Target price
What stock price does my call need to reach?
See which assumptions are needed to estimate the underlying price behind a call option target
Options field guide
Pricing & Greeks
An option can move the right way and still miss your target. These guides explain contract mechanics, strategy tradeoffs, and the stock, timing, and volatility assumptions behind an outcome
90 guides
Target price
What stock price does my put need to reach?
See how the checkpoint date and pricing assumptions shape the stock-price condition behind a put premium target
Why it changed
Why did my call fall after the stock rose?
Understand how time, implied volatility, and quote quality can outweigh the benefit of a rising stock for a call option
Why it changed
Why did my option target change since yesterday?
Learn why the underlying price required for the same option premium can change when time or volatility inputs move
Time decay
How does time decay change an option target?
Learn why the stock price required for an option target can rise as expiration approaches
Options mechanics
How to calculate an option break-even price
Use contract terms to find an expiration break-even and distinguish it from a profitable exit before expiration
Position planning
How to set an option profit target
Turn a percentage-return idea into a premium, checkpoint, stock-and-volatility condition, and executable exit plan
Why some complex models remain tractable
Affine Processes and Riccati Equations Explained
Learn how exponential-affine transforms, generalized Riccati equations, admissibility, and moment domains make affine finance models tractable
State-contingent claims
Arrow–Debreu State Prices Explained
Understand state-contingent securities, why state prices differ from probabilities, and how option-price curvature reveals a risk-neutral density
Calculate a bearish call credit spread
Bear call spread max profit, loss, and break-even
Calculate bear call spread maximum profit, maximum loss, expiration break-even, fees, buying power, dividend risk, and short-call assignment outcomes.
Calculate a bearish put debit spread
Bear put spread max profit, loss, and break-even
Calculate bear put spread maximum profit, maximum loss, expiration break-even, contract multiplier, costs, and short-put assignment scenarios.
Separate the two probability terms inside Black–Scholes
Black–Scholes d1 vs d2 Explained
Understand the d1 and d2 formulas, why N(d1) relates to call delta, why N(d2) relates to risk-neutral expiration ITM probability, and their limits.
Pricing foundations
Black–Scholes Model Assumptions and Limits Explained
Learn how Black–Scholes prices European options through replication, what its inputs and assumptions mean, and where market reality departs from the model
Convert a box price into a comparable financing yield
Box Spread Implied Interest Rate Calculation
Calculate a box spread implied simple or continuous interest rate from strike width, executable box price, days to expiration, multiplier, fees, and rate convention.
Calculate every body and tail outcome before trading
Broken-Wing Butterfly Max Profit and Loss Calculation
Calculate broken-wing butterfly maximum profit, tail losses, break-evens, signed entry credit, call and put formulas, contract multiplier, fees, and assignment scenarios.
Calculate a bullish call debit spread
Bull call spread max profit, loss, and break-even
Calculate bull call spread maximum profit, maximum loss, expiration break-even, contract value, fees, and outcomes before and at expiration.
Calculate a bullish put credit spread
Bull put spread max profit, loss, and break-even
Calculate bull put spread maximum profit, maximum loss, expiration break-even, buying power, fees, and assignment outcomes from the actual credit.
Which Gaussian path shifts are admissible
Cameron–Martin Theorem in Finance Explained
Learn which translations preserve equivalence of Wiener measure, what the Cameron–Martin space contains, and how path shifts relate to Girsanov and finance
Turn one transform into a strip of prices
Carr–Madan Fourier Option Pricing Explained
Learn how Carr–Madan damping, complex-shifted characteristic functions, FFT grids, and numerical error controls produce option prices
Calculate a cash-backed stock obligation
Cash-secured put max profit, loss, and break-even
Calculate cash-secured put maximum profit, maximum loss, break-even and effective stock cost, reserved cash, multiplier, fees, and assignment outcomes.
Price options without writing the density
Characteristic Function Option Pricing Explained
Understand how risk-neutral characteristic functions encode distributions, support Fourier inversion, and become option prices through payoff transforms
Turn a characteristic function into a cosine series
COS Method for Option Pricing Explained
Learn how the COS method combines a truncated density, Fourier-cosine coefficients, payoff coefficients, and convergence controls to price options
Calculate stock plus a short call
Covered call max profit, loss, and break-even
Calculate covered call maximum profit, maximum loss, break-even stock price, assigned return, contract multiplier, dividends, fees, and cost-basis choices.
