Computation of Portfolio Volatility - Simulating Efficient Frontier
A mathematical and simulation-driven look at the relationship between asset correlations, portfolio variance, and the efficient frontier.
This paper aims to break down how asset variance, covariance, and portfolio weights interact to determine overall portfolio risk. The paper builds the mathematical framework for portfolio volatility and then uses a large-scale simulation of random portfolios to illustrate how the efficient frontier forms. Through this, it shows how diversification and correlation shape the limits of achievable risk-return profiles.
Martingales and the importance of risk-neutral probabilities
An explanation of risk neutral probabilities and martingales. A look at their importance in the binomial model.
This paper explains the role of risk-neutral probabilities in the binomial model, showing how they ensure that the discounted stock price is a martingale and that arbitrage is avoided. It introduces martingales, submartingales, and supermartingales as key probabilistic tools for understanding expected price movements. By connecting these ideas, the paper provides an accessible foundation for option pricing and highlights why the risk-neutral measure is central in quantitative finance.
Expected Values: Cutting down on computational costs of the European binomial options pricing model
This paper shows how expected values and conditional probabilities not only reveal the intuition behind the binomial model but also lead to a faster, more efficient way to compute it.
This paper explains more probability theory and goes into the concept of expected values on finitie probability spaces. It goes into conditional expectations of a coin toss as a primer. Then it explains how expected values underly the binomial options pricing formual. It then demonstrates how this knowledge can be used to abreviated the binomial options pricing model for European options. This abreviation both goes to demonstrate the core concept and serves as means to cut down on computational costs.
European Option Pricing with the Binomial Tree: Convergence to Black Scholes
An introduction to the Binomial pricing model for European options, showing how it converges to the Black–Scholes formula.
This paper explores the Binomial pricing model as applied to European options. Originally developed as a teaching tool, the model has since become a powerful framework for pricing more complex derivatives, including American options.
The study outlines the mathematical foundations of the Binomial approach, explains its algorithmic implementation, and demonstrates how the model’s discrete step method converges to the continuous Black–Scholes formula. This highlights the deep connection between discrete-time and continuous-time option valuation methods.