Quantitative Trader — Competency Roadmap

You'll learn

• Difference of squares — algebraic identity (a+b)(a-b) used to multiply numbers equidistant from a round base
• Zetamac — standard timed arithmetic benchmark test used by proprietary trading firms
• Fast fraction-to-decimal conversions — instant memory mapping of 1/16, 1/32, and 1/8 increments

How to work through it

1. 1Learn algebraic identities used in mental math such as (a+b)(a-b) = a^2 - b^2
2. 2Memorize all integer squares up to 35 and cubes up to 12
3. 3Practice mental factoring and decimal-to-fraction conversions daily for 20 minutes

You'll learn

• Basis point (bp) — one hundredth of one percentage point (0.0001)
• Notional value — the total underlying value of a financial position
• Tick size — the minimum permitted price fluctuation of an instrument

How to work through it

1. 1Memorize standard basis point conversion scales (1 bp = 0.01% = 0.0001)
2. 2Practice calculating 1 bp and 0.5 bp changes on notional sizes of 1M, 10M, and 50M
3. 3Implement mental cross-rate multiplication for currency triples

You'll learn

• Fermi estimation — dimensional dimensional decomposition of large unknown quantities
• Confidence interval calibration — avoiding overconfidence when setting market-making bid/ask bands

How to work through it

1. 1Deconstruct broad estimation problems into chains of independent multiplicative factors
2. 2Establish dimensional sanity checks using powers of 10
3. 3Record 90% confidence intervals for real-world metrics and evaluate calibration error

You'll learn

• Binomial coefficient — the number of ways to choose k items from n possibilities without order
• Inclusion-Exclusion Principle — combinatorial method to compute size of overlapping set unions
• Sampling with/without replacement — the effect of prior draws on subsequent event probabilities

How to work through it

1. 1Review binomial coefficients, multinomials, and partition formulas
2. 2Solve problems involving selection with and without replacement
3. 3Derive discrete probability mass functions for card and dice scenarios

You'll learn

• Bayes' Theorem — mathematical formula for updating prior probabilities based on new evidence
• Prior and Posterior probability — the belief distribution before and after observing experimental data
• Law of Total Probability — partitioning sample spaces to calculate marginal event probabilities

How to work through it

1. 1Derive Bayes' Theorem from conditional probability fundamentals P(A|B) = P(B|A)P(A)/P(B)
2. 2Solve standard base-rate neglect scenarios and multi-stage medical/coin testing problems
3. 3Formulate posterior probability distributions after observing consecutive Bernoulli trials

You'll learn

• Linearity of Expectation — property where E[X+Y] = E[X] + E[Y] regardless of independence
• Variance and Covariance — measures of dispersion and joint linear variability between variables
• Fair Value — the expected payoff of a gamble without built-in spread or profit margin

How to work through it

1. 1Calculate expectation and variance for discrete random variables using linearity of expectation
2. 2Evaluate stopping-time games and optional stopping scenarios
3. 3Determine fair market prices for bets with asymmetric downside risk

You'll learn

• Limit Order Book (LOB) — the ledger of outstanding limit buy and sell orders sorted by price and time
• Maker vs Taker — providing liquidity to the book versus removing liquidity with aggressive orders
• Price-Time Priority (FIFO) — queue priority giving precedence to best price, then earliest arrival time

How to work through it

1. 1Learn order types: Limit, Market, Stop-Loss, Fill-or-Kill (FOK), and Immediate-or-Cancel (IOC)
2. 2Examine price-time priority (FIFO) and pro-rata queue allocation mechanisms
3. 3Analyze Level 1 (top of book), Level 2 (depth), and Level 3 (individual order) market data feeds

You'll learn

• Double Auction — mechanism where buyers and sellers submit bids and offers simultaneously
• Slippage — the difference between the expected price of a trade and the executed price
• Spread crossing — executing against resting liquidity when a bid meets or exceeds the best offer

