Actual Option Prices Research Articles

A Prediction of Option Price via Two Volatility Computations with Application to a 50ETF Option

Financial derivative trading is integral to stock markets, leading to high option price volatility due to increased trading volume. Determining a reasonable option price is complex and requires extensive research in fields like Economics, Applied Mathematics, and Finance Engineering. The Black-Scholes (BS) equation provides a scientific pricing tool for options by considering five parameters: stock price (S), option strike price (K), risk-free interest rate (r), time to expiration (τ), and volatility (σ). Notably, all model parameters except volatility σ can be directly observed from market data, necessitating the determination of this parameter from historical data when applying the BS model in practice.In this report, we examine two widely used computation methods for volatility. The first method involves a simple statistical calculation of historical data, resulting in the historical volatility (HV). The second method utilizes the BS model in a "backward" manner: given any previous option price and four other parameters, we solve the BS equation to obtain the implied volatility (IV). To determine such an IV parameter, we propose a generalized fixed-point iterative solver for solving a complex nonlinear equation. By employing a well-designed initial guess, we demonstrate that this fixed-point solver achieves global and rapid convergence.The two volatilities lead to different predictions of the option price. This report examines if the BS model accurately predicts the option price using these volatilities. It presents an empirical study on a 50ETF option, where we use t-tests to check the hypothesis and determine the relationship between predicted and actual option prices. The study finds that using implied volatility in the BS model provides significantly more accurate predictions compared to historical volatility.

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.

Analysis about the Black-Scholes asset price under the regime-switching framework

Assuming that the macroeconomic environment can be transformed into a two-district system, that is, the path of financial asset prices is uncertain, we track and study the motion of stocks and other asset price process under the conditional Black-Scholes model, and give the economical explanation of the mathematical formula. Further, we derive and analyze an option pricing formula for the Black-Scholes asset model under the condition that the risk-free interest rate is regime-switching too. The method in this article is applied to model the log rate of return of the Tencent stock in a two-district market environment. And the obtained parameter values are used to calculate the option price. In narrowing the gap with actual option prices, our method outperforms the classical option pricing model point by point. Compared with the general and pure mathematical model derived work and the empirical study work, our study does more work on the economic characteristics analysis and interpretation of the mathematical models, and plays a certain role in linking the results of mathematical models with empirical research.

A Study on the Applicability of Black-Scholes Option Pricing Model in Chinese Stock Index Option Market

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.

Comparative Analysis of the Application of Monte Carlo Model and BSM Model in European Option Pricing

At present, the expansion of China's domestic options market brings positive factors and risks, and in order to avoid risks, it is crucial to choose a suitable model for option pricing. This article provides an example of an option for the underlying asset of the SSE 50 ETF. Using the BSM model and the Monte Carlo model for the selected option pricing, and comparing the actual option price. It is found that the pricing efficiency of the Monte Carlo model is higher than that of the BSM model when the number of simulations reaches 30,000 times in the call option, and there is little difference between the two in the put option pricing. It is recommended to prefer the Monte Carlo model when pricing call options and the more convenient BSM model when pricing put options.

A Comparative Study of Support Vector Machine and Artificial Neural Network for Option Price Prediction

Option pricing has become one of the quite important parts of the financial market. As the market is always dynamic, it is really difficult to predict the option price accurately. For this reason, various machine learning techniques have been designed and developed to deal with the problem of predicting the future trend of option price. In this paper, we compare the effectiveness of Support Vector Machine (SVM) and Artificial Neural Network (ANN) models for the prediction of option price. Both models are tested with a benchmark publicly available dataset namely SPY option price-2015 in both testing and training phases. The converted data through Principal Component Analysis (PCA) is used in both models to achieve better prediction accuracy. On the other hand, the entire dataset is partitioned into two groups of training (70%) and test sets (30%) to avoid overfitting problem. The outcomes of the SVM model are compared with those of the ANN model based on the root mean square errors (RMSE). It is demonstrated by the experimental results that the ANN model performs better than the SVM model, and the predicted option prices are in good agreement with the corresponding actual option prices.

