Option Pricing Framework Research Articles

A rational finance explanation of the stock predictability puzzle

AbstractWe address the stock predictability puzzle, a challenge in the stock market often discussed in behavioral finance. Our approach formulates a statistical model within rational finance, avoiding reliance on behavioral finance assumptions, and integrates stock return predictability into the Black–Scholes option pricing framework. Empirical analysis focuses on the predictability of stock prices by option and spot traders, introducing a forward‐looking measure we term “implied excess predictability.” Results show that option traders' predictability of stock returns positively correlates with moneyness, whereas for spot traders, this relationship is inverse. These findings suggest a potential asymmetry in stock price predictability between spot and option traders. Additionally, we demonstrate the importance of incorporating stock return predictability into option pricing formulas, particularly for options with strike prices significantly different from the stock price. Conversely, when moneyness is close to unity, predictability is not integrated into option pricing, indicating equal information among spot and option traders. Comparison of volatility measures reveals the difference between implied and realized variances or variance risk premia as potential predictors of stock returns.

Conversion risk on 19th century French consols and embedded options: A simple exercise

This paper investigates the conversion risk exposure of French government consols on the Paris Stock Exchange market during the 19th century. Using the classic option pricing framework, our results indicate that market investors were able to detect the presence of an embedded conversion option and price it quite accurately almost two centuries ago.

Carr and Wu’s (2020) framework in the oil ETF option market

This paper studies the information inferred from the Carr and Wu’s (2020) formula based on a new option pricing framework in the United States Oil Fund (USO) options. We first document the term structure and dynamics of the risk-neutral variance and covariance rates which lead to a “U”-shaped implied volatility smile with a positive curvature. We then investigate the return predictability of the innovations in the risk-neutral variance and covariance rates (DRNV and DRNC) and their term structures (TRNV and TRNC) and find that DRNC is a significant and robust predictor to forecast daily, weekly and monthly USO excess returns in both statistical and economic terms based on in-sample and out-of-sample tests.

SABR equipped with AI wings

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.

Correcting for risk premium on Extended Generalised Leland Models: an empirical study on Dow Jones Industrial Average (DJIA) index options

The relative option pricing performance of Extended Generalised Leland models is examined in this study. We generalise the extended Leland models based on the implied adjusted volatility introduced in the models. Non-parametric framework is fitted into parametric option-pricing framework based on the Leland models to achieve a more realistic option pricing. To reflect the real probability measure, the implied adjusted information is corrected in terms of risk premium. This study concentrates mainly in examining the option-implied information produced by the models after correcting for risk-premium. Data extracted from DJIA index options are employed in this study, which covers the period from January 2009 until the end of 2015. We discovered that the option-implied volatility, which is priced using the Extended Generalised Leland models, especially after being corrected for risk-premium factor improves the option valuation accuracy significantly.

A Comparison of Artificial Neural Networks and Bootstrap Aggregating Ensembles in a Modern Financial Derivative Pricing Framework

In this paper, the pricing performances of two learning networks, namely an artificial neural network and a bootstrap aggregating ensemble network, were compared when pricing the Johannesburg Stock Exchange (JSE) Top 40 European call options in a modern option pricing framework using a constructed implied volatility surface. In addition to this, the numerical accuracy of the better performing network was compared to a Monte Carlo simulation in a separate numerical experiment. It was found that the bootstrap aggregating ensemble network outperformed the artificial neural network and produced price estimates within the error bounds of a Monte Carlo simulation when pricing derivatives in a multi-curve framework setting.

DeepOption: A novel option pricing framework based on deep learning with fused distilled data from multiple parametric methods

The remarkable performance of deep learning is based on its ability to learn high-level features by processing large amounts of data. This exceptionally superior performance has attracted the attention of researchers studying option pricing. However, option data are more expensive and less accessible than other types of data and are imbalanced because of the liquidity of options. This motivated us to propose a new option pricing and delta-hedging framework called DeepOption. This framework, which is based on deep learning, can improve the performance even when applying imbalanced real option data. In particular, the framework fuses simulated big data, known as distilled data, obtained using various traditional parametric methods. The proposed model employs the following three-stage training approach: Our model is pre-trained using big distilled data after it is fine-tuned using real option data through transfer learning. Finally, a delta branch is added to the model and trained. We experimentally evaluated the proposed method using three sets of real option data, namely S&P 500 European call options, EuroStoxx50 call options, and Hang Seng Index put options. Our experimental results on option pricing demonstrate that our proposed model outperforms parametric methods and other machine learning methods. Specifically, our model, which uses pre-training with distilled data, reduces the overall mean absolute percentage error (MAPE) by more than 50%, compared with that of a deep learning model using only real option data without pre-training.

