Mathematical models are commonly used to price financial
instruments, especially derivatives. One of the most famous financial models is
Black-Scholes which is often used to price a type of derivative known as an
option. An option is a contract that gives the buyer the option to buy a
security at a pre-specified price on a pre-specified day for a premium today.
Owning an option is similar to owning the underlying security, however with
protection against the downside risk. Financial models aim to calculate the
probability distribution of prices in a certain security at a specific point in
the future. If known precisely, the prices of the derivatives associated with
that security can be found easily. However, the mechanics of financial markets
do not follow a strict set of properties, unlike the mechanics of physical
objects for instance. This means that the future behavior of prices can never
be known. Black-Scholes models the changes in prices as Brownian Motion,
originally observed in fluid dynamics. The model is widely used due to its
simplicity and the existence of a closed-form solution. However, the
Black-Scholes model makes assumptions that do not always hold, leading to the existence
of more complex models such as Heston’s. Both of these models can be
implemented using Monte Carlo simulations, originally designed for use in
nuclear physics. Monte Carlo simulations work by randomly sampling a
probability distribution to create sample price paths. If simulated enough
times, the average payout of the price paths will converge on the true price of
a derivative.

This project aims to research both the Black-Scholes and Heston models in order to implement them with the Monte Carlo method. Then we will analyze the differences between the models as well as their practicality and accuracy. We will also analyze the translation of the Black-Scholes model to real-world market scenarios.

This project aims to research both the Black-Scholes and Heston models in order to implement them with the Monte Carlo method. Then we will analyze the differences between the models as well as their practicality and accuracy. We will also analyze the translation of the Black-Scholes model to real-world market scenarios.

## Contact

Conor HomscheidWashington University in St. Louis Systems Engineering and Finance Major chomscheid@wustl.edu |
Sam DonohueWashington University in St. Louis Systems Engineering and Computer Science Major sam.donohue@wustl.edu |