Black-Scholes Formalism

Derivations and calculations related to the Black-Scholes Formalism

The Black-Scholes model for financial derivatives has had an extraordinary effect on the financial world. It spawned new areas of business and as of 2016, the approximate notional value of derivative financial instruments valued by Black-Scholes and successor models was $500 trillion dollars.

This result also led to the Nobel Prize for Myron Scholes and Robert Merton in 1997 (Fisher Black died in 1995). https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model

The model has a representation in terms of a partial differential equation as well as the famous Black-Scholes formula for the fair value of options with European style exercise. Understanding this theoretical/mathematical framework for modeling options and other contingent claims is a fundamental requirement for those who work with financial derivatives and I spent much time on it during my years as a quant. I share some Mathematica notebooks related to various aspects of the Black-Scholes-Merton modeling formalism.

17-1 Derivation of classic Black-Scholes partial differential equation

Derivation of the Black-Scholes PDE for a single stock paying a continuous dividend (the classic case).

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17-2 Black-Scholes partial differential equation when a foreign currency is involved

Derivation of the Black-Scholes partial differential equation when a foreign equity is involved. The resulting equation is specialized to the case of a quanto option, which is an option of a foreign stock but with the strike price set in the domestic currency.

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17-3 Black-Scholes partial differential equation for two stocks

Derivation of the Black-Scholes partial differential equation for a contingent claim involving two stocks. The resulting equation is specialized to the case of a Margrabe exchange option to exchange one stock for another.

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17-9 Derivation of Black-Scholes formula by calculating an expectation

The Black-Scholes formula for the fair value of a European-style exercise call option is derived by calculating the expectation of the payoff under the risk-neutral probability distribution, an approach also known as the martingale method. The corresponding formula for a put option is derived by considering a symmetry relationship between call and put options. Compare this derivation with the derivation in 17-10.

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17-10 Solving BS PDE for call option 01-10-11

The Black-Scholes partial differential equation is solved t0 obtain the fair value of a European-style exercise call option. The PDE is transformed into the standard heat equation and the Green’s function solution is used. Compare this derivation with the derivation in 17-09. See also 17-5 Heat equation solved using Green’s function in the Mathematical Physics section.

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