European option pricing model with generalized Ornstein\u2013Uhlenbeck process under stochastic earning yield and stochastic dividend yield

This paper aims to examine and establish the models for European option pricing which include parameters of stochastic dividend yield and stochastic earning yield. We generalize the Ornstein–Uhlenbeck process and define it as generalized Ornstein–Uhlenbeck process. We have learned that the firm stocks, according to Black–Scholes–Merton structure, obey the geometric Brownian motion process. Under a stochastic earning yield, the dividend yield complies with the generalized Ornstein–Uhlenbeck process. The firm dividend randomly deviates from the earning yield flow because of the presence of stochastic components of dynamic Wiener process of generalized Ornstein–Uhlenbeck. In this study, we model the stock price with stochastic earning yield, and stochastic dividend yield to be taking account stochastic market price of risk parameter which is mean-reverting as well. We developed explicit formulae for European call option pricing calculations. From numerical simulation, we could evaluate the performance of our new model that could be compared with other notable option pricing models by using actual option price data. The outcomes prove that our new model performance is best when compared with others.

Option Market (In)efficiency and Implied Volatility Dynamics After Return Jumps

In informationally efficient financial markets, option prices and this implied volatility should immediately be adjusted to new information that arrives along with a jump in underlying's return, whereas gradual changes in implied volatility would indicate market inefficiency. Using minute-by-minute data on S&P 500 index options, we provide evidence regarding delayed and gradual movements in implied volatility after the arrival of return jumps. These movements are directed and persistent, especially in the case of negative return jumps. Our results are significant when the implied volatilities are extracted from at-the-money options and out-of-the-money puts, while the implied volatility obtained from out-of-the-money calls converges to its new level immediately rather than gradually. Thus, our analysis reveals that the implied volatility smile is adjusted to jumps in underlying's return asymmetrically. Finally, it would be possible to have statistical arbitrage in zero-transaction-cost option markets, but under actual option price spreads, our results do not imply abnormal option returns.

Third-order short-time expansions for close-to-the-money option prices under the CGMY model

ABSTRACTA third-order approximation for close-to-the-money European option prices under an infinite-variation CGMY Lévy model is derived, and is then extended to a model with an additional independent Brownian component. The asymptotic regime considered, in which the strike is made to converge to the spot stock price as the maturity approaches zero, is relevant in applications since the most liquid options have strikes that are close to the spot price. Our results shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behaviour of option prices near expiration when the strike is close to the spot price. In particular, a new type of transition phenomenon is uncovered in which the third-order term exhibits two distinct asymptotic regimes depending on whether or . Unlike second-order approximations, the expansions herein are shown to be remarkably accurate so that they can actually be used for calibrating some model parameters. For illustration, we calibrate the volatility of the Brownian component and the jump intensity C of the CGMY model to actual option prices.

Implied Binomial Trees with Cubic Spline Smoothing

The binomial model is a workhorse of numerical option pricing. But the plain vanilla model produces a risk-neutral probability density for the stock price at expiration that becomes lognormal in the limit. This is consistent with Black–Scholes’ assumptions but not with actual option prices in the market. An alternative is an implied tree, which begins with the risk-neutral density extracted from the market prices for options with the desired maturity. The procedure constructs a lattice that is consistent with the desired distribution. An N-step tree has N + 1 terminal nodes, so fitting the tree to the market prices can be quite time consuming. The author proposes a simplification that greatly increases efficiency with negligible cost in accuracy. The trick is to calibrate only a subset of the terminal nodes to prices in the options market and fill in the other nodes by cubic spline interpolation. <b>TOPICS:</b>Options, quantitative methods

The Application of Fractional Brownian Motion in Option Pricing

In this text, Fractional Brown Motion theory during random process is applied to research the option pricing problem. Firstly, Fractional Brown Motion theory and actuarial pricing method of option are utilized to derive Black-Scholes formula under Fractional Brown Motion and form corresponding mathematical model to describe option pricing. Secondly, based on BYD stock, estimation model on volatility of this stock is given to analyze and calculate the stock price volatility. Finally, make instance analysis for BYD’s option. Based on market data of BYD’s stock and option, calculate the actual option price and theoretical price of BYD by BlackScholes formula under Fractional Brown Motion. Compare the forecast price of this stock option given by model with actual price, relatively good effect is obtained, and then conclude that the model has relatively strong applicability.

