Option Pricing Research Articles

Analytical and numerical solutions for a special nonlinear equation

In the most of the option price dynamics in the financial markets, the transaction cost of option is ignored. Considering the transaction costs will lead to the emergence of models with nonlinear PDE. In this paper, transaction cost of option in the assumed market is considered, and the resulting dynamic is a nonlinear PDE, whose exact and numerical solutions have been computed in the present paper. To find the exact solution of the cited nonlinear equation, the Lie group algebra method has been used. The numerical solution has been given using the Chebyshev spectral method. In this method, the solution of the considered equation is approximated using Chebyshev polynomials. The convergence of the obtained polynomials to the solution of the differential equation has been shown, as well.

The Duality Principle for Multidimensional Optional Semimartingales

In option pricing, we often deal with options whose payoffs depend on multiple factors such as foreign exchange rates, stocks, etc. Usually, this leads to a knowledge of the joint distributions and complicated integration procedures. This paper develops an alternative approach that converts the option pricing problem into a dual one and presents a solution to the problem in the optional semimartingale setting. The paper contains several examples which illustrate its results in terms of the parameters of models and options.

Multi-regime foreign exchange rate model: Calibration and pricing

To price exotic foreign exchange (FX) options, a model needs to be selected for FX spot rate dynamics. The classic approach of modelling spot rates with Black–Scholes framework makes inappropriate assumptions of constant drift and volatility, resulting in mispricing. In this article, we investigate multi-regime Black–Scholes (MRBS) model for FX rate, with regime-switching behaviour of drift and volatility governed by a Markov chain. We derive an analytic formula for European FX call options via Fourier transform and present a Monte Carlo simulation algorithm for FX barrier option pricing. Further, we propose a calibration strategy with penalty. Finally, the empirical study shows that volatility smile can be recovered by MRBS model; calibrating MRBS with penalty gives stable calibrated parameters and small out-of-sample mean squared errors; combined use of our calibration methods and pricing algorithms provides reasonable prices for FX barrier options. Thus, MRBS model, together with proposed calibration-with-penalty strategy and pricing algorithm, provides a promising approach for FX option pricing.

Stochastic Modeling of Wind Derivatives with Application to the Alberta Energy Market

Wind-power generators around the world face two risks, one due to changes in wind intensity impacting energy production, and the second due to changes in electricity retail prices. To hedge these risks simultaneously, the quanto option is an ideal financial tool. The natural logarithm of electricity prices of the study will be modeled with a variance gamma (VG) and normal inverse Gaussian (NIG) processes, while wind speed and power series will be modeled with an Ornstein–Uhlenbeck (OU) process. Since the risk from changing wind-power production and spot prices is highly correlated, we must model this correlation as well. This is reproduced by replacing the small jumps of the Lévy process with a Brownian component and correlating it with wind power and speed OU processes. Then, we will study the income of the wind-energy company from a stochastic point of view, and finally, we will price the quanto option of European put style for the wind-energy producer. We will compare quanto option prices obtained from the VG process and NIG process. The novelty brought into this study is the use of a new dataset in a new geographic location and a new Lévy process, VG, apart from NIG.

Enterprise Value Assessment Based on Real Option Approach: The Case of SEMCORP

As the core power source of new energy vehicles, the lithium battery industry has ushered in a high-speed development, and has become the most widely used battery technology system. Due to the industry's high technical barriers, imperfect system, high volatility risk and other characteristics of the lithium battery enterprise value the overall assessment of the difficulty of ordinary investors to make a more accurate judgment. As an important representative of the lithium new energy materials industry, SEMCORP is in a golden development period. Based on the research results of previous researchers on enterprise value assessment, this paper examines the value assessment of n SEMCORP. Specifically, this paper applies a two-stage growth model to calculate the present value and evaluates the potential value separately with the help of an option pricing model to obtain its overall value. After analyzing, conclude that SEMCORP 's shares are overvalued. This paper can provide decision-making references for the investment and financing of Chinese lithium new energy materials enterprises, as well as further enriching the application of enterprise value assessment methods.

Martingale defects in the volatility surface and bubble conditions in the underlying

AbstractThe martingale theory of bubbles enables testing for asset price bubbles by analyzing option prices. As recently shown by Piiroinen et al. (Asset price bubbles: an option-based indicator, 2018), the SABR model is a strict local martingale when its parameterization implies a positive correlation between stock and option prices. We operationalize this theoretical result and analyze stock price bubbles in 2576 stocks over 26 years. Martingale defect conditions are absorbed quickly by options markets, but identify high proportions in significant and permanent changes in distribution of price returns, option trading activity, short interest in the underlying, and institutional ownership. These results confirm many common assumptions about stock price bubbles. These bubbles are temporally clustered, and tend to occur in periods of positive market development. Martingale defects are rare in market corrections, which indicates that they are a result of overoptimistic speculation.

