833 research outputs found
Executive stock option exercise with full and partial information on a drift change point
We analyse the optimal exercise of an executive stock option (ESO) written on
a stock whose drift parameter falls to a lower value at a change point, an
exponentially distributed random time independent of the Brownian motion
driving the stock. Two agents, who do not trade the stock, have differing
information on the change point, and seek to optimally exercise the option by
maximising its discounted payoff under the physical measure. The first agent
has full information, and observes the change point. The second agent has
partial information and filters the change point from price observations. This
scenario is designed to mimic the positions of two employees of varying
seniority, a fully informed executive and a partially informed less senior
employee, each of whom receives an ESO. The partial information scenario yields
a model under the observation filtration in which the
stock drift becomes a diffusion driven by the innovations process, an
-Brownian motion also driving the stock under
, and the partial information optimal stopping value
function has two spatial dimensions. We rigorously characterise the free
boundary PDEs for both agents, establish shape and regularity properties of the
associated optimal exercise boundaries, and prove the smooth pasting property
in both information scenarios, exploiting some stochastic flow ideas to do so
in the partial information case. We develop finite difference algorithms to
numerically solve both agents' exercise and valuation problems and illustrate
that the additional information of the fully informed agent can result in
exercise patterns which exploit the information on the change point, lending
credence to empirical studies which suggest that privileged information of bad
news is a factor leading to early exercise of ESOs prior to poor stock price
performance.Comment: 48 pages, final version, accepted for publication in SIAM Journal on
Financial Mathematic
Binomial approximation in financial models: computational simplicity and convergence
An exploration of the potential of transformation and other schemes in approximating diffusions (including those with boundaries) commonly seen in financial models. Convergence results are established for valuing both European and American contingent claims.Statistics
Three essays on option pricing
This thesis addresses option pricing problem in three separate and self-contained papers:
A. The Binomial CEV Model and the Greeks
This article compares alternative binomial approximation schemes for computing the option hedge ratios studied by Pelsser and Vorst (1994), Chung and Shackleton (2002), and Chung et al. (2011) under the lognormal assumption, but now considering the constant elasticity of variance (CEV) process proposed by Cox (1975) and using the continuous-time analytical Greeks recently offered by Larguinho et al. (2013) as the benchmarks. Among all the binomial models considered in this study, we conclude that an extended tree binomial CEV model with the smooth and monotonic convergence property is the most efficient method for computing Greeks under the CEV diffusion process because one can apply the two-point extrapolation formula suggested by Chung et al. (2011).
B. Valuing American-Style Options under the CEV Model: An Integral Representation Based Method
This article derives a new integral representation of the early exercise boundary for valuing American-style options under the constant elasticity of variance (CEV) model. An important feature of this novel early exercise boundary characterization is that it does not involve the usual (time) recursive procedure that is commonly employed in the so-called integral representation approach well known in the literature. Our non-time recursive pricing method is shown to be analytically tractable under the local volatility CEV process and the numerical experiments demonstrate its robustness and accuracy.
C. A Note on Options and Bubbles under the CEV Model: Implications for Pricing and Hedging
The discounted price process under the constant elasticity of variance (CEV) model is not a martingale for options markets with upward sloping implied volatility smiles.
The loss of the martingale property implies the existence of (at least) two option prices for the call option, that is the price for which the put-call parity holds and the price representing the lowest cost of replicating the payoff of the call. This article derives closed-form solutions for the Greeks of the risk-neutral call option pricing solution that are valid for any CEV process exhibiting forward skew volatility smile patterns. Using an extensive numerical analysis, we conclude that the differences between the call prices and Greeks of both solutions are substantial, which might yield significant errors of analysis for pricing and hedging purposes.Esta tese aborda a avaliação de opções em três artigos distintos:
A. The Binomial CEV Model and the Greeks
Este artigo compara diferentes aproximações binomiais para o cálculo dos Greeks das opções estudadas por Pelsser and Vorst (1994), Chung and Shackleton (2002), e Chung et al. (2011), no âmbito da distribuição lognormal, mas agora considerando o processo constant elasticity of variance (CEV) proposto por Cox (1975), utilizando os Greeks analÃticos em tempo contÃnuo, recentemente propostos por Larguinho et al. (2013) como referência. Entre os modelos binomiais considerados neste estudo, concluÃmos que um modelo extended tree binomial CEV com uma aproximação convergente e monótona é o método mais eficiente para o cálculo dos Greeks no âmbito do processo de difusão CEV porque podemos aplicar a fórmula de extrapolação de dois pontos, sugerido por Chung et al. (2011).
