José Fajardo

• Ph.D.
• Professor (Associate) at Getúlio Vargas Foundation

We study the skewness premium (SK) introduced by Bates (1991) in a general context using L ́evy Processes. Under a symmetry con-dition Fajardo and Mordecki (2006) obtain that SK is given by the Bate’sx% rule. In this paper we study SK under the absence of that symmetry condition. More exactly, we derive sufficient conditions for SK to be positive,...

In this paper, we analyze the determinants of CoCo bond issuance. We find evidence that banks that issue CoCos are typically large. Moreover, in the case of BRICS and other emerging economies, we find evidence that banks are also highly leveraged, aiming to meet the Basel III rules and replace debt with equity funding. Also, we study the strength o...

We study the equilibrium in the model proposed by Kyle (Econometrica 53(6):1315–1335, 1985) and extended to the continuous-time setting by Back (Rev Financ Stud 5(3):387–409, 1992). The novelty of this paper is that we consider a framework where the price pressure can be random. We also allow for a random release time of the fundamental value of th...

Issuing CoCo bonds is a possible way for banks to protect against economic uncertainty scenario. However, it remains unclear if CoCo bonds will be useful in loss absorption for issuers in the event of another financial distress. Using the model of Systemic Risk proposed by Brownlees and Engle (2016a), we estimated the expected capital shortfall for...

We introduce skewed Lévy models, characterized by a symmetric jump measure multiplied by a damping exponential factor. These models exhibit a clear implied volatility pattern, where the damping parameter controls the implied volatility curve’s skew, resulting in a measure of the model’s skewness. We show that the variation of this parameter produce...

We propose that a hyperinflation event has a long-lasting effect on household investment behavior. We want to investigate whether future stock market participation can be influenced by a single extreme macroeconomic instability episode. We use data from the Brazilian Institute of Geography and Statistics and find that households who experienced hyp...

We study the equilibrium in the model proposed by Kyle in 1985 and extended to the continuous time setting by Back in 1992. The novelty of this paper is that we consider a framework where the price pressure can be random. We also allow for a random release time of the fundamental value of the asset. This framework includes all the particular Kyle m...

In this article, we find empirical evidence of a new smirk factor, obtained from the jump structure of the risk neutral distribution of the underlying Lévy process. As an application we show how to price a barrier style contract.

In this paper we present new pricing formulas for some Barrier style contracts of European type when the underlying process is driven by an important class of Lévy processes, which includes CGMY model, generalized hyperbolic Model and Meixner Model, when no symmetry properties are assumed, complementing in this way previous findings in Fajardo (J B...

In 1988 Dybvig introduced the payoff distribution pricing model (PDPM) as an alternative to the capital asset pricing model (CAPM). Under this new paradigm agents preferences depend on the probability distribution of the payoff and for the same distribution agents prefer the payoff that requires less investment. In this context he gave the notion o...

In 1988 Dybvig introduced the payo distribution pricing model (PDPM) as an alternative to the capital asset pricing model (CAPM). Under this new paradigm agents preferences depend on the probability distribution of the payo and for the same distribution agents prefer the payo that requires less investment. In this context he gave the notion of ecie...

We study the optimal continuous trading strategy of an insider who is subject to the possibility of law penalties due to her illegal trading activity. Also, we discuss how to obtain the optimal penalty rule in order to maximize a welfare function.

http://dx.doi.org/10.2139/ssrn.2652396

In this paper we find empirical evidence of a new smirk factor, obtained from the jump structure of the risk neutral distribution of the underlying Lévy process. This new specification allow us to price digital call options for any barrier when no symmetry properties are assumed, complementing in this ways previous findings in the literature.

In this paper we find empirical evidence of a new smirk factor, obtained from the jump structure of the risk neutral distribution of the underlying Levy process. As an application we show how to price a barrier style contract.

In this paper we present new pricing formulas for some Power style contracts of European type when the underlying process is driven by an important class of L´evy processes, which includes CGMY model, generalized hyperbolic Model and Meixner Model, when no symmetry properties are assumed, extending and complementing in this way previous findings in...