Quantum Uncertainty and the Black-Scholes Formula

The publication of the Black-Scholes formula in 1973 appeared for the first time to put the pricing of financial options onto a rational and objective basis. Its adoption transformed the perception of option pricing from a form of gambling to scientific risk management, and led to a huge increase in options trading. However while the formula revolutionised the world of finance, and remains the industry-standard pricing model today, its proof relies on a number of assumptions about price behaviour which are often contested, such as that log prices follow a random walk with constant volatility, and that one can constantly buy or sell stocks and options without incurring transaction fees. This paper presents an alternative approach to option pricing, based on a quantum oscillator model of stock prices. In the quantum model, the bid/ask spread between buy and sell prices is treated as a fundamental measure of uncertainty, and volatility is not constant but exhibits a smile-like dependence on strike. We show how the Black-Scholes model and its assumptions lead to a systematic mispricing of commonly-traded options, while results can be improved by adopting the quantum model.

Linear extrapolation and model-free option implied moments

This study proposes an approach for assessing the effectiveness of linear extrapolation (LE) for the implied moment estimators even in cases in which the true values of implied moments are unknown. To this end, we develop truncation sensitivity functions for simulation and empirical analyses. LE proves effective for implied volatility, skewness, and kurtosis estimators. However, higher moment (i.e., implied skewness and kurtosis) estimators exhibit sensitivity to truncation, that is, the absence of option prices in the outermost region of the strike price domain, regardless of the use of LE to address truncation.

J&L Railroad: The Board Meeting

This short case could be handed out at the end of class discussion on “J&L Railroad” [UVA-F-1053] in preparation for the following class, or if students are more experienced with hedging and option pricing, the instructor may choose to cover both cases in a single class period. It is the companion case to “J&L Railroad” [UVA-F-1053], and presents more technical issues regarding the hedging problem by requiring students to understand option-pricing principles. The board likes the CFO's hedging recommendations, but it wants a more careful analysis of the bank's prices for its risk-management products: the caps and floors. Besides demanding an understanding of option pricing, this case puts particular emphasis on the calculation and use of implied volatility.

An obstacle problem arising from American options pricing: regularity of solutions

An obstacle problem arising from American options pricing: regularity of solutions

Lie symmetry, exact solutions and conservation laws of time fractional Black–Scholes equation derived by the fractional Brownian motion

Abstract The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation.

Reduced-Order Modeling for Heston Stochastic Volatility Model

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

Physics-informed convolutional transformer for predicting volatility surface

Predicting volatility is important for asset predicting, option pricing and hedging strategies because it cannot be directly observed in the financial market. The dynamics of the volatility surface is difficult to estimate. In this paper, we establish a novel architecture based on physics-informed neural networks and convolutional transformers. The performance of the new architecture is directly compared to other well-known deep-learning architectures, such as standard physics-informed neural networks, convolutional long-short term memory (ConvLSTM), and self-attention ConvLSTM. Numerical evidence indicates that the proposed physics-informed convolutional transformer network achieves a superior performance than other methods.

Application of propagating solitons to Ivancevic option pricing governing model and construction of first integral by Nucci's direct reduction approach

The current research aims to analyze the Ivancevic option pricing model and investigate its soliton solutions, which offer a comprehensive understanding of various related phenomena. The equation's Cauchy problem cannot be resolved through the inverse scattering transform, necessitating an analytical approach for accurate traveling wave solutions. To achieve this, the study employs the generalized logistic equation method and Nucci's reduction approaches to derive solitary wave solutions. Moreover, Nucci's reduction procedure is applied to formulate the first integral of a differential equation, ensuring conservation and exact solutions. In order to gauge the Ivancevic option pricing equation's sensitivity, a sensitivity analysis is conducted. At the end, the chaotic behavior of the considered model with the perturbed term is examined by using Lyapunov exponent approach.

How Integrated are Credit and Equity Markets? Evidence from Index Options

ABSTRACTWe study the extent to which credit index (CDX) options are priced consistent with S&P 500 (SPX) equity index options. We derive analytical expressions for CDX and SPX options within a structural credit‐risk model with stochastic volatility and jumps using new results for pricing compound options via multivariate affine transform analysis. The model captures many aspects of the joint dynamics of CDX and SPX options. However, it cannot reconcile the relative levels of option prices, suggesting that credit and equity markets are not fully integrated. A strategy of selling CDX volatility yields significantly higher excess returns than selling SPX volatility.

