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.

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.

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

Option pricing under stochastic volatility and stochastic interest rate in the Spanish case

Among the underlying assumptions of the Black-Scholes option pricingmodel, those of a fixed volatility of the underlying asset and of aconstantshort-term riskless interest rate, cause the largest empirical biases. Onlyrecently has attention been paid to the simultaneous effects of thestochasticnature of both variables on the pricing of options. This paper has tried toestimate the effects of a stochastic volatility and a stochastic interestrate inthe Spanish option market. A discrete approach was used. Symmetricand asymmetricGARCH models were tried. The presence of in-the-mean and seasonalityeffectswas allowed. The stochastic processes of the MIBOR90, a Spanishshort-terminterest rate, from March 19, 1990 to May 31, 1994 and of the volatilityofthe returns of the most important Spanish stock index (IBEX-35) fromOctober1, 1987 to January 20, 1994, were estimated. These estimators wereused onpricing Call options on the stock index, from November 30, 1993 to May30, 1994.Hull-White and Amin-Ng pricing formulas were used. These prices werecomparedwith actual prices and with those derived from the Black-Scholesformula,trying to detect the biases reported previously in the literature. Whereasthe conditional variance of the MIBOR90 interest rate seemed to be freeofARCH effects, an asymmetric GARCH with in-the-mean and seasonalityeffectsand some evidence of persistence in variance (IEGARCH(1,2)-M-S) wasfoundto be the model that best represent the behavior of the stochasticvolatilityof the IBEX-35 stock returns. All the biases reported previously in theliterature were found. All the formulas overpriced the options inNear-the-Moneycase and underpriced the options otherwise. Furthermore, in most optiontrading, Black-Scholes overpriced the options and, because of thetime-to-maturityeffect, implied volatility computed from the Black-Scholes formula,underestimatedthe actual volatility.

On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.

A decomposition formula for option prices in the Heston model and applications to option pricing approximation

By means of classical Itô's calculus we decompose option prices asthe sum of the classical Black-Scholes formula with volatility parameterequal to the root-mean-square future average volatility plus a term dueby correlation and a term due to the volatility of the volatility. Thisdecomposition allows us to develop first and second-order approximationformulas for option prices and implied volatilities in the Heston volatilityframework, as well as to study their accuracy. Numerical examples aregiven.

The Black Box of Business Dynamics

Research in business dynamics has been advancing rapidly in the last years but the translation of the new knowledge to industrial policy design is slow. One striking aspect in the policy area is that although research and analysis do not identify the existence of an specific optimal rate of business creation and business exit, governments everywhere have adopted business start-up support programs with the implicit principle that the more the better. The purpose of this article is to contribute to understand the implications of the available research for policy design. Economic analysis has identified firm heterogeneity as being the most salient characteristic of industrial dynamics, and so a better knowledge of the different types of entrepreneur, their behavior and their specific contribution to innovation and growth would enable us to see into the ‘black box’ of business dynamics and improve the design of appropriate public policies. The empirical analysis performed here shows that not all new business have the same impact on relevant economic variables, and that self-employment is of quite a different economic nature to that of firms with employees. It is argued that public programs should not promote indiscriminate entry but rather give priority to able entrants with survival capacities. Survival of entrants is positively related to their size at birth. Innovation and investment improve the likelihood of survival of new manufacturing start-ups. Investment in R&D increases the risk of failure in new firms, although it improves the competitiveness of incumbents.

Black holes by analytic continuation

In the context of a two-dimensional exactly solvable model, the dynamics of quantum black holes is obtained by analytically continuing the description of the regime where no black hole is formed. The resulting spectrum of outgoing radiation departs from the one predicted by the Hawking model in the region where the outgoing modes arise from the horizon with Planck-order frequencies. This occurs early in the evaporation process, and the resulting physical picture is unconventional. The theory predicts that black holes will only radiate out an energy of Planck mass order, stabilizing after a transitory period. The continuation from a regime without black hole formationaccessible in the 1+1 gravity theory consideredis implicit in an S-matrix approach and suggests in this way a possible solution to the problem of information loss.