In this paper we obtain some formulas for pricing contingent convertibles subject to what we call extension risk, i.e., the possibility that the bond issuer does not buy back the bond at pre-specified call dates. We follow a structural approach and we address the finite and infinite maturity cases.

We introduce skewed Lévy models, that have a symmetric jump measure multiplied by dumping exponential factor, in order to study the implied volatility smirk in Lévy markets. The dumping factor depends on a parameter beta, this results in a measure of the skewness of the model. We show that variation of this parameter produces the typical smirk obse...

In this paper we obtain some formulas for pricing contingent convertibles subject to what we call extension risk, i.e., the possibility that bond issuer does not buy back the bond at pre specified call dates and then new coupons rate are established until bond maturity. We follow a structural approach and we address the finite and infinite maturity...

In this paper we introduce the concept of Bad arbitrage opportunities and we show that when we avoid such strategies enforcement mechanisms in economies with default are effective, in the sense that agents can not anticipate payments exceeding the value of collateral requirement. More precisely, we prove that absence of Bad arbitrage opportunities...

We study the effects of psychological factors such as depression and wellness on women's financial decisions. More precisely, we find that the probability of a woman invest in stocks is reduced by 0.8%, if she has felt depressed recently, and increased by 1.1%, if she visits frequently her family or friends. Moreover, when the investment opportunit...

In this paper we present a very simple way to price a class of barrier options when the underlying process is driven by a huge class of L\'evy processes. To achieve our goal we assume that our market satisfies a symmetry property. In case of not satisfying that property some approximations can be obtained.

Contingent Convertibles (CoCos) are contingent capital instruments which convert into shares, or have a principal write down, if a trigger event takes place. CoCos exhibit the undesirable so-called death-spiral effect: by actively hedging the equity risk, investors can (unintentionally) force the conversion by making the share price deteriorate and...

In this paper we study to what extent extreme macroeconomic instability has a long-lasting effect in household investing behavior. Using data from the Brazilian National Household Survey (PNAD) and an additional survey based on the one used by Hong et al. (2009), we find evidence that individuals who were in their early adulthood during Brazil’s hy...

We study the optimal continuous trading strategy of an insider who is subject to the possibility of law penalties due to her illegal trading activity. This possibility was absent in previous works. Also, we discuss how to obtain the optimal penalty rule that maximize a welfare function.

This paper uses the Liu et al. (2007) approach to estimate the optionimplied Risk-Neutral Densities (RND), real-world density (RWD), and relative risk aversion from the Brazilian Real/US Dollar exchange rate distribution. Our empirical application uses a sample of exchange-traded Brazilian Real currency options from 1999 to 2011. Our estimated valu...

We find necessary and sufficient conditions for the market symmetry property, introduced by Fajardo and Mordecki (Quant Finance 6(3):219–227, 2006), to hold in the Ornstein–Uhlenbeck stochastic volatility model, henceforth OU–SV. In particular, we address the non-Gaussian OU–SV model proposed by Barndorff-Nielsen and Shephard (J R Stat Soc B 63(Par...

We study the skewness premium (SK) introduced by Bates [J. Finance, 1991, 46(3), 1009–1044] in a general context using Lévy processes. Under a symmetry condition, Fajardo and Mordecki [Quant. Finance, 2006, 6(3), 219–227] obtained that SK is given by Bates' x% rule. In this paper, we study SK in the absence of that symmetry condition. More exactly,...

This paper uses the Liu et al. (2007) approach to estimate the optionimplied Risk-Neutral Densities (RND), real-world density (RWD), and relative risk aversion from the Brazilian Real/US Dollar exchange rate distribution. Our empirical application uses a sample of exchange-traded Brazilian Real currency options from 1999 to 2011. Our estimated valu...

Well-known models that are extensively used by market traders, such as the Black-Scholes model, assume that the daily log-returns of assets follow a Normal distribution. Empirical evidences, however, show that return rates are frequently asymmetric and have fatter tails. Hence, this work aims to investigate if the Meixner distribution would be more...

