True martingales for upper bounds on Bermudan option prices under jump-diffusion processes

Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish finite error bounds of approximation formula of option prices under the rough Heston model for 0.1≤H<0.5 under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results that we manage to obtain is that the second-order approximation for implied volatility (derived by matching coefficients of the Taylor expansion) possesses explosive behavior for the short-time term structure of at-the-money implied volatility skew, which is also present in the short-time option prices under rough Heston dynamics. Numerical experiments were conducted to verify the effectiveness of the approximation formula of option prices and the formulas for the short-time term structure of at-the-money implied volatility skew.

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

This paper proposes an integrated pricing framework for convertible bonds, which comprises firm value evolving as an exponential jump diffusion, correlated stochastic interest rates movements and an efficient numerical pricing scheme. By construction, the proposed stochastic model fits in the framework of affine jump diffusion processes of Duffie et al. [Econometrica, 2000, 68, 1343–1376] with tractable behaviour. We define the firm’s optimal call policy and investigate its impact on the computed convertible bond prices. We illustrate the performance of the numerical scheme and highlight the effects originated by the inclusion of jumps, stochastic interest rates and a non-zero correlation structure between firm value and interest rates.

Pricing vulnerable power exchange options in an intensity based framework

In this paper we discuss an option pricing problem in a hidden Markovian regime-switching model with a stochastic interest rate and volatility. Regime switches are attributed to structural changes in an hidden economic environment and are described by a continuous-time, finite-state, unobservable Markov chain. The model is then applied to the valuation of a standard European option. By means of the standard separation principle, filtering and option valuation problems are separated. Robust filters for the hidden states of the economy and their robust filtered estimates of unknown parameters from the expectation maximization algorithm are presented based on standard techniques in filtering theory. Then an explicit expression of a conditional characteristic function relevant to option pricing is presented and the valuation of the option is discussed using the inverse Fourier transformation approach. Using the limiting behavior of the conditional characteristic function, an efficient implementation of the transform inversion integral is considered. Numerical experiments are given to illustrate the flexibility of filtering algorithms and the significance of regime-switching in option pricing.

Based on transformation techniques used in analysis of initial boundary value problems, we propose and discuss two numerical approaches for nonlinear models in option pricing. The first one exploits difference schemes for a degenerate parabolic problem on a finite interval. The second one solves the problem on infinite interval by Rothe's method. An appropriate substitution reduces the matter to a semilinear integral equation at each time step. Numerical experiments are discussed.

Partial integro-differential equation (PIDE) formulations are often preferable for pricing options under models with stochastic volatility and jumps, especially for American-style option contracts. We consider the pricing of options under such models, namely the Bates model and the so-called stochastic volatility with contemporaneous jumps (SVCJ) model. The nonlocality of the jump terms in these models leads to matrices with full matrix blocks. Standard discretization methods are not viable directly since they would require the inversion of such a matrix. Instead, we adopt a two-step implicit-explicit (IMEX) time discretization scheme, the IMEX-CNAB scheme, where the jump term is treated explicitly using the second-order Adams–Bashforth (AB) method, while the rest is treated implicitly using the Crank–Nicolson (CN) method. The resulting linear systems can then be solved directly by employing LU decomposition. Alternatively, the systems can be iterated under a scalable algebraic multigrid (AMG) method. For pricing American options, LU decomposition is employed with an operator splitting method for the early exercise constraint or, alternatively, a projected AMG method can be used to solve the resulting linear complementarity problems. We price European and American options in numerical experiments, which demonstrate the good efficiency of the proposed methods.

Supply disruption is a common phenomenon in practice. In this study, we investigate the firm's optimal procurement strategy through two kinds of option contracts under supply disruption risk. Through theoretical analysis and numerical simulation, we obtain some important managerial implications. Under AOPC, the unreliable supplier is always utilized due to the economic advantage. When the option price is low, the firm will always reserve options from the reliable supplier. With the increase of the disruption risk, the firm will rely more on the reliable supplier to mitigate supply risk. Under COPC, when the option price and disruption information cost is relatively large, the order strategy in this scenario can be represented by two single-sourcing optimal order points. Otherwise, the firm will decrease the normal order and rely more on the contingency option reservation. When the disruption information cost is relatively low, it is beneficial for the firm to obtain more disruption information and postpone her option reservation decision in most situations. When the disruption information cost is moderate, the firm will choose AOPC and COPC alternately, as the disruption risk increases.

In the regime switching extension of Black–Scholes–Merton model of asset price dynamics, one assumes that the volatility coefficient evolves as a hidden pure jump process. Under the assumption of Markov regime switching, we have considered the locally risk minimizing theoretical price of European vanilla options. By pretending these prices or their numerical approximations as traded prices, we have first computed the implied volatility (IV) of the underlying asset. Then by performing several numerical experiments on ternary regime models we have investigated the dependence of IV on the time to maturity (TTM) and strike price of the vanilla options. We have observed a clear dependence that is at par with the empirically observed stylized facts. Furthermore, we have experimentally validated that IV time series, obtained from contracts with moneyness and TTM varying in a particular narrow range, can recover the transition instances of the hidden Markov chain having three states. Such regime recovery for any arbitrary state‐space and transition parameters has also been established in a theoretical setting. Moreover, the novel scheme for computing option price is shown to be stable.

