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Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula
Abstract
Options traders use a pricing formula which they adapt by fudging and changing the tails and skewness by varying one parameter, the standard deviation of a Gaussian. Such formula is popularly called "Black-Scholes-Merton" owing to an attributed eponymous discovery (though changing the standard deviation parameter is in contradiction with it). However we have historical evidence that 1) Black, Scholes and Merton did not invent any formula, just found an argument to make a well known (and used) formula compatible with the economics establishment, by removing the "risk" parameter through "dynamic hedging", 2) Option traders use (and evidently have used since 1902) heuristics and tricks more compatible with the previous versions of the formula of Louis Bachelier and Edward O. Thorp (that allow a broad choice of probability distributions) and removed the risk parameter by using put-call parity. The Bachelier-Thorp approach is more robust (among other things) to the high impact rare event. It is time to stop calling the formula by the wrong name. The paper draws on historical trading methods and 19 th and early 20 th century references ignored by the finance literature.
... Thus, the prediction from BS and BSM models might underestimates the price of deep real and deep imaginary options correspondingly [20]. In addition to "volatility smile", there are other issues occurred in different cases, e.g., valuing bond options [21,22], interest-rate curve [23,24], Short stock rate [25], etc. ...
... In order to address above issues and other limitations caused by assumptions of the models, lots of extensions have been proposed, e.g., these extensions include models for American options [26]. Despite the proposals from large amounts of extension models, the applicability and operability for real assets are remain unknown according to previous literature [21,22]. In the future, more extensions with high accuracy and simple operations should be investigated and various empirical studies should be carried out in order to verify the practicability for all the extension models. ...
Black-Scholes (BS) model was first proposed in 1973, which has been modified by Robert Merton as the Black-Scholes-Merton (BSM) model subsequently. Contemporarily, these two models have been widely used and praised by financial scholars as well as employees. Plenty of scholars have tried to verify the accuracy of the and expressed their views on the existing defects in above models. Based on the existing literature, this article first introduces and derives the two models step by step and discusses the basic assumptions for these models. Subsequently, the applications of the two models are demonstrated separately. Specifically, the project valuation based on BS model is presented detaily while the applications of BSM model are introduced from four aspects (pricing of intangible assets, risk avoidance, default prediction and employee stock option’s pricing). Afterwards, the limitations and gaps of the models (e.g., volatility smile) ascribed to the ideal assumptions are discussed. In order to tackle the issue, improvement and suggestions are proposed including extending the models with different forms. These results offer a guideline for the option pricing, which can be widely applied in investment strategy.
... Many question fundamental model assumptions, such as liquidity (see, for example, Schönbucher and Wilmott 2000), market completeness (see, for example, Dumas and Lyasoff 2012), and transaction costs (see, for example, Dewynne et al. 1994;Leland 1985;Kennedy et al. 2009). In addition, the assumption of mesokurtosis has created a significant debate among both practitioners and theoreticians (see, for example, Haug and Taleb 2014). ...
... This yields a failure of dynamic replication-based determinism. In this sense, the work not only echoes model performance criticisms raised in Haug and Taleb (2014), but it offers an additional explanation for such failures. Constructively, it both provides a bridge between existing option pricing theory and the rich literature that attempts to optimally manage risky portfolios of assets, as exemplified in Maeso and Martellini (2020). ...
In this paper, we focus on an implicit assumption in the BSM framework that limits the scope of market network connections to seeking gains in the currency basis, i.e., on trading strategies between the numeraire and the stock and between the numeraire and the option, separately. We relax this assumption and derive the equivalent of the standard BSM approach under a more general market network framework in order to assess its implications. In doing so, we find that it is not possible to hedge on an implicit option that allows one to directly trade the option and stock. This represents a potential challenge to the BSM framework, since the missing market network connection provides a potentially useful mechanism for risk-bearing portfolio managers to alter their portfolios.
... An important consequence is that the price of options will be affected by supply and demand for options, a contrast to the assumptions of Black and Scholes (1973) and Merton (1973), where all risk can be removed with dynamic delta hedging. In practice we have supply-and-demand–based option pricing; see also O'Connell (2001), Derman and Taleb (2005), Haug (2007), and Haug and Taleb (2008Taleb ( , 2010). The situation where options are best hedged with other options that imply that supply and demand for options affect the price of the options was indirectly described over a century ago by Higgins (1902) and Nelson (1904). ...
... The standard is what is known as the Black and Scholes (1973) and Merton (1973) formula, or the Garman and Kohlhagen (1983) and Grabbe (1983) version of it. Haug (2007), Haug and Taleb (2008Taleb ( , 2010), and Haug (2009) discuss if the option formula and hedging methodology the market uses should be referred to as the Black-Scholes-Merton formula considering the long history of the method before the 1973 papers. The standard in the OTC FX market is to quote implied volatility before the option price. ...
In this paper we look at the distribution and statistical behavior of what is known as implied volatilities in the foreign exchange (FX) market. We describe the statistical properties of over-the-counter foreign exchange (OTC FX) options implied at-the-money volatilities. The data set is unique because it presents continuous daily obser-vations of implied volatility. Trying to do the same type of analysis for almost any other option market is very diffi cult because the data will be “unclean” for this purpose. This is explained in more detail. We find the distribution of implied volatilities has a high peak and fat tails relative to the Gaussian distribution. We also fi nd that the short-term implied volatilities are more volatile than long-term implied volatilities. The data show that the volatility of implied volatility is unstable over time and the correlation between the implied volatilities varies considerably over time. The paper discusses the various implications of these issues. An important question one needs to ask is: What are implied volatilities measuring? The answer to this question is not quite clear in the literature and will affect interpretation of the data.
... Creative industries are assumed to have positive role in creating innovations and jobs as well as benefits for urban regeneration [24]. Therefore, naturally, it is not surprising that the concept of creative industries is increasingly popular in transition economies CEEs [43], [37] including the Czech Republic [38], [25]. However, if the discussions of these concepts in the context of advanced economies produced certain ambiguities, then we can notice often uncritical adoption of these foreign theoretical postulates and examples of good practices without necessary contextualization in their discussions in the Czech Republic [46]. ...
... Kromû toho, jak zdÛrazÀují Bates [8], Haug a Taleb [43] ãi Berkowitz [10], obchodníci ve skuteãnosti fie‰í problém jinak. PouÏívají vzorec, kter˘ sice bûÏnû oznaãují za BlackÛv-ScholesÛv, ale místo aby do nûj dosazovali konstantní volatilitu, prÛbûÏnû ho rekalibrují podle implicitních volatilit. ...
Regional policy is one of the most important EU policies. Large amounts of resources used for support programmes automatically put forward the question of their effectiveness and efficiency. These aspects can be examined both from macro and micro perspectives. In this article we focus on one aspect of inefficiency – "deadweight". This effect occurs if public subsidy is spent on activities that would have happened even without these resources. We examine the effect on selected projects for small and medium-sized enterprises in the Czech and Slovak Republics. The article consists of four parts. First we discuss theoretical background for using the subsidies as development tool. Then we look more closely on the definition of deadweight effect and previous studies analyzing it. In the methodology part we discuss different approaches how to measure this effect and introduce methodology in our research. We also discuss potential problems with interpretation of the results and in the final part we present the results of our research and some policy implications. We found out that the deadweight effect is quite substantial and represents more than 35 % of public subsidies. The project characteristics itself (type of project, amount of budget) has the highest significance for the deadweight effect. The deadweight effect is higher for investments projects compared to the support of education or employment. The probability of the deadweight effect is also decreasing with the total amount of subsidies.
... This is the crucial point in the recent discussion about the Black Scholes model in the financial press. Veteran option traders Pablo Triana (2007Triana ( , 2008 and Nassim Taleb and Espen Haug (2009) argued that due to users' creativity in connection with implied volatility, the Black Scholes formula in its pure version has never been used. ...
