Neural network pricing of american put options

In this paper we use neural networks (NN), a machine learning method, to price American put options. We propose two distinct NN models – a simple one and a more complex one. The performance of two NN models is compared to the popular Least-Square Monte Carlo Method (LSM). This study relies on market American put option prices, with four large US companies as underlying – Bank of America Corp (BAC), General Motors (GM), Coca-Cola Company (KO) and Procter and Gamble Company (PG). Our dataset includes all options traded from December 2018 to March 2019. All methods show a good accuracy, however, once calibrated, NNs do better in terms of execution time and Root Mean Square Error (RMSE). Although on average both NN models perform better than LSM, the simpler model (NN model 1) performs quite close to LSM. On the other hand our NN model 2 substantially outperforms the other models, having a RMSE ca. 40% lower than that of the LSM. The lower RMSE is consistent across all companies, strike levels and maturities.info:eu-repo/semantics/publishedVersio

A Valuation of Options to Extend the Time Charter Period: The Application of Artificial Neural Networks

Options in the shipping market consist of paper freight options and physicaloptions attached to charterparties or newbuilding contracts. The options most frequently associated with the physical shipping market are options to extend the charter period on time charters and additional shipment options attached to contracts of affreightment. In both the paper market and the physical market, the value of freight options, in practice, is estimated mostly by referring to the forward curves of freight derivatives. The option on freight has different properties from its financial counterparts, and the straightforward adoption of theoretical models like the Black-Sholes option pricing model (BSM) has not produced promising results. So far, academic research in this field has also hardly made a meaningful contribution to practice and is in need of further elaboration. This research focuses on the period extension options attached to time charter contracts. In this paper, extension options, which have the property of options on futures, were conceptually transformed into regular European call options before the BSM was applied. The efficient market hypothesis (EMH), which justifies the parity of the performance of a long-term charter to that of repetitive short-term charters for the same period, worked as the basis of the conversion. The option values determined by the BSM were compared with the actual realized values to verify the applicability of the model. Additionally, a robust relationship mapping model, artificial neural networks (ANN), was employed to derive the option values, and then the results were compared with those of the BSM. The ANN is recently expanding its application to business, finance, and management, and is drawing attention in the areas of discrimination, pattern recognition, and forecasting. This study is meaningful as the first-time application of both the closed-form solution and the ANN to the valuation of physical freight options. In particular, the application of the ANN is expected to lead the active adoption of machine learning tools in the analysis of shipping market behavior. The result of this research can contribute to enhancing the quality of chartering decisions by providing criteria to determine option values. The decision rationality to be achieved by the model can be contrasted with the fact that, so far, decisions have been made with a ‘rule-ofthumb’ valuation of options. The extension option, in reality, tends to be granted to charterers with better credit, even free of charge when the market is at its trough. Hence, the results could also be used as a tool to quantify counterparty risk. This analysis is limited to the Panamax bulk market, which has long-term data consistency. The extension of the study to other segments of bulk shipping such as Cape, Supramax and even to wet bulk markets will help generalize the model’s performance. The result also implies the ‘forecasting’ performance of the ANN because the value of the extension options contains the information required to make freight market forecasts. Therefore, the study can be extended to the area of forecasting. In that case, the performances can be tested with additional input variables, such as forward market features, to the BSM input variables.List of Tables .................................................................................................................................. vi List of Figures ............................................................................................................................... vii 요 약 ............................................................................................................................................. viii Abstract .............................................................................................................................................. x Chapter 1 Introduction……………………………………………………………..…….1 1.1 Background .............................................................................................................................. 1 1.2 Research Purposes ................................................................................................................ 2 1.3 Research Scope ...................................................................................................................... 3 1.4 Research Procedures ............................................................................................................ 4 1.5 Contribution ............................................................................................................................. 6 1.6 Structure of the Paper ......................................................................................................... 7 Chapter 2 Bulk Shipping and Freight Options…………………………..……….8 2.1 Bulk Shipping as Freight Trading ................................................................................... 8 2.1.1 Freight trading ........................................................................................................ 8 2.1.2 Risk management ................................................................................................ 14 iv 2.2 Freight Options .................................................................................................................... 15 2.2.1 Paper freight options ......................................................................................... 16 2.2.2 Physical freight options .................................................................................... 18 Chapter 3 Literature Review…………………………………………………………..23 3.1 Asian Option Approximation .................................................................................... 23 3.2 Option on Futures ......................................................................................................... 25 3.3 European Options .......................................................................................................... 25 3.3.1 Binomial option pricing model ................................................................... 26 3.3.2 Black-Scholes option pricing model ....................................................... 26 3.4 Efficient Market Hypothesis and Expectations Theory ................................. 27 3.5 Artificial Neural Networks .......................................................................................... 30 Chapter 4 Data and Basic Assumptions…………………………………………..35 4.1 Data ...................................................................................................................................... 35 4.2 Basic Assumptions ......................................................................................................... 37 Chapter 5 Black-Scholes Option Pricing Model………………………………..40 5.1 The BSM ............................................................................................................................. 40 5.2 Input Variables ................................................................................................................. 43 v Chapter 6 Artificial Neural Networks………………………………………………44 6.1 Network Structure .......................................................................................................... 46 6.2 Normalization .................................................................................................................. 51 Chapter 7 Results………………………………………………………………………….53 7.1 Measurements ................................................................................................................. 53 7.2 Black-Scholes Option Pricing Model ..................................................................... 54 7.3 Artificial Neural Networks .......................................................................................... 55 7.4 Comparison ....................................................................................................................... 58 Chapter 8 Conclusion………………………………………………………………..60 References ...................................................................................................................................... 63 Appendix I ...................................................................................................................................... 68 Appendix II .................................................................................................................................... 74Docto

Statistical hedging with neural networks

This thesis investigates the problem of statistical hedging with artificial neural networks (ANNs). The statistical hedging is a data-driven approach that derives hedging strategy from data and hence does not rely on making assumptions of the underlying asset. Consider an investor who sells an option and wishes to hedge it with some amount of underlying asset. ANNs can be used to determine this number by minimising the discrete hedging error. In the first chapter, we provide a comprehensive literature review of papers on the topic of using ANNs for option pricing and hedging, as well as other related ones. Based on our research experience and summary of papers, we provide several advices that we believe are critical in using ANNs for option pricing and hedging problem. In particular, we point out an existing information leakage issue in the literature when preparing data. This review is invaluable for future researchers who are wish to work in this topic. In the second chapter, we consider the hedging problem in the single period case. The ANN is designed to output a hedging ratio directly, instead of first learning to prices. The experiments are taken on simulated Black-Scholes (BS), Heston, end-of-day S&P 500, and tick Euro Stoxx 50 datasets. The results show the ANN can significantly outperform the BS benchmark, but is only comparable to linear regressions on sensitivities. Hence, we illustrate that the edge of the two statistical hedging methods arises mainly from the existence of the leverage effect. Moreover, the information leakage found in the literature is reproduced. It’s shown that a wrong in- and out-of-sample split can overestimate the performance of statistical hedging methods. This leakage can be further exploited by tagging independent variables. Building on the previous chapter, sensitivity analysis are given in the third chapter. They concern data cleaning on the two historical datasets, different simulation parameters of the two simulated datasets, and data preparations. In particular, we show that the statistical hedging methods can also exploit drift and convexity apart from the leverage effect. In the last chapter, our model is extended to multiple periods on the Black-Scholes data. The replicating portfolio is rebalanced with a fixed frequency over the option’s life. We show again the ANN and linear regression methods outperform the BS benchmark, and their performance are comparable