation multistep methods are used in time. The parabolic equation is assumed to be linear, but the coefficients are allowed to depend on space and time. An error estimate is given for the combined space-time discretization scheme which contains the expected optional order space and time error contributions. However, another term appears which limits the accuracy of the method to second order (or fractionally better) in time if the time stepping method is of third or higher order. This latter term arises from the time dependence of the equation’s highest order coefficients, and hence does not appear in analyses of problems with only spatially dependent coefficients for which optimal order estimates are known to hold. It is shown through more detailed analysis of a specific problem, and the formulation of a lower error bound, that the error estimate is indeed sharp. Thus, it follows that standard higher order time stepping schemes cannot be expected to perform well for parabolic problems with time dependent coefficients in situations when nonsmooth initial data are encountered.

Solving complex PIDE systems for pricing American option under multi-state regime switching jump–diffusion model

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black–Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed form formula for integration. Moreover, these two methods solve the backward dynamic programing problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of the binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite that this model is only bidimensional, the whole history of the process impacts on the price, and how to handle all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.

American options require numerical methods, namely lattice models, to provide accurate price estimates. The computations can become expensive when more than one state variable is involved. Analytical upper bounds can therefore provide a useful guideline for how high American values can reach. In this paper, we derive analytical (closed-form) upper bounds for American option prices under stochastic interest rates, stochastic volatility, and jumps where American option prices are difficult to compute with accuracy. In a stochastic volatility model (Heston (1993) and Scott (1997)) that has two random factors, we demonstrate that the upper bound only takes a very small fraction of the time that the American option needs to compute.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

American options with stochastic dividends and volatility: A nonparametric investigation

Are There Critical Levels of Stochastic Volatility for Early Option Exercise?

Pricing of American options in discrete time by means of regression-based Monte Carlo methods is considered in this thesis. Thereby, the options are allowed to be based on several underlyings (so-called basket options). It is assumed that the underlyings price processes satisfy the Markov property and consequently are Markov processes. In this dissertation, Monte Carlo methods are used to generate artificial sample paths of these price processes, and subsequently adaptive nonparametric least squares regression estimates are used to estimate the so-called continuation values from these data. The continuation values describe the mean values of the American options for given values of the underlyings at time t, subject to the constraint that the options are not exercised at time t, but optimally exercised in the future. Adaptive least squares neural networks as well as adaptive least squares spline estimates are used as nonparametric regression estimates. In conjunction with the pricing of American options, these estimates are theoretically analysed, and results with respect to consistency and rate of convergence are derived. Finally, the pricing of American options on simulated data by means of the estimates is illustrated.

We present a quasi-analytical method for pricing multi-dimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson in J Financ Quant Anal 18(1):141–148 (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the Black-Scholes model, our method achieves the same efficiency as the quadratic approximation of Barone-Adesi and Whaley in J Financ 42:301–320 (1987), with our method being generally more accurate for out-of-the-money and long-maturity options. When applied to Heston’s stochastic volatility model, our method is shown to be extremely efficient and fairly accurate.

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v ∈ L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Stochastic volatility models capture the impact of time-varying volatility on the financial markets, and hence are heavily used in financial engineering. However, stochastic volatility is not directly observable in reality, but is only observable through the inference from the observed asset price. Most of the past research studied American option pricing in stochastic volatility models under the assumption that the volatility is fully observable, which often leads to overpricing of the option. In this paper, we treat the problem under the more realistic assumption of partially observable stochastic volatility, and propose a numerical solution method by extending the regression method and the martingale duality approach to the partially observable case. More specifically, we develop a filtering-based martingale duality approach that complements a lower bound on the option price with an approximate upper bound. Numerical experiments show that our method reduces overpricing of the option with a moderate computational cost.

The subdiffusion equation with a Caputo fractional derivative of order � 2 (0, 1) in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for dis- cretizing the Caputo fractional derivative in time. The scheme was analyzed earlier independently by Lin and Xu (2007) and Sun and Wu (2006), and an O(� 2−� ) convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an O(�) convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more gen- eral sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann-Liouville derivative of order � 2 (3/2,2) in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.

We derive rigorous error bounds for exponential Runge–Kutta discretizations of parabolic equations with nonsmooth initial data. Our analysis is carried out in the framework of abstract semilinear evolution equations, allowing for operators with non-dense domains. The results provide a foundation for establishing error estimates for nonsmooth data in prototypical problems such as the Allen-Cahn and Burgers equations. Furthermore, we apply these estimates to the analysis of split exponential integrators, yielding convergence results expressed explicitly in terms of the prescribed data.

In the previous section, we demonstrated that poles and branch points, rather than being two completely different cases that must be treated differently, are actually complementary in the contributions they provide to a contour integral. In this sense, the Residue Theorem is simply a shortcut because the integrals over the detours to the pole cancel. In the presence of branch points, these integrals do not cancel. This point of view allows us to consider cases avoided in first treatments of contour integration, namely, singularities that are both poles and branch points simultaneously. We now get to apply this point of view to some familiar types of integrals. This point of view also enables us to treat integrals that are usually not considered ones that anyone would want to evaluate using contour integration. In fact, the thought of using contour integration on some of these integrals may send chills down the spine. But using results from the previous chapter, we will see how this point of view pays dividends to those who treasure new ways of evaluating integrals. We will also illustrate applications to the evaluation of sums.

In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the two-dimensional Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty }$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in $\textbf {L}^2$-norm is derived for the discrete problem with nonsmooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.