A unified option pricing model with Markov regime-switching double stochastic volatility, stochastic interest rate and jumps

We consider a general option pricing framework incorporating the double Heston stochastic volatility, stochastic interest rate, jumps and Markov regime switching. Under the proposed framework, we derive the analytical pricing formulae for European options using Fourier transform technique. Numerical examples illustrate that the option prices and the implied volatility curves under different regimes vary clearly, and the effects of regime-switching and jumps on the option price differ.

The Chinese warrant bubble: A fundamental analysis

AbstractWe investigate the information content in Chinese warrant prices based on an option pricing framework that incorporates short‐selling and margin‐trading constraints in the underlying stock market. We show that Chinese warrant prices can be explained under this pricing framework. On the basis of this new model, we develop a price deviation measure to quantify stock market investors' unobserved demand for short selling or margin trading due to market constraints. We find that warrant‐price deviations are driven by underlying stock valuation to a great extent. Chinese warrant prices, save for the time around expiration dates, are better characterized as derivatives than as pure bubbles.

Optimal pooling strategies under heterogeneous risk classes

PurposeThe purpose of this paper is to determine optimal pooling strategies from the perspective of an insurer's shareholders underlying a default probability driven premium loading and convex price-demand functions.Design/methodology/approachThe authors use an option pricing framework for normally distributed claims to analyze the net present value for different pooling strategies and contrast multiple risk pools structured as a single legal entity with the case of multiple legal entities. To achieve the net present value maximizing default probability, the insurer adjusts the underlying equity capital.FindingsThe authors show with the theoretical considerations and numerical examples that multiple risk pools with multiple legal entities are optimal if the equity capital must be decreased. An equity capital increase implies that multiple risk pools in a single legal entity are generally optimal. Moreover, a single risk pool for multiple risk classes improves in relation to multiple risk pools with multiple legal entities whenever the standard deviation of the underlying claims increases.Originality/valueThe authors extend previous research on risk pooling by introducing a default probability driven premium loading and a relation between the premium level and demand through a convex price-demand function.

Application of Option Pricing Framework for Analyzing and Predicting Stock Prices in Energy Markets

In this study, the evaluation of the pricing framework for predicting West Texas Intermediate crude oil stock was implemented where detailed analysis with varying changepoint shows that an arbitrage-free forward price can be derived from the buy-and hold strategy in the energy market thereby enabling investors in the market willing to be salvage from the market uncertainties as well as Arrow-Debreu situations to execute a spot or forward contracts depending on the time and place the market becomes favorable.

Improving the option pricing performance of GARCH models in inefficient market

Understanding the relation between option pricing and market efficiency is important. Indeed, emphasizing this relation generates new insights that are appropriate in practice. These insights give a better understanding of the current limitations of the option pricing and hedging methods. This article thus aims to improve the performance of the option pricing approach. To start, the relation between the option pricing methodology and the informational market efficiency was discussed. It is, therefore, useful, before proceeding to apply the standard risk-neutral approach, to check the efficiency assumption. New modified GARCH processes were used to model the dynamics of the asset returns in the option pricing framework. The new considered approaches allow describing the dynamic of returns when the market is inefficient. Using real data on CAC 40 index, the performance of different models as a function of maturity and moneyness was studied. The in-sample analysis, interested in the stability of the pricing models across time, showed that the new approach, developed under the affine GARCH process, is the most accurate. The study of the out-of-sample performance, which aims to evaluate the forecasting ability of different approaches, confirmed the results of the in-sample analysis. For the optional portfolio hedging, always the best hedging approach is that obtained under the affine GARCH model. After a regression study, it was found that the difference between theoretical and observed option values can be explained by factors, which are not taken into account in the proposed pricing formulae.

