Symmetries of the Black-Scholes equation

The Black-Scholes Equation

The most important application of the Itô calculus, derived from the Itô lemma, in financial mathematics is the pricing of options. The most famous result in this area is the Black-Scholes formulae for pricing European vanilla call and put options. As a consequence of the formulae, both in theoretical and practical applications, Robert Merton and Myron Scholes were awarded the Nobel Prize for E...

Revisiting Black-Scholes Equation

In common finance literature, Black-Scholes partial differential equation of option pricing is usually derived with no-arbitrage principle. Considering an asset market, Merton applied the Hamilton-Jacobi-Bellman techniques of his continuous-time consumption-portfolio problem, deriving general equilibrium relationships among the securities in the asset market. In special case where the interest ...

The use of iterative methods for solving Black-Scholes equation

The Quantum Black-scholes Equation

Motivated by the work of Segal and Segal in [16] on the Black-Scholes pricing formula in the quantum context, we study a quantum extension of the BlackScholes equation within the context of Hudson-Parthasarathy quantum stochastic calculus,. Our model includes stock markets described by quantum Brownian motion and Poisson process. 1. The Merton-Black-Scholes Option Pricing Model An option is a t...

Symmetries, Mellin Transform and the Black-Scholes Equation (A Nonlinear Case)

Using the symmetry group that was found in [1] and further studied in [2], in this work we study the partial differential equation of the Black-Scholes model [3] and relate such symmetries with the Mellin transform to find the price of an european like investment option. We also consider the Black-Schoes-Merton equation in the non-linear case that models investments in which volatility is a fun...