I work through a simple Python bvimplementation of geometric Brownian motion and check it against the theoretical model.

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A mean–variance optimizer will hedge correlated assets. I explain why and then work through a simple example.

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In finance, the "Greeks" refer to the partial derivatives of an option pricing model with respect to its inputs. They are important for understanding how an option's price may change. I discuss the Black–Scholes Greeks in detail.

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The VIX is a benchmark for market-implied volatility. It is computed from a weighted average of variance swaps. I first derive the fair strike for a variance swap and then discuss the VIX's approximation of this formula.

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I work through a well-known approximation of the Black–Scholes price of at-the-money (ATM) options.

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The binomial options-pricing model converges to Black–Scholes as the number of steps in fixed physical time goes to infinity. I present Chi-Cheng Hsia's 1983 proof of this result.

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I present a simple yet useful model for pricing European-style options, called the binomial options-pricing model. It provides good intuition into pricing options without any advanced mathematics.

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In the options-pricing literature, the Carr–Madan formula equates a derivative's nonlinear payoff function with a portfolio of options. I describe and prove this relationship.

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The binomial options-pricing model is a numerical method for valuing options. I explore this model over a single time period and focus on two key ideas, the no-arbitrage condition and risk-neutral pricing.

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The Sharpe ratio measures a financial strategy's performance as the ratio of its reward to its variability. I discuss this metric in detail, particularly its relationship to the information ratio and $t$-statistics.

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A common heuristic for time-aggregating volatility is the square root of time rule. I discuss the big idea for this rule and then provide the mathematical assumptions underpinning it.

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I discuss multi-factor modeling, which generalizes many early financial models into a common prediction and risk framework.

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In finance, the capital asset pricing model (CAPM) was the first theory to measure systematic risk. The CAPM argues that there is a single type of risk, market risk. I derive the CAPM from the mean–variance framework of modern portfolio theory.

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I discuss prices, returns, cumulative returns, and log returns, with a special focus on some nice mathematical properties of log returns.

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Some important financial ideas are encoded in the geometry of the efficient frontier, such as the tangency portfolio and the Sharpe ratio. The goal of this post is to re-derive these ideas geometrically, showing that they arise from the mean–variance analysis framework.

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Drawdown measures the decline of a time series variable from a historical peak. I explore visualizing and computing drawdown-based metrics.

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The casual investor knows that diversification matters. This intuition is grounded in the mathematics of modern portfolio theory. I define diversification and formalize how diversification helps maximize risk-adjusted returns.

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