Generalized Hyperbolic Distributions and Brazilian Data

Fajardo, José; Farias, Aquiles

doi:10.2139/ssrn.338283

Cited by 15 publications

(18 citation statements)

References 34 publications

(12 reference statements)

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“…Formula (18) is obtained taking the supremum over all τ ∈ M T in equation (14). Let us see then (19).…”

Section: Proof Of Corollary 22 (American Options)mentioning

confidence: 99%

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Symmetry and duality in Lévy markets

Fajardo

Mordecki

2006

Quantitative Finance

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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, that is also reviewed in the paper, and that we call put-call duality. Symmetric Lévy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices.Put-Call Duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanov's Theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanov's Theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Lévy process, answering a question raised by Carr and Chesney (1996). Some empirical evidence is shown, supporting that in general markets are not symmetric.

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“…Formula (18) is obtained taking the supremum over all τ ∈ M T in equation (14). Let us see then (19).…”

Section: Proof Of Corollary 22 (American Options)mentioning

confidence: 99%

“…The processS t represents the price of KS 0 dollars measured in units of stock S. This market is the auxiliary market in Detemple (2001), and we call it dual market; accordingly, we call Put-Call duality the relation (14). It must be noticed that Peskir and Shiryaev (2001) propose the same denomination for a different relation in [26].…”

Section: Dual Marketsmentioning

confidence: 99%

Symmetry and duality in Lévy markets

Fajardo

Mordecki

2006

Quantitative Finance

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“…The one dimensional hyperbolic distribution is widely used in the modeling of univariate financial data, for example in Eberlein and Keller [1995] and Farjado and Farias [2003].…”

Section: Some Special Casesmentioning

confidence: 99%

Portfolio optimization for studenttand skewedtreturns

Hu¹,

Kercheval

2010

Quantitative Finance

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It is well-established that equity returns are not Normally distributed, but what should the portfolio manager do about this, and is it worth the effort? It is now feasible to employ better multivariate distribution families that capture heavy tails and skewness in the data; we argue that among the best are the Student t and skewed t distributions. These can be efficiently fitted to data, and show a much better fit to real returns than the Normal distribution. By examining efficient frontiers computed using different distributional assumptions, we show, using for illustration 5 stocks chosen from the Dow index, that the choice of distribution has a significant effect on how much available return can be captured by an optimal portfolio on the efficient frontier.Portfolio optimization requires balancing risk and return; for this purpose one needs to employ some precise concept of "risk". Already in 1952, Markowitz used the standard deviation (StD) of portfolio return as a risk measure, and, thinking of returns as Normally distributed, described the efficient frontier of fully invested portfolios having minimum risk among those with a specified return. This concept has been extremely valuable in portfolio management because a rational portfolio manager will always choose to invest on this frontier.The construction of an efficient frontier depends on two inputs: a choice of risk measure (such as StD, V aR, or ES, described below), and a probability distribution used to model returns.Using StD (or equivalently, variance) as the risk measure has the drawback that it is generally insensitive to extreme events, and sometimes these are of most interest to the investor. Value at Risk (V aR) better reflects extreme events, but it does not aggregate risk in the sense of being subadditive on portfolios. This is a well-known difficulty addressed by the concept of a "coherent risk measure" in the sense of Artzner, et. al. [1999]. A popular example of a coherent risk measure is expected shortfall (ES), though V aR is still more commonly seen in practice.Perhaps unexpectedly, the choice of risk measure has no effect on the actual efficient frontier when the underlying distribution of returns is Normal -or more generally any "elliptical" distribution. Embrechts, McNeil, and Straumann [2001] show that when returns are elliptically distributed, the minimum risk portfolio for a given return is the same whether the risk measure is standard deviation, V aR, ES, or any other positive, homogeneous, translation-invariant risk measure.This fact suggests that the portfolio manager should pay at least as much attention to the family of probability distributions chosen to model returns as to the choice of which risk measure to use.It is now commonly understood that the multivariate Normal distribution is a poor model of generally acknowledged "stylized facts" of equity returns:• return distributions are fat-tailed and skewed• volatility is time-varying and clustered 1• returns are serially uncorrelated, but squared returns are seri...

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“…The presence of jumps is very important since it provides a better adjustment with real data and when we price derivatives, as European call, we can find a significant difference with the price obtained using the EMM of the BS. As was pointed out by Fajardo and Farias (2004). They used Generalized Hyperbolic distributions to study Brazilian data.…”

Section: Brownian Motion and Two Jumps Processmentioning

confidence: 99%

“…To obtain parameter θ, we use the parameters obtained by Fajardo and Farias (2004) for Brazilian Bovespa index . The parameters are, (α, β, µ, δ) = (31.9096, −0.0035, 0.0233, 0.0012) Assuming r = 13%, the solution to equation (10) Now in (5), we have the density of the EMM:…”

Section: Normal Inverse Gaussian Processmentioning

confidence: 99%