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 asks how to price a European claim that pays the natural logarithm of a stock price at maturity, including the possibility of a negative payoff when the stock finishes below one. It presents the general risk-neutral valuation rule: discount the…
The document raises a question about constructing a variance-swap replication portfolio from options. It describes an apparent jump in implied volatility when the option wing switches from puts below a strike to calls above it. The quoted explanation is that…
The document investigates whether an FX option’s time to expiry in an at-the-money delta-neutral strike calculation includes hours and minutes or counts only whole days. It applies the standard forward-based strike formula to a one-day example, using the…
The document reconciles several formulations of Durrleman’s condition for preventing butterfly arbitrage in an implied volatility surface. It explains how expressions differ when volatility is represented as raw volatility, total volatility, or total…
The document compares ways to choose the drift parameter in a recombining binomial tree. It describes the Cox–Ross–Rubinstein construction, which uses symmetric log-price steps, and an alternative that adds a fixed drift to the up and down multipliers. It…
The document asks how to price a one-year call when a stock can move to either of two values, given a risk-free rate and equal real-world probabilities. It uses a one-step binomial model: calculate the call payoff in each state, form a replicating hedge, or…
The document asks why volatility models often treat out-of-the-money calls and puts as a group even though their implied volatilities can differ. It identifies liquidity as a reason for focusing on OTM contracts and notes that OTM puts commonly carry higher…
The document asks why an American call on a stock paying continuous dividends has a finite stock price above which early exercise is optimal. It contrasts the option’s value, which cannot fall below intrinsic value, with the possibility that the two values…
The document explains how to convert a perturbative approximation for a vanilla call's equivalent Black volatility into an expansion centered at the midpoint between the current forward and strike. The original approximation expands the local volatility…
The document describes a daily-rolled backtest of EURUSD options using one-month and two-month straddles, 25-delta risk reversals, and 25-delta butterflies. Each position is held for one trading day, with the next day's valuation using the full volatility…
The document sketches why a market with stochastic volatility may be incomplete when volatility risk is driven by an independent source of randomness that cannot be traded directly. In a one-stock setting, changing the risk premium associated with the…
The document investigates a Python QuantLib TypeError raised while constructing Heston model calibration helpers for options. The replies point to mismatched input types or values as the likely cause and identify code issues to check, including an…
The response distinguishes two interpretations of a share purchase plan in which the bank can terminate during the purchase period. If the customer’s price is the average only through the termination date, the answer argues that termination does not create…
The document explains why a Black–Scholes option price is not the discounted expected payoff under real-world probabilities. That calculation uses the stock’s physical drift, which reflects expected returns, but ignores how traded assets and hedging…
The document clarifies how SVI parameterizations describe implied volatility data. Raw and natural SVI parameterizations represent total implied variance across strikes for a single expiry, while SVI-Jump-Wings represents implied variance for a single…
The document explains why binary option quotes that appear inconsistent with a simple Black–Scholes volatility range may still be reasonable. It gives the digital call price in terms of the forward price, strike, time to expiry, rates, and volatility, then…
The document examines a terminal payoff formed by taking the positive part of the stock’s gain over its current price, subtracting a fixed notional, and capping the result at zero. It asks how to price this payoff under risk-neutral valuation, while noting…
The document considers whether large traders can manipulate e-mini futures and whether that possibility makes systematic trading impractical. Its answers distinguish ordinary market impact from deliberate manipulation: large buy orders can push prices up,…
The document considers how to adjust a fitted vanilla-option implied volatility surface to represent a lower-volatility market or an abrupt volatility shock. It cautions that blindly adding a constant to implied volatility or multiplying it by a scale factor…
The document asks how to compare the distributions of swaption market observations, such as volatility or liquidity spreads, across two time periods. It considers whether all USD swaption observations should be treated as two samples or whether observations…
The document gives risk-neutral valuation formulas for continuously monitored geometric-average Asian options with a floating strike. It uses a symmetry between a floating-strike call and a fixed-strike put: after substituting the relevant parameters into…
The document asks whether no-arbitrage conditions are known for Bachelier, or normal, implied volatility surfaces. Normal volatility is defined through the parameter that reproduces an option price under the Bachelier formula. The author contrasts this with…
The note explains why two formulas for risk-neutral probabilities in a binomial option-pricing model may appear unequal. One expression gives the up-move probability directly, while the cited algorithm stores discounted up and down weights. Comparing them…
The document explains how a delta-hedging argument leads to the Black–Scholes partial differential equation. A portfolio holds one derivative and shorts an amount of stock equal to the derivative’s sensitivity to the stock price. Applying Itô’s formula…