Derivation of Black-Scholes Equation Using Itô's Lemma

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 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...

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...

The use of iterative methods for solving Black-Scholes equation

A non-linear Black-Scholes equation

The field of mathematical finance has gained significant attention since Black and Scholes (1973) published their Nobel Prize work in 1973. Using some simplifying economic assumptions, they derived a linear partial differential equation (PDE) of convection–diffusion type which can be applied to the pricing of options. The solution of the linear PDE can be obtained analytically. In this paper we...