Robust Upper Bounds For American Options And Exotic Options

Option pricing has been playing a very important role in the finance.The traditional model-based pricing research has successfully derived analytical or numerically efficient pricing formulas,and these pricing theories can partially explain some typical characteristics and phenomena observed in financial mar-kets.However,there is no single model or pricing formula which can perfectly explain all the empirical facts in financial markets.Model-specification risk has always been an inevitable problem in this kind of methods,and attracts more and more attention from researchers.The more assumptions the option pricing depends on,the more risk of the model mis-specification.Therefore,the option pricing theory should reduce the number of assumptions depends on.Specially,when an option pricing theory only rely on no-arbitrage assumption,we call it a robust one.This paper mainly focus on how to determine model-free upper bounds for the price of options under the no-arbitrage assumption.Concern about model-specification risk has inspired a large number of s-tudies on model-independent upper or lower bounds for the price of options.These researches mainly focus on European-type options.In existing litera-ture on standard American options and exotic options pricing,most of them derive model-independent upper bounds for American option and exotic op-tions based on standard European options.Few studies in this field determine model-free and closed-form upper bounds for standard American options and exotic options with as few assumptions as possible.The fewer assumptions the option pricing depends on,the less the risk of model mis-specification is,and the more accurate the result is.Furthermore,the upper bounds for the prices of standard American options and exotic options with precise analyt-ical expression have more extensive application scenarios.They can provide quality and quantity control for option valuation under multi-dimensional s-tate variables,develop approximate option pricing formula,set trading limit and manage the market risk of option portfolios.Last but not the least,the analytical upper bounds can also be used to obtain the information implied by market prices of options,such as implied volatility and implied market beta,so as to provide a reference for the investment decision and risk management.Therefore,under the only assumptions of no-arbitrage,it is of great theoret-ical and practical significance to determine model-independent,robust,and analytical upper bounds for prices of standard American options and exotic options.Based on the existing literature,this paper applies no-regret learning the-ory to option pricing.By exploiting the gradient trading strategy proposed by Demarzo(2016),robust,model-independent and analytical upper bounds on standard American options and common exotic options'prices are established.Robustness here means that the upper bounds rely only on no-arbitrage prin-ciple,and do not require the specific dynamics of the underlying asset price,the complete market,continuous trading,and any pricing kernel.This paper adopts the method of combining theory and empirical analysis to carry out the research.Specifically,this article first generalize the result in Demarzo(2016)and derive robust,model-free and analytical upper bounds for stan-dard European options with risk-free interest rate r=0.The extended results can not only be used for the pricing of standard European options in a wider range,but also help to determine robust and analytical upper bounds for stan-dard American options and exotic options.Furthermore,using similar gradient trading strategies,direct and indirect valuation methods are used to price s-tandard American options in the discrete time trading model,respectively,and robust and model-free upper bounds with exact analytical expression are de-termined for them.Last but not the least,the article focuses on exotic options and derives robust and closed-form upper bounds for five exotic options,i.e.chooser options,forward start options,European-type and American-type ex-change options,shout options and Asian options.In addition to the theoretical proof,this article also examines the performance of the upper bounds for the prices of standard European options and American options by numerical simu-lation and empirical analysis based on the market price of 50ET F index option and S&P 100 index option,which show that our upper bounds are empirically meaningful.Seven chapters are contained in this paper.The first chapter is the intro-duction.which draws out the problems to be studied in this paper,explains several important concepts in detail,and describes the research methods,ba-sic framework and major contributions of this paper.The second chapter is the theoretical basis and literature review.The theoretical basis of options is introduced,and then the current research status of option pricing and online learning theory is summarized and evaluated.The third chapter is the corner-stone of theoretical research in full.A simple discrete-time trading model is constructed and the gradient trading strategies is introduced.With r>0,the performance of a dynamically adjusted portfolio with the gradient strategy at time T is proved.In the fourth chapter,the result in De Marzo et al.(2016)on the upper bound for standard European call options is generalized by al-lowing r>0 and underlying assets to pay dividend.Numerical simulations and empirical analysis based on 50ET F index options show that these upper bounds are empirically meaningful.In chapter five,a similar gradient trading strategy is utilized to determine robust and model-independent upper bounds for standard American options,whose underlying asset pays a known contin-uously compounding dividend yield or cash dividends.Numerical simulation and empirical analysis based on S&P 100 index options show that these upper bounds are empirically meaningful.In chapter six,with r>0 and the known continuously compounding dividend yield paid by the underlying asset,robust and closed-form upper bounds for five exotic options,i.e.chooser options,for-ward start options,European-type and American-type exchange options,shout options and Asian options,are derived.Chapter seven concludes the paper,which summarizes the research content and prospects the possible research directions in the future.The main contributions of this article include the following four aspects:First,Demarzo et al.