Three essays on the valuation of American-style options

Classificação: G13Esta tese aborda a avaliac¸ ˜ao de opc¸ ˜oes de estilo Americano, com e sem barreira, em tr ˆes artigos distintos: A. Pricing and Static Hedging of American-style Options under the Jump to Default Extended CEV Model Este artigo avalia (e faz o hedging) de opc¸ ˜oes de estilo Americano atrav´es do static hedge approach (SHP) proposto por Chung and Shih (2009) e estende a literatura em duas direc¸ ˜oes. Primeiramente, o SHP ´e adaptado ao modelo jump to default extended CEV (JDCEV) de Carr and Linetsky (2006), e s˜ao avaliadas opc¸ ˜oes de estilo Americano sem barreira sobre activos com possibilidade de fal ˆ encia. A robustez e a efici ˆencia das soluc¸ ˜oes de avaliac¸ ˜ao propostas, s˜ao comparadas com o optimal stopping approach de Nunes (2009), no ˆambito dos modelos JDCEV e constant elasticity of variance (CEV) de Cox (1975), considerando diferentes valores para o parˆametro de elasticidade. Em segundo lugar, tanto o SHP como o optimal stopping approach s˜ao estendidos para a avaliac¸ ˜ao de opc¸ ˜oes de estilo Americano com um cap. B. General Put-Call Symmetry for American-style Barrier Options Este artigo deriva relac¸ ˜oes de simetria put-call para opc¸ ˜oes de estilo Americano com uma e duas barreiras. Usando a t ´ecnica de mudanc¸a de numer´ ario proposta por Geman et al. (1995) e Schroder (1999) estas simetrias s˜ao derivadas sem impor restric¸ ˜oes pr ´evias sobre o processo estoc´ astico seguido pelo activo subjacente. Os resultados s˜ao testados atrav´es de uma extensa an´ alise num´ erica sob o modelo constant elasticity of variance. C. In-Out Parity Relations and Early Exercise Boundaries for American-style Barrier Options Este artigo deriva novas relac¸ ˜oes de paridade in-out para puts de estilo Americano com uma barreira inferior e calls de estilo Americano com uma barreira superior. Mais importante, ´e proposta uma nova representac¸ ˜ao da fronteira de exerc´ıcio antecipado para opc¸ ˜oes de estilo Americano com dupla barreira knock-out, em termos da fronteira de exerc´ıcio ´optimo de uma opc¸ ˜ao de estilo Americano com uma s´o barreira. Assim sendo, o m´etodo static hedge portfolio ´e estendido para a avaliac¸ ˜ao de opc¸ ˜oes de estilo Americano com dupla barreira knock-out. Os resultados s˜ao testados atrav´es de uma extensa an´ alise num´ erica sob os modelos geometric Brownian motion e constant elasticity of variance.This thesis addresses the valuation of American-style standard and barrier options in three separate and self-contained papers: A. Pricing and Static Hedging of American-style Options under the Jump to Default Extended CEV Model This paper prices (and hedges) American-style options through the static hedge approach (SHP) proposed by Chung and Shih (2009) and extends the literature in two directions. First, the SHP approach is adapted to the jump to default extended CEV (JDCEV) model of Carr and Linetsky (2006), and plain-vanilla American-style options on defaultable equity are priced. The robustness and efficiency of the proposed pricing solutions are compared with the optimal stopping approach offered by Nunes (2009), under both the JDCEV framework and the nested constant elasticity of variance (CEV) model of Cox (1975), using different elasticity parameter values. Second, both the SHP and the optimal stopping approaches are extended to the valuation of American-style capped options. B. General Put-Call Symmetry for American-style Barrier Options This paper derives put-call symmetries for American-style single and double barrier options. Using the change of numeraire technique proposed by Geman et al. (1995) and Schroder (1999) we are able to derive these symmetries without imposing previous assumptions on the process followed by the underlying asset. Our results are tested through an extensive numerical analysis run under the constant elasticity of variance model. C. In-Out Parity Relations and Early Exercise Boundaries for American-style Barrier Options This paper derives new in-out parity relations for American-style puts with a down barrier and American-style calls with an up barrier. More importantly, we also propose a novel representation for the early exercise boundary of American-style double knock-out options in terms of the simpler optimal stopping boundary for a nested single barrier American-style option. Therefore, we are able to extend the static hedge portfolio approach to the valuation of American-style double barrier knockout options. Our results are tested through an extensive numerical analysis run under the geometric Brownian motion (GBM) and the constant elasticity of variance models

