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This study proposes a pricing model for American options that incorporates the empirically observed dependence of regime transitions on sojourn time. Conventional regime switching models, where model parameters vary with economic regimes, generally assume time homogeneous transition probabilities, independent of how long the economic regime has remained in the current regime. However, empirical studies have shown that this assumption cannot adequately describe the positive sojourn time dependence observed in real financial markets. To address this limitation, we formulate the pricing problem within a semi-Markov decision process (SMDP) framework that explicitly accounts for sojourn time. The proposed model integrates two types of decisions: predetermined periodic decisions and regime transition triggered decisions that occur when the economic regime changes. Numerical experiments confirm the expected relationships between option prices, asset prices, and time to maturity. Furthermore, comparative analysis demonstrates that ignoring sojourn time leads to a systematic undervaluation of American options, with the largest difference observed when the option is at-themoney and has a medium-term maturity.

Pricing high-dimensional basket options poses significant challenges, especially when dealing with nonlinear payoffs. Previous research has demonstrated the effectiveness of neural networks in pricing Bermudan basket call options, particularly under the assumption that the underlying assets follow geometric Brownian motion (GBM). Building on these, the contribution of this study is twofold. First, we extended the scope of options to include both call and put options, as well as payoffs based on the maximum and minimum of the underlying assets. Second, and more importantly, we addressed the practical relevance of market conditions by modeling the underlying assets using a high-dimensional Heston stochastic volatility model with a full correlation structure. Using a least-squares Monte Carlo approach, we approximated the continuation value of the options across a large number of underlying assets using a shallow neural network. We demonstrated that our model yields accurate results in low-dimensional Heston settings and high-dimensional GBM settings, aligning with existing literature and providing confidence in its validity. While high-dimensional pricing has been explored under GBM, our contribution lies in extending this capability to the Heston model, for which we presented numerical experiments involving up to 50 assets, a setting that, to the best of our knowledge, has not been previously studied.

An age-replacement policy provides an optimal preventive replacement interval to minimize the maintenance costs. Our lives are supported by various systems. Failure of such infrastructural systems must be avoided as it causes financial and temporal damage to society. Therefore, maintenance, which ensures system reliability and recovery from failure conditions, plays an important role in ensuring the longer-term use of systems. In general, there are two types of maintenance: preventive maintenance, which is performed before a system failure, and corrective maintenance, which is performed after a failure. This chapter describes the series system model and provides three optimization problems for this model. It also describes the conditions for the existence of an optimal replacement interval. The chapter then describes the numerical examples of the proposed model and the effect of risk on the optimal replacement interval. It summarizes the main points of this study and points out its limitations. 

In this paper, a numerical forward-backward stochastic differential equations (FBSDEs) approach is used for pricing American options under the Heston model. The motivation behind this approach lies in Heston model’s ability to capture significant market features, including stochastic volatility and the correlation between the underlying asset and the volatility. Yet, to the best of our knowledge, there is a lack of dedicated investigation on American option pricing using FBSDE approach under this model. In the present study, the FBSDE representation is obtained and utilized to develop a numerical scheme, which enables the approximation of option prices and hedge ratios. The primary focus lies in approximating American option price as the solution to the backward stochastic differential equation (BSDE) within the FBSDE representation using a θ discretization, which we name as the BSDE-θ Scheme. This BSDE-θ scheme offers computational advantages and flexibility for handling both lower and higher dimensions. Finally, numerical experimental studies are conducted to assess the accuracy and stability of the scheme across various choices of θ. Additionally, the impact of the choice of basis functions in the form of global polynomials on the estimation of conditional expectation functions is investigated. The findings indicate that the choice of θ influences the accuracy of the numerical scheme in particular cases. Moreover, the choice of basis functions under consideration does not exhibit any observable impact on the results.

In this chapter, we consider the problem of American option pricing when the asset price dynamics follow a binomial model dominated by a varying economic situation. We assume that the economic situation transits based on a semi-Markov chain. The pricing procedure is formulated using a semi-Markov decision process, as the decision-maker decides whether to exercise early or hold the option. Some properties of the optimal exercise regions and the monotonicity of option prices are discussed based on simulation results when each parameter is changed, and the optimal strategies are investigated.

We study the double-mean-reverting model by Gatheral. Our previous results concerning the asymptotic expansion of the implied volatility of a European call option, are improved up to order 3, that is, the error of the approximation is ultimately smaller that the 1.5th power of time to maturity plus the cube of the absolute value of the difference between the logarithmic security price and the logarithmic strike price.

This contribution deals with an extension to our developed novel cubature methods of degrees 5 on Wiener space. In our previous studies, we studied cubature formulae that are exact for all multiple Stratonovich integrals up to dimension equal to the degree. In fact, cubature method reduces solving a stochastic differential equation to solving a finite set of ordinary differential equations. Now, we apply the above methods to construct trinomial models and to price different financial derivatives. We will compare our numerical solutions with the Black’s and Black-Scholes models’ analytical solutions. The constructed model has practical usage in pricing American options and American-style derivatives.