... See the discussion on this topic in Pablo Triana(2007, 2008) as well as in Nassim Taleb and EspenHaug (2009). ...
Die Ausdrücke ‚Grammatik‘ und ‚Rhetorik‘ scheinen klare Bedeutungen zu haben. Grammatik beschreibt die Regeln richtigen Sprechens,
umfasst Regeln zur Wort- und Satzbildung und scheidet schließlich richtige Sprachverwendung von falscher. Seit einiger Zeit
genießen Bücher große Popularität, die ‚falsche ‘ Sprachverwendung brandmarken; Sprachwächterinnen schwingen sich zum Schutze
des Genitivs auf, dessen Überleben durch Umtriebe des Dativs bedroht sei. Rhetorik nun wird assoziiert mit Präsentationskursen,
mit überzeugendem Reden (gern auch ohne gute Argumente) und mit ‚schöner ‘ Ausdrucksweise. Gelegentlich – und historisch gar
nicht falsch – wird rhetorisches Können den Rechtsanwältinnen und den Poltikerinnen zugesprochen: sie agieren erfolgreich
als Wortverdreherinnen, um (so unterstellen wir in der antiken Tradition der mulier bona) eine gute Sache voranzutreiben.
... This is the crucial point in the recent discussion about the Black Scholes model in the financial press. Veteran option traders Pablo Triana (2007Triana ( , 2008 and Nassim Taleb and Espen Haug (2009) argued that due to users' creativity in connection with implied volatility, the Black Scholes formula in its pure version has never been used. ...
... See the discussion on this topic in Pablo Triana(2007, 2008) as well as in Nassim Taleb and EspenHaug (2009). ...
Creative innovation is defined as the ability to understand and express novel orderly relationships. Novelty can be understood as the coalescence of any two or more different entities or thoughts, and creativity involves many complex processes such as preparation, incubation, illumination and verification. High levels of general intelligence, domain-specific knowledge and special skills are important and necessary components of creative behaviour. Some researchers propose that, beyond intelligence, there must also be “wisdom” which evaluates novel ideas according to their appropriateness. Recent studies consider creativity as a series of complex cognitive processes followed by some sort of process that is not precisely known.
... This is the crucial point in the recent discussion about the Black Scholes model in the financial press. Veteran option traders Pablo Triana (2007Triana ( , 2008 and Nassim Taleb and Espen Haug (2009) argued that due to users' creativity in connection with implied volatility, the Black Scholes formula in its pure version has never been used. ...
... See the discussion on this topic in Pablo Triana(2007, 2008) as well as in Nassim Taleb and EspenHaug (2009). ...
Städte sind die Stätte der Globalisierung. Kleinstädte werden belächelt. Großstädte wurden und werden kritisiert. Städte sind
seit den 1990er Jahren zu einer neuen Stätte der Regional- und Raumwissenschaften, Soziologie, Politikökonomie, Wirtschaftsgeographie
und eben auch zunehmend der Wirtschaftswissenschaften geworden. Spätestens zur Jahrtausendwende hat die Globalisierung das
Thema der Metropole zum faszinierenden Populärthema werden lassen. Die Mega Citys oder Global Citys sind nun wieder problematisierend in der Diskussion und die Debatte „Mythos Metropole“ macht erneut die Runde (vgl. Fuchs/Moltmann/Prigge
1995; Häußermann 2000; Sohn/Weber 2000; Sassen 2001a, 2001 b, 2004; Bronger 2004).
... The model discovery problem is the task of finding parameters for which a given model satisfies a given specification. This problem is known to be a key challenge for modelers of stochastic systems including those in systems biology [LHFH08] and in finance [HT08]. Very often, the modeler uses her intuition to make an educated guess about the parameters and then performs extensive manual validation to discover whether her educated guess was correct. ...
... Example 4 (Stochastic Models in Finance) The price of a stock is often modeled by a geometric stochastic differential equation [HT08,BS73,Hes93,Hul06]. This is also the model for stock prices used in the famous Black-Scholes-Merton equation: In Fig. 2 ...
... We could not calculate implied volatilities and implied rates of return at the same time. So, Espen Haug and Nassim Taleb [7] are wrong about never using the Black-Scholes formula. If you trade with someone who uses the notion of implied volatility, you use Black-Scholes. ...
In this article, I try to show that a given market segment needs at least two models: one as a pricing convention and one to run the portfolio of securities. These models should be related in such a way that the model to manage the books is an extension of the pricing convention. Financial modeling succeeded in the case of options, but it failed with the CDOs, resulting in the credit crunch.
... The practicality of this approach to pricing an option has often been criticised; see e.g. Derman and Taleb (2005), Jankova (2018), Haug and Taleb (2007) or Hurvich (2021). Nevertheless, this risk-neutral approach and the BS formula together with its myriad extensions, form the basis of much of the derivatives financial market (see e.g. ...
... Many question fundamental model assumptions, such as liquidity [see, for example, Schonbucher and Wilmott (2000)], market completeness [see, for example, Dumas and Lyasoff (2012)], and transaction costs [see, for example, Dewynne, Whalley, and Wilmot (1994), Leland (1985), Kennedy, Forsyth, and Vetzal (2009)]. In addition, the assumption of mesokurtosis has created a significant debate among both practitioners and theoreticians [see, for example, Haug and Taleb (2014)]. ...
In this paper we focus on an implicit assumption in the BSM framework that limits the scope of market network connections to seeking gains in the currency basis, i.e., on trading strategies between the numeraire and the stock and between the numeraire and the option, separately. We relax this assumption and derive the equivalent of the standard BSM approach under a more general market network framework in order to assess its implications. In doing so, we find that it is not possible to hedge the implicit option that allows one to directly trade the option and stock. This represents a potential challenge to the BSM framework, since the missing market network connection provides a potentially useful mechanism for risk-bearing portfolio managers to alter their portfolios.
... The firm was badly hit by the Russian financial crisis. As a matter of fact, the Black-Scholes-Merton model assumes that events follow a thin-tailed distribution (Haug and Taleb 2008: 2) which explains why it failed in the presence of a massive tail event like the Russian collapse. ...
... Under this model it can be shown that there is a continuous hedging program (or replicating portfolio) that can be executed to eliminate arbitrage possibilities, thus implying "risk neutrality". The practicality of this approach to pricing an option has often been criticized (see, for example, Derman and Taleb 2005;Jankova 2018;Haug and Taleb 2007;Hurvich 2000). Nevertheless, this risk-neutral approach and the BS formula, together with its myriad extensions, form the basis of much of the derivatives financial market (see, for example, Millo and MacKenzie 2009). ...
The pricing of options is discussed, using an approach based on expected profit (EP) and expected loss (EL) as measures of the reward and risk of trades, respectively. It is shown that the EL/EP ratio is an important indicator of the quality of trades.
Formulas are derived for these measures for European call and put options under the traditional geometric Brownian motion assumption for the price movement of the underlying security. The Black–Scholes notion of implied volatility is generalized to consensus implied volatility for options chains. Optimal portfolios for trading in option chains are introduced and illustrated with practical data. It is shown that the EP–EL approach yields much useful information to option traders.
... Después de 2008, al menos algunos de los mismos agentes tuvieron que dar la vuelta y reeducarse para dejar de pretender que sus cerebros podrían ser reemplazados por algoritmos automáticos. 60 Esta reeducación puede tomarle más tiempo a los apologistas de las finanzas mecanizadas como Alan Greenspan y Gordon Brown. ...
