Mathematics

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18 pages, 1761 KiB

Open AccessArticle

Pricing European Vulnerable Options with Jumps and Stochastic Default Obstacles Barrier under Regime Switching

by Xiangdong Liu and Zanbin Zhang

Mathematics 2023, 11(19), 4155; https://doi.org/10.3390/math11194155 - 3 Oct 2023

Viewed by 714

In this paper, we propose an enhanced model for pricing vulnerable options. Specifically, our model assumes that parameters such as interest rates, jump intensity, and asset value volatility are governed by an observable continuous-time finite-state Markov chain. We take into account European vulnerable [...] Read more.

In this paper, we propose an enhanced model for pricing vulnerable options. Specifically, our model assumes that parameters such as interest rates, jump intensity, and asset value volatility are governed by an observable continuous-time finite-state Markov chain. We take into account European vulnerable options that are exposed to both default risk and rare shocks from underlying and counterparty assets. We also consider stochastic default barriers driven by a regime-switching model and geometric Brownian motion, thus improving upon the assumption of fixed default barriers. The risky assets follow a related jump-diffusion process, whereas the default barriers are influenced by a geometric Brownian motion correlated with the risky assets. Within the framework of our model, we derive an explicit pricing formula for European vulnerable options. Furthermore, we conduct numerical simulations to examine the effects of default barriers and other related parameters on option prices. Our findings indicate that stochastic default barriers increase credit risk, resulting in a decrease in option prices. By considering the aforementioned factors, our research contributes to a better understanding of pricing vulnerable options in the context of counterparty credit risk in over-the-counter trading. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

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12 pages, 293 KiB

Open AccessArticle

Pricing Equity-Indexed Annuities under a Stochastic Dividend Model

by Yuanchuang Shan, Huisheng Shu and Haoran Yi

Mathematics 2023, 11(3), 603; https://doi.org/10.3390/math11030603 - 25 Jan 2023

Viewed by 1060

Abstract

In this paper, we examine the valuations of equity-indexed annuities (EIAs) when their reference stocks distribute stochastic dividends. Due to the fact that stocks typically pay dividends at discrete times after the payment dates are announced, pricing EIAs with dividends is deemed to [...] Read more.

In this paper, we examine the valuations of equity-indexed annuities (EIAs) when their reference stocks distribute stochastic dividends. Due to the fact that stocks typically pay dividends at discrete times after the payment dates are announced, pricing EIAs with dividends is deemed to be practically significant. We directly model the discrete dividend payments using the jump diffusion process with regime switching, and then determine the dynamics of the stock price. The equivalent martingale measure of fair valuation in incomplete markets is determined by employing the Esscher transform. Finally, the pricing formulas of several of the most common EIAs in the market under the stochastic dividend model are obtained. Our model incorporates and extends the present literature on EIAs with accurate and effective valuation methods. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

26 pages, 621 KiB

Open AccessArticle

Portfolio Selection Problem Using CVaR Risk Measures Equipped with DEA, PSO, and ICA Algorithms

by Abdelouahed Hamdi, Arezou Karimi, Farshid Mehrdoust and Samir Brahim Belhaouari

Mathematics 2022, 10(15), 2808; https://doi.org/10.3390/math10152808 - 8 Aug 2022

Cited by 6 | Viewed by 2137

Abstract

Investors always pay attention to the two factors of return and risk in portfolio optimization. There are different metrics for the calculation of the risk factor, among which the most important one is the Conditional Value at Risk (CVaR). On the other hand, [...] Read more.

Investors always pay attention to the two factors of return and risk in portfolio optimization. There are different metrics for the calculation of the risk factor, among which the most important one is the Conditional Value at Risk (CVaR). On the other hand, Data Envelopment Analysis (DEA) can be used to form the optimal portfolio and evaluate its efficiency. In these models, the optimal portfolio is created by stocks or companies with high efficiency. Since the search space is vast in actual markets and there are limitations such as the number of assets and their weight, the optimization problem becomes difficult. Evolutionary algorithms are a powerful tool to deal with these difficulties. The automotive industry in Iran involves international automotive manufacturers. Hence, it is essential to investigate the market related to this industry and invest in it. Therefore, in this study we examined this market based on the price index of the automotive group, then optimized a portfolio of automotive companies using two methods. In the first method, the CVaR measurement was modeled by means of DEA, then Particle Swarm Optimization (PSO) and the Imperial Competitive Algorithm (ICA) were used to solve the proposed model. In the second method, PSO and ICA were applied to solve the CVaR model, and the efficiency of the portfolios of the automotive companies was analyzed. Then, these methods were compared with the classic Mean-CVaR model. The results showed that the automotive price index was skewed to the right, and there was a possibility of an increase in return. Most companies showed favorable efficiency. This was displayed the return of the portfolio produced using the DEA-Mean-CVaR model increased because the investment proposal was basedon the stock with the highest expected return and was effective at three risk levels. It was found that when solving the Mean-CVaR model with evolutionary algorithms, the risk decreased. The efficient boundary of the PSO algorithm was higher than that of the ICA algorithm, and it displayed more efficient portfolios.Therefore, this algorithm was more successful in optimizing the portfolio. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

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15 pages, 741 KiB

Open AccessArticle

Spread Option Pricing in Regime-Switching Jump Diffusion Models

by Alessandro Ramponi

Mathematics 2022, 10(9), 1574; https://doi.org/10.3390/math10091574 - 6 May 2022

Cited by 2 | Viewed by 1895

Abstract

In this paper, we consider the problem of pricing a spread option when the underlying assets follow a bivariate regime-switching jump diffusion model. We exploit an approximation technique which is based on the univariate Fourier transform representation of the option price. The method [...] Read more.

