Option Pricing Research Articles

Persistent and transient variance components in option pricing models with variance-dependent Kernel

This paper examines theoretically and empirically a variance-dependent pricing kernel in the continuous-time two-factor stochastic volatility (SV) model. We investigate the relevance of such a kernel in the joint modeling of index returns and option prices. We contrast the pricing performance of this model in capturing the term structure effects and smile/smirk patterns to discrete-time GARCH models with similar variance-dependent kernels. We find negative and significant risk premium for both volatility factors, implying that investors are willing to pay for insurance against increases in volatility risk, even if it has little persistence. In-sample, the component GARCH model exhibits a slightly better fit overall and across all maturity buckets than the two-factor SV model. However, the two-factor SV model reduces strike price bias, giving rise to the model’s ability in reconciling the physical and risk-neutral distribution. Out-of-sample, the two-factor SV model has better fit to data.

On the pricing of vulnerable Parisian options

Parisian options, serving as substitutes for barrier options, are frequently embedded in complex financial derivatives. Despite the considerable mathematical challenges in pricing, their significant applications in finance and insurance have generated extensive studies in the literature. However, the impact of counterparty risk has not been taken into account so far. To address this gap, we develop a closed-form pricing framework for vulnerable Parisian options. Utilizing the Laplace transform and measure-change technique, we derive closed-form pricing formulas of Parisian options incorporating counterparty credit risk. Finally, we conduct numerical analyses to verify our pricing formulas’ accuracy and efficiency.

Option Pricing in an Incomplete Market

The purpose of this paper is to illustrate the pricing of options in an incomplete market using the new consistent uplifted martingale measure methodology introduced by Grigorian and Jarrow [2024, Filtration Reduction and Incomplete Markets, Frontiers of Mathematical Finance, 3(1), 78–105; 2023, Filtration Reduction and Completeness in Brownian Motion Models. Working Paper, Cornell University; 2024, Filtration Reduction and Completeness in Jump-Diffusion Models. Working Paper, Cornell University]. We apply it to an incomplete market where a stock has stochastic volatility. Two valuation formulas are generated, depending upon whether the trader is more concerned about volatility or price risk in the construction of a partial replicating portfolio for the option’s payoff.

Pricing multi-asset options with tempered stable distributions

We derive methods for risk-neutral pricing of multi-asset options, when log-returns jointly follow a multivariate tempered stable distribution. These lead to processes that are more realistic than the better known Brownian motion and stable processes. Further, we introduce the diagonal tempered stable model, which is parsimonious but allows for rich dependence between assets. Here, the number of parameters only grows linearly as the dimension increases, which makes it tractable in higher dimensions and avoids the so-called “curse of dimensionality.” As an illustration, we apply the model to price multi-asset options in two, three, and four dimensions. Detailed goodness-of-fit methods show that our model fits the data very well.

The option pricing problem based on the uncertain fractional volatility stock model

The option pricing problem based on the uncertain fractional volatility stock model

Pricing VIX Futures and Options With Good and Bad Volatility of Volatility

ABSTRACTThis article studies the pricing of VIX futures and options by directly modeling the dynamics of VIX, based on realized semivariances computed from high‐frequency data of VIX. We derive the closed‐form pricing formula for both the VIX futures and options. The empirical results show that the new model provides superior pricing performance compared with the model based on conventional unsigned realized variance and the classic Heston‐Nandi GARCH model, both in sample and out of sample. Our study confirms that the decomposition of realized variance into upside and downside components helps to improve the pricing performance for VIX futures and options.

Considering Appropriate Input Features of Neural Network to Calibrate Option Pricing Models

Considering Appropriate Input Features of Neural Network to Calibrate Option Pricing Models

A Study of Black-Scholes-Merton Model

This study introduces the Option Pricing Theory improved by Black, Scholes, and Merton, which is also known as the Black-Scholes-Merton model, in detail and investigates its derivation process and historical development process. Despite the content above, it also includes its characteristics and mechanism. The newest Option Pricing formula proposed by Black, Scholes, and Merton was derived from mathematical methods such as the stochastic integral equation and Ito theorem. This essay concludes that the Black Scholes Merton model paved the way for the development and innovation of the subject of Financial Mathematics, which was found by the person who first came up with the Option Pricing Theory, which also contributes to the investment field to reduce the risk.