The last 10 years have seen unprecedented efforts to automate whole new ranges of human and nonhuman activity: trust, recognition, identification, care, respect, translation and interpretation itself. It may be helpful to look at these developments -- which include Bitcoin and blockchain -- in the light of 19th-century mechanization. Although the new tide of automation recruits technologies that have become available only in the 21st century, it is no less dependent on the living work of human and more-then-human beings. Nor is it any less prone to exhaust or "max out" that work, wreaking ecological destruction and necessitating the organization of new frontiers of extraction. The new mechanization is also entwined with some of the same fantasies and rituals that have animated industrial capitalism since its beginnings.
... For example, in the theory of option pricing due to Black-Scholes and Merton, a model for movement of stock prices is posited, and in conjunction with basis theory which states that a riskless investment will receive the risk free rate of return, the researchers reason that a value can be assign to an option that is independent of the expected future value of the stock [15] . The Black option pricing model is under strong criticism: [16] argued that the Black model merely recast existing widely used models in terms of practically impossible "dynamics hedging" rather than 'risk' in order to make them more compatible with mainstream neoclassical economic theory. Added they opined that "Reliance on models based on incorrect axioms has clear and large effects; while [17] argued that Black models assume that the probability of extreme price changes is negligible, when in reality, stock prices are much Jerkier than this". ...
The Black Option Pricing Model in recent times has been under strong criticism. The Black Models assume that the probability of extreme price changes is negligible, when in reality stock prices in the stock market are subject to fluctuations. Again, the modern formulation of statistical models is based on the description of the physical system by an ensemble that represents all possible configurations of the system and the probability of realizing each configuration. Therefore, the probability of extreme price changes should not be neglected. Consequently, this work set out to develop a model (the Modified Geometric Brownian Motion Model (MGBMM)) for perpetual warrant option for prices of assets (shares of stock) traded in a perfect market with an arbitrary stock price in the warrant. Reasons that support the Modified Geometric Brownian Motion Model as an appropriate model in a perfect market are presented with a completely specified strategy for managing option investment which permits practical testing of the model's efficacy. Thus, illustrating how the notion of Statistical Mechanics is applied in economics to model the prices of assets in the financial market.
... Bachelier's trailblazing study on La Boursa was based on cotton prices [17], which laid the foundation of many a discipline in the future. Even award-winning Black-Scholes-Miller model was a variant of his model only [15]. Two American researchers furthered the work of Bachelier in true sense, while showcasing the scaling behaviour of power law matches that of financial markets [22]. ...
Financial Reynolds number (Re) has been proven to have the capacity to predict volatility, herd behaviour and nascent bubble in any stock market (bourse) across the geographical boundaries. This study examines forty two bourses (representing same number of countries) for the evidence of the same. This study finds specific clusters of stock markets based on embedded volatility, herd behaviour and nascent bubble. Overall the volatility distribution has been found to be Gaussian in nature. Information asymmetry hinted towards a well-discussed parameter of 'financial literacy' as well. More than eighty percent of indices under consideration showed traces of mild herd as well as bubble. The same indices were all found to be predictable, despite being stochastic time series. In the end, financial Reynolds number (Re) has been proved to be universal in nature, as far as volatility, herd behaviour and nascent bubble are concerned.
... Delta hedging as an argument for risk-neutral valuation works particular poorly when there are jumps in the underlying asset. With jumps in the asset price, Carr and Wu (2002) shows how hedging options with options is superior to delta hedging, see also Derman and Taleb (2005), Haug (2007b), Haug and Taleb (2008) and Hyungsok and Wilmott (2008) for a detailed analysis and discussion on this topic. According to Carr and Wu simulations indicate that the inferior performance of the delta hedge in the presence of jumps cannot be improved upon by increasing the rebalancing frequency, see also Hyungsok and Wilmott (2007). ...
This article introduces the concept of embedded options in the world's physical monies, both coin and paper. The option value for base metal coins is presented. The various strategies for redemption by the owner and the prevention of redemption by the issuer (central banks) are discussed. The market values of gold coins are discussed in light of the embedded option valuation. In conclusion, the rational behavior of both individuals and central banks in light of these valuations is described.
... Post-2008, at least a few of the same traders had to turn around and reeducate themselves to stop pretending that their brains could be pretty much replaced by automated algorithms. 60 The re-education can take longer for apologists for mechanized finance such as Alan Greenspan and Gordon Brown. ...
The last 10 years have seen unprecedented efforts to automate whole new ranges of human and nonhuman activity: trust, recognition, identification, care, respect, translation and interpretation itself.
It may be helpful to look at these developments -- which include Bitcoin and blockchain -- in the light of 19th-century mechanization. Although the new tide of automation recruits technologies that have become available only in the 21st century, it is no less dependent on the living work of human and more-then-human beings. Nor is it any less prone to exhaust or "max out" that work, wreaking ecological destruction and necessitating the organization of new frontiers of extraction. The new mechanization is also entwined with some of the same fantasies and rituals that have animated industrial capitalism since its beginnings.
A powerpoint companion piece to this paper, "The Ecology of Trust Mechanization," can be downloaded in English and Spanish at http://www.thecornerhouse.org.uk/resource/ecology-trust-mechanization.
... However, in reality this is not always what happens in case of exercising the options and the reality is much more deviated from the theoretical explanation of the BSOPM. The deviation occurs because there is risk in trading and factors like supply and demand, bank rate, speculative pressure etc., whereas Black-Scholes developed a model in which they argued to value options by removing "risk" parameter through dynamic hedging [11]. So including the risk back in means deviation of the theoretical price from the real trading price. ...
The Black-Scholes Option pricing model (BSOPM) has long been in use for valuation of equity options to find the prices of stocks. In this work, using BSOPM, we have come up with a comparative analytical approach and numerical technique to find the price of call option and put option and considered these two prices as buying price and selling price of stocks of frontier markets so that we can predict the stock price (close price). Changes have been made to the model to find the parameters 'strike price' and the 'time of expiration' for calculating stock price of frontier markets. To verify the result obtained using modified BSOPM we have used machine learning approach using the software Rapidminer, where we have adopted different algorithms like the decision tree, ensemble learning method and neural network. It has been observed that, the prediction of close price using machine learning is very similar to the one obtained using BSOPM. Machine learning approach stands out to be a better predictor over BSOPM, because Black-Scholes-Merton equation includes risk and dividend parameter, which changes continuously. We have also numerically calculated volatility. As the prices of the stocks goes high due to overpricing, volatility increases at a tremendous rate and when volatility becomes very high market tends to fall, which can be observed and determined using our modified BSOPM. The proposed modified BSOPM has also been explained based on the analogy of Schrodinger equation (and heat equation) of quantum physics.
... However, in reality this is not always what happens in case of exercising the options and the reality is much more deviated from the theoretical explanation of the BSOPM. The deviation occurs because there is risk in trading and factors like supply and demand, bank rate, speculative pressure etc., whereas Black-Scholes developed a model in which they argued to value options by removing "risk" parameter through dynamic hedging [11]. So including the risk back in means deviation of the theoretical price from the real trading price. ...
The Black-Scholes Option pricing model (BSOPM) has long been in use for valuation of equity options to find the prices of stocks. In this work, using BSOPM, we have come up with a comparative analytical approach and numerical technique to find the price of call option and put option and considered these two prices as buying price and selling price of stocks of frontier markets so that we can predict the stock price (close price). Changes have been made to the model to find the parameters strike price and the time of expiration for calculating stock price of frontier markets. To verify the result obtained using modified BSOPM we have used machine learning approach using the software Rapidminer, where we have adopted different algorithms like the decision tree, ensemble learning method and neural network. It has been observed that, the prediction of close price using machine learning is very similar to the one obtained using BSOPM. Machine learning approach stands out to be a better predictor over BSOPM, because Black-Scholes-Merton equation includes risk and dividend parameter, which changes continuously. We have also numerically calculated volatility. As the prices of the stocks goes high due to overpricing, volatility increases at a tremendous rate and when volatility becomes very high market tends to fall, which can be observed and determined using our modified BSOPM. The proposed modified BSOPM has also been explained based on the analogy of Schrodinger equation (and heat equation) of quantum physics.