A Tractable Framework for Option Pricing with Dynamic Market Maker Inventory and Wealth

We develop a tractable dynamic model of an index option market maker with limited capital. We solve for the variance risk premium and option prices as a function of the asset dynamics and market maker option holdings and wealth. The market maker absorbs end users’ positive demand and requires a more negative variance risk premium when she incurs losses. We estimate the model using returns, options, and inventory and find that it performs well, especially during the financial crisis. The restrictions imposed by nested existing reduced-form stochastic-volatility models are strongly rejected in favor of the model with a market maker.

A Stochastic Volatility Model With Realized Measures for Option Pricing

Based on the fact that realized measures of volatility are affected by measurement errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both observed returns and realized measures to the latent conditional variance. A semi-analytical option pricing framework is developed for this class of models. In addition, we provide analytical filtering and smoothing recursions for the basic specification of the model, and an effective MCMC algorithm for its richer variants. The empirical analysis shows the effectiveness of filtering and smoothing realized measures in inflating the latent volatility persistence—the crucial parameter in pricing Standard and Poor’s 500 Index options.

Option Pricing for TGARCH-M with GED Based on Improved EEMD

ABSTRACTAlthough option pricing plays an important role in risk management and investments, accurately pricing options remains challenging because of the increasingly complicated fluctuations in asset price processes. This article proposes a new option pricing model, the threshold GARCH with generalized error distribution (TGARCH-M with GED), based on an improved EEMD. By considering three key factors in the option pricing framework: different frequency risks, information asymmetry and non-normality, we show this novel model can capture more volatility features. Furthermore, the empirical results indicate we obtain better parameter estimation results and fewer pricing errors through comparative analysis. Our research provides meaningful guidance and new insights in the fields of risk management and investment.

Valuing strategic investments under stochastic interest rates: A real option approach

One of the most challenging issues in management is the valuation of strategic investments. In particular, when undertaking projects such as an expansion or the launch of a new brand, or an investment in R&D and intellectual capital, which are characterized by a long-term horizon, a firm has also to face the risk due to the interest rate. In this work, we propose to value investments subject to interest rate risk using a real options approach (Schulmerich, 2010). This task requires the typical technicalities of option pricing, which often rely on complex and time-consuming techniques to value investment projects. For instance, Schulmerich (2010) is, to the best of our knowledge, the first work where the interest rate risk is considered for real option analysis. Nevertheless, the valuation of investment projects is done by employing binomial trees, which are computationally very expensive. In the current paper, a different modeling framework (in continuous-time) for real option pricing is proposed which allows one to account for interest rate risk and, at the same time, to reduce computational complexity. In particular, the net present value of the cash inflows is specified by a geometric Brownian motion and the interest rate is modeled by using a process of Vasicek type, which is calibrated to real market data. Such an approach yields an explicit formula for valuing various kinds of investment strategies, such as the option to defer and the option to expand. Therefore, the one proposed is the first model in the field of real options that accounts for the interest rate risk and, at the same time, offers an easy to implement formula which makes the model itself very suitable for practitioners. An empirical analysis is presented which illustrates the proposed approach from the practical point-of-view and highlights the impact of stochastic interest rates in investment valuation.

PRICING SOVEREIGN CONTINGENT CONVERTIBLE DEBT

We develop a pricing model for Sovereign Contingent Convertible bonds (S-CoCo) with payment standstills triggered by a sovereign’s Credit Default Swap (CDS) spread. We model CDS spread regime switching, which is prevalent during crises, as a hidden Markov process, coupled with a mean-reverting stochastic process of spread levels under fixed regimes, in order to obtain S-CoCo prices through simulation. The paper uses the pricing model in a Longstaff–Schwartz American option pricing framework to compute future state contingent S-CoCo prices for risk management. Dual trigger pricing is also discussed using the idiosyncratic CDS spread for the sovereign debt together with a broad market index. Numerical results are reported using S-CoCo designs for Greece, Italy and Germany with both the pricing and contingent pricing models.