(2016)established the upper bound of robustness for European call option prices under the assumption that the risk-free interest rate r=0 and no dividends are paid.This article generalizes their conclusions.Under the conditions of r>0,robust,model-independent and analytical upper bounds for standard European options with the underlying assets paying known dividends or cash dividends are derived.The expanded results can not only be used for a wider range of standard European option pricing,but also help to determine robust and analytical upper bounds for the prices of standard American options and exotic options that follow in this article.Second,this article applies the no-regret learning theory to standard American option pricing.Using a similar gradient trading strategy in the discrete-time trading model,robust,model-independent and analytical upper bounds for standard American options with the underlying assets paying known dividends or cash dividends are derived.When pricing American options,this paper uses two methods to determine the upper bound of the price.The first method is the direct valuation method,which uses a gradient trading strate-gy to construct a dynamically adjusted investment portfolio to directly obtain the upper bounds for the prices of American options.Different from the first method that directly constructs the investment portfolio,the second method is the indirect valuation method.The proof idea is:we first establish a In-equality relationship between the price of an American option and that of its corresponding European option under the assumption of risk-free arbitrage.Then,using this inequality relationship as a bridge,the existing upper bound for the European option price is used to indirectly obtain the upper bound for the American option price.Under special circumstances,the upper bounds obtained by the two methods will coincide with each other.For more gener-al cases,directly take the minimum of the two as the tighter upper bounds for the prices of American options.In the existing model-independent Amer-ican option pricing research,the upper bounds for standard American option proposed by Chen and Yeh(2002)and Chung and Chang(2007)need to cal-culate conditional expectations under a risk-neutral measure.Therefore,the specification of the risk-neutral measure of the underlying asset is required to determine the upper bounds for American options.This will undoubtedly bring the risk of model mis-specification.Chaudhury(2006)and Hobson and Neuberger(2017)use a series of European option prices to determine the up-per bound for standard American options.When European option prices are unobservable,the upper bounds for American options prices are no longer in-dependent of the model.Different from the existing upper bounds for standard American options,the upper bounds for standard American options obtained by the no-regret learning strategy in this paper have an accurate analytical formula.The only unobservable parameter in the analytical upper bound ex-pression is the maximal quadratic variation of the excess log return q~2(?_T).These new upper bounds for standard American options enrich the research on model-independent American option pricing.Third,this article applies the no-regret learning theory to exotic option pricing.Using the expanded gradient trading strategy,under the condition that r>0 and the underlying asset pays a known dividend rate,robust,model-independent and analytical upper bounds for chooser options,forward start options,European-type and American-type exchange options,shout op-tions,and Asian options are determined.In the existing model-independent pricing research on exotic options,related studies usually use standard Euro-pean option prices to determine the upper bounds for exotic option based on static or dynamic super-replication strategies.These studies mainly focus on European-style exotic options.Few studies have conducted model-independent option pricing for chooser options,American-type exchange options,and shout options.When standard European option prices are unobservable,these up-per bounds are no longer model-independent.Different from the existing upper bounds for exotic options,the upper bounds for exotic options obtained by the no-regret learning strategy in this paper have an accurate analytical formula.The only unobservable parameter in the analytical upper bound expression is the maximal quadratic variation of the underlying asset return paths.These new upper bounds for exotic options enrich the research on model-independent exotic option pricing.Fourth,in addition to theoretical proofs,this paper also examines the performance of our upper bounds for standard European options and American options through numerical simulation and empirical analysis based on market prices of options.These numerical and empirical results show that our upper bounds are not only valuable in theory,but also empirically meaningful.This article has derived robust and model-independent upper bounds with analytical expression for standard European options,standard American op-tions and common exotic options.However,there are still three aspects that need to be improved.First,numerical simulation and empirical analysis sug-gest that the upper bounds are relatively loose when standard European op-tions and American options are out of the money.Secondly,the upper bounds derived in this paper is not the only one.Can we find the optimal upper bound with the no-regret learning strategy?Finally,this paper does not de-rive robust lower bounds for various options by gradient strategy.Improving the upper bounds for the prices of out of the money options,deriving robust lower bounds,and finding the optimal upper bound for various options will be the direction of further research in the future.