Model uncertainty and the pricing of American options

The virtue of an American option is that it can be exercised at any time. This right is particularly valuable when there is model uncertainty. Yet almost all the extensive literature on American options assumes away model uncertainty. This paper quantifies the potential value of this flexibility by identifying the supremum on the price of an American option when we do not impose a model, but rather consider the class of all models which are consistent with a family of European call prices. The bound is enforced by a hedging strategy involving these call options which is robust to model error

On Arbitrage and Duality under Model Uncertainty and Portfolio Constraints

We consider the fundamental theorem of asset pricing (FTAP) and hedging prices of options under non-dominated model uncertainty and portfolio constrains in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value process is a local super-martingale under these measures. We also get the non-dominated optional decomposition with constraints. From this decomposition, we get duality of the super-hedging prices of European options, as well as the sub- and super-hedging prices of American options. Finally, we get the FTAP and duality of super-hedging prices in a market where stocks are traded dynamically and options are traded statically.Comment: Final version. To appear in Mathematical Financ

INCREASING THE ACCURACY OF OPTION PRICING BY USING IMPLIED PARAMETERS RELATED TO HIGHER MOMENTS

The inaccuracy of the Black-Scholes formula arises from two aspects: the formula is for European options while most real option contracts are American; the formula is based on the assumption that underlying asset prices follow a lognormal distribution while in the real world asset prices cannot be described well by a lognormal distribution. We develop an American option pricing model that allows non-normality. The theoretical basis of the model is Gaussian quadrature and dynamic programming. The usual binomial and trinomial models are special cases. We use the Jarrow-Rudd formula and the relaxed binomial and trinomial tree models to imply the parameters related to the higher moments. The results demonstrate that using implied parameters related to the higher moments is more accurate than the restricted binomial and trinomial models that are commonly used.option pricing, volatility smile, Edgeworth series, Gaussian Quadrature, relaxed binomial and trinomial tree models, Marketing, Risk and Uncertainty,

Nonparametric predictive inference for option pricing based on the binomial tree model

Nonparametric Predictive Inference (NPI) is a frequentist statistical method based on only fewer assumptions, which has been developed for and applied to, several areas in statistics, reliability and finance. In this thesis, we introduce NPI for option pricing in discrete time models. NPI option pricing is applied to vanilla options and some types of exotic options. We first set up the NPI method for the European option pricing based on the binomial tree model. Rather than using the risk-neutral probability, we apply NPI to get the imprecise probabilities of underlying asset price movements, reflecting more uncertainty than the classic models with the constant probability while learning from data. As we assign imprecise probabilities to the option pricing procedure, surely, we get an interval expected option price with the upper and lower expected option prices as the boundaries, and we named the boundaries the minimum selling price and the maximum buying price. The put-call parity property of the classic model is also proved to be followed by the NPI boundary option prices. To study its performance, we price the same European options utilizing both the NPI method and the Cox, Ross, and Rubinstein binomial tree model (CRR) and compare the results in two different scenarios, first where the CRR assumptions are right, and second where the CRR model assumptions deviate from the real market. It turns out that our NPI method, as expected, cannot perform better than the CRR in the first scenario with small size historical data, but as enlarging the history data size, the NPI method's performance gets better. For the second scenario, the NPI method performs better than the CRR model. The American option pricing procedure is also presented from an imprecise statistical aspect. We propose a novel method based on the binomial tree. We prove through this method that it may be optimal for an American call option without dividends to be exercised early, and some influences of the stopping time toward option price prediction are investigated in some simulation examples. The conditions of the early exercise for both American call and put options are derived. The performance study of the NPI pricing method for American options is evaluated via simulation in the same two scenarios as the European options. Through the performance study, we conclude that the investor using the NPI method behaves more wisely in the second scenario than the investor using the CRR model, and faces to more profit and less loss than what it does in the first scenario. The NPI method can be applied to exotic options if the option payoffs are a monotone function of the number of upward movements in the binomial tree, like the digital option and the barrier option discussed in this thesis. Otherwise, either we can manipulate the binomial tree in order to assign the upper and lower probabilities, for instance, the look-back option with the float strike price, or a new probability mass is needed to be assigned to the payoff binomial tree according to the option definition which is attractive and challenging for future study

Topics in Optimal Stopping and Fundamental Theorem of Asset Pricing.