Variation in Option Returns: The Importance of Moneyness, Maturity and Skewness

This paper examines the differences in average options returns for a variety of option maturities. A simulation which assumes that the returns in the underlying asset are normally distributed and that options are priced according to Black-Scholes produces average returns which are positive and decreasing in the time to maturity for Call options of all moneyness levels and negative and increasing in the time to maturity for Put options of all moneyness levels. However, a simulation based on actual S&P 500 Index returns, which include a sub-period of high, positively skewed returns (bull market) and a sub-period of lower negatively skewed returns (bear market), produces some different results. Specifically, short-dated out-the-money (OTM) Calls exhibit negative average returns in the bull market, and Calls of all maturities deliver negative average returns in the bear market. Long-dated Put options deliver positive average returns in the bear market. I conclude with some empirical work on actual option prices, which must be interpreted with caution because of the illiquidity in all but the shortest-dated at-the-money options. The empirical evidence suggests short-dated OTM Calls actually produced positive returns in the bull market because of relative underpricing consistent with the well-known implied volatility skew.

Optimal forecasting of option prices using particle filters and neural networks

A novel online forecasting system on financial option prices is implemented in this paper by combining a particle filter with a neural network. The particle filter (PF) model is based on the Black-Sholes (BS) model for prediction, while the neural network is employed to capture the nonlinear residuals between the actual option prices and the PF predictions. Taking the transaction data of option prices on the Taiwan composite stock index (TWSI), we found that the forecasting performance of the hybrid model is superior to the traditional extended Kalman filter model and the PF model. Our results can help investors make a good hedge on their option positions.

Accuracy Tests Of American Put Valuation Models For Pharmaceutical Equity Options

This study compares the performance of the Macmillan (1986), Barone-Adesi and Whaley (1987) MBAW model, Ju and Zhong (1999) MQuad model, Black-Scholes model and Put-Call Parity in pricing American put options of pharmaceutical companies. These are evaluated using actual option prices for three companies over 2000 to 2005, as opposed to the previous use of generated binomial option pricing data. We compare the forecasting accuracy by maturity, moneyness, and variance estimate. Contrary to Ju and Zhong (1999), we find that the MBAW outperforms the other models for at-the-money, and out-of-the-money options. The MQuad model performs best for in-the-money options. However, in this case both the MBAW and MQuad models estimates are very similar. Our results are consistent irrespective of option maturities and volatility estimates. These findings raise questions regarding the practice of using actual prices as the true value, compared to the previous results that use simulated prices.

A Cumulative Prospect Theory Approach to Option Pricing

It is a well known from the empirical option pricing literature, that actual option prices show persistent and systematic deviations from theoretical values under standard pricing assumptions. While a substantial number of enhancements have been proposed, these approaches typically leave investor’s preferences towards risk unmodified. In this paper, we study option prices under cumulative prospect theory. We distinguish two prospect option pricing models, based on whether cash flows are either considered to be segregated or aggregated over time. A numerical example is provided to highlight the effect of different degrees of prospect behavior on option prices. In an empirical application to S&P 500 index options, our models are compared with benchmark models that are frequently used in the option pricing literature. Results suggest that our prospect option pricing models significantly improve the fitting performance over the Black-Scholes model and that especially the aggregated version’s performance is at least equivalent to the stochastic volatility model of Heston.