In the present paper we analyze the characteristics of women that invest in the Brazilian capital market, directly or indirectly, through stock funds. Our goal is to verify if, in addition to the usual factors like age, years of education, marital status and wealth, there is evidence that social interactions increase the participation of women in c...

In this paper we find that Fama and French factors can explain the future behavior of three macroeconomic variables of the Brazilian economy: GDP, industrial production and inflation. The results show that these three factors explain the future behavior of the macroeconomic variables with a six-month lag, that SMB is positively correlated with econ...

In this paper we use multivariate affine generalized hyperbolic (MAGH) distributions, introduced by Schmidt et al. (2006), to show how to price multidimensional derivatives when the underlying asset follows a MAGH distribution. We also illustrate the approach using market data from the BOVESPA (São Paulo Stock Exchange) and the exchange rate of the...

This paper studies the implications of the absence of statistical arbitrage opportunities in a two-period incomplete market economy where default is allowed but there are collateral requirements. Modified versions of the fundamental theorem of asset pricing are obtained.

In this paper we study the implications of the absence of Behavioral arbitrage opportunities (BAO) in a two period incomplete markets economy where default is allowed but there are collateral requirements and utility penalties due to uncertainty on deliveries. We obtain a modified version of the Fundamental Theorem of asset pricing with default whe...

In this paper we examine which Brownian subordination with drift exhibits the symmetry property introduced by Fajardo and Mordecki [2006. Quantitative Finance 6, 219-227]. We find that when the subordination results in a Lévy process, a necessary and sufficient condition for the symmetry to hold is that the drift must be equal to-1/2. Also, we deri...

The aim of this paper is to estimate multivariate affine generalized distributions (MAGH) using market data. We use the Ibovespa, CAC, DAX, FTSE, NIKKEI and S&P500 indexes. We estimate the univariate distributions, bi-variate distributions and six-dimensional distribution. Then we assess their goodness of fit using Kolmogorov distances. As an appli...

We establish the existence of subgame perfect equilibria in general menu games, known to be sufficient to analyze common agency problems. Our main result states that every menu game satisfying enough continuity properties has a subgame perfect equilibrium. Despite the continuity assumptions that we make, discontinuities naturally arise due to the a...

In this paper we study the asset pricing and individual optimality problem in a two period incomplete markets economy where default is allowed but there are utility penalties and collateral requirements.

We study the implications of the absence of symmetry in a Levy market. Then, we address option price monotonicity with respect to a symmetry parameter. As applications we construct a semi-static superhedge for options and we obtain characteristics of the shape of the implied volatility. Extending in this way some results obtained by Fajardo and Mor...

The purpose of this paper is to investigate three anomalies in the São Paulo Stock Exchange (BOVESPA) index: the day-of-the-week effect, the twist on the Monday effect and the holiday effect. The period from Jan/1995 to Dec/2007 is analyzed, with subperiods established according to presidential terms. The paper addresses the theories on market effi...

We study the skewness premium (SK) introduced by Bates (1991) in a general context using L\'evy Processes. Under a symmetry condition Fajardo and Mordecki (2006) obtain that SK is given by the Bate's $x%$ rule. In this paper we study SK under the absence of that symmetry condition. More exactly, we derive sufficient conditions for SK to be positive...

The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economist...

To verify whether an empirical distribution has a specific theoretical distribution, several tests have been used like the Kolmogorov-Smirnov and the Kuiper tests. These tests try to analyze if all parts of the empirical distribution has a specific theoretical shape. But, in a Risk Management framework, the focus of analysis should be on the tails of...

To verify whether an empirical distribution has a specific theoretical distribution, several tests have been used, for example: Kolmogorov-Smirnov and Kuiper. These tests try to analyze if all parts of the empirical distribution has a specific theoretical shape. But, in a Risk Management framework, the focus of analysis should be on the tails of th...

