An Option Pricing Formula for Active Hedging Under Logarithmic Investment Strategy

Option pricing model is a wildly interested topic in an area of financial Mathematics. The pioneer model was introduced by Fischer Black and Myron Scholes which is known as the Black-Scholes model. This model was derived under various assumptions such as liquidity and no transaction costs for which a underlying asset price in stock market might not be satisfied. With this fact, the underlying asset price models were remodeled, in order to determine an option value. This research aims to extend the Black-Scholes model by relaxing the assumption of no transaction costs in illiquid markets. Also, jumps of asset price are considered in this work. To do this, a differential form of asset price with transaction costs and jumps in illiquid markets is introduced and then used to construct the extended option pricing model. Furthermore, a numerical result of a call option price under a new situation is provided.

The objective of this article is to evaluate the performance of the option pricing model at the cross-sectional level. For that purpose, the authors propose a statistical framework, in which they in particular account for the uncertainty associated with the reported pricing performance. Instead of a single figure, the authors determine an entire probability distribution function for the loss function that is used to measure the performance of the option pricing model. This method enables them to visualize the effect of parameter uncertainty on the reported pricing performance. Using a data-driven approach, the authors confirm previous evidence that standard volatility models with clustering and leverage effects are sufficient for the option pricing purpose. In addition, they demonstrate that there is short-term persistence but long-term heterogeneity in cross-sectional option pricing information. This finding has two important implications. First, it justifies the practitioner’s routine to refrain from time series approaches and instead estimate option pricing models on a cross section by cross section basis. Second, the long-term heterogeneity in option prices pinpoints the importance of measuring, comparing, and testing the option pricing model for each cross section separately. To the authors’ knowledge no statistical testing framework has previously been applied to a single cross section of option prices. They propose a method that addresses that need. The proposed framework can be applied to a broad set of models and data. In the empirical part of the article, they show by means of example, an application that uses a discrete time volatility model on S&P 500 index options. TOPICS:Options, volatility measures Key Findings • Absolute pricing performance, measured by a loss-function, is an inappropriate criteria to benchmark competing option pricing models at the cross-sectional level. • The long-term heterogeneity in cross-sectional option pricing information and the uncertainty of reported pricing performance calls for the necessity to rely on an entire probability distribution function of the loss function when comparing models performance. • This article proposes a statistical framework, based on a data-driven approach, to compare model performance accounting for model uncertainty applicable to a single cross-section of option prices.

Fractional Brownian motion in option pricing and dynamic delta hedging: Experimental simulations

Approximating option prices under large changes of underlying asset prices

The option pricing model can predict the future trend of the financial market. In order to more accurately describe the changing process of the financial market, the Hurst index which can describe the characteristics of long-term memory is introduced into the traditional Heston model. Under the assumption that the underlying asset price follows fractional Brownian motion, the fractional stochastic volatility pricing of European option pricing model (Hurst-Heston model) is constructed, and the closed solution of the model is obtained according to the partial differential equation satisfied by the model. By analyzing the relationship between Hurst index and asset price, it is found that the movement process of asset price under the hypothesis of this model is more consistent with the real market change law, which verifies the rationality of the model. In the process of empirical analysis of SSE 50ETF put option data, Self-adaptive Differential Evolution algorithm is used to estimate parameters. The results showed that the error of Hurst-Heston model is smaller than other models, and the prediction error for consecutive trading days is similar. It showed that the pricing results of Hurst-Heston model are more accurate and stable.

In an efficient market, the no‐arbitrage condition implies that the price difference between any two assets must be the market value of all differences in their cash flows. We use this logic to deduce the price of the prepayment option embedded in fixed‐rate Government National Mortgage Association (GNMA) mortgage‐backed securities. The option price equals the difference between an observed GNMA price and the cost of a synthetic, nonprepayable GNMA constructed from the least expensive portfolio of Treasury securities that exactly replicates the promised GNMA cash flow stream, assuming prepayment is precluded. We regress the option prices on variables found significant in previous prepayment studies, finding that five key regressors explain more than 90% of the prepayment option value in pooled time‐series cross‐sectional analysis. We also show that the time value of the prepayment option calculated by our method displays a pattern similar to that produced by the Black‐Scholes (1973) option pricing model. An additional empirical result is the existence of negative option prices and negative time value of the option prices. We attribute these to the fact that homeowners sometimes exercise their prepayment options when they are out‐of‐the‐money, and to refinancing transaction costs. Our method is independent of assumptions regarding interest rate processes and the homeowner's prepayment behavior, and it provides a benchmark for testing theoretical prepayment models.

<p>This study discusses the pricing of European call options by considering transaction costs using the Barles-Soner model. The method used in this study is an analytical and asymptotic approach based on literature studies. Stock and option price data were obtained from Yahoo Finance, while the risk-free interest rate was taken from the Federal Reserve Bank of St. Louis. Volatility was calculated based on historical stock return data, while transaction costs and risk aversion parameters were determined based on previous studies. The Barles-Soner model introduces non-linear effective volatility, which is estimated using an asymptotic approach to obtain effective volatility. The option price calculated using the Barles-Soner model is , higher than the option price using the Black-Scholes model of , with a difference of . These results confirm that transaction costs have a significant effect on option prices. Therefore, the Barles-Soner model is more comprehensive in calculating stock option prices than the Black-Scholes model.</p>