Understand time decay while markets are closed
Does Option Theta Decay Over Weekends and Holidays?
Learn how option time value changes over weekends and holidays, why theta conventions differ, and why Monday prices are not a fixed calendar-day deduction.
Predictable compensation
Doob–Meyer Decomposition in Finance Explained
Learn how Doob–Meyer separates a submartingale into martingale innovation and predictable accumulation, with applications to intensity and American options
Generators become expected change
Dynkin's Formula for Stochastic Processes Explained
Learn how Dynkin's formula connects Markov generators, expected change, stopping times, martingales, pricing equations, and moment calculations
Two normal-based expansions that are often confused
Edgeworth vs Gram–Charlier Expansions Explained
Compare how Edgeworth and Gram–Charlier expansions use cumulants, why their ordering differs, and what can go wrong in option pricing and tail estimation
Exponential measure tilting
Esscher Transform in Option Pricing Explained
Understand exponential tilting, cumulant shifts, Lévy measure changes, martingale parameter selection, incomplete markets, and existence checks
Can square-root variance reach zero
Feller Condition in CIR and Heston Models Explained
Understand the Feller condition, zero-boundary behavior, nonnegativity versus strict positivity, and numerical consequences for CIR and Heston models
PDE and expectation bridge
Feynman–Kac Formula for Option Pricing Explained
Learn how Feynman–Kac links linear pricing PDEs with discounted conditional expectations, including generators, cash flows, numerics, and limits
Numerical PDEs
Finite Difference Methods for Option Pricing Explained
Learn how finite differences solve an option pricing PDE on a grid, how explicit, implicit, and Crank–Nicolson schemes differ, and where numerical error enters
A fast transform still needs an error budget
Fourier Option Pricing Error Control Explained
Understand truncation, quadrature, aliasing, interpolation, damping, and complex-arithmetic errors in Fourier option pricing and how to diagnose them
No-arbitrage theory
Fundamental Theorem of Asset Pricing Explained
Learn how no arbitrage, equivalent martingale measures, and market completeness fit together in the fundamental theorem of asset pricing
Measure transformation
Girsanov Theorem in Option Pricing Explained
Learn how Girsanov's theorem changes Brownian drift between probability measures, why volatility stays fixed, and how risk-neutral dynamics arise
Continuous decisions become an equation
Hamilton–Jacobi–Bellman Equation in Finance Explained
Understand how dynamic programming produces the HJB equation for stochastic control, portfolio choice, verification, constraints, and numerical policy design
Invert an option price into model-implied volatility
How Implied Volatility Is Calculated
Learn option-price bounds, vega and monotonicity, Newton–Raphson and bisection, low-vega instability, quote selection, and IV solver validation.
Simulating rare payoffs without waiting for them
Importance Sampling for Rare Events and Options
Learn how likelihood-ratio reweighting preserves expectations, how exponential tilting reduces rare-event variance, and how weight degeneracy can ruin an option simulation
Calculate a centered four-option expiration payoff
Iron Butterfly Max Profit, Loss, and Break-Even
Calculate short iron butterfly maximum profit, maximum loss, two break-even prices, wing width, net credit, multiplier, fees, asymmetric wings, and expiration payoff.
Calculate the four-strike expiration payoff
Iron Condor Max Profit, Loss, and Break-Even
Calculate iron condor maximum profit, maximum loss, lower and upper break-even prices, unequal-wing risk, contract multiplier, fees, and expiration outcomes.
Stochastic calculus
Itô's Lemma in Option Pricing Explained
Learn why stochastic functions need an extra second-order term, how Itô's lemma expands option value, and how delta hedging leads to a pricing equation
Two directions of Markov evolution
Kolmogorov Forward and Backward Equations Explained
Understand how generators evolve conditional values backward and probability densities forward, with links to Fokker–Planck, pricing, calibration, and boundaries
Finding the small region that dominates a large-parameter integral
Laplace’s Method vs Steepest Descent Explained
Compare real Laplace asymptotics with complex contour steepest descent, learn the leading saddle formula, and see how singularities and competing saddles affect option calculations
Calculate a standalone call purchase
Long call max profit, loss, and break-even
Calculate long call maximum profit, maximum loss, expiration break-even, contract value, return on premium, fees, time value, IV, and automatic exercise outcomes.
