Concept# Black–Scholes model

Summary

The Black–Scholes ˌblæk_ˈʃoʊlz or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments, using various underlying assumptions. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The equation and model are named after economists Fischer Black and Myron Scholes; Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.
The main principle behind the model is to hedge the option by buying and selling the underlying asset in a specific way to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is th

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Participants of this course will master computational techniques frequently used in mathematical finance applications. Emphasis will be put on the implementation and practical aspects.

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The objective of this course is to provide a detailed coverage of the standard models for the valuation and hedging of derivatives products such as European options, American options, forward contracts, futures contract and exotic options.

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This article derives a closed-form pricing formula for European exchange options under a non-Gaussianframework for the underlying assets, intending to resolve mispricing associated with a geometric Brownianmotion. The dynamics of each of the two correlated underlying assets are assumed to be governed by theexponential of a skew-Brownian motion, which is specified as a sum of a standard Brownian motion and anindependent reflected Brownian motion. The proposed pricing formula does not incur additional computationalcosts than the standard Black-Scholes framework, which one can quickly recover as a particular case of theproposed framework. Finally, we present some numerical experiments followed by a valuable discussion onthe results

In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. We provide sufficient conditions to guarantee convergence of the series. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however we show numerically that the bias can safely be ignored in practice.

This thesis studies the valuation and hedging of financial derivatives, which is fundamental for trading and risk-management operations in financial institutions. The three chapters in this thesis deal with derivatives whose payoffs are linked to interest rates, equity prices, and dividend payments.
The first chapter introduces a flexible framework based on polynomial jump-diffusions (PJD) to jointly price the term structures of dividends and interest rates. Prices for dividend futures, bonds, and the dividend paying stock are given in closed form. Option prices are approximated efficiently using a moment matching technique based on the principle of maximum entropy. An extensive calibration exercise shows that a parsimonious model specification has a good fit with Euribor interest rate swaps and swaptions, Euro Stoxx 50 index dividend futures and dividend options, and Euro Stoxx 50 index options.
The second chapter revisits the problem of pricing a continuously sampled arithmetic Asian option in the classical Black-Scholes setting. An identity in law links the integrated stock price to a one-dimensional polynomial diffusion, a particular instance of the PJD encountered in the first chapter. The Asian option price is approximated by a series expansion based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however numerical experiments show that the bias can safely be ignored in practice.
The last chapter presents a non-parametric method to construct a maximally smooth discount curve from observed market prices of linear interest rate products such as swaps, forward rate agreements, or coupon bonds. The discount curve is given in closed form and only requires basic linear algebra operations. The method is illustrated with several practical examples.