The quantitative finance syllabus, layer by layer

Reference article

A quant syllabus is a stack, not a list. Option pricing needs stochastic calculus, which needs probability, which needs calculus and counting, which need school algebra. Start a layer before the one under it and every page feels hard for a reason that is not on that page. So this syllabus is in seven layers, each with a short test that says whether you are ready for the next.

Quantitative finance uses maths, statistics and code to price things, measure risk and test trading ideas in markets. This page is a full syllabus for quantitative finance, in seven layers. Each layer lists its topics, what it needs from the layers below, and a self-test: a few questions with short answers, so you can check yourself before you move up.

The search word "quant" also means the maths section of business-school entrance exams such as the CAT and the GMAT. This page is not about those.

How to read this syllabus

• Layers, not years. A layer can take a few weeks or a full university year, depending on where you start.
• Code runs alongside. Start writing code at Layer 2 and keep going. Every later layer is easier to learn when you can simulate it.
• Which layers you need depends on the job. A quant developer needs Layer 5 numerical methods more than Layer 6 statistics. A quant researcher needs both. Our quant roadmap by role covers that choice.

Layer 1: school maths, rebuilt

Topics: fractions, percentages and ratios; powers, roots and logarithms; algebra and rearranging equations; graphs of lines and curves; reading mathematical notation; simple and compound growth.

Needs: nothing.

Self-test:

1. A price falls 20%. What rise takes it back to where it started? Answer: 25%, because $0.8 \times 1.25 = 1$.
2. Solve $100 (1 + r)^2 = 121$ for $r$. Answer: $r = 0.1$, or 10%.
3. What does $\sum_{i=1}^{n} x_i$ mean in words? Answer: add up $x_1$ to $x_n$.

Layer 2: probability and statistics

Probability is the maths of chance. Statistics is the craft of learning from data. Quant work runs on both.

Topics: counting; conditional probability and Bayes' rule; random variables; expectation, variance and covariance; the binomial, Poisson, normal and exponential distributions; the law of large numbers and the central limit theorem; sampling, estimators, confidence intervals and hypothesis tests; linear regression.

Needs: Layer 1.

Self-test:

1. Roll two fair dice. What is the chance that the total is 7? Answer: $6/36 = 1/6$.
2. $X$ has mean 2 and variance 9. What are the mean and variance of $3X + 1$? Answer: 7 and 81.
3. A test is 99% accurate, and 1 person in 1,000 has the condition. A test is positive. Is the chance that the person has the condition closer to 9% or to 99%? Answer: about 9%, by Bayes' rule: $0.99 \times 0.001 / (0.99 \times 0.001 + 0.01 \times 0.999) \approx 0.09$.

Layer 3: the university maths core

Topics: calculus in one and several variables, including Taylor series; linear algebra (vectors, matrices, eigenvalues, the singular value decomposition); ordinary and partial differential equations at an introductory level; optimisation with and without constraints; and, for researchers, measure-theoretic probability.

Needs: Layers 1 and 2.

Self-test:

1. What is the second-order Taylor expansion of $e^x$ around 0? Answer: $1 + x + x^2/2$.
2. A covariance matrix has eigenvalues 3 and 1. What share of the total variance does the first principal component explain? Answer: $3/4$, or 75%.
3. Minimise $x^2 + y^2$ subject to $x + y = 2$. Answer: $x = y = 1$, value 2.

Layer 4: markets and instruments

Topics: what an exchange does; shares, bonds, funds and indices; interest rates, discounting and compounding; forwards and futures; options and their payoffs; put-call parity; the limit order book and how a price forms; what Volatility measures.

Needs: Layer 1. It can be learned beside Layers 2 and 3.

Self-test:

1. A forward on a share that pays no dividend, with spot price 100, one year to delivery and a 5% continuously compounded rate. What is the forward price? Answer: $100 e^{0.05} \approx 105.13$.
2. State put-call parity for European options on a share with no dividends. Answer: $C - P = S - K e^{-rT}$.
3. Why must a call option's price never fall as its strike falls? Answer: a lower strike pays at least as much in every outcome.

