LEARN THE MODEL.
CHALLENGE THE HEDGE.
Choose a track, keep the essential lesson visible, and open derivations or implementation only when you need the depth.
Probability, measures & pricing
A rigorous route from distributions and stochastic calculus through Black–Scholes, conditional expectation, Girsanov and forward-measure pricing.
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Volatility
A sequenced path through measurement, option-implied coordinates, surface construction, dynamics, calibration and hedge risk.
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Rates & curves
A sequenced rates path through conventions, overnight compounding, products, curve construction, multi-curve valuation, risk and no-arbitrage dynamics.
View all 13 stages
Numerical finance
Reproducible Monte Carlo, SDE schemes, variance reduction and transform pricing with visible error controls.
Greeks & hedging
Desk-unit Greeks, nonlinear cross-risk and a discrete hedging loop with explicit costs and residuals.
Risk & xVA
Portfolio exposure, counterparty and funding adjustments, tail metrics and the controls that make model outputs defensible.
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The existing typed catalog remains intact beneath the sequenced flagship curriculum.
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Random Variables
Map uncertain outcomes to measurable numerical values.
Brownian Motion
The continuous-time noise behind classical diffusion models.
Risk-Neutral Pricing
Value discounted payoffs under a measure that removes risk premia.
Bid Ask and Mid
Separate executable sides from the midpoint used for analysis.
Market Price vs Model Price
Keep observed quotes separate from calculated fair values.
Streaming Quotes and Staleness
Treat market data as a timed state, not a timeless number.
Reference vs Real-Time Data
Distinguish official observations, delayed feeds and executable streams.
Prediction Market Probabilities
Interpret binary contract prices without treating them as certainty.
Prediction Event Market Outcome and Token
Keep event, market, outcome and CLOB token identifiers distinct.
Prediction Market Order Books
Read bid, ask, midpoint, spread, depth and imbalance for outcome tokens.
Prediction Market Resolution and Negative Risk
Connect settlement rules, oracle resolution and linked mutually exclusive markets.
Prediction Market Liquidity Open Interest and Volume
Separate trading activity, outstanding exposure and available book depth.
Realized vs Implied Volatility
Compare measured return dispersion with option-implied volatility.
Equity Forward
Lock a future equity purchase price after funding and dividends.
Black-Scholes
Closed-form European option pricing under lognormal diffusion.
Greeks
Local sensitivities that translate model parameters into hedge language.
Implied Volatility
The volatility input that makes a model reproduce a market option price.
Volatility Smile
Strike-dependent implied volatility at a single expiry.
Volatility Surface
Implied volatility across strike and maturity.
FX Spot
The exchange rate for near-immediate delivery between two currencies.
FX Forward
A future exchange rate implied by two funding curves.
FX Swap
Exchange currencies now and reverse the exchange later.
Garman-Kohlhagen
Black-Scholes adapted to domestic and foreign interest rates.
FX Delta Conventions
Premium-adjusted, forward and spot delta quotation choices.
Risk Reversal
Call-minus-put volatility at matched absolute delta.
Butterfly
A convexity quote combining wing and ATM volatilities.
FX Volatility Smile
Construct an FX smile from ATM, risk reversal and butterfly quotes.
Discount Factors
Present value of one unit of currency paid at a future date.
Zero Rates
Single-period rates implied by discount factors.
Forward Rates
Rates implied today for borrowing over a future interval.
Yield Curves
Term structures linking maturity to discounting or yield.
OIS
Fixed-versus-compounded overnight indexed swaps.
FRAs
Contracts fixing a future simple interest rate.
Interest Rate Swaps
Exchange fixed coupons for floating-rate cashflows.
Curve Bootstrapping
Solve discount factors sequentially from market instruments.
Spot vs Futures
Separate immediate physical value from exchange-traded future delivery.
Forward Curves
Commodity delivery prices across maturities.
Contango
A forward curve whose later deliveries trade above nearby prices.
Backwardation
A forward curve whose later deliveries trade below nearby prices.
Convenience Yield
The non-cash benefit of holding physical inventory.
Black-76
Option pricing on forwards under lognormal forward dynamics.
Asian Options
Options whose payoff depends on an average price.
Rough Volatility
Very low regularity volatility models aligned with observed short-scale behaviour.
AAD
Adjoint algorithmic differentiation for many sensitivities at near-constant reverse cost.
Differentiable Pricing
Pricing systems designed for gradients across models and parameters.
Machine Learning Surrogates
Fast learned approximations to expensive pricing maps.
Deep Hedging
Learn hedging policies under frictions and non-quadratic objectives.
Neural SDEs
Stochastic differential equations with learned functional components.