Volatility, Uncertainty, and Option Pricing

Amidst global and Chinese uncertainties, this paper delves into stock market volatility and option pricing within the current economic policy context. Focusing on China's stock market, it calculates returns and volatility changes in key indices, analyzing differences among indices and funds, along with temporal variations. Shared data characteristics and reflections on new market trends emerge. Employing methods such as the GARCH model, Black-Scholes formula, and Monte Carlo algorithm, it analyzes volatility and option pricing using extensive annual and monthly data. Visualizations showcase market change patterns. The study defines volatility as a measure of financial asset price fluctuation extent, reflecting asset risk. Higher volatility indicates pronounced price fluctuations and uncertainty, while lower volatility signifies smoother fluctuations and greater certainty. Merging data with volatility's significance, the study probes China's securities market uncertainty, investigating the link between option pricing and volatility. It concludes by identifying the connection between volatility, uncertainty, and option pricing, pointing to future research directions and challenges. Future work will track the latest market trends to enrich understanding.

Conformal prediction of option prices

The uncertainty associated with option price predictions has largely been overlooked in the literature. This paper aims to fill this gap by quantifying such uncertainty using conformal prediction. Conformal prediction is a model-agnostic procedure that constructs prediction intervals, ensuring valid coverage in finite samples without relying on distributional assumptions. Through the simulation of synthetic option prices, we find that conformal prediction generates prediction intervals for gradient boosting machines with an empirical coverage close to the nominal level. Conversely, non-conformal prediction intervals exhibit empirical coverage levels that fall short of the nominal target. In other words, they fail to contain the actual option price more frequently than expected for a given coverage level. As anticipated, we also observe a decrease in the width of prediction intervals as the size of the training data increases. However, we uncover significant variations in the width of these intervals across different options. Specifically, out-of-the-money options and those with a short time-to-maturity exhibit relatively wider prediction intervals. Then, we perform an empirical study using American call and put options on individual stocks. We find that the empirical results replicate those obtained in the simulation experiment.

Locally risk minimizing pricing of Asian option in a semi-Markov modulated market

We consider a regime-switching model where the stock volatility dynamics is a semi-Markov process. Under this model assumption, we find the locally risk-minimizing price of some Asian options with European-style exercise. The price function is shown to satisfy a non-local degenerate system of parabolic PDEs in dimension two with a terminal condition. We show this by deriving the F-S decomposition of the discounted contingent claim. The related Cauchy problem involving the PDE is shown to be equivalent to an integral equation (IE). The existence and uniqueness of the classical solution to the PDE are determined by studying the IE and using semigroup theory. To be more precise, we first obtain the mild solution of the PDE and then we show that the mild solution has sufficient regularity. The locally risk-minimizing hedging for the option has also been identified in this work. Finally, the computational aspects of Asian option prices have been discussed by solving the equation numerically.

Subdiffusive option price model with Inverse Gaussian subordinator

The paper focuses on the option price subdiffusive model under the unusual behavior of the market, when the price may not be changed for some time, which is a quite common situation in modern illiquid financial markets or during global crises. In the model, the risk-free bond motion and classical geometrical Brownian motion (GBM) are time-changed by an inverted inverse Gaussian($\mathit{IG}$) subordinator. We explore the correlation structure of the subdiffusive GBM stock returns process, discuss option pricing techniques based on the martingale option pricing method and the fractal Dupire equation, and demonstrate how it applies in the case of the $\mathit{IG}$ subordinator.

The fractional soliton solutions: shaping future finances with innovative wave profiles in option pricing system

<p>Financial engineering problems hold considerable significance in the academic realm, where there remains a continued demand for efficient methods to scrutinize and analyze these models. Within this investigation, we delved into a fractional nonlinear coupled system for option pricing and volatility. The model we examined can be conceptualized as a fractional nonlinear coupled wave alternative to the governing system of Black-Scholes option pricing. This introduced a leveraging effect, wherein stock volatility aligns with stock returns. To generate novel solitonic wave structures in the system, the present article introduced a generalized Ricatti mapping method and new Kudryashov method. Graphical representations, both in 3D and 2D formats, were employed to elucidate the system's response to pulse propagation. These visualizations enabled the anticipation of appropriate parameter values that align with the observed data. Furthermore, a comparative analysis of solutions was presented for different fractional order values. Additionally, the article showcases the comparison of wave profiles through 2D graphs. The results of this investigation suggested that the proposed method served as a highly reliable and flexible alternative for problem-solving, preserving the physical attributes inherent in realistic processes. To sum up, the main objective of our work was to conceptualize a fractional nonlinear coupled wave system as an alternative to the Black-Scholes option pricing model and investigate its implications on stock volatility and returns. Additionally, we aimed to apply and analyze methods for generating solitonic wave structures and compare their solutions for different fractional order values.</p>