# Option pricing using the Black-Scholes model, without the formula

## An alternative perspective on the basics of quantitative finance, the Black-Scholes formula.

Every university student taking a module on finance has seen the Black-Scholes-Merton option pricing formula. It is long, ugly, and confusing. It doesn’t even give an intuition for pricing options. The derivation of it is so difficult that Scholes and Merton received a Nobel prize for it in 1997 (Black died in 1995). It relies on the Feynman-Kac theorem and risk-neutral measures, but I will not get into it.

# Black-Scholes PDE

Pricing an option can be done using the Black-Scholes partial differential equation (BS PDE). …

# The Best Numerical Derivative Approximation Formulas

Approximating derivatives is a very important part of any numerical simulation. When it is no longer possible to analytically obtain a value for the derivative, for example when trying to simulate a complicated ODE. It is of much importance though, as getting it wrong can have detrimental effects on the solution. In this article, I will demonstrate the finite difference approximation schemes and show their accuracy.

Let me start with the standard definition of the derivative:

## 1. Forward difference

Taking the limit of the above function as `h`goes to 0 is numerically infeasible (a computer can’t do it), so the first thing that…

# Option pricing using the Black-Scholes model, without the formula

Every university student taking a module on finance has seen the Black-Scholes-Merton option pricing formula. It is long, ugly, and confusing. It doesn’t even give an intuition for pricing options. The derivation of it is so difficult that Scholes and Merton received a Nobel prize for it in 1997 (Black died in 1995). It relies on the Feynman-Kac theorem and risk-neutral measures, but I will not get into it

# Black-Scholes PDE

Pricing an option can be done using the Black-Scholes partial differential equation (BS PDE). …

# The Time Evolution of a Particle’s Position

To determine the position of a particle can be a very challenging task. The obvious way is to look at it and see where it is at that time. However, as one can expect, measuring the position of a particle at every instance of time is very difficult (impossible). (A particle could have a deterministic trajectory, for example in space without any collisions, but imagine dust particles in the frequently colliding with each other.)

To overcome this obstacle, we can turn to probability theory and try to model the particle’s position in a probabilistic manner. Let’s say that we measure… ## Daniel Reti

Interested in applied maths, quantitative finance, and game theory.