In this thesis, we investigate several problems in optimal stopping and fundamental theorem of asset pricing (FTAP). In Chapter II, we study the controller-stopper problems with jumps. By a backward induction, we decompose the original problem with jumps into controller-stopper problems without jumps. Then we apply the decomposition result to indifference pricing of American options under multiple default risk. In Chapters III and IV, we consider zero-sum stopping games, where each player can adjust her own stopping strategies according to the other’s behavior. We show that the values of the games and optimal stopping strategies can be characterized by corresponding Dynkin games. We work in discrete time in Chapter III and continuous time in Chapter IV. In Chapter V, we analyze an optimal stopping problem, in which the investor can peek epsilon amount of time into the future before making her stopping decision. We characterize the solution of this problem by a path-dependent reflected backward stochastic differential equation. We also obtain the order of the value as epsilon goes to zero. In Chapters VI-VIII, we investigate arbitrage and hedging under non-dominated model uncertainty in discrete time, where stocks are traded dynamically and liquid European-style options are traded statically. In Chapter VI we obtain the FTAP and hedging dualities under some convex and closed portfolio constraints. In Chapter VII we study arbitrage and super-hedging in the case when the liquid options are quoted with bid-ask spreads. In Chapter VIII we investigate the dualities for sub and super-hedging prices of American options. Note that for these three chapters, since we work in the frameworks lacking dominating measures, many classical tools in probability theory cannot be applied. In Chapter IX, we consider arbitrage, hedging, and utility maximization in a given model, where stocks are available for dynamic trading, and both European and American options are available for static trading. Using a separating hyperplane argument, we get the result of FTAP, which implies the dualities of hedging prices. Then the hedging dualities lead to the duality for the utility maximization.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111416/1/zhouzhou_1.pd

From Black-Scholes to Online Learning: Dynamic Hedging under Adversarial Environments

We consider a non-stochastic online learning approach to price financial options by modeling the market dynamic as a repeated game between the nature (adversary) and the investor. We demonstrate that such framework yields analogous structure as the Black-Scholes model, the widely popular option pricing model in stochastic finance, for both European and American options with convex payoffs. In the case of non-convex options, we construct approximate pricing algorithms, and demonstrate that their efficiency can be analyzed through the introduction of an artificial probability measure, in parallel to the so-called risk-neutral measure in the finance literature, even though our framework is completely adversarial. Continuous-time convergence results and extensions to incorporate price jumps are also presented

Real Option Valuation of a Portfolio of Oil Projects

Various methodologies exist for valuing companies and their projects. We address the problem of valuing a portfolio of projects within companies that have infrequent, large and volatile cash flows. Examples of this type of company exist in oil exploration and development and we will use this example to illustrate our analysis throughout the thesis. The theoretical interest in this problem lies in modeling the sources of risk in the projects and their different interactions within each project. Initially we look at the advantages of real options analysis and compare this approach with more traditional valuation methods, highlighting strengths and weaknesses ofeach approach in the light ofthe thesis problem. We give the background to the stages in an oil exploration and development project and identify the main common sources of risk, for example commodity prices. We discuss the appropriate representation for oil prices; in short, do oil prices behave more like equities or more like interest rates? The appropriate representation is used to model oil price as a source ofrisk. A real option valuation model based on market uncertainty (in the form of oil price risk) and geological uncertainty (reserve volume uncertainty) is presented and tested for two different oil projects. Finally, a methodology to measure the inter-relationship between oil price and other sources of risk such as interest rates is proposed using copula methods.Imperial Users onl

Robust pricing--hedging duality for American options in discrete time financial markets

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglb\"ock et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.Comment: 29 page