How to work through it

1. 1Define Order and PriceLevel classes using linked lists or collections.deque for FIFO order queues
2. 2Implement add_limit_order, cancel_order, and execute_market_order methods
3. 3Write unit tests verifying correct execution prices, partial fills, and spread updates

You'll learn

• Central Limit Theorem (CLT) — sum of independent random variables tends toward a normal distribution
• Fat Tails / Leptokurtosis — higher probability of extreme outlier events compared to normal distributions
• Lognormal distribution — continuous distribution where the logarithm of the variable is normally distributed

How to work through it

1. 1Derive mean, variance, and moment generating functions for Normal and Lognormal variables
2. 2Simulate the Central Limit Theorem across non-Gaussian distributions in Python
3. 3Compute skewness and kurtosis metrics on historical equity return series

You'll learn

• Geometric Brownian Motion (GBM) — continuous-time stochastic process used to model stock price dynamics
• First Passage Time — the time it takes for a stochastic process to reach a specified threshold
• Markov Property — property where future states depend only upon the present state, not the historical path

How to work through it

1. 1Implement 1D symmetric and biased random walks and compute first-passage times
2. 2Construct Markov transition matrices and solve for stationary distributions using linear algebra
3. 3Discretize and simulate Geometric Brownian Motion using Ito's lemma formulation (Euler-Maruyama method)

You'll learn

• Cost of Carry — net cost of holding an underlying asset (financing cost minus dividend/convenience yield)
• Contango and Backwardation — market states where futures prices are higher or lower than the spot price
• Basis — the price difference between the cash/spot price and the futures contract price

How to work through it

1. 1Derive the Cost of Carry formula: F = S * exp((r - q + u) * T)
2. 2Calculate the basis and identify contango vs backwardation market structures
3. 3Model cash-and-carry and reverse cash-and-carry arbitrage execution loops

You'll learn

• Yield to Maturity (YTM) — the total rate of return anticipated on a bond if held until maturity
• Modified Duration — percentage price sensitivity of a bond relative to a yield change
• DV01 / PV01 — the dollar change in bond value resulting from a 1 basis point (0.01%) shift in yield

How to work through it

1. 1Calculate bond clean price, dirty price, and accrued interest
2. 2Compute Modified Duration and Convexity to estimate price changes under yield shifts
3. 3Implement DV01 (Dollar Value of an 01) hedging calculations for a basket of interest rate bonds

You'll learn

• Put-Call Parity — static no-arbitrage relationship linking European call and put prices
• Synthetic position — replicating the payoff of an asset or option using a combination of other instruments
• Straddle & Strangle — non-directional options strategies profiting from large volatility expansions

How to work through it

1. 1Prove Put-Call Parity: C - P = S - K * exp(-rT)
2. 2Identify arbitrage opportunities when Put-Call Parity is violated by market quotes
3. 3Implement visual payoff functions for multi-leg strategies (straddles, butterflies, vertical spreads)

You'll learn

• Delta — rate of change of option price with respect to underlying asset price
• Gamma — rate of change of Delta with respect to underlying asset price
• Vega — sensitivity of option price to changes in implied volatility
• Theta — rate of change of option price with respect to time decay

How to work through it

1. 1Implement standard Black-Scholes call and put analytical pricing formulas in Python
2. 2Derive formulas for Delta, Gamma, Vega, Theta, and Rho
3. 3Verify pricing calculations against known market benchmark datasets

You'll learn

• Implied Volatility (IV) — the market's forecast of underlying asset price movement implied by option prices
• Volatility Skew / Smile — the empirical phenomenon where OTM puts trade at higher IV than ATM/OTM calls
• Newton-Raphson method — iterative root-finding algorithm using function values and derivatives

How to work through it

1. 1Implement Newton-Raphson solver using analytical Vega to find implied volatility from option price
2. 2Handle edge cases such as deep in-the-money options using bisection or Brent's method
3. 3Plot the Volatility Skew across strikes and term structure across expiration dates

