Option Pricing Research Articles

Is Reinforcement Learning Good at American Option Valuation?

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of “Decisions” made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates “Decisions” at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy.

Pricing exchange options under hybrid stochastic volatility and interest rate models

This paper investigates the pricing of exchange options under hybrid models integrating stochastic volatility and stochastic interest rates. It aims to achieve two primary objectives. First, we derive a closed-form pricing formula for exchange options under a two-factor Heston–Hull–White hybrid model, which accounts for long-term volatility and exhibits relatively broad correlations among the dynamics of asset prices, volatilities, and interest rates. Second, we explore the Heston model’s integration with a generalized single-factor stochastic interest rate model, illustrating that the price is not dependent on the specific form of the interest rate process. A closed-form pricing formula for exchange options under this framework is also derived. Our numerical experiments support the proposed formulas and elucidate the effects of various parameters on option prices.

Energy models in service of aquifer specific groundwater irrigation expansion in India

Abstract Pumping energy is a key component of the groundwater governance challenge, yet it is largely missing in the discourse on agricultural use of groundwater. A sub-category of literature studying groundwater-energy nexus tends to focus on groundwater depletion hotspots where entrenched interests and long-standing histories restrict the range of feasible energy pricing options. Using an agent based model, we estimate expected impacts of expanding groundwater irrigation under five different energy provision models in Odisha, an Indian state with among the lowest irrigation coverage, and therefore, free of path dependent policies. We find aquifer properties play a crucial role in mediating the groundwater-energy nexus. For this study region, on average, the maximum volume of water that can be pumped from a well of a specific depth in an alluvial aquifer is approximately 150 times the volume that can be pumped from a well in a hard-rock aquifer. Therefore, the risk of over-consumption and aquifer depletion is a far greater challenge in alluvial than hard-rock aquifers. Risks of groundwater consumption and depletion can be limited in hard-rock aquifers provided the number and depths of wells can be controlled. Capital subsidies for well construction could be an effective policy to increase irrigated area so long as economic incentives for digging deeper are not distorted. Our results imply that solar pumps are a relatively safe option for hard-rock regions where deep drawdowns naturally limit the extent of over-extraction. Solar pumps are also estimated to be among the most economical for expanding irrigation. Using a novel dataset comprising of biophysical and socioeconomic data, we find hard-rock regions to have limited irrigation coverage, high availability of annually replenishable groundwater, and high concentrations of marginalized farmers. Therefore, groundwater irrigation expansion in hard-rock areas could have dual benefits of ensuring future food security and targeting poverty reduction.

Alternatives to classical option pricing

Alternatives to classical option pricing

Current Developments and Limitations for Pricing Options

Valuing options holds significant importance within the derivative market, serving as a cornerstone for investors, traders, and financial institutions. Over time, economists have dedicated efforts to develop models that accurately reflect the complexities of this market environment. The evolution of economic theories can be observed through the examination of key option pricing models, which have undergone refinements and enhancements over the decades. By exploring models such as the Black-Scholes model, the Merton model, and the Cox-Ross-Rubinstein model, it is possible to gain valuable insights into the progression of economic thought in this field. Each of these models offers unique perspectives on option pricing, illustrating various principles and methodologies employed to assess and value options. Moreover, understanding the extensions and limitations of these models provides crucial insights into their applicability and effectiveness in real-world scenarios. By conducting an in-depth examination of these models, the objective of this article is to offer a nuanced comprehension of how economic theories have developed and adjusted to the ever-changing conditions of the derivative market.

A Binomial Tree-Based Empirical Study on the Biases of American Call Option

The application of financial derivatives, particularly options, has become pervasive in modern finance, offering effective tools for risk management. Among these, American options, allowing flexibility in exercising rights prior to expiration, dominate the derivatives market. Accurate pricing of American options is crucial for informed investment decisions and risk assessments. While various pricing models exist, the binomial tree model is welcomed for its simplicity and accessibility. However, inherent biases in pricing models raise questions about their efficacy across different sectors of the market. Based on the binomial tree model, this study empirically examines the pricing accuracy of American call options in both the technology sector and the industrial sector, by collecting data from top five largest companies in each sector. For the technology sector, samples are mainly from semiconductor manufacturers, software companies, etc., which are all situated in thriving fields; while those five companies from the industrial sector mainly belong to traditional industries, such as energy and engine suppliers. By analyzing prediction biases between two sectors, this research tries to give out reasons behind it and grasps a deeper understanding of option pricing nuances between sectors, and also the limitations of the binomial tree model.