The Role of FPGAs in Modern Option Pricing Techniques: A Survey

In financial computation, Field Programmable Gate Arrays (FPGAs) have emerged as a transformative technology, particularly in the domain of option pricing. This study presents the impact of Field Programmable Gate Arrays (FPGAs) on computational methods in finance, with an emphasis on option pricing. Our review examined 99 selected studies from an initial pool of 131, revealing how FPGAs substantially enhance both the speed and energy efficiency of various financial models, particularly Black–Scholes and Monte Carlo simulations. Notably, the performance gains—ranging from 270- to 5400-times faster than conventional CPU implementations—are highly dependent on the specific option pricing model employed. These findings illustrate FPGAs’ capability to efficiently process complex financial computations while consuming less energy. Despite these benefits, this paper highlights persistent challenges in FPGA design optimization and programming complexity. This study not only emphasises the potential of FPGAs to further innovate financial computing but also outlines the critical areas for future research to overcome existing barriers and fully leverage FPGA technology in future financial applications.

METODE BLACK-SCHOLES DALAM PENENTUAN HARGA OPSI

The Black-Scholes Method is one of the option pricing method that introduced by Fischer Black and Myron Scholes in 1973. This study aim to review the determination of the price of an option with stocks as the underlying assets and based on Black and Scholes assumptions. These assumptions lead to construct an equation named Black-Scholes differential equations, which is the equation that must be satisfied for option as the derivative instrument and non-dividend giving stocks as the underlying assets. After the Black-Scholes differential equations formed successfully, the next step is to find the solution of that equation. Consider the option is call option, the solution that will be obtained from solving the equation is the price of call option. Then substitute it to the put-call parity equation, which is the equation that shows the relationship between call and put option prices, so the price of put option can be obtained too. The estimation that obtained from the Black-Scholes method is the highest price for a contract is said to be fair and worth to buy for the holder.

A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models

Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean–Vlasov equation dXt=σ(t,Xt)XtvtE[vt|Xt]dWt,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ d X_{t}= \\sigma (t,X_{t}) X_{t} \\frac{\\sqrt{v}_{t}}{\\sqrt{\\mathbb{E}[v_{t}|X_{t}]}}dW_{t}, $$\\end{document} where W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W$\\end{document} is a Brownian motion and v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$v$\\end{document} is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. While such models are quite popular among practitioners, its well-posedness has unfortunately not yet been fully understood and in general is possibly not guaranteed at all. We develop a novel regularisation approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularised model is well posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularised model is able to perfectly replicate option prices coming from typical local volatility models. Our results are also applicable to more general McKean–Vlasov equations.

Higher order approximation of option prices in Barndorff-Nielsen and Shephard models

We present an approximation method based on the mixing formula [Hull, J. and White, A., The pricing of options on assets with stochastic volatilities. J. Finance, 1987, 42, 281–300; Romano, M. and Touzi, N., Contingent claims and market completeness in a stochastic volatility model. Math. Finance, 1997, 7, 399–412] for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené [Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility. Stochastics, 2022, 94, 745–788]. We obtain asymptotic results for the error of the N th order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process.

Analytically pricing European options in dynamic markets: Incorporating liquidity variations and economic cycles

This paper discusses the European option pricing problem in the context of asset prices being influenced by liquidity risks and economic cycles. We employ regime switching for asset volatility, and liquidity risks are captured by market-wide liquidity. We obtain an analytical formulation of European option prices, allowing for fast model calibration using real-market data. By estimating model parameters using real option data, we show that pricing errors can be significantly reduced using our model that considers stochastic liquidity, indicating that our model has real-world applications. Our results can help investors and regulators better understand the market and provide a potentially effective risk management tool.

‘Most Favored Nation’ Clauses in the Online Food Delivery Platform Market: Beyond Booking.com Cases

The COVID-19 pandemic catalyzed a transformative shift in the European food service industry, with local restaurants increasingly turning to online food delivery platforms (OFDPs) to sustain their operations during lockdowns. However, this symbiotic relationship has raised concerns regarding the prevalence of Most Favoured Nation clauses (MFNs) within contractual agreements between local restaurants and OFDPs These clauses, designed to ensure price parity across different sales channels, have sparked debates over their impact on competition and consumer welfare. This is due to their potential effect of hindering local restaurants’ ability to compete effectively, thus, consumers may face reduced price options and higher food costs. The Bundeskartellamt’s landmark decisions regarding MFNs in the online accommodation sector concerning Booking.com have provided initial guidance, but the application of this approach to OFDP markets remains complex. Against this backdrop, and following the reasoning laid down in a more recent approach from the Bundeskartellamt, we propose a systematic methodology for evaluating the effects of MFNs, taking as a starting point of analysis OFDP market conditions and business models. Ultimately, we aim to address the balance between positive and negative effects of usingMFNs byOFDPs to incentivize platform investment and protect the efficiencies passed on to consumers and local restaurants.