You'll learn

• Vectorized backtesting — calculating strategy performance across time series without iterative row loops
• Lookahead bias — inadvertently using information in testing that was not available at signal generation time
• Turnover — the percentage of a portfolio that is replaced in a given time period

How to work through it

1. 1Set up data structures using pandas and numpy for vectorized signal generation
2. 2Implement transaction cost models accounting for fixed fees and bid-ask spread crossing
3. 3Compute strategy return series, cumulative equity curve, and turnover metrics

You'll learn

• Sharpe Ratio — measure of excess return per unit of total risk (standard deviation)
• Sortino Ratio — variation of Sharpe ratio measuring return against downside/harmful volatility only
• Maximum Drawdown (MDD) — the maximum observed loss from a peak to a trough of a portfolio before a new peak

How to work through it

1. 1Implement formulas for annualized return, annualized volatility, and annualized Sharpe Ratio
2. 2Calculate Maximum Drawdown (MDD) and Drawdown Duration series
3. 3Compute Sortino Ratio (penalizing only downside deviation) and Calmar Ratio

You'll learn

• Cointegration — stationary linear combination of two or more non-stationary time series
• Z-Score — statistical measure of how many standard deviations an observation is from the mean
• Mean Reversion — financial theory suggesting asset prices and historical returns return to long-run averages

How to work through it

1. 1Test asset pairs for stationarity and cointegration using the Augmented Dickey-Fuller (ADF) test
2. 2Calculate the hedge ratio via Ordinary Least Squares (OLS) regression on asset prices
3. 3Generate trading signals based on Z-score normalized spread deviations with stop-loss bounds

You'll learn

• Avellaneda-Stoikov Model — canonical quantitative model for optimal high-frequency quoting with inventory risk
• Reservation Price — a market maker's subjective fair value adjusted downward/upward for existing long/short inventory
• Adverse Selection — the risk of trading with market participants who have superior private information

How to work through it

1. 1Implement the reservation price equation: r(s, q, t) = s - q * gamma * sigma^2 * (T - t)
2. 2Compute optimal spread around the reservation price based on order arrival intensity
3. 3Simulate order fills using Poisson process arrivals and plot PnL vs inventory trajectory

You'll learn

• Delta Hedging — trading underlying assets to maintain a net delta of zero and eliminate directional risk
• Gamma Scalping — dynamically buying low and selling high to capture profit from large price oscillations
• Realized vs Implied Volatility — the actual observed historical volatility versus the price paid for option volatility

How to work through it

1. 1Simulate an underlying stock path following Geometric Brownian Motion
2. 2Calculate option Delta and execute rebalancing trades in the underlying asset at discrete intervals
3. 3Quantify PnL variance caused by discrete rebalancing and compare realized volatility to implied volatility

You'll learn

• Kelly Criterion — mathematical formula for sizing bets to maximize the expected geometric growth rate of capital
• Fractional Kelly — betting a fraction (e.g., half) of the theoretical Kelly size to dramatically reduce drawdown risk
• Gambler's Ruin — the statistical certainty of going bankrupt when playing a negative expectation or over-leveraged game

How to work through it

1. 1Derive the basic Kelly formula: f* = (bp - q) / b for binary payoffs
2. 2Model ruin probability and drawdown distributions for 1.0x Kelly vs 0.5x Kelly
3. 3Implement multi-asset Kelly sizing with covariance matrices in Python

You'll learn

• Value at Risk (VaR) — statistical estimate of the maximum expected loss over a specific time horizon at a given confidence level
• Expected Shortfall (CVaR) — average loss incurred in the tail scenarios beyond the VaR threshold
• Stress Testing — evaluating portfolio behavior under historical crisis scenarios (e.g., 2008 crash, 2020 liquidity shock)

How to work through it

1. 1Compute Historical Simulation VaR by sorting historical portfolio returns
2. 2Implement Parametric VaR using portfolio variance and normal distribution z-scores
3. 3Calculate Conditional VaR (Expected Shortfall) measuring expected loss conditional on exceeding the VaR threshold