Layer 5: stochastic calculus and pricing

Topics: Brownian motion and quadratic variation; Ito's lemma; stochastic differential equations; martingales; no-arbitrage and risk-neutral valuation; the Black-Scholes model and the Greeks; binomial trees; Monte Carlo simulation; finite-difference methods; implied volatility and the volatility surface; interest-rate models.

Needs: Layers 2, 3 and 4.

Self-test:

1. What is the variance of $W_t$ for a standard Brownian motion? Answer: $t$.
2. Under Black-Scholes, what is $d(\ln S_t)$? Answer: $(\mu - \tfrac12 \sigma^2) dt + \sigma dW_t$.
3. A Monte Carlo price has standard error 0.3 with 40,000 paths. About how many paths give 0.1? Answer: 360,000, because the error shrinks like one over the square root of the path count.

Our free stochastic calculus interview guide works through this layer's core tools.

Layer 6: statistics for research, and risk

Topics for research: time series (autocorrelation, stationarity, ARMA and GARCH models); factor models; regression in practice; machine learning and its failure modes; backtesting, overfitting and multiple testing.

Topics for risk: value at risk and expected shortfall; fat tails and extreme values; credit and counterparty risk; liquidity; the regulatory framework.

Needs: Layers 2 and 3, plus Layer 4 for the risk topics.

Self-test:

1. You test 100 trading rules that have no edge at the 5% level. About how many pass by chance? Answer: about 5.
2. A one-day 99% value at risk is 1 million. What does that number say, and what does it not say? Answer: losses above 1 million are expected on about 1 day in 100; it says nothing about how large those losses are.
3. Daily returns show no autocorrelation, but squared returns do. What does that suggest? Answer: volatility clusters, which a GARCH model is built for.

Layer 7: code

Code is a layer of its own, but it runs alongside the others.

Topics: Python with NumPy and pandas; writing tests; SQL for time series; enough C++ to read and speed up a pricing library; profiling; version control with Git.

Needs: Layer 1 to start. It grows with every layer.

Self-test:

1. Simulate 10,000 rolls of two dice in Python and estimate the chance of a 7. Is it close to $1/6$?
2. Price a European call by Monte Carlo and compare it with the Black-Scholes formula. Do they agree within two standard errors?
3. Compute rolling 20-day volatility of a price series in pandas without a Python loop.

Our quant finance projects page turns this layer into eight things to build.

How Thuztra Learn maps to these layers

Thuztra Learn is our product. Its Core quant curriculum follows this order. Level 0 has eight modules, from what quants do to school maths, money maths, probability, data and a first Python notebook. Level 1 covers probability and expectation, analysis and linear algebra, optimisation, estimates and tails, and stopping problems and games. Level 2 has a model-making module, a rates module, and role tracks for the researcher, trader, developer and risk quant, which cover Layers 5 and 6. A separate coding path covers most of Layer 7: Python, C++, SQL, profiling, and checks for AI-assisted code. Level 3 covers econophysics and machine learning.

What Learn does not cover: low-latency trading systems, network code and exchange connections. The roadmap by role says more about what Learn teaches and what it does not. You can read the full curriculum on the home page. Membership is a paid subscription: see what Thuztra Learn is and what it costs.

Using this syllabus

Pick your current layer by its self-test, not by your degree. A degree tells you what was taught, and the self-test tells you what you can still do. Fix the lowest layer you fail first: every layer above it gets faster.

Education only, not advice. Nothing here promises a job.

In short

Work from the bottom layer up: school maths, probability and statistics, the university maths core, markets, stochastic calculus and pricing, then statistics for research and risk, with code alongside from the second layer. Move up a layer only when you pass the self-test of the one below it.

Terms used here

• Quantitative finance
• Probability
• Option
• Volatility
• Backtest
• Time series

Related

More on these topics

• Getting into quant from any college
• How to become a quant in India, in quantitative finance
• How to become a quant in the UK and Europe, in quantitative finance
• Quant vs data scientist, in quantitative finance
• Quant vs software engineer, and the quant developer in between
• Quantitative finance courses compared, and how to choose one
• The best books for quantitative finance interviews

Topics

• syllabus
• getting started
• prerequisites
• curriculum
• careers

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