Calculate a standalone put purchase
Long put max profit, loss, and break-even
Calculate long put maximum profit at a zero stock price, maximum loss, expiration break-even, contract value, fees, time value, IV, and exercise outcomes.
Calculate a two-sided long-volatility payoff
Long straddle max profit, loss, and break-even
Calculate long straddle maximum loss, unlimited upside, finite downside profit, two break-even prices, multiplier, fees, IV, time decay, and expiration outcomes.
Calculate two separated long option wings
Long strangle max profit, loss, and break-even
Calculate long strangle maximum loss, unlimited upside, finite downside profit, call and put break-evens, contract value, fees, volatility, and expiration outcomes.
Risk representation
Martingale Representation Theorem in Finance Explained
Learn how martingales become stochastic integrals, why the integrand becomes a hedge, and how representation determines market completeness
Conditional valuation
Martingales in Option Pricing Explained
Understand conditional expectation, why discounted tradable prices become martingales under a pricing measure, and what the concept does not imply
Numerical pricing
Monte Carlo Simulation for Option Pricing Explained
Learn how Monte Carlo option pricing turns simulated risk-neutral paths into a value, how sampling error behaves, and how path dependence and early exercise are handled
When an exponential defines a measure
Novikov and Kazamaki Conditions Explained
Learn how Novikov and Kazamaki criteria turn a stochastic exponential from a local martingale into a valid density for measure change
Relative valuation
Numeraire and Change of Measure Explained
Understand how choosing a numeraire changes the probability measure, which relative prices become martingales, and why forward measures simplify pricing
Sensitivity computation
Numerical Greeks for Option Pricing Explained
Learn how bump-and-revalue, pathwise, likelihood-ratio, and adjoint methods estimate option Greeks and how to diagnose numerical error
Stopping a fair process
Optional Stopping Theorem in Finance Explained
Learn when stopping a martingale preserves its expectation, why doubling strategies fail, and how stopping times enter barrier and American option valuation
Calculate a stock floor and ceiling
Options collar max profit, loss, and break-even
Calculate collar maximum profit, maximum loss, break-even, signed net debit or credit, stock floor and ceiling, multiplier, dividends, and assignment outcomes.
Two views of optimal control
Pontryagin Maximum Principle vs HJB Explained
Compare costates and Hamiltonian maximization with value functions and dynamic programming in deterministic and stochastic financial control
Calculate a stock downside floor
Protective put max profit, loss, and break-even
Calculate protective put maximum loss, unlimited upside, married-put break-even, floor value, contract coverage, hedge cost, expiration, and basis choices.
Second-order path variation
Quadratic Variation in Finance Explained
Learn why squared path increments survive at fine scales, produce the Itô correction, and connect diffusion volatility with realized variance
Measure density
Radon–Nikodym Derivative in Finance Explained
Learn how the Radon–Nikodym derivative reweights probabilities, converts expectations, forms a density process, and supports risk-neutral valuation
Dynamic replication
Replicating Portfolios and Self-Financing Strategies Explained
Learn why identical payoffs imply identical prices, what self-financing means, and how discrete and continuous replication create option values
Turning cumulants into accurate tail probabilities
Saddlepoint Approximation in Option Pricing Explained
Understand the saddlepoint equation, density and tail approximations, exponential tilting, option-pricing uses, and the numerical checks that matter
Calculate an uncovered call sale
Short call max profit, loss, and break-even
Calculate uncovered short call maximum profit, unlimited loss, expiration break-even, contract exposure, margin, fees, IV, early assignment, and stock delivery risk.
Calculate an uncovered put sale
Short put max profit, loss, and break-even
Calculate uncovered short put maximum profit, stock-to-zero loss, expiration break-even, effective purchase price, margin, fees, IV, assignment, and funding risk.
Calculate a one-strike short-volatility liability
Short straddle max profit, loss, and break-even
Calculate short straddle maximum profit, unlimited upside loss, stock-zero downside loss, two break-evens, multiplier, margin, fees, assignment, and early P&L.
