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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. 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 19th and early 20th century references ignored by the finance literature. It is time to stop calling the formula by the wrong name.

Option pricing, put-call parity, delta hedging, Black-Scholes-Merton, Bachelier, Thorp

Abstract:
Closed form formulae for European barrier options are well known from the literature. This is not the case for American barrier options, for which no closed form formulae have been published. One has therefore had to resort to numerical methods. Using lattice models like a binomial or a trinomial tree for valuation of barrier options is known to converge extremely slowly, compared to plain vanilla options. Methods for improving the algorithms have been described by several authors. However, these are still numerical methods that are quite computer intensive. In this paper we show how American barrier options can be valued analytically in a very simple way. This speeds up the valuation dramatically as well as give new insight into barrier option valuation.

Abstract:
In this article we show a simple but important relationship between the put-call transformation and the put-call symmetry as well as extend the relationship to also hold for single and double barrier options. These new barrier transformations give new insight in barrier option valuation. Using the transformation it is possible to value a barrier put option from a barrier call option formula and vice versa. Our results also extend the possibilities for static hedging and closed form valuation for many new exotic options. The new relationships also make us able to value a double barrier option in a simple and intuitive way, only using a few single barrier options.

Abstract:
Embedded options in the worlds physical monies, both coin and paper, are introduced. 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.

Coins, bills, paper money, gold and silver coins, central