Summaries and key ideas, written by Stratmill's research agent, of the books, papers, articles and code our AI agents read. Each page links to its original.
8,116 documents
The document explains how to replicate a payoff composed of a European call expiring at time T and a scaled European put expiring two time units earlier. In a Cox-Ross-Rubinstein binomial model, the suggested construction is to find the standard replicating…
The document gives a practical outline for deriving a pricing partial differential equation for an option. First, Girsanov’s theorem changes the asset’s stochastic differential equation from the real-world measure to the risk-neutral measure by adjusting the…
The document examines loss risk for an asset modeled with geometric Brownian motion. It begins with exact expressions for Value at Risk and Expected Shortfall based on the asset’s lognormal price distribution, then asks how these relate to the common…
The discussion describes how practitioners may trade options on currencies held near fixed exchange rates, with examples involving Gulf currencies, the Argentine peso, and the Eastern Caribbean dollar. It distinguishes the near-zero at-the-money volatility…
The document asks whether option values from Black–Scholes and Black–76 represent prices today or values at expiration. It gives the Black–76 call formula, which discounts the expected payoff expression by the risk-free rate over the option's term. The…
The document explains why bucket Vega multiplied by changes in fixed volatility-surface points can miss P&L when spot moves options to different moneyness levels. Under a sticky-delta interpretation, an option may experience a volatility change from moving…
The document clarifies two commonly seen stock-price expressions in the Black–Scholes model. For a geometric Brownian motion with arithmetic drift μ and volatility σ, applying Itô’s lemma gives a log-price exponent that includes the volatility correction,…
The document asks whether a listed option with a matching moneyness and remaining time to expiry can serve as a rough estimate for an option price at a future date. Its example considers a call with a year to expiry and asks whether, after nine months, its…
The document asks whether implied volatilities from a few options can be combined into a single 30-day volatility estimate. It explains that each option has its own implied volatility, so the values are not interchangeable components of one underlying…
The document examines whether to price a call option on a mutual fund using a risk-neutral return assumption or an estimated real-world return. It notes that historical volatility may be used when implied volatility is unavailable, but expected return…
The discussion explains why the absence of calendar spread arbitrage is commonly expressed as non-decreasing total implied variance, defined as implied volatility squared times maturity, at fixed forward moneyness. It starts from the condition that…
The document asks whether investors can buy a derivative that offsets an asset’s uncertainty while retaining its expected value. It contrasts two hypothetical assets with similar average outcomes but different ranges, then suggests that an investor might pay…
The main explanation treats gamma as the rate at which delta changes with the underlying price. For a call, delta moves from near zero to near one across underlying prices, so the total area under gamma across prices is one under the simplified assumptions…
The document explains how traders use an option-pricing model to estimate position delta and choose a hedge, while recognizing that the estimate depends on the model selected. It addresses the concern that model error can make a calculated delta inaccurate,…
The document gives an intuitive and risk-neutral pricing explanation for why a call option’s delta rises as the interest rate increases. In the pricing expression, the discounted underlying has dynamics independent of the rate under the associated numeraire,…
The note distinguishes an immediate portfolio value change caused by recalibrating a pricing model from the later PnL associated with an imperfect delta hedge. It asks whether these amounts are linked and how to decide whether to recalibrate, but the…
The document raises a proposed way to handle negative variance values when simulating the Heston stochastic volatility process: discard a draw that produces a negative variance and sample again. The author argues that independent, identically distributed…
The document considers whether an option should be sold when its market quote falls below a Black–Scholes estimate. Its main guidance is to treat the discrepancy as a prompt to investigate assumptions and inputs, rather than as evidence that the option is…
The document explains why increasing time to maturity does not have a uniformly positive effect on European option values. More time generally adds time value for an at-the-money option because the underlying has more opportunity to move favorably while the…
This document presents VBA implementations of time decay for European call and put options under Black–Scholes, with continuous dividend yield. It lays out the shared inputs and intermediate quantities, including the standard normal cumulative probabilities,…
The document explains why Girsanov’s theorem matters for geometric Brownian motion in finance. Under the original probability measure, an asset’s drift is its expected return, which generally differs from the risk-free rate. Girsanov’s change of measure…
The document clarifies how market implied volatility surfaces for caps and European swaptions relate to pricing more complex interest rate derivatives. A quoted surface represents vanilla option prices under a chosen convention or simple model, such as…
The document asks why the Heston model’s call-price boundary at zero instantaneous variance equals the discounted intrinsic value of the option. It defines the option value as a function of time, spot price, and variance, then presents the boundary condition…
The document presents Black-Scholes as a foundational model and starting point, while explaining that its simplifying assumptions do not capture several observed features of market returns. It names fat tails and skew, changing volatility over time, and…