You'll learn

• Nash Equilibrium — stable state where no player can benefit by unilaterally changing strategy
• Vickrey Auction — second-price sealed-bid auction where bidding true valuation is a dominant strategy
• Winner's Curse — tendency for winning bids in common-value auctions to exceed intrinsic asset value due to estimation noise

How to work through it

1. 1Find pure and mixed strategy Nash equilibria in 2x2 and continuous payoff matrices
2. 2Analyze bidding incentives in First-Price vs Second-Price (Vickrey) auctions
3. 3Model the Winner's Curse in common-value auctions with noisy private signals

You'll learn

• Prospect Theory — behavioral model describing how people decide between alternatives involving risk and probability
• Loss Aversion — the psychological reality that losses feel roughly twice as painful as equivalent gains feel pleasurable
• Outcome Bias — judging the quality of a decision based solely on its eventual outcome rather than the process

How to work through it

1. 1Examine Prospect Theory: S-shaped value function, risk-seeking in losses, and risk-averse in gains
2. 2Identify the Disposition Effect (selling winners too early, holding losers too long)
3. 3Review risk management case studies (e.g., Nick Leeson, LTCM, London Whale)

You'll learn

• Two-sided market — simultaneously quoting both a buy price (bid) and a sell price (ask)
• Inventory skewing — shifting your bid and ask lower (if long) or higher (if short) to attract offsetting flow
• Information leakage — inferring the private holdings of other traders from the trades and quotes they make

How to work through it

1. 1Learn the rules of standard trading games (e.g., bidding on the sum of 2 unseen cards in a standard deck)
2. 2Calculate dynamic conditional expectations as partial information is revealed on the table
3. 3Quote tight bid-ask spreads, adjust quotes when hit, and flatten toxic inventory quickly

You'll learn

• Execution API — programmatic interface for sending, modifying, and canceling live orders
• Circuit Breaker — automated safety mechanism that halts trading upon exceeding predetermined loss parameters
• Order Invalidation — handling rejected orders, network drops, and state desynchronization

How to work through it

1. 1Connect your algorithm to exchange API websockets for live order book streaming and order execution
2. 2Implement automated circuit breakers that halt trading on excessive drawdown or position limits
3. 3Deploy and monitor bot performance across high-volatility and ranging market regimes

You'll learn

• The Green Book — 'A Practical Guide to Quantitative Finance Interviews', the industry standard quant question bank
• Martingale stopping theorem — powerful mathematical tool for finding expected stopping times in fair games
• Symmetry arguments — simplifying complex probabilistic systems by identifying geometric or game symmetries

How to work through it

1. 1Solve probability problems covering stopping times, dice games, and coin toss sequences
2. 2Work through logic puzzles, river-crossing invariants, and symmetry-based game theory problems
3. 3Write out clean verbal explanations for every solution as if explaining live to an interviewer

You'll learn

• Gamma-Theta Trade-off — the mathematical reality that positive Gamma (convexity) requires paying negative Theta (time decay)
• Rule of 16 — dividing annual volatility by 16 (approx sqrt(252)) to get expected daily price moves
• Sticky Strike vs Sticky Delta — models of how implied volatility curves shift when underlying spot price moves

How to work through it

1. 1Practice explaining the relationship between Gamma and Theta (the cost of holding convexity)
2. 2Solve for break-even straddle moves at expiration given ATM implied volatility
3. 3Answer qualitative questions on how changes in interest rates, dividends, and spot price shift the smile

You'll learn

• Thinking Out Loud — communicating intuition, assumptions, and partial progress clearly during live derivations
• Composure Under Uncertainty — maintaining systematic problem-solving ability after making a calculation error
• Mock Review Matrix — structured rubric evaluating speed, mathematical correctness, and risk calibration

How to work through it

1. 1Conduct structured 45-minute interviews covering 1 mental math screen, 2 probability puzzles, and 1 trading game
2. 2Practice articulating hypotheses and thought processes out loud while calculating on a whiteboard
3. 3Document errors, speed bottlenecks, and communication blind spots in an interview retrospective log