Efficient option pricing in the rough Heston model using weak simulation schemes

We provide an efficient and accurate simulation scheme for the rough Heston model in the standard (H>0) as well as the hyper-rough regime ( -1/2 $ ]]> H > − 1 / 2 ). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [Bayer and Breneis, arXiv:2309.07023], and provides a weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second-order weak convergence, while the computational cost increases linearly with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral's HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method's accuracy and efficiency.

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

This paper concerns a local volatility model in which the volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power-law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper, we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black–Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

A Two-Step Longstaff Schwartz Monte Carlo Approach to Game Option Pricing

We proposed a two-step Longstaff Schwartz Monte Carlo (LSMC) method with two regression models fitted at each time step to price game options. Although the original LSMC can be used to price game options with an enlarged range of path in regression and a modified cashflow updating rule, we identified a drawback of such approach, which motivated us to propose our approach. We implemented numerical examples with benchmarks using binomial tree and numerical PDE, and it showed that our method produces more reliable results comparing to the original LSMC.

The risk and return of equity and credit index options

We develop a structural credit risk model, which allows us to price equity/credit indices and their options through the asset dynamics of index constituents. We estimate the model via MLE and find that equity and credit index option prices are well explained out-of-sample. Contrary to recent empirical findings, the two option markets are not inconsistently priced through the lens of our model. Returns on both options, while extreme, do not indicate any evidence of mispricing. Our analysis suggests that jointly addressing the pricing of various instruments requires properly attributing three different sources of systematic risk: asset, variance, and jump risks.

Analysis of Put and Call Option Pricing on BCA Stock Using the Black-Scholes Model: Financial and Risk Management Perspective

Analysis of Put and Call Option Pricing on BCA Stock Using the Black-Scholes Model: Financial and Risk Management Perspective

Pricing options on the maximum or the minimum of several assets with default risk

This paper presents analytical solutions for options on the maximum or the minimum of several assets with counterparty default risk before maturity, including derivations of several specific Greeks. To derive the solutions, we provide the joint distribution of the minimum value of one Brownian motion and the terminal values of all Brownian motions for correlated multidimensional Brownian motion. We then conduct a numerical analysis to examine the effects of counterparty risk on option prices.

Deterministic modelling of implied volatility in cryptocurrency options with underlying multiple resolution momentum indicator and non-linear machine learning regression algorithm

Modeling implied volatility (IV) is important for option pricing, hedging, and risk management. Previous studies of deterministic implied volatility functions (DIVFs) propose two parameters, moneyness and time to maturity, to estimate implied volatility. Recent DIVF models have included factors such as a moving average ratio and relative bid-ask spread but fail to enhance modeling accuracy. The current study offers a generalized DIVF model by including a momentum indicator for the underlying asset using a relative strength index (RSI) covering multiple time resolutions as a factor, as momentum is often used by investors and speculators in their trading decisions, and in contrast to volatility, RSI can distinguish between bull and bear markets. To the best of our knowledge, prior studies have not included RSI as a predictive factor in modeling IV. Instead of using a simple linear regression as in previous studies, we use a machine learning regression algorithm, namely random forest, to model a nonlinear IV. Previous studies apply DVIF modeling to options on traditional financial assets, such as stock and foreign exchange markets. Here, we study options on the largest cryptocurrency, Bitcoin, which poses greater modeling challenges due to its extreme volatility and the fact that it is not as well studied as traditional financial assets. Recent Bitcoin option chain data were collected from a leading cryptocurrency option exchange over a four-month period for model development and validation. Our dataset includes short-maturity options with expiry in less than six days, as well as a full range of moneyness, both of which are often excluded in existing studies as prices for options with these characteristics are often highly volatile and pose challenges to model building. Our in-sample and out-sample results indicate that including our proposed momentum indicator significantly enhances the model’s accuracy in pricing options. The nonlinear machine learning random forest algorithm also performed better than a simple linear regression. Compared to prevailing option pricing models that employ stochastic variables, our DIVF model does not include stochastic factors but exhibits reasonably good performance. It is also easy to compute due to the availability of real-time RSIs. Our findings indicate our enhanced DIVF model offers significant improvements and may be an excellent alternative to existing option pricing models that are primarily stochastic in nature.