Employee Stock Options: Much More Valuable Than You Thought

Previous papers have argued that trading restrictions can result in a typical employee stock option having a subjective value (certainty equivalent value) that is substantially less than its Black-Scholes value. However, these analyses ignore the manager’s ability to (at least partially) control the risk level within the firm. In this paper, we show how managerial control can lead to such options having much larger certainty equivalent values for employees who can exercise control. We also show that the potential for early exercise is substantially less valuable with managerial control. The certainty equivalent value for a European option with managerial control can easily exceed the Black-Scholes value for a comparable option without control. However, it is questionable whether Black-Scholes is an appropriate benchmark for an option where the underlying process exhibits controlled volatility. We show how to obtain a risk-neutral valuation for such an option. That risk-neutral value can be substantially greater or less than the Black- Scholes value. Furthermore, the option’s certainty equivalent value can also be greater or less than its risk-neutral value.

Exercise Strategies for American Exotic Options under Ambiguity

We analyze several exotic options of American style in a multiple prior setting and study the optimal exercise strategy from the perspective of an ambiguity averse buyer in a discrete time model of Cox–Ross–Rubinstein style. The multiple prior model relaxes the assumption of a known distribution of the stock price process and takes into account decision maker’s inability to completely determine the underlying asset’s price dynamics. In order to evaluate the American option the decision maker needs to solve a stopping problem. Unlike the classical approach ambiguity averse decision maker uses a class of measures to evaluate her expected payoffs instead of a unique prior. Given time-consistency of the set of priors an appropriate version of backward induction leads to the solution as in the classical case. Using a duality result the multiple prior stopping problem can be related to the classical stopping problem for a certain probability measure – the worst-case measure. Therefore, the problem can be reduced to identifying the worst-case measure. We obtain the form of the worstcase measure for different classes of exotic options explicitly exploiting the observation that the options can be decomposed in simpler event-driven claims.American option, optimal stopping, ambiguity, uncertainty aversion

Essays on American Options

This thesis deals with the pricing of American equity options exposed to correlated interest rate and equity risks. The first article, American options on high dividend securities: a numerical investigation by F. Rotondi, investigates the Monte Carlo-based algorithm proposed by Longstaff and Schwartz (2001) to price American options. I show how this algorithm might deliver biased results when valuing American options that start out of the money, especially if the dividend yield of the underlying is high. I propose two workarounds to correct for this bias and I numerically show their strength. The second article, American options and stochastic interest rates by A. Battauz and F. Rotondi introduces a novel lattice-based approach to evaluate American option within the Vasicek model, namely a market model with mean-reverting stochastic interest rates. Interestingly, interest rates are not assumed to be necessarily positive and non standard optimal exercise policy of American call and put options arise when interest rates are just mildly negative. The third article, Barrier options under correlated equity and interest rate risks by F. Rotondi deals with derivatives with barrier features within a market model with both equity and interest rate risk. Exploiting latticebased algorithm, I price European and American knock-in and knock-out contracts with both a discrete and a continuous monitoring. Then, I calibrate the model to current European data and I document how models that assume either a constant interest rate, or strictly positive stochastic interest rates or uncorrelated interest rates deliver sizeable pricing errors

Managerial Responses to Incentives: Control of Firm Risk, Derivative Pricing Implications, and Outside Wealth Management

We model a firm’s value process controlled by a manager maximizing expected utility from restricted shares and employee stock options. The manager also dynamically controls allocation of his outside wealth. We explore interactions between those controls as he partially hedges his exposure to firm risk. Conditioning on his optimal behavior, control of firm risk increases the expected time to exercise for his employee stock options. It also reduces the percentage gap between his certainty equivalent and the firm’s fair value for his compensation, but that gap remains substantial. Managerial control also causes traded options to exhibit an implied volatility smile.

Managerial Responses to Incentives: Control of Firm Risk, Derivative Pricing Implications, and Outside Wealth Management

We model a firm’s value process controlled by a manager maximizing expected utility from restricted shares and employee stock options. The manager also dynamically controls allocation of his outside wealth. We explore interactions between those controls as he partially hedges his exposure to firm risk. Conditioning on his optimal behavior, control of firm risk increases the expected time to exercise for his employee stock options. It also reduces the percentage gap between his certainty equivalent and the firm’s fair value for his compensation, but that gap remains substantial. Managerial control also causes traded options to exhibit an implied volatility smile.Risk; Wealth Management; Derivative

The early exercise boundary under the jump to default extended CEV model

This paper proves the existence, uniqueness, monotonicity and continuity of the early exercise boundary attached to American-style standard options under the jump to default extended constant elasticity of variance model of Carr and Linetsky (Financ Stoch 10(3):303–330, 2006).info:eu-repo/semantics/acceptedVersio

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