Math for Quantitative Trading

Interpret the variables, constants, operators, and units in a quantitative-trading formula.

Evaluate a multivariable formula by substituting given numerical values in the correct order.

Rearrange a formula to isolate a specified variable.

Convert among fractions, decimals, percentages, growth multipliers, and basis points.

Calculate net profit or loss for a long or short position from its entry price, exit price, quantity, and stated fees.

Calculate the simple return of an asset over one period.

Calculate return on posted capital for a leveraged position.

Check a numerical result for consistent units and a reasonable order of magnitude.

Apply the laws of exponents to simplify a numerical or algebraic expression.

Calculate the future value of an amount that grows at a fixed rate over repeated periods.

Convert between a constant per-period growth rate and cumulative growth, including finding the equivalent geometric average rate.

Define a logarithm as the inverse of exponentiation and evaluate basic logarithmic expressions.

Solve for the number of periods in a compound-growth equation using logarithms.

Calculate the logarithmic return between two prices.

Aggregate consecutive logarithmic returns by addition.

Compare simple return with logarithmic return for a given price change.

Identify the input, output, domain, and range of a function.

Plot a linear function and interpret its slope and intercept.

Determine the equation of a straight line from two points.

Graph an absolute-value function and interpret its value as distance from zero.

Evaluate a piecewise-defined function at a specified input.

Sketch and interpret the key shapes of exponential and logarithmic functions.

Apply a horizontal or vertical shift to a function graph.

Find and interpret the break-even point where two value functions intersect.

Solve a linear inequality and express its solution on a number line.

Solve an absolute-value inequality representing a price-distance condition.

Solve a system of two linear equations in two unknowns.

Solve a quadratic equation using the quadratic formula.

Determine the nth term of an arithmetic sequence.

Determine the nth term of a geometric sequence.

Translate a finite sum into and out of sigma notation.

Calculate the sum of a finite geometric series.

Calculate the arithmetic mean of a set of returns.

Calculate a weighted mean from values and their assigned weights.

Calculate specified quantiles and the interquartile range of a dataset.

Calculate population or sample variance using the stated denominator convention.

Calculate standard deviation and interpret it as dispersion in returns.

Standardize an observation by calculating its z-score.

Calculate covariance between two return series.

Calculate a correlation coefficient, interpret its sign and magnitude, and explain why correlation alone does not establish causation.

Assign probabilities to outcomes using equally likely cases or observed relative frequencies.

Calculate the probability of a complement or union of events, including mutually exclusive events.

Count unordered selections using combinations.

Calculate a conditional probability from joint and marginal probabilities.

Determine whether two events are independent.

Apply the multiplication rule to a sequence of events.

Update a probability using Bayes’ theorem.

Calculate the expected value of a discrete trading payoff.

Represent a discrete random variable with a probability table.

Calculate the variance and standard deviation of a discrete random variable.

Calculate a binomial probability for a specified number of successes.

Interpret probability as area under a continuous density curve.

Read probabilities and percentiles from a cumulative distribution function.

Convert a normally distributed value into a standard normal z-score.

Calculate an interval or tail probability under a normal distribution.

Explain why exponentiating a normally distributed variable produces a positive lognormal model for prices.

Distinguish a population, sample, parameter, and statistic in a market-data example.

Calculate the standard error of a sample mean.

Use the central limit theorem to approximate the sampling distribution of a mean.

Construct and interpret a normal-approximation confidence interval for a population mean.

Identify null and alternative hypotheses and correctly interpret a supplied p-value without treating it as the probability that a hypothesis is true.

Fit a least-squares line to a small paired dataset and calculate its residuals.

Interpret the slope and intercept of a fitted regression line.

Interpret R-squared as explained variation without treating it as evidence of causation.

Represent asset returns or portfolio weights as vectors.

Add vectors and multiply a vector by a scalar.

Calculate and interpret the dot product of two vectors.

Identify the dimensions and entries of a matrix.

Calculate a matrix-vector product.

Transpose a vector or matrix and use row-column orientation to determine whether a matrix product is defined.

Calculate expected portfolio return as a weighted sum.

Calculate portfolio variance from weights and a covariance matrix.

Evaluate the limit of an elementary function at a specified point.

Calculate the derivative of a linear or quadratic function from the limit definition.

Differentiate a power function.

Differentiate an exponential or logarithmic function.

Apply the product rule to differentiate a product of functions.

Apply the chain rule to differentiate a composite function.

Locate and classify a local maximum or minimum using first and second derivatives.

Approximate a definite integral from a graph or table and interpret it as accumulated quantity or probability.

Create a lagged version of a time-indexed price or return series.

Calculate first differences of a time series.

Transform a sequence of prices into a sequence of simple or logarithmic returns.

Calculate a rolling mean over a specified window.

Calculate rolling volatility over a specified window.

Assess whether the mean and variance of a series appear stable over time.

Calculate and interpret a lag-one autocorrelation.

Distinguish the mathematical behavior of a random walk from that of a mean-reverting process.

Annualize a periodic arithmetic mean using a stated convention and distinguish it from a compounded annual return.

Annualize volatility using the square-root-of-time rule and state its key assumption.

Calculate and interpret a Sharpe ratio from return, benchmark, and volatility data.

Calculate maximum drawdown from an equity curve.

Calculate historical value at risk at a specified confidence level.

Calculate notional exposure and profit or loss for a linear perpetual-futures position.

Estimate the liquidation price of a long or short position using a simplified isolated-margin and maintenance-margin model.

Calculate a perpetual-futures funding payment and interpret the sign of the funding rate.