Calculate two separated short option liabilities
Short strangle max profit, loss, and break-even
Calculate short strangle maximum profit, unlimited upside loss, stock-zero downside loss, both break-evens, multiplier, margin, fees, assignment, and early valuation.
Optimal stopping value
Snell Envelope and Optimal Stopping Explained
Understand the smallest supermartingale behind optimal stopping, American option exercise, backward induction, and reliable numerical bounds
Pricing kernel
Stochastic Discount Factor Explained
Learn how a stochastic discount factor prices payoffs across time and states, connects risk premia to covariance, and relates to risk-neutral valuation
Multiplicative stochastic growth
Stochastic Exponential and Doléans–Dade Exponential Explained
Understand the stochastic exponential, its jump correction, positivity, martingale tests, measure changes, and role in asset-value dynamics
Discrete replication
The Binomial Option Pricing Model Explained
Learn how a binomial tree uses one-step replication, risk-neutral probabilities, backward induction, and early-exercise checks to value options
Choosing a distribution from incomplete constraints
The Maximum Entropy Principle in Finance
Understand entropy maximization under moment and option-price constraints, its exponential-family dual, the role of a reference measure, and the checks needed for an implied distribution
Exercise as a free boundary
Variational Inequalities and Free Boundaries in Options
Learn how American option values combine payoff dominance, continuation PDEs, complementarity, free boundaries, smooth fit, and numerical checks
Market structure metrics
Gamma exposure (GEX) explained without treating it as a price forecast
Understand what aggregate gamma-exposure estimates attempt to measure, the dealer assumptions they require, and why models can disagree
Option probability
Probability of touch vs expiring ITM vs profit: three different events
Learn why touching a strike, expiring in the money, and finishing a trade profitably require different events, barriers, assumptions, and probability calculations
Option pricing relationships
Put-call parity explained with practical checks
Understand how matching calls, puts, stock, and financing are linked and why a visible price gap is not automatically an arbitrage
Probability in option prices
Risk-neutral probability: why option-implied odds are not forecasts
Understand the pricing probability behind options, how it differs from real-world probability, and why delta and option-implied distributions need careful interpretation
Second-order Greeks
Vanna and Charm: how option Delta changes with volatility and time
Learn what Vanna and Charm measure, why Delta can drift without a stock move, and how to use second-order Greeks without false precision
Second-order Greeks
Volga or Vomma Explained: How Vega Changes with IV
Learn what Vomma or Volga measures, how it adds curvature to a vega estimate, why its sign varies, and how to aggregate it across an option position
Pricing inputs
How dividends and interest rates affect options
See how dividends and rates enter option values, forward prices, early-exercise decisions, and target-price scenarios
Option valuation
Theoretical option value vs. market price
Understand what an option-pricing model estimates, what bid and ask prices represent, and why a model output is not an executable quote
Greek interpretation
Why option delta is not a guaranteed probability
Understand when delta can serve as a rough probability proxy and why it differs from profit probability, touch probability, and a forecast
Options decision guide
Option Greeks explained
Understand option Greeks, the decision it supports, and the pricing and execution risks to check before acting
Options decision guide
Option premium explained
Understand option premium, the decision it supports, and the pricing and execution risks to check before acting
Options decision guide
Probability of profit vs. break-even
Understand probability of profit and break-even, the decision it supports, and the pricing and execution risks to check before acting
Position management
Using delta, theta, and vega to manage an options position
Read directional, time, and volatility sensitivity together instead of relying on one Greek
Options Greeks
What is rho in options?
Understand how rho estimates an option's sensitivity to interest rates and when the effect matters more
Options Greeks
What is theta in options?
Learn what option theta estimates, why time decay is not linear, and how expiration changes the result
Options pricing
What is an option break-even point?
Learn why a break-even point is normally stated at expiration and can differ from an earlier checkpoint
Options Greeks
What is option delta?
Learn what delta estimates, why it is not a guarantee, and why it changes with the option
Options Greeks
What is option gamma?
Understand gamma as the rate at which delta changes when the underlying price moves
Options Greeks
What is option vega?
Learn how vega estimates an option's sensitivity to a change in implied volatility
Options pricing
What are intrinsic value and time value in options?
Understand the two components that can make up an option premium before expiration