Bayesian Calibration
Infer parameter distributions rather than one best-fit point.
GPU Monte Carlo
Parallel simulation and payoff evaluation on graphics processors.
Conditional Expectation
Update expected values using the information currently available.
Martingales
Processes whose conditional future value equals their current value.
Change of Measure
Reweight probabilities to move between pricing numeraires.
Itô Calculus
Differential calculus for stochastic processes with quadratic variation.
Monte Carlo
Estimate prices and risk by simulating many model paths.
Variance Reduction
Improve simulation precision without merely adding paths.
Finite Differences
Approximate derivatives and solve pricing PDEs on a grid.
PDE Methods
Translate no-arbitrage dynamics into boundary-value problems.
Calibration Basics
Choose parameters that reconcile a model with observed instruments.
Interpolation Basics
Estimate values between liquid market pillars without inventing arbitrage.
Numerical Stability
Keep computed outputs reliable under finite precision and difficult inputs.
Call-Put Parity
Connect European calls, puts, forwards and discounting by no-arbitrage.
Dividend Carry
Separate funding and distributions in equity forward value.
Historical Volatility
Estimate dispersion from a time series of past returns.
Realized Volatility
Measure variance accumulated over an observed period.
Term Structure
Track implied volatility across option maturities.
Local Volatility
Infer state-dependent instantaneous variance from a vanilla surface.
Stochastic Volatility
Model volatility itself as a random process.
Heston
Model variance as a mean-reverting square-root diffusion.
SABR
Model forward and volatility jointly for smile dynamics.
Gamma Scalping
Monetise convexity through repeated delta rebalancing.
Variance Swaps
Trade future realised variance against a fixed strike.
Barrier Options
Activate or extinguish payoff when an underlying crosses a level.
Digital Options
Pay a fixed amount when a terminal condition is met.
American Options
Allow exercise before expiry and introduce an optimal stopping problem.
Early Exercise
Compare continuation value with immediate exercise value.
Volatility Arbitrage
Express relative-value views across implied and realised volatility.
Forward Points
Quote the forward-minus-spot adjustment implied by two currencies.
ATM Conventions
Define where at-the-money sits under pair-specific FX rules.
Premium-Adjusted Delta
Account for option premium in the hedge-ratio convention.
Smile Construction
Turn market quote coordinates into a strike-volatility curve.
FX Digitals
Price fixed cashflows conditional on an exchange-rate event.
FX Barriers
Add path-dependent trigger levels to FX option payoffs.
Dual-Currency Notes
Combine fixed-income cashflows with embedded FX optionality.
Quanto Effects
Value an asset payoff translated at a fixed exchange rate.
Triangular Arbitrage
Enforce consistency across three quoted currency pairs.
Cross-Currency Basis
Measure funding dislocations not explained by covered interest parity.
Day Count Conventions
Convert calendar dates into contractual accrual fractions.
Compounding Conventions
Translate rates consistently across simple, periodic and continuous forms.
Multi-Curve Framework
Separate discounting from tenor-specific projection curves.
DV01
Measure value change for a one-basis-point rate shift.
Key-Rate Duration
Allocate curve sensitivity to selected maturity nodes.
Carry and Rolldown
Decompose expected horizon P&L with an unchanged curve.
Caps
Limit floating-rate payments through a strip of caplets.
Floors
Protect minimum floating-rate receipts through floorlets.
Swaptions
Grant the right to enter an interest-rate swap.
Short-Rate Models
Model the instantaneous funding rate to generate a term structure.
Hull-White
Use a mean-reverting Gaussian short rate fitted to today’s curve.
LMM
Model a family of market forward rates under linked measures.
Convexity Adjustments
Correct linear forward intuition when payoff and discounting are nonlinear.
Negative Rates
Handle rate distributions and quotation when strikes can cross zero.
Storage Costs
Embed physical warehousing, insurance and financing in commodity carry.
Seasonality
Model recurring calendar patterns in supply, demand and forward prices.
Roll Yield
Measure the return from moving exposure along a forward curve.
Calendar Spreads
Trade relative value between delivery months.
Commodity Swaps
Exchange floating commodity prices for fixed contractual levels.
Commodity Options
Attach optionality to forwards, futures or physical indices.
Mean Reversion
Model commodity prices returning toward an equilibrium level.
Spread Options
Option the difference between related prices.
Swing Options
Optimise repeated exercise volumes under operational constraints.
Weather Derivatives
Link payoffs to temperature or other weather indices.
Crack Spreads
Track refinery margin between crude and products.
Spark Spreads
Track power-generation margin between electricity and fuel.
Real Options
Value operational flexibility using option-pricing logic.