Buy the near option and sell the farther same-strike option17 min read
Short Calendar Spread Options Strategy
Learn how a short or reverse calendar spread works with calls or puts, including credit, large-move thesis, unknown loss region, negative theta, assignment, margin, and exits.
Read guide →Calculate an uncovered call sale15 minute read
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.
Read guide →Calculate an uncovered put sale15 minute read
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.
Read guide →Calculate a one-strike short-volatility liability15 minute read
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.
Read guide →Sell two at-the-money volatility exposures14 min read
Short Straddle Options Strategy Explained
Learn the short straddle payoff, break-even formulas, theta and volatility exposure, unlimited upside loss, assignment, margin, and expiration risks.
Read guide →Compare two undefined-risk premium-selling ranges13 min read
Short Straddle vs Short Strangle
Compare short straddle and short strangle strikes, premium, break-even width, Greeks, probability claims, margin, assignment, and unlimited tail risk.
Read guide →Calculate two separated short option liabilities15 minute read
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.
Read guide →Sell a range with two out-of-the-money options14 min read
Short Strangle Options Strategy Explained
Learn the short strangle payoff, strike selection, break-even range, limited premium, unlimited tail risk, theta, volatility, margin, and assignment.
Read guide →Optimal stopping value13 min read
Snell Envelope and Optimal Stopping Explained
Understand the smallest supermartingale behind optimal stopping, American option exercise, backward induction, and reliable numerical bounds
Read guide →What replaces the bell curve when variance does not exist15 min read
Stable Distributions and Infinite Variance in Finance
Understand alpha-stable laws, the generalized Central Limit Theorem, infinite variance, tail parameters, aggregation, early financial evidence, and the diagnostics needed before using stable models
Read guide →Three ideas often treated as synonyms but answering different questions16 min read
Stationarity, Unit Roots, and Mean Reversion in Finance
Understand strict and weak stationarity, random walks, unit roots, persistence, mean reversion, half-life, spurious regression, ADF and KPSS tests, structural breaks, and financial diagnostics
Read guide →Pricing kernel12 min read
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
Read guide →Multiplicative stochastic growth14 min read
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
Read guide →Compare two option underlyings15 minute read
Stock options vs index options: key differences
Compare stock and index options by underlying, deliverable, multiplier, cash settlement, exercise style, dividends, basis risk, notional size, and expiration.
Read guide →Lower a recovery target without adding initial shares15 min read
Stock Repair Options Strategy Explained
Learn the stock repair strategy, its 1-by-2 call structure, lowered break-even goal, capped recovery, continued stock downside, assignment, and expiration risk.
Read guide →One-sided replication12 min read
Superhedging Price and No-Arbitrage Bounds Explained
Learn how superhedging guarantees claim coverage, creates upper and lower price bounds, and connects incomplete markets to martingale-measure duality
Read guide →Discrete replication12 min read
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
Read guide →Why many small risks can look normal without being normal15 min read
The Central Limit Theorem in Finance
Understand what the Central Limit Theorem says about standardized sums, which assumptions financial returns can violate, why convergence is not exact, and when square-root-of-time scaling fails
Read guide →Correcting normal quantiles for skewness and kurtosis15 min read
The Cornish–Fisher Expansion for Risk Quantiles
Understand how Cornish–Fisher adjusts normal quantiles with cumulants, how it differs from Edgeworth, and why tail risk estimates need monotonicity and stability checks
Read guide →From scaled cumulants to a large-deviation rate15 min read
The Gärtner–Ellis Theorem Explained
Learn how a limiting scaled cumulant-generating function can produce a large-deviation rate function, which regularity conditions matter, and how the result reaches option tails
Read guide →Choosing a distribution from incomplete constraints15 min read
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
Read guide →Smile dynamics12 min read
The SABR Volatility Model Explained
Learn how the SABR model links a random forward and volatility, how alpha, beta, rho, and vol-of-vol shape the smile, and where its approximation can fail
Read guide →Build one counterfactual from a transparent donor pool18 min read
The Synthetic Control Method
Understand donor weights, pretreatment fit, convex-hull limits, placebo inference, spillovers, and how synthetic controls evaluate a policy affecting one market or firm.
Read guide →A loss threshold and the average beyond it15 min read
Value at Risk vs Expected Shortfall Explained
Compare VaR and Expected Shortfall definitions, calculations, diversification properties, estimation error, and the backtests needed before using either tail-risk measure
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