... Without loss of generality, these assumptions make it relatively easy to compute competitive equilibrium prices. 35 See also Haug and Taleb (2008). 36 Appendix C spells out the model in detail. ...
... What is known as Value-at-Risk (VaR) has been heavily used in risk management by many financial institutions, and other common measures and models of modern finance including the Sharpe Ratio, the capital asset pricing model (CAPM), and the famous Black, Scholes, and Merton model [10,11] are all based on the assumption of Gaussian-distributed returns. For in-depth discussions regarding the weakness of the Gaussian distribution and its many implications for finance and beyond see [4,5,12,13], and [14]. ...
For many stochastic phenomena, we observe statistical distributions that have fat-tails and high-peaks compared to the Gaussian distribution. In this paper, we will explain how observable statistical distributions in the macroscopic world could be related to the randomness in the subatomic world. We show that fat-tailed (leptokurtic) phenomena in our everyday macroscopic world are ultimately rooted in Gaussian – or very close to Gaussian-distributed subatomic particle randomness, but they are not, in a strict sense, Gaussian distributions.
By running a truly random experiment over a three and a half-year period, we observed a type of random behavior in trillions of photons. Combining our results with simple logic, we find that fat-tailed and high-peaked statistical distributions are exactly what we would expect to observe if the subatomic world is quantized and not continuously divisible. We extend our analysis to the fact that one typically observes fat-tails and high-peaks relative to the Gaussian distribution in stocks and commodity prices and many aspects of the natural world; these instances are all observable and documentable macro phenomena that strongly suggest that the ultimate building blocks of nature are discrete (e.g. they appear in quanta).
... Espen Gaarder Haug and Nassim Nicholas Taleb argue that the BlackScholes model merely recast existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk," to make them more compatible with mainstream neoclassical economic theory. [30] Similar arguments were made in an earlier paper by Emanuel Derman and Nassim Taleb. [11] In response, Paul Wilmott has defended the model, [60], [51]. ...
This current thesis aims to survey recent development on certain problems in Mathematical
Finance. The geometric Brownian motion model for stock price was first proposed by the
renowned economist Samuelson in 1965. Later in 1973 Black, Scholes and Merton used that
model to find a formula for price of European options. This work commences the application
of Stochastic calculus in the research field of quantitative finance. But this model assumes
that the basic market parameters, namely, growth rate, volatility and bank interest rate
remain constant during the entire period of the option. Numerical data from actual market
does not support these assumptions. To overcome these drawbacks, several alternative models are still being proposed in the literature and thereby new mathematical challenges are
arising. In recent years a large amount of research is being carried out by considering the
market parameters as Markov chains which evolve according to a prescribed transition rate.
Markov modulated GBM model is one of that kind. This model can be regarded as straight
forward generalization of B-S-M (Black, Scholes and Merton) model. Although such market
is proved to have no arbitrage, but the cost paid for this generalization includes features like
incompleteness of market, lack of analytic solution, non-uniqueness of option pricing etc.
Nevertheless, consideration of the above model opens up a wide range of research topics.
The existing literature, those assume above model and related to locally risk minimizing
pricing, optimal hedging, portfolio optimization with risk sensitive cost, stability of numerical solutions of associated PDEs, computation of complexity of numerical schemes etc. are
thoroughly being surveyed in this current project. Besides, a number of numerical experiments are carried out based on the theoretical results. During thorough study of Springer
lecture note on Introduction to stochastic Calculus for Finance" by Dieter Sondermann, as
part of prerequisite, a list of errata along with few corrections/suggestion is prepared and
enclosed to this thesis.
... 19 Moreover, the defects of the formulas that were the engine of commodification were routinely compensated for and concealed by traders' use of a "dark twin" 20 of older "heuristics and tricks" as well as a vernacular understanding of possible scenarios and narratives that they had acquired through long, everyday practice, none of which relied so heavily on spurious assumptions of normal or Gaussian distributions. 21 Of course, top managers and economists at a distance from the trading floor had learned to acknowledge that "a model is inherently wrong, because a model only looks backwards". 22 But because they believed that models were nevertheless useful approximations or heuristic devices, this obligatory admission did little more than inoculate them against a loss of confidence in the "inherently wrong" mechanisms that were continuing to play a key role in churning out uncertainty commodities. ...
The controversy about how financial derivatives markets are to be regulated that has been opened up by the credit crunch in many ways parallels and overlaps the widening debate over regulation of carbon markets. Both markets involve hitherto untried attempts at commodification: in the case of the financial markets, commodification of an unprecedented range of uncertainties, and in the case of the carbon markets, commodification of climate benefits or the earth's carbon-cycling capacity. Policy responses to the crises facing both markets can be divided roughly into two streams. One, inspired by neoclassical economics and doctrines of "market failure", tends to assume that the production and exchange of the new proto-commodities can be successfully regulated or corrected by "internalizing externalities". Another, more pragmatically oriented, looks to partial or full decommodification as a way of tackling the problems engendered by the new markets. In the financial markets, a decommodification approach emphasizes measures such as removing certain instruments from trade, preventing the exchange of commercial bank deposits in some uncertainty markets, limiting securitisation, and so forth. In the carbon markets, a decommodification approach might, for example, prohibit offsets from being exchanged with emissions reductions, or challenge the supposed climatic equivalence, and thus the fungibility, among emissions reductions undertaken in different locations and technological contexts. Interestingly, both the calculative, "internalizing externalities" approach and the decommodification approach have supporters from wide ranges of the political spectrum, although an increasing number of policy analysts are adopting elements from the decommodification approach.
... The Black-Scholes-Merton framework is a cornerstone of contemporary finance. This model which is nowadays well-known either practitioners or theoreticians, also generates a lot of debates between critics and supporters of the Black and Scholes model: while Kalotay (1995), Derman and Taleb (2005) or Haug and Taleb (2008) called the Black and Scholes model (BSM) into question, other authors tried "to save" the model (Duffie, 1998;Wilmott, 2008aWilmott, , 2008b by replying that the model is "correct on average" (Wilmott, 2008, p.2). These debates shed light on the distinction between practices (i.e. ...
The debates on the Black and Scholes model shed light on the distinction between practices (i.e. inductive know-how or techne) and theory (i.e. deductive know-why or episteme) in finance. We revisit the classical distinction, still accepted widely in the literature, between episteme and techne and develop a nuanced view by introducing two other levels of knowledge we will call “commanding knowledge” (epitaktike) and “practical wisdom” (phronesis). The major contribution of this paper is to use these four levels of knowledge (episteme, epitaktike, techne and phronesis) in order to highlight how this model subtly influenced financial practices by shaping the microstructure of the emerging Chicago Board Options Exchange (CBOE). Our analysis will then be completed by a re-interpretation of the existing literature about the performativity of the BSM model to show how these levels of knowledge combined each other in the evolution post-crash (1987) financial practices.
... To correct for these effects, most traders 'fudge' the volatility parameter in the model, leading to the phenomenon of a volatility smile or skew for different option strikes and maturities [20]. Furthermore, studies on realized volatility indicate that, far from being stationary, asset price volatilities evolve in a random way, exhibiting such phenomena as volatility clustering (autocorrelation in the 'volatility process') and the leverage effect (negative returns cause the volatility to increase). ...
... For criticism on the Black-Scholes option pricing model, see e.g.Haug and Taleb (2009). ...
