QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

This bestselling and thoroughly classroom-tested textbook is a complete resource for finance students. A comprehensive and illustrated discussion of the most common empirical approaches in finance prepares students for using econometrics in practice, while detailed case studies help them understand how the techniques are used in relevant financial contexts. Worked examples from the latest version of the popular statistical software EViews guide students to implement their own models and interpret results. Learning outcomes, key concepts and end-of-chapter review questions (with full solutions online) highlight the main chapter takeaways and allow students to self-assess their understanding. Building on the successful data- and problem-driven approach of previous editions, this third edition has been updated with new data, extensive examples and additional introductory material on mathematics, making the book more accessible to students encountering econometrics for the first time. A companion website, with numerous student and instructor resources, completes the learning package.

Introductory Econometrics for Finance

/pdf/the-black-swan-the-impact-of-the-highly-improbable-578rzpihbf.pdf

The Black Swan: The Impact of the Highly Improbable

The Random Character of Stock Market Prices.

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Gravitation And Cosmology

The Complete Guide to Option Pricing Formulas

Option traders use a heuristically derived 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 contra- diction with it). However, we have historical evidence that: (1) the said 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) sophisticated 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 using put-call parity, (3) option traders did not use the Black-Scholes-Merton formula or similar formulas after 1973 but continued their bottom-up heuristics more robust to the high impact rare event. The paper draws on historical trading methods and 19th and early 20th century references ignored by the finance literature. It is time to stop using the wrong designation for option pricing.

/pdf/option-traders-use-very-sophisticated-heuristics-never-the-3p65jvqjya.pdf

Option traders use (very) sophisticated heuristics, never the Black- Scholes-Merton formula 1

This article discusses the valuation and hedging of volatility swaps within the frame of a GARCH(1,1) stochastic volatility model. First we use a general and flexible partial differential equation (PDE) approach to determine the first two moments of the realized variance in a continuous or discrete context. Next, and also the main contribution of the paper, is a closed-form approximate solution for the so-called convexity correction, when the risk-neutral process for the instantaneous variance is a continuous time limit of a GARCH (1,1) model. Following this, we provide a numerical example using S&P 500 data.

GARCH and volatility swaps

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. 3) Option traders did not use formulas after 1973 but continued their bottom-up heuristics. The Bachelier-Thorp approach is more robust (among other things) to the high impact rare event. The paper draws on historical trading methods and 19 th and early 20 th century references ignored by the finance literature. It is time to stop calling the formula by the wrong name.

/pdf/why-we-have-never-used-the-black-scholes-merton-option-38rtdszbdo.pdf

Espen Gaarder Haug

Papers

The Complete Guide to Option Pricing Formulas

Option traders use (very) sophisticated heuristics, never the Black- Scholes-Merton formula 1

GARCH and volatility swaps

Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula

Algorithms. Back to basics: A new approach to the discrete dividend problem