Comportamento asintotico della volatilità implicita vicino a scadenza

La tesi presenta una descrizione completa del comportamento asintotico della volatilità implicita vicino a scadenza, sotto le condizioni di non arbitraggio. I risultati ottenuti, che non dipendono dalla scelta del modello per il sottostante, saranno applicati nel caso di un modello a volatilità locale.

Volatilità implicita. Comportamento vicino alla scadenza.

In generale, ottenere un'espressione analitica per la volatilità implicita è impossibile, lo scopo di questa tesi infatti è trovare una sua approssimazione. Si mostra come è possibile descrivere il comportamento di tale parametro vicino alla scadenza, analizzando il corrispondente comportamento della superficie del prezzo della Call. Attraverso processi analitici di approssimazione, mettiamo in evidenza l'andamento asintotico del prezzo di Black-Sholes di un'opzione Call, sfruttando tale risultato, presentiamo il comportamento vicino alla scadenza della volatilità implicita.

Impact of Black-Scholes assumptions on Delta Hedging

In this work we are going to evaluate the different assumptions used in the Black- Scholes-Merton pricing model, namely log-normality of returns, continuous interest rates, inexistence of dividends and transaction costs, and the consequences of using them to hedge different options in real markets, where they often fail to verify. We are going to conduct a series of tests in simulated underlying price series, where alternatively each assumption will be violated and every option delta hedging profit and loss analysed. Ultimately we will monitor how the aggressiveness of an option payoff causes its hedging to be more vulnerable to profit and loss variations, caused by the referred assumptions.

Derivats financers al mercat espanyol: el Model de Black-Scholes i corbes de volatilitat per opcions sobre l'índex niniIBEX-35

El nostre treball es centrarà en conèixer i aprendre les nocions bàsiques del mercat financer espanyol, primer; i aplicar uns coneixements per veure si es verifica unahipòtesi plantejada, després. La incògnita que volem resoldre és la següent: comprovarsi tots els supòsits i resultats que faciliten els models teòrics emprats en l’estudi dels mercats financers a l’hora de la veritat es compleixen.D’entre els múltiples conceptes que ens proporcionen els estudis de mercatsfinancers ens centrarem sobretot en el model de Black-Scholes i els somriures devolatilitat per desenvolupar el nostre treball. Després de cercar les dades necessàries a través de la web del M.E.F.F., entrevistar-nos amb professionals del sector i fer un seguiment d’aproximadament dos mesos dels moviments de les opcions sobre l’Índex Mini-Íbex 35, amb l’ajuda d’un programa informàtic en llenguatge C, hem calculat les corbes de volatilitat de les opcions sobre l’Índex Mini-Íbex 35.Les conclusions més importants que hem extret són que el Model de Black-Scholes, malgrat va revolucionar el món dels mercats financers, està basat en 2 supòsits que no es compleixen a la realitat: la distribució lognormal del preu de les accions i unavolatilitat constant. Tal i com hem pogut comprovar, la corba de volatilitat de lesopcions sobre l’Índex Mini-Íbex 35 és decreixent amb el preu d’exercici i laMoneyness, tal i com sostenen les teories dels somriures de volatilitat; per tant, no és constant. A més, hem comprovat que a mesura que s’apropa el venciment d’una opció,el preu acordat de l’actiu subjacent a l’opció s’apropa al preu de mercat.

Local volatility changes in the black-scholes model

In this paper we address a problem arising in risk management; namely the study of price variations of different contingent claims in the Black-Scholes model due to anticipating future events. The method we propose to use is an extension of the classical Vega index, i.e. the price derivative with respect to the constant volatility, in thesense that we perturb the volatility in different directions. Thisdirectional derivative, which we denote the local Vega index, will serve as the main object in the paper and one of the purposes is to relate it to the classical Vega index. We show that for all contingent claims studied in this paper the local Vega index can be expressed as a weighted average of the perturbation in volatility. In the particular case where the interest rate and the volatility are constant and the perturbation is deterministic, the local Vega index is an average of this perturbation multiplied by the classical Vega index. We also study the well-known goal problem of maximizing the probability of a perfect hedge and show that the speed of convergence is in fact dependent of the local Vega index.