B. Valuing American-Style Options under the CEV Model: An Integral Representation Based Method
Este artigo deriva uma nova representação integral da barreira de exercÃcio antecipado para a avaliação das opções Americanas no âmbito do modelo constant elasticity of variance (CEV), um importante aspecto desta nova caracterização da barreira de exercÃcio antecipado é que este não envolve o usual processo recursivo que é habitualmente aplicado e conhecido na literatura como a abordagem de representação integral. O nosso método de avaliação não recursivo é de fácil tratamento analÃtico sob o processo de difusão CEV e os resultados numéricos demonstram a sua robustez e precisão.
C. A Note on Options and Bubbles under the CEV Model: Implications for Pricing and Hedging
O processo de desconto de preço no âmbito do modelo constant elasticity of variance (CEV) não é um martingale para os mercados de opções com uma volatility smile de inclinação ascendente. A perda da propriedade martingale implica a existência de (pelo menos) dois preços de opção para a opção de compra, que é o preço para qual se verifica a paridade put-call e este preço representa o menor custo de replicação do payoff da call. Este artigo deriva as soluções em fórmula fechada para os Greeks da opção call no risco neutral que são válidas para qualquer processo CEV que possui padrões de enviesamento ascendentes. Tendo por base uma analise numérica extensiva, concluÃmos que a diferença entre os preços da call e os Greeks de ambas as soluções são substanciais, o que pode gerar erros significativos de análises no cálculo do preço da call e dos Greeks
Pricing American Options on Jump-Diffusion Processes using Fourier Hermite Series Expansions
This paper presents a numerical method for pricing American call options where the underlying asset price follows a jump-diffusion process. The method is based on the Fourier-Hermite series expansions of Chiarella, El-Hassan & Kucera (1999), which we extend to allow for Poisson jumps, in the case where the jump sizes are log-normally distributed. The series approximation is applied to both European and American call options, and algorithms are presented for calculating the option price in each case. Since the series expansions only require discretisation in time to be implemented, the resulting price approximations require no asset price interpolation, and for certain maturities are demonstrated to produce both accurate and efficient solutions when compared with alternative methods, such as numerical integration, the method of lines and finite difference schemes.American options; jump-diusion; Fourier-Hermite series expansions; free boundary problem
Pricing American Options by Exercise Rate Optimization
We present a novel method for the numerical pricing of American options based
on Monte Carlo simulation and the optimization of exercise strategies. Previous
solutions to this problem either explicitly or implicitly determine so-called
optimal exercise regions, which consist of points in time and space at which a
given option is exercised. In contrast, our method determines the exercise
rates of randomized exercise strategies. We show that the supremum of the
corresponding stochastic optimization problem provides the correct option
price. By integrating analytically over the random exercise decision, we obtain
an objective function that is differentiable with respect to perturbations of
the exercise rate even for finitely many sample paths. The global optimum of
this function can be approached gradually when starting from a constant
exercise rate.
Numerical experiments on vanilla put options in the multivariate
Black-Scholes model and a preliminary theoretical analysis underline the
efficiency of our method, both with respect to the number of
time-discretization steps and the required number of degrees of freedom in the
parametrization of the exercise rates. Finally, we demonstrate the flexibility
of our method through numerical experiments on max call options in the
classical Black-Scholes model, and vanilla put options in both the Heston model
and the non-Markovian rough Bergomi model
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
Pricing an European gas storage facility using a continuous-time spot price model with GARCH diffusion
In this article we present both a theoretical framework and a solved example for pricing an European gas storage facility and computing the optimal strategy for its operation. As a representative price index we choose the Dutch TTF day-ahead gas price. We present statistical evidence that the volatility of this index is time-varying, so we introduce a new continuous-time model by incorporating GARCH diffusion into an Ornstein-Uhlenbeck process. Based on this price process we use dynamic programming methods to derive partial differential equations for pricing a storage facility. As an example we apply our methodology to a storage site located in Epe at the German-Dutch border. In this context we investigate the effects of multiple contract types, and perform a sensitivity analysis for all model parameters. We obtain a value surface displaying the properties of a financial straddle. Both volatility and mean reversion influence the facility value - but only around the long-run mean of the gas price. The terminal condition, which includes information about the contract provisions, is of importance if it contains e.g. penalty terms for low inventory levels. Otherwise its influence is diminishing for increasing lease periods. --TTF gas price,GARCH diffusion,natural gas storage,dynamic computing
Efficient tree methods for option pricing
The aim of this dissertation is the study of efficient algorithms based on lattice procedures for dealing with two relevant issues arising in the recent literature on option pricing: the pricing of complex barrier-type options and the pricing of options when the equity model takes into account a stochastic interest rate. This research is developed with a twofold perspective: first, we propose a good solution from a numerical point of view through the introduction of efficient lattice procedures and secondly, we study the theoretical aspects related to the tackled problems (such as the convergence and the rate of convergence of the scheme proposed)
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