Analytical approximation method of option pricing under geometric mean-reverting process

A Closed-Form Solution to American Options under General Diffusions

This paper considers the pricing issue of vulnerable European option when the dynamics of the underlying asset value and counterparty’s asset value follow two correlated exponential Lévy processes with stochastic volatility, and the stochastic volatility is divided into the long-term and short-term volatility. A mean-reverting process is introduced to describe the common long-term volatility risk in underlying asset price and counterparty’s asset value. The short-term fluctuation of stochastic volatility is governed by a mean-reverting process. Based on the proposed model, the joint moment generating function of underlying log-asset price and counterparty’s log-asset value is explicitly derived. We derive a closed-form solution for the vulnerable European option price by using the Fourier inversion formula for distribution functions. Finally, numerical simulations are provided to illustrate the effects of stochastic volatility, jump risk, and counterparty credit risk on the vulnerable option price.

Approximating option prices under large changes of underlying asset prices

6 - Bayesian analysis of the Black-Scholes option price

European call option issued on a bond governed by a modified geometric Ornstein-Uhlenbeck process, is investigated. Objective price of such option as a function of the mean and the variance of a geometric Ornstein-Uhlenbeck process is studied. It is proved that the “Ornstein-Uhlenbeck” market is arbitrage-free and complete. We obtain risk-neutral measure and calculate the fair price of a call option. We consider also the bond price, governed by a modified fractional geometric Ornstein-Uhlenbeck process with Hurst index $H\in (1/2,1)$. Limit behaviour of the variance of the process as $H\to 1/2$ and $H\to 1$ is studied, the monotonicity of the variance and the objective price of the option as a function of Hurst index is established.

<abstract><p>In this paper, the pricing of European options under a new two-factor non-affine stochastic volatility model is studied. In order to reduce the computational complexity, we use the Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices. Numerical experiments prove that the proposed formula is fast and efficient for pricing European options compared with Monte Carlo simulations. The sensitivity of the parameters is analyzed to explain the rationality of the model. Finally, we present some preliminary empirical analysis revealing that the pricing performance of our proposed model is superior to that of the single-factor model.</p></abstract>

Classic options can no longer meet the diversified needs of investors; thus, it is of great significance to construct and price new options for enriching the financial market. This paper proposes a new option pricing model that integrates the logarithmic investment strategy with the classic Black–Scholes theory. Specifically, this paper focus on put options, introducing a threshold-based strategy whereby investors sell stocks when prices fall to a certain value. This approach mitigates losses from adverse price movements, enhancing risk management capabilities. After deriving an analytical solution, we utilized mathematical software to visualize the factors influencing new option prices in three-dimensional space. The findings suggest that the pricing of these new options is influenced not only by standard factors such as the underlying asset price, volatility, risk-free rate of interest, and time to expiration, but also by investment strategy parameters such as the investment strategy index, investment sensitivity, and holding ratios. Most importantly, the pricing of new put options is generally lower than that of classic options, with numerical simulations demonstrating that under optimal parameters the new options can achieve similar hedging effectiveness at approximately three-quarters the cost of standard options. These findings highlight the potential of logarithmic investment strategies as effective tools for risk management in volatile markets. To validate our theoretical model, numerical simulations using data from Shanghai 50 ETF options were used to confirm its accuracy, aligning well with theoretical predictions. The new option model proposed in this paper contributes to enhancing the efficiency of resource allocation in capital markets at a macro level, while at a micro level, it helps investors to apply investment strategies more flexibly and reduce decision-making errors.

ABSTRACTThe classical linear Black–Scholes model for pricing derivative securities is a popular model in the financial industry. It relies on several restrictive assumptions such as completeness, and frictionless of the market as well as the assumption on the underlying asset price dynamics following a geometric Brownian motion. The main purpose of this paper is to generalize the classical Black–Scholes model for pricing derivative securities by taking into account feedback effects due to an influence of a large trader on the underlying asset price dynamics exhibiting random jumps. The assumption that an investor can trade large amounts of assets without affecting the underlying asset price itself is usually not satisfied, especially in illiquid markets. We generalize the Frey–Stremme nonlinear option pricing model for the case the underlying asset follows a Lévy stochastic process with jumps. We derive and analyze a fully nonlinear parabolic partial-integro differential equation for the price of the option contract. We propose a semi-implicit numerical discretization scheme and perform various numerical experiments showing the influence of a large trader and intensity of jumps on the option price.

This paper develops a binomial-Wiener hybrid model to the pricing of iron ore options in the China-Australia market, addressing the limitations of two classical models: the Black-Scholes-Merton (BSM) model and Cox-Ross-Rubinstein (CRR) model. This hybrid model combines the discrete flexibility of the CRR model with the continuous stochastic dynamics of the Wiener process, enabling it to capture both early exercise features and intra-period volatility clustering. Through an analytical approximation method, the model incorporates the full distributional effects of Wiener process without relying on computationally intensive numerical simulations. The empirical section selects options contracts from the Dalian Commodity Exchange (DCE), covering the high volatility in 2021 and low volatility in 2023. Results show that the BSM model yields the largest pricing deviations, the CRR model improves accuracy but remains structurally constrained. The proposed hybrid model consistently delivers option values closest to market prices and demonstrates robustness across different volatility scenarios. These findings suggest that the binomial-Wiener hybrid model provides a more reliable and practical tool for commodity option valuation, with potential applications beyond the iron ore trade.

A Bayesian estimation of exponential Lévy models for implied volatility smile

Margrabe's option to exchange in a paretian-stable subordinated market

This study proposes an artificial neural network (ANN) based option pricing framework under the SABR (stochastic alpha beta rho) and free boundary SABR volatility models. Unlike previous research, we do not directly apply the ANN technique to train and predict option implied volatilities. Instead, we use the technique to train and predict the differences between the implied volatilities calculated by asymptotic approximation and those computed by Monte Carlo simulation. Since the analytical solution for an option written on an underlying asset price in the SABR model is intractable, approximation techniques are used to derive an approximate closed-form solution in practice. However, these approximation formulas worsen as maturity increases and the underlying asset's volatility is high. The accuracy decreases rapidly in wings, which represents deep in-the-money and out-of-the-money cases. By combining the approximation formulas and ANN framework, our new option pricing method offers the following improvements: (1) the training becomes more robust, and the predictions produce more stable and accurate results: (2) it significantly speeds up the training procedure: and (3) the accuracy and efficiency of approximation are improved in the wings without sacrificing performance.

This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

Using Relative Returns to Accommodate Fat-Tailed Innovations in Processes and Option Pricing

In this article, we generalize the classical Edgeworth series expansion used in the option pricing literature. We obtain a closed-form pricing formula for European options by employing a generalized Hermite expansion for the risk neutral density. The main advantage of the generalized expansion is that it can be applied to heavy-tailed return distributions, a case for which the standard Edgeworth expansions are not suitable. We also show how the expansion coefficients can be inferred directly from market option prices.