Numerical Solution of European Put Option for Black-Scholes Model Using Keller Box Method

In this study, we propose to determine option pricing by using Black-Scholes model numerically. The Keller box method, a numerical method with a box-shaped implicit scheme, is chosen to solve the problem of pricing stock options, especially European-put option. This option pricing involves several parameters such as stock price volatility, risk-free interest rate and strike price. The numerical stability of the method is checked using Von Neumann stability before the simulation is conducted. The influence of interest rates, volatility, and strike price on the option price state that the higher the value of the interest rate parameter, the lower the option price value, while the greater the value of stock price volatility and strike price, the higher the option price.

RISK ANALYSIS OF GOOGL & AMZN STOCK CALL OPTIONS USING DELTA GAMMA THETA NORMAL APPROACH

Stocks, as investment products, tend to carry risks due to fluctuations. The tendency of stock prices to rise over time leads investors to opt for call options, which are one of the derivative investment products. However, call options are influenced by several factors that can pose risks and have nonlinear dependence on market risk factors. Therefore, methods are needed to measure the risk of call options, such as Delta Normal Value at Risk and Delta Gamma Normal Value at Risk. Delta and Gamma are part of Option Greeks, parameters that measure the sensitivity of options to various factors used in determining option prices with the Black-Scholes model. This study uses an approach with the addition of Theta, which can measure the sensitivity of options to time. This study aims to analyze Value at Risk with the Delta Gamma Theta Normal approach for call options on Google (GOOGL) and Amazon (AMZN) stocks. The analysis uses closing stock price data from September 7, 2022, to September 7, 2023, and three in-the-money and out-of-the-money call option prices. The study begins by collecting closing stock prices and call option contract components, testing the normality of stock returns, calculating volatility, , Delta, Gamma, and Theta, then calculating the Value at Risk. Based on the analysis, it is found that GOOGL and AMZN call options have a Value at Risk of $0.89588 and $0.92760, respectively, at a 99% confidence level with a strike price of $120. Furthermore, based on the comparison of Value at Risk between in-the-money and out-of-the-money call options, it can be concluded that out-of-the-money call options tend to have larger estimated losses.

European Call Option under Stochastic Interest Rate in a Fractional Brownian Motion with Transaction Cost

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price.

Option pricing for Barndorff–Nielsen and Shephard model by supervised deep learning

This paper aims to develop a supervised deep learning scheme to compute call option prices for the Barndorff-Nielsen and Shephard model with a non-martingale asset price process having infinite active jumps. In our deep learning scheme, teaching data are generated through the Monte Carlo method developed by Arai and Imai [(2024). Monte Carlo simulation for Barndorff-Nielsen and Shephard model under change of measure, Mathematics and Computers in Simulation, 218, 223–234]. Moreover, the BNS model includes many parameters, which makes the deep learning accuracy worse. Therefore, we will create another input parameter using the Black–Scholes formula. As a result, the accuracy is improved dramatically.

Application of Extended Normal Distribution in Option Price Sensitivities

Application of Extended Normal Distribution in Option Price Sensitivities

Numerical methods base on trinomial trees for option pricing

This paper investigates the approach to pricing European options, starting with one-step binomial tree pricing (a relatively simple way to calculate option value). In the next step, an additional possible rate of change of stock price is added to make the model more realistic, resulting in the one-step trinomial tree model. The model bounds the option price under the no-arbitrary principle. The paper then analyzes the circumstances under which options have a xed price by completing the market and giving the solution formula of option price through the model. Last, put-call parity is used to prove the rationality of one-step trinomial model so that the model eectively prevents the occurrence of risk-free arbitrage in the market. This helps traders to price options reasonably in the market and maintains the stability of the options market.