Opción de responsabilidad limitada y opción de abandonar: una integración para el análisis del coste de capital

El objetivo de este documento es formalizar el valor de las acciones de una empresa endeudada, la responsabilidad limitada de los accionistas de una sociedad anónima y la rentabilidad exigida en un horizonte perpetuo, aplicando la teoría de opciones. El modelo que se elabora parte del establecido por Fisher Black y Myron Scholes en 1973 sobre el valor de las acciones de una empresa endeudada en un horizonte monoperiódico. En este modelo se encuentra un problema: la limitación del horizonte a un solo período. El modelo que se desarrolla en este documento, se basa en que el horizonte de la empresa es, en principio, ilimitado, y es frecuente que se mantenga un determinado grado de endeudamiento a lo largo de su vida. Es decir, se considera el endeudamiento de la empresa de horizonte perpetuo. Como consecuencia, los accionistas pueden declarar la quiebra en cualquier momento independientemente del vencimiento de la deuda, dejando la empresa en manos de los acreedores. Basándose en este modelo de horizonte perpetuo, se introduce la opción de abandonar. Se utiliza el valor de venta que tiene el activo de la empresa en el mercado, los accionistas lo comparan con el valor de las deudas, y deciden si es más rentable seguir con la empresa o vender los elementos del activo en el mercado. Para la valoración, se utiliza la fórmula deducida por Merton (1990) de la opción de venta americana perpetua. También se utiliza las opciones de barrera. Una vez valorada la responsabilidad limitada de los accionistas en un horizonte perpetuo, se puede calcular el interés efectivo en condiciones de riesgo.

How tight should one's hands be tied? Fear of floating and credibility of exchange rate regimes

This paper analyzes the linkages between the credibility of a target zone regime, the volatility of the exchange rate, and the width of the band where the exchange rate is allowed to fluctuate. These three concepts should be related since the band width induces a trade-off between credibility and volatility. Narrower bands should give less scope for the exchange rate to fluctuate but may make agents perceive a larger probability of realignment which by itself should increase the volatility of the exchange rate. We build a model where this trade-off is made explicit. The model is used to understand the reduction in volatility experienced by most EMS countries after their target zones were widened on August 1993. As a natural extension, the model also rationalizes the existence of non-official, implicit target zones (or fear of floating), suggested by some authors.

Are one factor logarithmic volatility models useful to fit the features of financial data? An application to microsoft data.

This paper provides empirical evidence that continuous time models with one factor of volatility, in some conditions, are able to fit the main characteristics of financial data. It also reports the importance of the feedback factor in capturing the strong volatility clustering of data, caused by a possible change in the pattern of volatility in the last part of the sample. We use the Efficient Method of Moments (EMM) by Gallant and Tauchen (1996) to estimate logarithmic models with one and two stochastic volatility factors (with and without feedback) and to select among them.

Forecasting volatility using a continuous time model

This paper evaluates the forecasting performance of a continuous stochastic volatility model with two factors of volatility (SV2F) and compares it to those of GARCH and ARFIMA models. The empirical results show that the volatility forecasting ability of the SV2F model is better than that of the GARCH and ARFIMA models, especially when volatility seems to change pattern. We use ex-post volatility as a proxy of the realized volatility obtained from intraday data and the forecasts from the SV2F are calculated using the reprojection technique proposed by Gallant and Tauchen (1998).

The black-white test score gap widens with age?

We re-examine the theoretical concept of a production function for cognitive achievement, and argue that an indirect production function that depends upon the variables that constrain parents' choices is both moretractable from an econometric point of view, and more interesting from an economic point of view than is a direct production function that depends upon a detailed list of direct inputs such as number of books in the household. We estimate flexible econometric models of indirect production functions for two achievement measures from the Woodcock-Johnson Revised battery, using data from two waves of the Child Development Supplement to the PSID. Elasticities of achievement measures with respect to family income and parents' educational levels are positive and significant. Gaps between scores of black and white children narrow or remain constant as children grow older, a result that differs from previous findings in the literature. The elasticities of achievement scores with respect to family income are substantially higher for children of black families, and there are some notable difference in elasticities with respect to parents' educational levels across blacks and whites.