In this paper we review several relationships between prices of put and call options, of both the European and the American type, obtained mainly through Girsanov Theorem, when the asset price is driven by a time-changed Lévy process. This relation is called put-call duality, and includes the relation known as put-call symmetry as a particular case...

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a two-dimensional Levy process. The main idea is to apply Girsanov's theorem for Levy processes, in order to reduce the problem to the pricing of a one Levy driven stock in an auxiliary market, baptized as the "dual market". In th...

In this paper we compute equivalent martingale measures when the asset price return is modeled by a Lévy process. We follow the approach introduced by Gerber and Shiu (1994).

We establish the existence of subgame perfect equilibria in general menu games, known to be sufficient to analyze common agency problems. Our main result states that every menu game satisfying enough continuity properties has a subgame perfect equilibrium. Despite the continuity assumptions that we make, discontinuities naturally arise due to the a...

A common statistical problem in finance is measuring the goodness-of-fit of a given distribution to real world data. This can be done using distances to measure how close an empirical distribution is from a theoretical distribution. The tails of the distribution should receive special importance if the focus is on Value-at-Risk (VaR) calculations....

Uma grande dificuldade de testar o CAPM, como apontado na crítica de Roll, é selecionar uma "proxy" adequada para carteira de mercado. A literatura recente tem buscado alternativas para a construção de uma carteira de mercado, as quais procuram incorporar os efeitos de ativos não transacionados em bolsa, como o capital humano. Este trabalho segue a...

We consider general menu games, known to be sufficient to analyze common agency problems. For these games, we compare the equilib-rium concepts of Page and Monteiro (2003) and Monteiro and Page (2005) to the more familiar notion of sequential equilibrium.

The aim of this paper is to introduce the notion of symmetry in a Lévy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, which is also reviewed in the paper, and that we call put–call duality. Symmetric Lévy markets have the distinctive fe...

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to a problem with one Lévy driven stock in an auxiliary market, baptized as "dual market". In this...

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to the pricing of a one Lévy driven stock in an auxiliary market, baptized as "dual market". In th...

In this paper we study the pricing problem of derivatives written in terms of a two dimensional time–changed Lévy processes. Then, we examine an existing relation between prices of put and call options, of both the European and the American type. This relation is called put–call duality. It includes as a particular case, the relation known as put–c...

We study an economy where there are two types of assets. Consumers’ promises are the primitive defaultable assets secured by collateral chosen by the consumers themselves. These personalized assets are purchased by financial intermediaries who finance these purchases by selling back derivatives to consumers. We show that non-arbitrage prices of pri...

In this paper, we study the implications of the absence of arbitrage in a two-period incomplete markets economy with default and exogenous collateral.

The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economist...

The aim of this paper is to discuss the use of the Generalized Hyperbolic Distributions to fit Brazilian assets returns. Selected subclasses are compared regarding goodness of fit statistics and distances. Empirical results show that these distributions fit data well. Then we show how to use these distributions in value at risk estimation and deriv...

Neste artigo analisamos o apreçamento de contratos que tenham seus resultados atrelados a mais de um ativo subjacente, em especial, opções bidimensionais. Para isto usamos a fórmula desenvolvida por Margrabe (1978) e o modelo de árvores de Rubinstein (1991a). Em seguida apresentamos exemplos práticos de opções bidimensionais e apreçamos estas opçõe...

The aim of this work is to study the pricing problem for derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to the pricing of a one Lévy driven stock in an auxiliary market, baptized as ``dual market''. In this way, we exte...

It is well known that, under uniform impatience, positive net supply assets are free of bubbles for non-arbitrage kernel deflators that yield finite present values of wealth. However, this does not mean that prices cannot be above the series of deflated dividends for the deflators given by the agents' marginal rates of substitution, which also yiel...

It is well known that, under uniform impatience, positive net supply assets are free of bubbles for non-arbitrage kernel deflators that yield finite present values of wealth. However, this does not mean that prices cannot be above the series of deflated dividends for the deflators given by the agents' marginal rates of substitution, which also yiel...