Option valuation as an expectation in the complex domain: The Black-Scholes case

It is very well known that the first succesful valuation of a stock option was done by solving a deterministic partial differential equation (PDE) of the parabolic type with some complementary conditions specific for the option. In this approach, the randomness in the option value process is eliminated through a no-arbitrage argument. An alternative approach is to construct a replicating portfolio for the option. From this viewpoint the payoff function for the option is a random process which, under a new probabilistic measure, turns out to be of a special type, a martingale. Accordingly, the value of the replicating portfolio (equivalently, of the option) is calculated as an expectation, with respect to this new measure, of the discounted value of the payoff function. Since the expectation is, by definition, an integral, its calculation can be made simpler by resorting to powerful methods already available in the theory of analytic functions. In this paper we use precisely two of those techniques to find the well-known value of a European call

Option valuation as an expectation in the complex domain: The Black-Scholes case

It is very well known that the first succesful valuation of a stock option was done by solving a deterministic partial differential equation (PDE) of the parabolic type with some complementary conditions specific for the option. In this approach, the randomness in the option value process is eliminated through a no-arbitrage argument. An alternative approach is to construct a replicating portfolio for the option. From this viewpoint the payoff function for the option is a random process which, under a new probabilistic measure, turns out to be of a special type, a martingale. Accordingly, the value of the replicating portfolio (equivalently, of the option) is calculated as an expectation, with respect to this new measure, of the discounted value of the payoff function. Since the expectation is, by definition, an integral, its calculation can be made simpler by resorting to powerful methods already available in the theory of analytic functions. In this paper we use precisely two of those techniques to find the well-known value of a European call

Aplicações do modelo black-scholes ao mercado de opções de compra, utilizando-se metodologias diferenciais para o cálculo de volatilidade

Este trabalho cuida de avaliar a eficiência do mercado de opções de ações da bolsa de valores de são Paulo (BOVESPA). A avaliação é feita através do modelo Black-Scholes, e traz como principal novidade diversas estimativas de volatilidade. Portanto torna-se um teste conjunto da eficiência do mercado, do modelo Black-Scholes e das diversas estimativas de volatilidade. O objetivo principal ~ determinar a volatilidade que gera o melhor retorno , isto é , aponta a maior ineficiência do mercado. Foram utilizadas opções de Paranapanema-pp e Petrobr's-pp no per(odo de novembro de 1987 a outubro de 1988. Dois testes de eficiência foram realizados para cada volatilidade estimada . Em ambos observou-se que o mercado é ineficiente, e no segundo obtivemos evidência de que uma das estimativas de volatilidade gera um retorno maio

Black-scholes type equations

In this work we are concerned with the analysis and numerical solution of Black-Scholes type equations arising in the modeling of incomplete financial markets and an inverse problem of determining the local volatility function in a generalized Black-Scholes model from observed option prices. In the first chapter a fully nonlinear Black-Scholes equation which models transaction costs arising in option pricing is discretized by a new high order compact scheme. The compact scheme is proved to be unconditionally stable and non-oscillatory and is very efficient compared to classical schemes. Moreover, it is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. In the next chapter we turn to the calibration problem of computing local volatility functions from market data in a generalized Black-Scholes setting. We follow an optimal control approach in a Lagrangian framework. We show the existence of a global solution and study first- and second-order optimality conditions. Furthermore, we propose an algorithm that is based on a globalized sequential quadratic programming method and a primal-dual active set strategy, and present numerical results. In the last chapter we consider a quasilinear parabolic equation with quadratic gradient terms, which arises in the modeling of an optimal portfolio in incomplete markets. The existence of weak solutions is shown by considering a sequence of approximate solutions. The main difficulty of the proof is to infer the strong convergence of the sequence. Furthermore, we prove the uniqueness of weak solutions under a smallness condition on the derivatives of the covariance matrices with respect to the solution, but without additional regularity assumptions on the solution. The results are illustrated by a numerical example.