In this paper, we consider the problem of pricing a spread option when the underlying assets follow a bivariate regime-switching jump diffusion model. We exploit an approximation technique which is based on the univariate Fourier transform representation of the option price. The method proves to be computationally very effective with respect to benchmark Monte Carlo estimators and permits the use of several kinds of jump models other than the standard Gaussian setting. As a by-product, the exact price of an Exchange Option may be efficiently computed within this framework. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

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12 pages, 269 KiB

Open AccessArticle

Power Exchange Option with a Hybrid Credit Risk under Jump-Diffusion Model

by Junkee Jeon and Geonwoo Kim

Mathematics 2022, 10(1), 53; https://doi.org/10.3390/math10010053 - 24 Dec 2021

Cited by 1 | Viewed by 1667

Abstract

In this paper, we study the valuation of power exchange options with a correlated hybrid credit risk when the underlying assets follow the jump-diffusion processes. The hybrid credit risk model is constructed using two credit risk models (the reduced-form model and the structural [...] Read more.

In this paper, we study the valuation of power exchange options with a correlated hybrid credit risk when the underlying assets follow the jump-diffusion processes. The hybrid credit risk model is constructed using two credit risk models (the reduced-form model and the structural model), and the jump-diffusion processes are proposed based on the assumptions of Merton. We assume that the dynamics of underlying assets have correlated continuous terms as well as idiosyncratic and common jump terms. Under the proposed model, we derive the explicit pricing formula of the power exchange option using the measure change technique with multidimensional Girsanov’s theorem. Finally, the formula is presented as the normal cumulative functions and the infinite sums. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

23 pages, 1693 KiB

Open AccessArticle

An Asymptotic Solution for Call Options on Zero-Coupon Bonds

by Michael J. Tomas III and Jun Yu

Mathematics 2021, 9(16), 1940; https://doi.org/10.3390/math9161940 - 14 Aug 2021

Cited by 4 | Viewed by 1826

Abstract

We present an asymptotic solution for call options on zero-coupon bonds, assuming a stochastic process for the price of the bond, rather than for interest rates in general. The stochastic process for the bond price incorporates dampening of the price return volatility based [...] Read more.

We present an asymptotic solution for call options on zero-coupon bonds, assuming a stochastic process for the price of the bond, rather than for interest rates in general. The stochastic process for the bond price incorporates dampening of the price return volatility based on the maturity of the bond. We derive the PDE in a similar way to Black and Scholes. Using a perturbation approach, we derive an asymptotic solution for the value of a call option. The result is interesting, as the leading order terms are equivalent to the Black–Scholes model and the additional next order terms provide an adjustment to Black–Scholes that results from the stochastic process for the price of the bond. In addition, based on the asymptotic solution, we derive delta, gamma, vega and theta solutions. We present some comparison values for the solution and the Greeks. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

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14 pages, 301 KiB

Open AccessArticle

Market and Liquidity Risks Using Transaction-by-Transaction Information

by Mariano González-Sánchez, Eva M. Ibáñez Jiménez and Ana I. Segovia San Juan

Mathematics 2021, 9(14), 1678; https://doi.org/10.3390/math9141678 - 16 Jul 2021

Cited by 1 | Viewed by 1667

Abstract

The usual measures of market risk are based on the axiom of positive homogeneity while neglecting an important element of market information—liquidity. To analyze the effects of this omission, in the present study, we define the behavior of prices and volume via stochastic [...] Read more.

The usual measures of market risk are based on the axiom of positive homogeneity while neglecting an important element of market information—liquidity. To analyze the effects of this omission, in the present study, we define the behavior of prices and volume via stochastic processes subordinated to the time elapsing between two consecutive transactions in the market. Using simulated data and market data from companies of different sizes and capitalization levels, we compare the results of measuring risk using prices compared to using both prices and volumes. The results indicate that traditional measures of market risk behave inversely to the degree of liquidity of the asset, thereby underestimating the risk of liquid assets and overestimating the risk of less liquid assets. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

27 pages, 939 KiB

Open AccessArticle

Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

by Florin Avram, Dan Goreac, Juan Li and Xiaochi Wu

Mathematics 2021, 9(9), 931; https://doi.org/10.3390/math9090931 - 22 Apr 2021

Cited by 3 | Viewed by 1619

Abstract

We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using [...] Read more.

We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections

(- a, 0, b)

policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a, and only pay dividends when the reserve reaches an upper barrier b. An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti (

a = 0

) or Shreve-Lehoczy-Gaver (

a = \infty

) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition Løkka-Zervos (2008). In the first case, companies start paying dividends at the barrier

b^{*} = 0

, while in the second they must wait for reserves to build up to some (fully determined)

b^{*} > 0

before paying dividends. Full article

(This article belongs to the Special Issue Stochastic Processes Applied to Modelling in Finance: Latest Advances and Prospects)

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