We study the estimation of volatility using the Fractional Brownian Motion (FBM) to model asset returns. Then, we price some European options using a Black-Scholes type formula derived for the FBM market model.

The aim of this work is to examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov’s Theorem, is called put–call duality. It includes as a particular case, the relation known as...

The goal of this paper is to analyze the use of the Generalized Hyperbolic (GH) Distributions to model the US Dollar/Brazilian Real exchange rate in a way to produce more accurate VaR (Value at Risk) measurements. After the GH parameters estimation, several distances were calculated to verify the fitting quality of Normal distribution and GH distri...

RESUMO A opção de IDI da BM&F possui características peculiares que torna o seu apreçamento diferente das opções de taxa de juros mais comuns, como as de títulos de renda fixa. Este artigo desenvolve uma fórmula para apreçamento dessas opções de IDI, utilizando a precificação livre de arbitragem. O modelo utilizado considera apenas um fator estocás...

We study the intertemporal consumption and investment problem in a continuous time setting when the security prices follow a Geometric Lévy process. Using stochastic calculus for semimartingales we obtain conditions for the existence of optimal consumption policies. Also, we give a charaterization of the equivalent martingale measures.Estudamos o p...

The aim of this paper is to discuss the use of the Generalized Hyperbolic Distributions to fit Brazilian assets returns. Selected subclasses are compared regarding goodness of fit statistics and distances. Empirical results show that these distributions fit data well. Then we show how to use these distributions in value at risk estimation and deriv...

We construct a representative agent supporting regular equilibria in a stochastic economy with more than two agents. Then we give conditions for the existence of equilibria. In this way we extend the results of Cuoco and He (1994). Resumo Construfmos urn agente representativQ que suporta equilibrios regulares numa economia estocastica com mais de d...

The present paper presents the Levy processes used in the literature for the modeling of the returns of financial assets, which are generated by stable Paretian and hyperbolic distributions. Some properties of these distributions, especially the time-scale invariance, are analyzed. In the end, empirical evidence of the applicability of these proces...

We study the implications of the absence of arbitrage in an two period economy where default is allowed and assets are secured by collateral choosen by the borrowers. We show that non arbitrage sale prices of assets are submartingales, whereas non arbitrage purchase prices of the derivatives (secured by the pool of collaterals) are super-martingale...

This work presents the implications of the absence of arbitrage in a two period incomplete markets economy where default is allowed, but it is required that all the assets be backed by a collateral bundle. This collateral can be exogenously given or can be determined by the sellers of assets, as in the collateralized mortgage obligation markets. Al...

We solve the intertemporal consumption and investment problem in a con­ tinuous time setting assuming that the security prices follow a Hyperbolic Levy lvIotion. Using Stochastic Calculus for Levy processes, we give sufficient condi­ tions for the existence of optimal consumption and investment policies. Resumo Resolvemos a problema do consuma e in...

We establish the existence of sequential equilibrium in general catalog games, known to be su-cient to analyze common agency problems. In particular, we show that our result solves some unpleasant features of the early approaches of Page and Monteiro (2003) and Monteiro and Page (2003).

We study the skewness premium (SK) introduced by Bates (1991) in a general context using Levy Processes. Under a symmetry con- dition Fajardo and Mordecki (2006) obtain that SK is given by the Bate's x% rule. In this paper we stdy SK under the absence of that symmetry condition. More exactly, we derive su-cient conditions for SK to be positive, in...

We introduce a symmetry concept and study its relation with the Skewness premium introduced by Bates (1991) in a more general con- text using Levy Processes. We obtain su-cient and necessary condi- tions for Bate's x% rule to hold. Then, we derive su-cient conditions for SK to be positive, in terms of the characteristic triplet of the Levy Process...

We study different notions of skewness through a parameter that we find relevant in order to quantify and explain the different types of skewness encountered in market models. Then, we study the implied volatility skew. We assume that stock is driven by a time-dependent Lévy process.