... Pero los operadores que realmente comprendieron los modelos aprendidos para compensar su inoperancia (y algunas veces, consciente o inconscientemente, para ocultar su inexperiencia técnica de la vista de los supervisores y de clientes), apoyándose en el "gemelo oscuro" 133 de la antigua "heurística y sus trucos" y una comprensión en lengua vernácula de los escenarios posibles que habían adquirido a través de tiempo, en la práctica diaria. 134 Conscientes de que las fórmulas quantistas para cultivar el futuro, lo simplificaban y lo desestabilizaban de forma peligrosa, los inteligentes, operadores experimentados que trabajan con las manos en la masa, han señalado desde el principio que, si bien en gran medida el modelo de mercantilización de incertidumbres impulsado podría ampliar las oportunidades de beneficio, en última instancia haría las crisis inevitables. ...
Cuando los mercados son veneno. Aprender sobre política climática de la crisis financiera. Larry Lohmann. THE CORNER HOUSE "Las formas de práctica social que han dado lugar a nuevos tipos de la calculabilidad, y que los cálculos intentan formar, también continuamente han convertido al mundo en más móvil, incierto e incalculable." Timothy Mitchell Rule of Experts: Egypt, Technopolitics, Modernity 1 En todo el mundo, los grupos progresistas se han apresurado a asociar la crisis financiera en curso con las crisis simultáneas del clima, alimentación, energía, atención de la salud y el militarismo. Acogiendo con satisfacción la aparente ruptura del experimento neoliberal, han llamado a la construcción de movimientos populares integrados para una mayor "control democrático de las instituciones financieras y económicas" 2 -un "nuevo paradigma", que: "Ponga el sistema financiero al servicio de un nuevo sistema democrático internacional basado en la satisfacción de los derechos humanos, el trabajo decente, la soberanía alimentaria, el respeto por el medio ambiente, la diversidad cultural, la economía social y solidaria y un nuevo concepto de riqueza" 3 "Las crisis más evidente a las que nos enfrentamos colectivamente hoy están vinculadas y las soluciones a ellas deben estar vinculadas, también" va un manifiesto. "Adecuadamente dirigidas y utilizadas,"la crisis financiera "podría abrir las puertas del salto cuantitativo y cualitativo que debemos hacer." 4 "La crisis financiera de 2008", insiste otro: "Presenta la mejor oportunidad en más de un siglo, para al mismo tiempo reformar los sistemas del dinero y crear medios adicionales de intercambio y mecanismos de financiación para acelerar el cambio de la era de los combustibles fósiles/nuclear-Industrial a una más verde, rica en información, Edad Solar " 5 .
... However, despite the relative sophistication of pricing and risk analysis options have been around for centuries. Haug and Taleb [2008] provides a good discussion on options in a historical setting. ...
... To this we would make two responses. First, the models used are diverse and involve improvisation by reflexive market actors (Beunza & Stark, 2010;Haug & Taleb, 2009;MacKenzie, 2003). Haug (2006) lists 60 models for pricing options alone, while Merton (1995) lists 11 strategies for taking the same basic leveraged long position. ...
This paper is about knowledge limits and the financial crisis. It begins by examining various existing accounts of crisis which disagree about the causes, but share the belief that the crisis represents a problem of socio-technical malfunction which requires some kind of technocratic fix: the three variants on this explanation are the crisis as accident, conspiracy or calculative failure. This paper proposes an alternative explanation which frames the crisis differently as an elite political debacle. Political and technocratic elites were hubristically detached from the process of financial innovation as it took the form of ‘bricolage’, which put finance beyond technical control or management. The paper raises fundamental questions about the politicized role of technocrats after the 1980s and emphasizes the need to bring private finance and its public regulators under democratic political control whose technical precondition is a dramatic simplification of finance.
... For instance, the right panel of Figure 3 depicts that the three cases' simulated returns may be approximated through a linear fit, which corresponds to the Gaussian or normality assumption. Each case displays a particular slope, where the higher the dependence the lower the slope; this is, the more persistent (antipersistent) the Such interesting results regarding Figure 3 correspond to an attribute typical of the Gaussian distribution highlighted by Haug and Taleb (2009): it is possible to express any probability distribution in terms of Gaussian, even if it has fat tails, by varying the standard deviation at the level of the density of the random variable. In the case in hand, the Gaussian can express a broad type of stochastic processes by changing how volatility behaves with respect to the time horizon. ...
As a natural extension to León and Vivas (2010) and León and Reveiz (2010) this paper briefly describes the Cholesky method for simulating Geometric Brownian Motion processes with long-term dependence, also referred as Fractional Geometric Brownian Motion (FBM). Results show that this method generates random numbers capable of replicating independent, persistent or antipersistent time-series depending on the value of the chosen Hurst exponent. Simulating FBM via the Cholesky method is (i) convenient since it grants the ability to replicate intense and enduring returns, which allows for reproducing well-documented financial returns’ slow convergence in distribution to a Gaussian law, and (ii) straightforward since it takes advantage of the Gaussian distribution ability to express a broad type of stochastic processes by changing how volatility behaves with respect to the time horizon. However, Cholesky method is computationally demanding, which may be its main drawback. Potential applications of FBM simulation include market, credit and liquidity risk models, option valuation techniques, portfolio optimization models and payments systems dynamics. All can benefit from the availability of a stochastic process that provides the ability to explicitly model how volatility behaves with respect to the time horizon in order to simulate severe and sustained price and quantity changes. These applications are more pertinent than ever because of the consensus regarding the limitations of customary models for valuation, risk and asset allocation after the most recent episode of global financial crisis.
... First, traders who actually understood the models compensated for their unworkability by relying on the 'dark twin' 61 of older 'heuristics and tricks' and a vernacular understanding of possible scenarios that they had acquired through long, everyday practice. 62 This had the effect, intended or not, of hiding the shortcomings of the prominently-displayed model-engines from technically-inexperienced higher-ups, clients, governments and the public. Second, many traders used the failures of the models as money-making opportunities, thus ironically shoring up the dominance of the models by becoming trading partners of more gullible quantist true-believers. ...
Enormous new markets in uncertainty and in carbon have been created recently, ostensibly to enhance the cost-effectiveness of both finance and climate action. In both cases, however, creating the abstract commodity framework necessary to make sense of the notion of 'cost-effectiveness' has entailed losing touch with what was supposedly being costed, helping to engender systemic crisis. The new financial markets expanded credit and multiplied leverage by isolating, quantifying, slicing, dicing and circulating diverse types of uncertainty; the resulting unchecked pursuit of liquidity led to a catastrophic drying up of liquidity. The carbon markets, meanwhile, by identifying global warming solutions with reductions in an abstract pool of tradable pollution rights and linking them with 'offsets' manufactured through quantitative techniques, ended up blocking prospective historical pathways toward less fossil fuel dependence and thus exacerbated the climate problem. Unsurprisingly, both markets have provoked strong, if diverse and confused, movements of societal self-defence. This pattern of action and reaction constitutes a chapter in the political history of commodification as significant in some ways as that describing the movements to commodify land and labour analysed by Karl Polanyi.
... There are further examples of such cases in the literature. The recent discussions initiated by option-trading veterans Nassim Taleb (2007), Espen Haug (with Taleb, 2009) and Pablo Triana (2008 suggest that even the paragon of the performativity concept, the Black-Scholes formula, could not produce the markets it described because the model was transformed and manipulated when applied in practice. This statement refers, above all, to the issue of implied volatility. ...
Financial models vary according to their ability to shape markets. This ability
depends on the way models are used in institutional settings. In contrast to the
prevalent view that involvement in organizational practices almost automatically
makes models performative, this paper argues that institutional design might
obstruct potential model performativity. This crucial issue determines whether
models are strongly or only ‘generically’ or ‘effectively’ performative. The empirical
study of the usage of the discounted cash flow (DCF) model presented in this paper
offers an example of an application practice that significantly limits the model’s
performative power.