Pricing Average Price Advertising Options When Underlying Spot Market Prices Are Discontinuous

Advertising options have been recently studied as a special type of guaranteed contracts in online advertising, which are an alternative sales mechanism to real-time auctions. An advertising option is a contract which gives its buyer a right but not obligation to enter into transactions to purchase page views or link clicks at one or multiple pre-specified prices in a specific future period. Different from typical guaranteed contracts, the option buyer pays a lower upfront fee but can have greater flexibility and more control of advertising. Many studies on advertising options so far have been restricted to the situations where the option payoff is determined by the underlying spot market price at a specific time point and the price evolution over time is assumed to be continuous. The former leads to a biased calculation of option payoff and the latter is invalid empirically for many online advertising slots. This paper addresses these two limitations by proposing a new advertising option pricing framework. First, the option payoff is calculated based on an average price over a specific future period. Therefore, the option becomes path-dependent. The average price is measured by the power mean, which contains several existing option payoff functions as its special cases. Second, jump-diffusion stochastic models are used to describe the movement of the underlying spot market price, which incorporate several important statistical properties including jumps and spikes, non-normality, and absence of autocorrelations. A general option pricing algorithm is obtained based on Monte Carlo simulation. In addition, an explicit pricing formula is derived for the case when the option payoff is based on the geometric mean. This pricing formula is also a generalized version of several other option pricing models discussed in related studies.

Taxes and risk-taking behavior: evidence from mergers and acquisitions in the G7 nations

PurposeThis study aims to examine how corporate taxes affect corporate risk-taking decisions.Design/methodology/approachThis study examines corporate risk-taking by analyzing how a firm’s asset risk changes following an acquisition carried out by publicly listed companies in the G7 nations. To measure the asset risk of a firm, this study uses the option pricing framework in Merton (1974).FindingsConsistent with an implication of the Merton (1974) framework, the findings show that firms take more risk in their investment decisions when tax rates are high. Moreover, the tax effects wane for firms with a relatively large borrowing opportunity and this suggests that the risk-taking incentive from taxes is moderated by the reputation concern in the debt market, lending support to the Diamond (1989) reputation-building model. The empirical results also show that the tax-induced risk-taking incentive is restrained by creditor rights. Overall, the study reveals an important role of taxes in the structure of corporate investment decisions.Practical implicationsThe implications of this study can be beneficial to policymakers when considering the alteration of tax rates, as it will affect the riskiness of firm investment decisions.Originality/valueThis study provides a better understanding of the role of taxes on risk-taking and also contributes to the growing body of evidence supporting tax effects of risk-taking. The relationship between taxes and risk-taking has proven that the corporate taxation is one of the key factors that firms consider during their selection of risky investments. Unlike previous studies, this research is the first to investigate the change in asset risk, estimating by the option pricing framework, through studying a particular event: mergers and acquisitions.

Analytical valuation for geometric Asian options in illiquid markets

Classical option pricing theories are usually built on the paradigm of competitive and frictionless markets, while ignoring the impact of market liquidity on asset prices. In this paper, we study the pricing problems of the continuously monitored fixed and floating strike Asian options in imperfectly liquid markets. First, we assume that the dynamics of the underlying asset price follow a liquidity-adjusted Black–Scholes model. And then we derive the partial differential equation for the continuous fixed and floating strike geometric Asian options based on the delta-hedging strategy. Meanwhile, we also present the PDE governing the price of the continuous fixed and floating strike arithmetic Asian options. Second, the analytical pricing formulas for the continuous fixed and floating strike geometric Asian call options are derived by using the PDE method. Besides, we demonstrate the put–call parity relations for the continuous fixed and floating strike geometric Asian options. Finally, numerical experiments are performed to illustrate the accuracy and efficiency of the proposed liquidity-adjusted option pricing model through comparing the analytical solution with Monte Carlo simulation. Furthermore, we investigate the sensitivity of the continuous arithmetic and geometric Asian option prices to the liquidity factors. The numerical results support our idea of introducing market liquidity effect into option pricing framework.