A Black-Scholes-integrated Gaussian Process Model for American Option Pricing
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Acknowledging the lack of option pricing models that simultaneously have high prediction power, high computational efficiency, and interpretations that abide by financial principles, we suggest a Black-Scholes-integrated Gaussian process (BSGP) learning model that is capable of making accurate predictions backed with fundamental financial principles. Most data-driven models boast strong computational power at the expense of inferential results that can be explained with financial principles. Vice versa, most closed-form stochastic models (principle-driven) exhibit inferential results at the cost of computational efficiency. By integrating the Black-Scholes computed price for an equivalent European option into the mean function of the Gaussian process, we can design a learning model that emphasizes the strengths of both data- driven and principle-driven approaches. Using American (SPY) call and put option price data from 2019 May to June, we condition the Black-Scholes mean Gaussian Process prior with observed data to derive the posterior distribution that is used to predict American option prices. Not only does the proposed BSGP model provide accurate predictions, high computational efficiency, and interpretable results, but it also captures the discrepancy between a theoretical option price approximation derived by the Black-Scholes and predicted price from the BSGP model.
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Kim, Chiwan (2020). A Black-Scholes-integrated Gaussian Process Model for American Option Pricing . Honors thesis, Duke University. Retrieved from https://hdl.handle.net/10161/21549 .
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Options and their pricing : the Black-Scholes model
- Masters Thesis
- Moss, Cynthia
- Gold, Jerrold M.
- Vakilian, Ramin
- Breen, Stephen
- Mathematics
- California State University, Northridge
- Dissertations, Academic -- CSUN -- Mathematics.
- 2016-08-25T14:38:02Z
- http://hdl.handle.net/10211.3/175808
- by Cynthia Moss
- Includes bibliographical references (leaf 62)
- California State University, Northridge. Department of Mathematics.
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University of Wollongong Thesis Collection 2017+
A comprehensive study of option pricing with transaction costs.
Dong Yan , University of Wollongong
Degree Name
Doctor of Philosophy
School of Mathematics and Applied Statistics
Option pricing has become a key problem studied in academia as well as in finance industry ever since the publication of the seminal papers by Black and Scholes (1973) and Merton (1973). The Black-Scholes model has laid a solid foundation for the rapid development of various option pricing theories in the next half a century. However, the Black-Scholes model imposes some unrealistic assumptions in order to achieve analytical tractability, among which, the assumption of no transaction costs when trading stocks contradicts the fact there would always be costs associated with transactions of stocks in real markets. Although significant development has been made in studying the effects of transaction costs on option pricing in recent years, there are still gaps to fill in the literature.
In general, there are two different approaches to tackle the problem of pricing options with transactions costs: the hedging strategies and utility indifference pricing. Each of these two approaches has its own merits for pricing options under incomplete markets. The former method is easy to implement, but does not take investors’ preferences into consideration, the latter is very computational intensive. This thesis aims to provide a comprehensive study of option pricing with transaction costs under both the hedging strategy and the utility maximization theory, where the effects of transaction costs and stochastic volatility on option prices are analyzed, with the emphasis on American option prices and their optimal exercise boundaries.
The thesis is composed of seven chapters, with Chapter 1 being the introduction, Chapter 2 providing a review of preliminary knowledge which are necessary for our works in later chapters, and Chapter 3 presents a pricing model for European options with transaction costs under Heston-type stochastic volatility. This approach is formulated using the hedging strategy with some approximation in order to simplify the calculation of the expected transaction costs in hedging. This new approach is different from the existing literature (Mariani and SenGupta 2012) in two different aspects: Heston volatility is used and the option price does not depend on another option. The solution of the non-linear partial differential equation is obtained by a finite-difference scheme, proving a fair price range.
Then we focus on the hedging strategy and utility indifference method for pricing options with transaction costs under constant volatility in Chapter 4-6. Our ultimate goal is to study the American option pricing problem with transaction costs via utility indifference approach. Due to the nonlinearity resulted from the early exercise right of an American option, pricing American options via utility indifference approach raised two key issues: the optimal exercise boundary which needs to be solved as part of the solution and the heavy computational need for the none-linear problem. To solve such a complicated option pricing problem, we start with dealing two fundamental problems as the base of our ultimate goal. Firstly, we prove that the utility indifference approach is equivalent to the hedging strategy for the American option pricing problem in a complete market in Chapter 4. This problem is not well addressed due to the nonlinearity of the problem resulted from the optimal exercise boundary. A numerical study is conducted to deal with such an important problem. Then, in Chapter 5, we derive a new pricing approach for European options with transaction costs, where the ideas of hedging strategy and utility indifference approach are combined to achieve a balance of efficiency and accuracy. Our utility indifference approach reduces the dimension of the portfolio problem without options, thus, achieves better efficiency than the standard utility approach. In Chapter 6, we price American options with transaction costs via these two approaches. Since the impact of transaction costs on the American option price particularly on the most important feature of American options, is much less investigated, we provide a supplement study in this area by analyzing the effect of transaction costs to the optimal exercise price of an American option in addition to the option price itself through a utility-based approach. With a computationally efficient numerical scheme, we are able to demonstrate clearly how the optimal exercise price should be calculated and consequently how the option prices for the buyer and writer as well as the early exercise decision are affected by the inclusion of transaction cost.