A fuzzy analogue of the binomial option pricing tree model introduced by Cox et.al [1] was proposed by Yoshida [2] to study fuzzy American put option model in an uncertain environment using discrete time fuzzy stochastic process. Later, Muzzioli [3] used non overlapping triangular and trapezoidal fuzzy numbers to model the jump factors in American put option pricing model admitting impreciseness only in volatility while Xcaojian Yu [4] persumed impreciseness in both risk-free interest rate and volatility of the underlying stock involving non-overlapping trapezoidal fuzzy numbers. K. Meenakshi et al. [5] defined new fuzzy risk-neutral probability measures using general trapezoidal fuzzy numbers to study "Problem of Pricing American Fuzzy Put Option Buyer’s Model". The two up and down jump factors and the risk-free interest rate fluctuates oftenly and hence are uncertain in nature. When financial investors come across a high volatile or low volatile (up and down jump factors) fuzzy stocks, non-overlapping type of fuzzy numbers will not be sufficient to predict the underlying fuzzy stock prices as the fuzzy stock prices would go up only in the up state and would go down only in the down state of the fuzzy binomial tree. The uncertainty associated in the above stated fuzzy option pricing parameters could not be captured completely here. Such a scenario needs attention. To handle such a situation, we need to consider fuzzy numbers which are not only non - overlapping but partially or fully overlapping and/ or contained in too. Also not much consideration had been given to American fuzzy put option model involving fuzzy martingales. This paper will consider the study of the fuzzy analogue of martingale pricig theory in the context of fuzzy option pricing theory. In this study, we discuss American Fuzzy Put Option Seller’s Model (AFPOSM) based on fuzzy future contract involving general linear octagonal fuzzy numbers (GLOFN); general in the sense that they could be either overlapping or non-overlapping partially or completely and/or contained in using the fuzzy risk-neutral probability measures introduced by us. We record a computational procedure to obtain the fuzzy profit and loss (PL) values of sellers using two-period fuzzy binomial tree model wherein the fuzzy stock price and the fuzzy future price following discrete time fuzzy stochastic process. The same is performed by deploying the real stock market data obtained from the website [6] that includes American style Microsoft Corporation (MSFT) shares of the year 2018 under the bullish (the constant initial fuzzy stock price goes above the constant fuzzy strike price) scenario. Utilizing the PL values, we estimate an optimal exercise time and an optimal exercise price of sellers. Also the discounted fuzzy intrinsic values of AFPOSM on fuzzy future contract are validated. The computations carried out in AFPOSM are performed using MATLAB 2016a software.    

6 - Bayesian analysis of the Black-Scholes option price

Option pricing is essential for market participants as it facilitates risk management and price discovery. This study provides a comprehensive elucidation of diverse option pricing models. The author delineates the development of option pricing models from the fundamental Black-Scholes-Merton model (BSM) to more intricate methodologies. The importance of comprehending option pricing for hedging strategies and risk management is crucial in financial mathematics. The study analyses several models, including the BS model, Jump Diffusion model (JDM), Binomial Tree model (BTM), Monte Carlo simulation (MCS), and stochastic volatility models (SVM), while addressing their applications, limitations, advantages, and disadvantages. This work delineates the history and distinctions between other models and the Black-Scholes-Merton model. The Black-Scholes-Merton model exerts the most substantial influence on financial markets. The alternative models exhibit greater flexibility and enhanced accuracy when confronting complicated and uncertain market trends. The trend of future development is bound to be the continuous updating of models to adapt to more complex situations.

Compared with the developed countries, Chinese stock index option market developed late and accumulated less historical data. At the same time, it has the characteristics that many developed countries' stock option markets do not have. Black-Scholes Model is an important model in the field of option pricing. It is not enough for a financial model to be reasonable in theory, but it needs to be tested by the actual market. Therefore, based on the main transaction data of the CSI 300 stock index option contracts from 2010 to 2020, this paper analyzes the compliance of the CSI 300 stock index option market to the basic assumptions of the B-S option pricing model, the main parameters of the CSI 300 stock index options in the B-S option pricing model, and the fitting degree of the B-S model's prediction and actual prices. The result shows that there is a certain deviation between the pricing conclusions of the B-S model and the actual option prices. This paper analyzes the reasons for the deviation from many aspects, and puts forward the corresponding improvement strategies that the users of the model should adopt, so as to improve the applicability of the B-S option pricing model in Chinese stock index option market.

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.

<title>Abstract</title> This study explores the put-call duality under the Truncated Normal Distribution (TND) model in option pricing which assumes that the log-prices of the underlying assets are bounded above and below. The model is compared with famous Black \&amp; Scholes Model which has been consistently been considered as the benchmark model for option pricing. Using the model, we derived a model-consistent put-call duality identity and applied it in the construction of an objective function that improves the accuracy and consistency of option pricing. This is in conjunction with the various modifications that have been put forward to "improve" the option price. The purpose of this study was to extend Zhu \&amp; He's work by studying the the put-call duality relation of the proposed model and compare it with the results obtained using the classical Black \&amp; Scholes (BS) Model. Similarly, this study investigated the implied volatility of the proposed model and compared it with the BS model. From the duality constraint calibration, the proposed model performed significantly better than the traditional Black-Scholes model. A numerical analysis was done using real market data for S\&amp;P 500 stock price data retrieved on June 27, 2025. Model performance check such as AIC, BIC and pricing errors indicated that outstanding performance of the TND model. From the methodology and numerical analysis, there is evidence that the duality relation holds for the Truncated Normal Distribution model as is the case with the Black \&amp; Scholes model. The implied volatility indicated that the TND model was sensitive for in-the-money and out-of-the money options. Therefore, the enforcement of the duality constraint in the pricing of European option improved the market consistency ensures that no arbitrage opportunities exist in the market that cab provide enormous risk-free earnings in the market.

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.

In this paper, we analyze the pricing of European option when the riskfree interest rate follows a jump process. An expression for European call price is first obtained in the case of constant volatility. Then we present a general formulation which takes care of the variation of volatility. In this general formulation we utilize a point process filtering technique to estimate the state process. Finally we carry out some numerical simulation of our results