Finance is one domain where mutually exciting Hawkes processes are particularly interesting. The global financial crisis was a painful lesson in the unobserved interdependencies of our financial systems. In the literature, there is the term ‘financial contagion’ to describe the effect of a financial event in one firm or asset impacting other firms or assets like a virus. The mutually exciting Hawkes model is obviously a useful model for any financial risk manager.
Random walk was an early descriptive phrase and plank in what has come to be called modern portfolio theory. It includes the notions of efficient markets and equilibrium economics. This chapter reviews the subject in detail, not because the theory is necessarily correct and useful, but because it provides a benchmark against which newer and better approaches can be developed. Volatility is a key factor in all bubbles and crashes and an extreme events line based on option volatility is introduced.
We investigate whether there are consistent misspecifications of volatility in the freight options markets that can be exploited in profitable trading strategies. We derive smooth forward freight rate curves from observed market prices and convert these to term structures of historical volatility for the Capesize, Panamax and Supramax segments of the drybulk shipping market. The differences between the historical and implied volatility term structures form the basis for executing option trading strategies. We find that there exist statistical arbitrage profits in the freight option markets, suggesting a degree of market inefficiency.
This book is mainly about the mathematics of Professor Vinzenz Bronzin and his remarkable book on option pricing, published in 1908. This chapter concerns the wider history of option pricing and hedging where Bronzin’s work shines out as a beautiful diamond. The study of the history of option pricing and hedging is much more than simply a study of the ancient past. It reveals more than this: It tells us where we came from, where we are, and possibly even gives us some hints about where we are going or, at least, what direction we should following. The put-call-parity, hedging options with options and some types of market-neutral delta hedging were understood and used at least a hundred years ago and is, in my view, still the foundation of what knowledgeable option traders use today. A careful study of the history, including several somewhat forgotten and ignored ancient sources, several of which have been recently rediscovered, tells us that many of the option traders as well as academics from history were much more sophisticated than most of us would have thought. Here, I will try to give a short (but still incomplete) and, hopefully, useful summary of the history of option pricing and hedging from my viewpoint today. The history of option pricing and hedging is far too complex and profound to be fully described within a few pages or even a book or two, but, hopefully, this contribution will encourage readers to search out more old books and papers and question the premises of modern text books that are often not revised with regard to the history option pricing.
We sketch a large-scale computable general equilibrium model of the macroeconomy that includes modern features such as financial derivatives. This model can be used to examine proposed new economic policies that involve large structural changes in the economy. To simulate and study the model, considerable computational power is required for extensive Monte Carlo simulations. We propose using a grid supercomputer to do these Monte Carlo simulations so that the results can be obtained in a reasonable amount of time. To evaluate the new policy, the supercomputer will run two sets of Monte Carlo simulations: (1) Baseline (2) Supercharged. Both sets contain trillions of stochastic simulations. After running both the baseline and supercharged simulations, the social welfare in the two possible scenarios can be compared to see if economic welfare was improved by the proposed supercharged economic policy.
Integration of financial markets due to globalization generates new paradigms of financialization. And with HFT i.e. the high-frequency trading, financialization has distorted the relations of financial markets. HFT is based on highly complex financial products such as Index Options which are linked with volatility and it‘s forecasting. After the introduction of Volatility, VIX Index of Chicago Board of Options Exchange becomes the effective benchmark for stock market volatility now a day. Although VIX Index is a volatility measure derived from Standard and Poor 500 Index (SPX) option prices, traders are unaware of the inverse relationship between these markets. This study purpose is to understand the relationship between the two trading vehicles and increase the market awareness based on high order moment models which are used to mimic the behavior of these index options. It also explains the logic versus perception perspective in option pricing theory to develop theoretical foundations and consider it in future theory erection. Finding shows that SPX index is negatively correlated with VIX Index and financial markets have an inverse relationship between them.
Random walk was an early descriptive phrase and plank in what has come to be called modern portfolio theory. It includes the notions of efficient markets and equilibrium economics. This chapter reviews the subject in detail, not because the theory is necessarily correct and useful, but because it provides a benchmark against which newer and better approaches can be compared. Volatility is a key factor in all bubbles and crashes and an extreme events line based on option volatility is introduced.
Two cornerstones in Behavioral Finance, the limits of arbitrage and investor psychology, can explain the formation of implied volatility skew existing in Taiwan stock index option (TXO) market well. Adopting the real-time data which exhibits the limits of arbitrage in the futures market and designing two speculation models which describe the trading behavior of the market maker in the option market, this study successfully calibrates out the market maker's perceived volatility with a view to exhibit the similar pattern to the volatility asymmetry in spot market. The trading behavior of the market maker in the prevalence of positive feedback traders is based upon the argument of Destabilizing Rational Speculation of (De Long et al., 1990) that is well suitable to a market full of noise traders like the TXO market. one thing deserves to mention is that the calibration of our second speculation model, Shifted Speculation Model, on the implied volatility curve involving merely single volatility parameter shows that it is a self-consistent model, improving the often unjustly maligned defect of Black-Scholes Model and conforming to the market practice.
I argue that the culture of speculative capitalism is created in the Gilded Age through innovative financial practices, contemporaneous with industrial capitalism. Two key trajectories led to this normative formation: (a) increasing technicality and isolation of economics as a field of knowledge, and (b) legitimation of futures trading as valid, valuable labor. I focus on 1875–1936, revisiting debates resulting in the legal-discursive construction of difference between “gamblers” and “investors,” and also account for the “sciencing” of the discipline of economics. The claim is that the excesses of speculative capitalism are not to be pinned on 21st century Wall Street nor capitalism writ large but on the culture of speculative capitalism and the gambling spirit that it creates and/or harnesses.
Doreurs et joueurs : la culture du capitalisme spéculatif aux États-Unis
Résumé : Je soutiens que la culture du capitalisme spéculatif est créée dans l'ère dorée par des pratiques financières innovatrices, contemporaines au capitalisme industriel. Deux trajectoires clés ont mené à cette formation normative : (1) la technicalité et l'isolement croissants de l'économie comme champ de connaissances, et (2) la légitimation des transactions à terme comme un travail valide et de grande valeur. Je me concentre sur la période 1875-1936, réexaminant les débats ayant pour résultat la construction légale-discursive de la différence entre les 〈 joueurs 〉 et les 〈 investisseurs 〉, et j'explique aussi la 〈 sciencisation 〉 de la discipline de l'économie. L'affirmation est que les excès du capitalisme spéculatif ne doivent pas être imputés au Wall Street au 21e siècle ni au capitalisme exagéré, mais sur la culture du capitalisme spéculatif et l'esprit de jeu d'argent qu'il crée ou exploite.
Mots clés : économie politique, culture et politique, discours, études culturelles critiques, sphère publique
Vergolder und Zocker: Die Kultur des spekulativen Kapitalismus in den USA
Ich argumentiere, dass sich die Kultur des spekulativen Kapitalismus im Goldenen Zeitalter durch innovative finanzielle Praktiken zeitgleich mit dem industriellen Kapitalismus entwickelt hat. Zwei zentrale Bewegungen führten zu dieser normativen Formation: 1) eine zunehmende Formalisierung und Isolation der Wirtschaftswissenschaft als Wissensgebiet und 2) die Legitimation von Terminhandel als anerkannte und erlaubte Beschäftigung. Ich fokussiere auf den Zeitraum zwischen 1987 und 1936 und betrachte Debatten, die in einer rechtlich-diskursiven Konstruktion von Unterschieden zwischen Spielern“ und Investoren“ resultieren, und zudem auf die Verwissenschaftlichung der Wirtschaftswissenschaft als Disziplin Bezug nehmen. Die Kernaussage lautet, dass die Auswüchse des spekulativen Kapitalismus nicht an der Wallstreet des 21. Jhd. oder dem Kapitalismus an sich festgemacht werden können, sondern an einer Kultur des spekulativen Kapitalismus und dem Glückspielgeist, die aus ihm erwachsen und/oder den er sich zunutze macht.