Effective options trading strategies based on volatility forecasting recruiting investor sentiment

This study investigates an algorithm for an effective option trading strategy based on superior volatility forecasts using actual option price data for the Taiwan stock market. The forecast evaluation supports the significant incremental explanatory power of investor sentiment in the fitting and forecasting of future volatility in relation to its adversarial multiple-factor model, especially the market turnover and volatility index which are referred to as the investors’ mood gauge and proxy for overreaction. After taking into consideration the margin-based transaction cost, the simulated trading indicates that a long or short straddle 15days before the options’ final settlement day based on the 60-day in-sample-period volatility forecasting recruiting market turnover achieves the best average monthly return of 15.84%. This study bridges the gap between option trading, market volatility, and the signal of the investors’ overreaction through the simulation of the option trading strategy. The trading algorithm based on the volatility forecasting recruiting investor sentiment could be further applied in electronic trading and other artificial intelligence decision support systems.

A Prospect Approach to Option Pricing

It's a well known empirical fact that actual option prices show persistent and systematic deviations from Black-Scholes option values. While a substantial number of enhancements have been proposed in the literature, these approaches typically leave investor's preferences towards risk unmodified. Recently, empirical studies using option prices find support for non-concave utility functions proposed by Kahneman and Tversky. In this paper, we study option prices in an economy where investors are loss averse over fluctuations in the value of their financial wealth. The design of our pricing model is influenced by prospect theory, including behavioral aspects like risk attitude, mental accounting and probability perception. The theoretical marginal prospect option writer is risk averse in the domain of gains, risk taking in the domain of losses; (s)he overestimates small probabilities and underestimates large probabilities of the option being exercised in-the-money. The pricing framework can help to explain the implied volatility pattern typically observed from actual option prices. Empirical analysis on European call options on the S&P 500 index shows that prospect option pricing models significantly improve the fitting performance in in-sample, as well as in out-of-sample analysis. Further, the analysis shows that the marginal investor's behavior is different from the prospect theoretical observations. In stead of being risk averse in the domain of gains, risk taking in the domain of losses, and overestimate small and underestimate large probabilities of the option expiring in-the-money, marginal writers of call options on the S&P 500 index display the reverse behavior.

Online option price forecasting by using unscented Kalman filters and support vector machines

This study develops a hybrid model that combines unscented Kalman filters (UKFs) and support vector machines (SVMs) to implement an online option price predictor. In the hybrid model, the UKF is used to infer latent variables and make a prediction based on the Black–Scholes formula, while the SVM is employed to model the nonlinear residuals between the actual option prices and the UKF predictions. Taking option data traded in Taiwan Futures Exchange, this study examined the forecasting accuracy of the proposed model, and found that the new hybrid model is superior to pure SVM models or hybrid neural network models in terms of three types of options. This model can help investors for reducing their risk in online trading.

Implied Binomial Trees in Excel without VBA

We show how to implement a Rubinstein (1994) implied binomial tree using an Excel spreadsheet, but without having to use visual basic in Excel (VBA). We demonstrate both the optimization needed to generate implied ending risk-neutral probabilities from a set of actual option prices and the backwards recursion needed to solve for the entire implied tree. By using only standard Excel functions, and not resorting to VBA, we make this option pricing technique immediately accessible to both practitioners and academics. With minimal preparation, this technique can also be introduced to the undergraduate classroom.

Study on option pricing in an incomplete market with stochastic volatility based on risk premium analysis

We focus on an option pricing mechanism in a market, for which the underlying asset has a stochastic volatility. This model generally belongs to the class of incomplete market models, and hence, a no-arbitrage option price is not uniquely determined. However, in the actual market, options are traded at some price. This fact may suggest that the market participant compromises on the risk caused by stochastic volatility and that the balance of risk aversion of sellers and buyers determines the market price. To analyze this mechanism, we introduce a concept of the risk premium for stochastic volatility (RPSV) and develop a method to estimate the RPSV implied in actual option prices. In the method, homogeneous RPSV is not required to allow the segmentation of the market. Estimated RPSV of Nikkei 225 options are almost positive, and certainly depend on the strike price and time to maturity.