The comparison presented in the thesis proves that utility indifference methods compared to hedging strategies produce more realistic option prices in the presence of transaction costs. The utility indifference approach proposed in this study could provide a useful, computational efficient way for pricing options with transaction costs.
Recommended Citation
Yan, Dong, A comprehensive study of option pricing with transaction costs, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2021. https://ro.uow.edu.au/theses1/1136
Since October 20, 2021
Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.
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Option pricing with stochastic volatility models
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- March 21, 2019
- Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
- Despite the success and the user-friendly features of Black-Scholes (BS) pricing, many empirical results in the option pricing literature have shown the departures from the BS model. The motivation of this dissertation starts from these departures. In the first part of dissertation, we take the popular approach of stochastic volatility and jump models that are known to give good explanations to the empirical phenomenon. In order to keep analytic tractability, we derive the Generalized Black-Scholes (GBS) formula by a proper conditioning in a general mixture framework. By taking advantage of this new version of option pricing formula, we propose an approximation scheme that is well suited for the conditional Monte Carlo method. The simulation study and Markov Chain Monte Carlo (MCMC) algorithm give an evidence of a huge computational time reduction without much loss of accuracy. In the second part, we provide a new prospective on the forecasting ability and information content of the BS implied volatility in the presence of nonzero leverage effect. The leverage effect, which is the correlation between the return and volatility process, is introduced to model the observed Black-Scholes implied volatility (BSIV) smile and its skewness. We provide a simple theoretical framework that explains and justifies the use of BSIV from at-the-money option for the volatility forecast. Based on this and simulation study, which show the sensitivity of the concavity of option price with respect to the underlying stock price (the gamma effect), we propose a new approach to improve option pricing accuracy by a proper account for the gamma effect.
- August 2008
- https://doi.org/10.17615/nsad-mr96
- Dissertation
- In Copyright
- Ji, Chuanshu
- University of North Carolina at Chapel Hill
- Open access
- October 11, 2010
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for Option Pricing by Ziqun Ye A thesis presented to the University of Waterloo in ful llment of the thesis requirement for the degree of Master of Mathematics in Statistics Waterloo, Ontario, Canada, 2013 c Ziqun eY 2013. I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,
The semiparametric pricing approach initially proposed by Lai and Wong (2004) provides a rst attempt to bridge the gap between model and market option prices. However, its application to the S&P 500 futures options is not a success, when the original additive regression splines are used for the nonparametric part of the pricing formula.
thesis aims to provide a comprehensive study of option pricing with transaction costs under both the hedging strategy and the utility maximisation theory, where the effects ... on the American option price, particularly on the most important feature of American options, is much less investigated, we provide a supplementary study in this area by ...
The paper reviews evidence from time series analysis, option prices and option price evolution regarding those risks, and discusses required compensation. David S. Bates Henry B. Tippie College of Business Department of Finance University of Iowa Iowa City, IA 52242-1000 and NBER [email protected].
et al introduced the "homogeneity hint" to constrain the set of possible outputs such that the option pricing function is homogeneous in asset price and strike price with degree 1 [9]. Considering option pricing and volatility estimation as a supervised learning problem, the Multi-Layer Perceptron (MLP) has been the workhorse neural network [15].
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Acknowledging the lack of option pricing models that simultaneously have high prediction power, high computational efficiency, and interpretations that abide by financial principles, we suggest a Black-Scholes-integrated Gaussian process (BSGP) learning model that is capable of making accurate predictions backed with fundamental financial principles.
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The thesis is composed of seven chapters, with Chapter 1 being the introduction, Chapter 2 providing a review of preliminary knowledge which are necessary for our works in later chapters, and Chapter 3 presents a pricing model for European options with transaction costs under Heston-type stochastic volatility. ... Option pricing has become a ...
of option pricing using arti cial neural networks in order to motivate the purpose and addition of this thesis in relation to existing literature. The earliest attempt at pricing options using non-parametric machine learning algorithms is byHutchinson et al.(1994). They investigate if any of four non-parametric machine learning
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We provide a simple theoretical framework that explains and justifies the use of BSIV from at-the-money option for the volatility forecast. Based on this and simulation study, which show the sensitivity of the concavity of option price with respect to the underlying stock price (the gamma effect), we propose a new approach to improve option ...
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