Schlüsselbegriffe: Politische Ökonomie, Kultur und Politik, Diskurs, Kritische Kulturstudien, Öffentlichkeit
Los Dorados y los Jugadores: La Cultura del Capitalismo Especulativo en los Estados Unidos
Resumen: Arguyo que la cultura especulativa del capitalismo está creada en la Edad Dorada a través de las prácticas financieras innovadoras, contemporáneas al capitalismo industrial. Dos trayectorias claves llevaron a esta formación normativa: 1) el incremento del tecnicismo y el aislamiento de la economía como campo de conocimiento, y 2) la legitimación del comercio futuro como labor válida y valioso. Me focalizo en 1875-1936, revisitando los debates resultantes de la construcción legal discursiva de la diferencia entre los ‘jugadores’ y los ‘inversores,’ y explicando el ‘cientificismo’ de la disciplina económica. La alegación es que los excesos del capitalismo especulativo no están prendidos al Wall Street del siglo 21 ni al capitalismo ordenado largamente sino a la cultura especulativa del capitalismo y el espíritu de juego que lo crea y/o lo aprovecha.
Palabras claves: Economía Política, Cultura y política, Discurso, Estudios culturales críticos, Esfera pública.
The focus of this paper is the identification, and more importantly, sustainable management, of risks embedded in guarantees attaching to unit linked savings and retirement contracts (as commonly referred to as GMxBs). In developing customer centric guarantees that are not readily transferrable to the capital markets, insurance undertakings require the skills and resources to hedge the guarantees within their own balance sheet (or with a temporary use of packaged solutions such as reinsurance). In taking on the guarantee manufacture task insurers are departing from areas of historic competence and need to develop a comprehensive understanding of all elements of market risk replication. These include both first order market exposures as well as the material second order risks associated with market micro structure. The paper seeks to integrate this comprehensive analysis within a practitioner focused framework and concludes with a senior executive summary of “Seven key considerations in successful guarantee manufacture”.
Does it make sense to talk about creativity in financial markets? At first glance, the title of the article might appear paradoxical.
Financial markets are considered to be a place where rationality governs. Economic agents behave rationally as they maximize
their utility; they choose the action that helps them most to pursue defined goals. Financial market participants are just
such rational agents; they follow strict rules, and their behavior is unambiguously determined and thus calculable and predictable.
What is the reason then to talk about creativity in this supposedly most rational field of economic life?
I contend that rational expectations theory totally misinterprets how financial markets operate. Although rational expectations theory is no longer taken seriously outside academic circles, the idea that financial markets are self-correcting and tend towards equilibrium remains the prevailing paradigm on which the various synthetic instruments and valuation models which have come to play such a dominant role in financial markets are based. I contend that the prevailing paradigm is false and urgently needs to be replaced." George Soros. 2008. This conceptual paper will take on the Soros challenge, using the concepts and methodologies of real options 'in' economic systems and adding fresh thinking about data sources and the meaning of risk. It will examine the shortcomings and fallacies associated with modern portfolio and capital market theory and neo-classical economics and ask many disquieting questions. General solutions, based in real options thinking, will be proposed and extended to the outer edge of global economic systems as found in the underground economies of the developing world.
Author begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new ″down-and-out″ option are derived. Other results.
Au cours de la dernière décade on a constaté que le joueur pouvait avoir l'avantage dans certains jeux de hasard. On verra que le "blackjack", la mise latérale au Baccara - tel qu'il est joué dans le Névada - la roulette et la "roue de la fortune", peuvent tous offrir au joueur une espérance de gain positive. La Bourse a beaucoup de traits communs avec ces jeux de hasard [5]. Elle offre des situations particulières avec des gains attendus allant au-delà d'un taux annuel de 25% [23]. Dès que la théorie particulière d'un jeu a été utilisée pour identifier des situations favorables, se pose le problème de savoir comment répartir au mieux nos ressources. Parallèlement à la découverte de situations favorables dans certains jeux, les grandes lignes d'une théorie mathématique générale pour exploiter ces opportunités s'est développée [2. 3. 10. 13.]. On décrira d'abord les jeux favorables mentionnés ci-dessus: ce sont ceux que l'auteur connaît le mieux. On discutera ensuite la théorie mathématique générale, telle qu'elle s'est développée jusqu'à maintenant, et son application à ces jeux. Une connaissance détaillée d'un jeu particulier n'est pas nécessaire pour suivre l'explication. Chaque discussion portant un jeu favorable dans la partie I est suivie d'un résumé donnant les probabilités correspondantes. Ces résumés sont suffisants pour la discussion de la partie II de sorte qu'un lecteur qui n'a aucun intérêt dans un jeu particulier peut passer directement au résumé. Des références sont données pour ceux qui désirent étudier certains jeux en détail. Pour l'instant, "jeu favorable" veut dire, jeu dans lequel la stratégie est telle que $P({\rm lim}\,S_{n}=\infty)>0$ où Sn est le capital du joueur après n essais.
The validity of the classic Black-Scholes option pricing formula depends on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the more-general case when the underlying stock returns are generated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black-Scholes formula in that it does not depend on investor preferences or knowledge of the expected return on the underlying stock. Moreover, the same analysis applied to the options can be extended to the pricing of corporate liabilities.
The essays in this volume are written by a distinguished and adventurous set of historians and economists who have been willing, in many cases, to step beyond their typical field of inquiry and explore the historical foundations of financial innovation. The essays are motivated by the need to place our current age of finanical revolution in historical perspective. The continuing process of financial innovation, as sophisticated as it may seem to most of the modern world, is in fact built on surprisingly few basic principles: the inter-temporal transfer of value through time, the ability to contract on future outcomes, and the negotiability of claims. This book traces the evolution of these basic principles of finance through 3,000 years of history - to the dawn of writing. The methodology that is used can be thought of as financial archaeology in the sense that the authors focus on primary survived financial documents to draw their conclusions such as clay tablets, notched sticks, sealed parchment and printed paper. The analysis of original documents is a means for economists to focus on the primary text, to analyze and interpret the object and to move interpretation and understanding of its relationship to modern financial instruments and markets. The result is a collection of interdisciplinary studies of the key innovations in finance from the Old Babylonion loan tablets, to the 1953 London Debt Agreement that span regions in Asia, Africa, North America and Europe.
Insights into the dynamics of a complex system are often gained by focusing on large fluctuations. For the financial system, huge databases now exist that facilitate the analysis of large fluctuations and the characterization of their statistical behaviour. Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades. Surprisingly, the exponents that characterize these power laws are similar for different types and sizes of markets, for different market trends and even for different countries--suggesting that a generic theoretical basis may underlie these phenomena. Here we propose a model, based on a plausible set of assumptions, which provides an explanation for these empirical power laws. Our model is based on the hypothesis that large movements in stock market activity arise from the trades of large participants. Starting from an empirical characterization of the size distribution of those large market participants (mutual funds), we show that the power laws observed in financial data arise when the trading behaviour is performed in an optimal way. Our model additionally explains certain striking empirical regularities that describe the relationship between large fluctuations in prices, trading volume and the number of trades.