In this paper, we consider a numerical pricing of European call and put options under the Heston-Cox-Ingersoll-Ross (HCIR) model. Based on this model, the prices of options are derived by solving a three-dimensional partial differential equation. We generalize a componentwise splitting scheme for solving this equation. The idea of this scheme is to decompose the discretized HCIR partial differential equation into six one-dimensional equations in six fractional time steps. These equations are represented in tridiagonal systems, which are solved by the Thomas algorithm. Moreover, the numerical experiments show that the European option prices are affected by changes in volatility, interest rate, strike price, and correlation factors. Furthermore, numerical experiments compare the calculated prices based on our scheme with the prices reported in the literature.

We consider a continuous-time Markov process on a large continuous or discrete state space. The process is assumed to have strong enough ergodicity properties and to exhibit a number of metastable sets. Markov state models (MSMs) are designed to represent the effective dynamics of such a process by a Markov chain that jumps between the metastable sets with the transition rates of the original process. MSMs have been used for a number of applications, including molecular dynamics, for more than a decade. Their approximation quality, however, has not yet been fully understood. In particular, it would be desirable to have a sharp error bound for the difference in propagation of probability densities between the MSM and the original process on long timescales. Here, we provide such a bound for a rather general class of Markov processes ranging from diffusions in energy landscapes to Markov jump processes on large discrete spaces. Furthermore, we discuss how this result provides formal support or shows the limitations of algorithmic strategies that have been found to be useful for the construction of MSMs. Our findings are illustrated by numerical experiments.

This thesis is concerned with presenting convex optimization based tractable solutions for three fundamental problems: 1. Planted subgraph problem: Given two graphs, identifying the subset of vertices of the larger graph corresponding to the smaller one. 2. Graph edit distance problem: Given two graphs, calculating the number of edge/vertex additions and deletions required to transform one graph into the other. 3. Affine inverse eigenvalue problem: Given a subspace e ⊂ 𝕊ⁿ and a vector of eigenvalues λ ∈ ℝⁿ, finding a symmetric matrix with spectrum λ contained in e. These combinatorial and algebraic problems frequently arise in various application domains such as social networks, computational biology, chemoinformatics, and control theory. Nevertheless, exactly solving them in practice is only possible for very small instances due to their complexity. For each of these problems, we introduce convex relaxations which succeed in providing exact or approximate solutions in a computationally tractable manner. Our relaxations for the two graph problems are based on convex graph invariants, which are functions of graphs that do not depend on a particular labeling. One of these convex relaxations, coined the Schur-Horn orbitope, corresponds to the convex hull of all matrices with a given spectrum, and plays a prominent role in this thesis. Specifically, we utilize relaxations based on the Schur-Horn orbitope in the context of the planted subgraph problem and the graph edit distance problem. For both of these problems, we identify conditions under which the Schur-Horn orbitope based relaxations exactly solve the corresponding problem with overwhelming probability. Specifically, we demonstrate that these relaxations turn out to be particularly effective when the underlying graph has a spectrum comprised of few distinct eigenvalues with high multiplicities. In addition to relaxations based on the Schur-Horn orbitope, we also consider outer-approximations based on other convex graph invariants such as the stability number and the maximum-cut value for the graph edit distance problem. On the other hand, for the inverse eigenvalue problem, we investigate two relaxations arising from a sum of squares hierarchy. These relaxations have different approximation qualities, and accordingly induce different computational costs. We utilize our framework to generate solutions for, or certify unsolvability of the underlying inverse eigenvalue problem. We particularly emphasize the computational aspect of our relaxations throughout this thesis. We corroborate the utility of our methods with various numerical experiments.

<p>Total variation (TV) regularization is a powerful tool in image denoising, but it often exhibits limited performance in preserving edges. In contrast, non-convex TV regularization can more effectively preserve edges and contours, albeit posing challenges when solving. Recently, the convex non-convex (CNC) strategy has emerged as a potent approach that allows incorporating non-convex TV regularization terms while maintaining the overall convexity of the objective function. Its superior performance has been validated through various numerical experiments; however, theoretical analysis remains lacking. In this paper, we provided theoretical analysis of the performance of the CNC-TV denoising model. By utilizing the oracle inequality, we derived an improved upper bound on its performance compared to TV regularization. In addition, we devised an alternating direction method of multipliers (ADMM) algorithm to address the proposed model and verified its convergence properties. Our proposed model has been validated through numerical experiments in 1D and 2D denoising, demonstrating its exceptional performance.</p>

The time filter for the non-stationary coupled Stokes/Darcy model

This paper investigates lowest-order weak Galerkin finite element methods for solving linear elasticity problems on rectangular and brick meshes. Specifically, constant vectors are used in element interiors and on element interfaces respectively for approximating displacement. For these constant basis functions, their discrete weak gradients are calculated in the local Raviart–Thomas spaces $$ RT_{[0]}^d $$ ( $$ d=2 $$ or 3), whereas their discrete weak divergences are calculated as elementwise constants. Discrete weak strains are calculated accordingly. Then these quantities are used to develop finite element schemes in both strain-div and grad-div formulations, on both rectangular and brick meshes. A theoretical analysis supported by numerical experiments in both 2-dim and 3-dim reveal that the methods are locking-free and have optimal 1st order convergence in displacement, stress, and dilation (divergence of displacement), when the exact solution has full regularity. The methods can also capture low-regularity solutions very well. Strategies for efficient implementation including Schur complement are presented. Extension to quadrilateral and hexahedral meshes, in both theoretical analysis and numerical experiments, is also examined.