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Lecture

Black-Scholes Formula: Risk-neutral Pricing and Option Pricing

In course

MGT-301: Foundations in financial economics

The aim of this course is to expose EPFL bachelor students to some of the main areas in financial economics. The course will be organized around six themes. Students will obtain both practical insights through real-world examples and understand how one can ...

Description

This lecture covers the Black-Scholes formula in financial economics, focusing on risk-neutral pricing and option pricing. The instructor explains the model assumptions, risk-neutrality constraint, and the derivation of the Black-Scholes formula. Topics include implied volatility, option pricing, and pricing by replication.

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Instructors (3)

Julien Hugonnier

Pierre Collin-Dufresne

Rüdiger Fahlenbrach

RUEDIGER FAHLENBRACH is Full Professor at Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland. Formerly on the faculty of the Fisher College of Business of the Ohio State University (USA), he holds a Ph.D. in Finance from the University of Pennsylvania (Wharton). He holds a senior research chair from the Swiss Finance Institute, and is research member at the European Corporate Governance Institute (ECGI). He has research interests in empirical corporate finance, in particular corporate governance and entrepreneurship. Ruediger Fahlenbrach has published in the leading academic journals in finance, including the Journal of Finance, the Journal of Financial Economics, the Review of Financial Studies and the Journal of Financial and Quantitative Analysis. Ruediger currently serves a three year term as elected director of the European Finance Association (2018-2020). He is Associate Editor of the Review of Finance, and former associate editor of the Review of Financial Studies and Financial Management. His research has been reported in many large-circulation newspapers such as The New York Times, The Wall Street Journal, The Economist, Le Temps, NZZ, Handelsblatt, Forbes Magazine, USA Today, and Fortune Magazine.

Official source

Ontological neighbourhood

Economics

Microeconomics: Financial economics

Related concepts (32)

Binomial options pricing model

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting. The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (), and formalized by Cox, Ross and Rubinstein in 1979 and by Rendleman and Bartter in that same year.

Local volatility

A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level and of time . As such, it is a generalisation of the Black–Scholes model, where the volatility is a constant (i.e. a trivial function of and ). Local volatility models are often compared with stochastic volatility models, where the instantaneous volatility is not just a function of the asset level but depends also on a new "global" randomness coming from an additional random component.

Volatility smile

Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices (and thus implied volatilities) than what is suggested by standard option pricing models. These options are said to be either deep in-the-money or out-of-the-money.

Implied volatility

In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equal to the current market price of said option. A non-option financial instrument that has embedded optionality, such as an interest rate cap, can also have an implied volatility. Implied volatility, a forward-looking and subjective measure, differs from historical volatility because the latter is calculated from known past returns of a security.

Stochastic volatility

In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others.

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