In recent years, physicists have begun to apply concepts and methods of statistical physics to study economic problems, and the neologism "econophysics" is increasingly used to refer to this work. Much recent work is focused on understanding the statistical properties of time series. One reason for this interest is that economic systems are examples of complex interacting systems for which a huge amount of data exist, and it is possible that economic time series viewed from a di#erent perspective might yield new results. This manuscript is a brief summary of a talk that was designed to address the question of whether two of the pillars of the #eld of phase transitions and critical phenomena -- scale invariance and universality -- can be useful in guiding research on economics. We shall see that while scale invariance has been tested for many years, universality is relatively less frequently discussed. This article reviews the results of two recent studies -- (i) The probability distribution of stock price #uctuations: Stock price #uctuations occur in all magnitudes, in analogy to earthquakes -- from tiny #uctuations to drastic events, such as market crashes. The distribution of price #uctuations decays with a power-law tail well outside the L#evy stable regime and describes #uctuations that di#er in size by as much as eight orders of magnitude. (ii) Quantifying business #rm #uctuations: We analyze the Computstat database comprising all publicly traded United States manufacturing companies within the years 1974--1993. We #nd that the distributions of growth rates is di#erent for di#erent bins of #rm size, with a width that varies inversely with a power of #rm size. Similar variation is found for other complex organizations, including country size, university research bud...
This book takes a theoretical and practical look at some of the latest and most important ideas behind derivatives pricing models. In each chapter the author highlights the latest thinking and trends in the area. A wide range of topics is covered, including valuation methods on stocks paying discrete dividend, Asian options, American barrier options, Complex barrier options, reset options, and electricity derivatives. The book also discusses the latest ideas surrounding finance like the robustness of dynamic delta hedging, option hedging, negative probabilities and space-time finance.
The accompanying CD with additional Excel sheets includes the mathematical models covered in the book.
The book also includes interviews with some of the world’s top names in the industry, and an insight into the history behind some of the greatest discoveries in quantitative finance. Interviewees include:
Nassim Taleb on Black Swans,
Edward Thorp on Gambling and Trading,
Alan Lewis on Stochastic Volatility and Jumps,
Emanuel Derman, the Wall Street Quant,
Peter Carr, the Wall Street Wizard of Option Symmetry and Volatility,
Clive Granger, Nobel Prize winner in Economics 2003, on Cointegration,
Stephen Ross on Arbitrage Pricing Theory,
Bruno Dupire on Local and Stochastic Volatility Models,
Eduardo Schwartz the Yoga Master of Quantitative Finance,
Aaron Brown on Gambling, Poker and Trading,
Knut Aase on Catastrophes and Financial Economics,
Elie Ayache on Modeling,
Paul Wilmott on Paul Wilmott,
Andrei Khrennikov on Negative Probabilities,
David Bates on Crash and Jumps,
Peter Jäckel on Monte Carlo Simulation
Rank-size plots, also called Zipf plots, have a role to play in representing statistical data. The method is somewhat peculiar, but throws light on one aspect of the notions of concentration. This chapter’s first goals are to define those plots and show that they are of two kinds. Some are simply an analytic restatement of standard tail distributions but other cases stand by themselves. For example, in the context of word frequencies in natural discourse, rank-size plots provide the most natural and most direct way of expressing scaling. Of greatest interest are the rank-size plots that are rectilinear in log-log coordinates. In most cases, this rectilinearity is shown to simply rephrase an underlying scaling distribution, by exchanging its coordinate axes. This rephrasing would hardly seem to deserve attention, but continually proves its attractiveness. Unfortunately, it is all too often misinterpreted and viewed as significant beyond the scaling distribution drawn in the usual axes. These are negative but strong reasons why rank-size plots deserve to be discussed in some detail. They throw fresh light on the meaning and the pitfalls of infinite expectation, and occasionally help understand upper and lower cutoffs to scaling.
URL: http://www-spht.cea.fr/articles/s00/202
This is a compact report on desultory researches stretching over more than a decade.
Long-established as a definitive resource by Wall Street professionals, The Complete Guide to Option Pricing Formulas has been revised and updated to reflect the realities of today's options markets. The Second Edition contains a complete listing of virtually every pricing formula_ all presented in an easy-to-use dictionary format, with expert author commentary and ready-to-use programming code.
The Second Edition of this classic guide now includes more than 60 new option models and formulas…extensive tables providing an overview of all formulas…new examples and applications…and an updated CD containing all pricing formulas, with VBA code and ready-to-use Excel spreadsheets.
The volume also features several new chapters covering such things as: option sensitivities, discrete dividend, commodity options, and two chapters on numerical methods covering trees, finite difference and Monte Carlo Simulation.
Jean-Phillipe Bouchaud and Marc Potters, citing option markets and risk awareness, challenge the view that the Black-Scholes model needs little improvement - in fact, it should be seen as a special case of a more general theory.
We model demand-pressure effects on option prices. The model shows that demand pressure in one option contract increases its price by an amount proportional to the variance of the unhedgeable part of the option. Similarly, the demand pressure increases the price of any other option by an amount proportional to the covariance of the unhedgeable parts of the two options. Empirically, we identify aggregate positions of dealers and end-users using a unique dataset, and show that demand-pressure effects make a contribution to well-known option-pricing puzzles. Indeed, time-series tests show that demand helps explain the overall expensiveness and skew patterns of index options, and cross-sectional tests show that demand impacts the expensiveness of single-stock options as well. The Author 2009. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.
If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.
This paper implements the time-state preference model in a multi-period economy, deriving the prices of primitive securities from the prices of call options on aggregate consumption. These prices permit an equilibrium valuation of assets with uncertain payoffs at many future dates. Furthermore, for any given portfolio, the price of a $1.00 claim received at a future date, if the portfolio's value is between two given levels at that time, is derived explicitly from a second partial derivative of its call-option pricing function. An intertemporal capital asset pricing model is derived for payoffs that are jointly lognormally distributed with aggregate consumption. It is shown that using the Black-Scholes equation for options on aggregate consumption implies that individuals' preferences aggregate to isoelastic utility.
We propose an extension of Harsanyi's Impartial Observer Theorem based on the representation of ignorance as the set of all possible probability distributions over individuals. We obtain a characterization of the observer's preferences that, under our most restrictive conditions, is a convex combination of Harsanyi's utilitarian and Rawls' egalitarian criteria. This representation is ethically meaningful, in the sense that individuals' utilities are cardinally measurable and fully comparable. This allows us to conclude that the impartiality requirement cannot be used to decide between Rawls' and Harsanyi's positions.
The introduction of exchange-traded options in 1973 led to explosive growth in the stock options market, but put and call options on equity securities have existed for more than a century. Prior to the listing of option contracts, trading was conducted in an order-driven over-the-counter market. From 1873 to 1875, quotes for options contracts were published weekly in The Commercial and Financial Chronicle during a period that saw extensive marketing efforts by a number of brokerage firms. In this article, the authors examine these quotes to determine why this seemingly sophisticated market existed for only a brief period in financial history. Copyright 1997 by American Finance Association.
Understanding Put and Call Options
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The Illusion of Dynamic Delta Replication
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Quantitative Finance, 5(4), 323-326.
Quantitative Finance
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How to Make Profits in Puts and Calls
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Option Writing and Hedging Strategies
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Re-printed in the book: Extraordinary Popular Delusions and the Madness of Crowds & Confusión de Confusiones
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Extraordinary Popular Delusions and the Madness of Crowds & Confusión de
Confusiones edited by Martin S. Fridson, 1996, New York: Wiley Publishing.
A New Approach for Understanding the. Impact of Volatility on Option Prices
- B Dupire
Dupire, B. (1998): "A New Approach for Understanding the. Impact of Volatility on
Option Prices", Discussion paper Nikko Financial Products.
The Business of Options
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- P Martin
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Volatility derivatives modeling
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A History of The Theory of Investments
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Thorp, E. O. (2007): "Edward Thorp on Gambling and Trading," in Haug (2007).
The A B C of Options and Arbitrage
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The Nature of Puts & Calls
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Scale-Invariance in Practice: Some Questions and Workable Patches
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Patches", working paper.
The volatility Surface
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The Behaviour of Prices
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