The Impact of Stochastic Volatility on Option Pricing Using the Black-Scholes Model: Empirical Evidence from NVIDIA

This paper investigates the impact of volatility on option pricing using the Black-Scholes model. Utilizing historical stock and option data from NVIDIA, this study demonstrates the limitations of the Black-Scholes model in capturing dynamic market conditions. The empirical analysis reveals significant discrepancies between model predictions and actual market prices, highlighting the importance of accurate volatility estimation. The findings suggest that incorporating better volatility measures can improve the accuracy of option pricing models. Additionally, the implications of these findings for traders and risk managers are discussed, emphasizing the need for more sophisticated approaches to volatility estimation in financial markets. This study adds to the body of knowledge by presenting actual data regarding the efficiency of different volatility metrics in option pricing. Moreover, it underscores the necessity of continuous improvement in financial modeling techniques to adapt to changing market dynamics. The study's results align with previous research that has shown the limitations of the Black-Scholes model and the benefits of alternative volatility estimation methods. This paper also discusses the potential application of these findings to improving trading strategies and risk management practices. The practical implications are significant, suggesting that traders and risk managers should consider incorporating advanced volatility measures into their models to enhance decision-making processes and financial outcomes. The conclusions drawn from this study highlight the critical role of accurate volatility estimation in achieving more reliable option pricing and underscore the need for ongoing research and innovation in this field.

Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model

We present a unified, market-complete model that integrates both Bachelier and Black–Scholes–Merton frameworks for asset pricing. The model allows for the study, within a unified framework, of asset pricing in a natural world that experiences the possibility of negative security prices or riskless rates. Unlike the classical Black–Scholes–Merton, we show that option pricing in the unified model differs depending on whether the replicating, self-financing portfolio uses riskless bonds or a single riskless bank account. We derive option price formulas and extend our analysis to the term structure of interest rates by deriving the pricing of zero-coupon bonds, forward contracts, and futures contracts. We identify a necessary condition for the unified model to support a perpetual derivative. Discrete binomial pricing under the unified model is also developed. In every scenario analyzed, we show that the unified model simplifies to the standard Black–Scholes–Merton pricing under specific limits and provides pricing in the Bachelier model limit. We note that the Bachelier limit within the unified model allows for positive riskless rates. The unified model prompts us to speculate on the possibility of a mixed multiplicative and additive deflator model for risk-neutral option pricing.

Closed-Form Formula for the Conditional Moment-Generating Function Under a Regime-Switching, Nonlinear Drift CEV Process, with Applications to Option Pricing

An analytical derivation of the conditional moment-generating function (MGF) for a regime-switching nonlinear drift constant elasticity of variance process is established. The proposed model incorporates both regime-switching mechanisms and nonlinear drift components to better capture market phenomena such as volatility smiles and leverage effects. Regime-switching models can match the tendency of financial markets to often change their behavior abruptly and the phenomenon that the new behavior of financial variables often persists for several periods after such a change. Closed-form formulas for the MGF under various conditions, which are then applied for option pricing, are also derived. The efficacy and accuracy of the results are validated through a discrete Markov chain simulation. The results obtained from the proposed formulas completely match with those from MC simulations, while requiring significantly less computational time.

Option pricing in a stochastic delay volatility model

This work introduces a new stochastic volatility model with delay parameters in the volatility process, extending the Barndorff–Nielsen and Shephard model. It establishes an analytical expression for the log price characteristic function, which can be applied to price European options. Empirical analysis on S&P500 European call options shows that adding delay parameters reduces mean squared error. This is the first instance of providing an analytical formula for the log price characteristic function in a stochastic volatility model with multiple delay parameters. We also provide a Monte Carlo scheme that can be used to simulate the model.

Option pricing in the Heston model with physics inspired neural networks

Option pricing in the Heston model with physics inspired neural networks

An inter-provincial coordinate model under Renewable Portfolio Standards policy based on tradable green certificate options trading

The current territorial management model under Renewable Portfolio Standards in China cannot fully develop renewable energy due to the mismatch between high energy-consumption regions and renewable energy-rich regions. Therefore, we proposed an inter-provincial coordinate model based on tradable green certificate options trading (CMTOT), which includes four components: (1) A dual-objective optimization model is established to minimize the RPS target realization cost and maximize the pollutant emission reduction based on Blacke-Scholes option pricing model. (2) The identification of buyer and seller provinces of tradable green certificate (TGC) options based on the above model. (3) An optimization model in buyers' or sellers' cooperation alliance is established. (4) The cost allocation models by Shapley value and Nucleolus are established to ensure the fairness in different scenarios. A case study of Tianjin, Shandong, Shanxi and Inner Mongolia shows that the CMTOT can effectively reduce the RPS target realization cost by 16.79 % and increase the pollutant emission reduction by 4.32 %. A sensitivity analysis confirms the robustness of the CMTOT. The results show that the CMTOT can both fully develop renewable energy and effectively reduce pollutant emission. Therefore, we proposed some policy recommendations to support the implementation of the CMTOT.

A Time-Varying Feature Rearranged Convolutional Neural Network for Option Pricing

A Time-Varying Feature Rearranged Convolutional Neural Network for Option Pricing