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 a practical way to quote a binary call by approximating its payoff with a call spread. A seller of the binary can buy a spread around the strike; the spread’s cost provides a replication-based estimate of the option’s price. The…
The question derives a replication-error expression in which the difference between assumed local volatility and realized volatility is multiplied by option gamma over time. It interprets the sign: for positive gamma, assuming volatility above realized…
The discussion clarifies how risk-neutral pricing connects to delta hedging and the Black–Scholes equation. In a standard setup, a self-financing hedged portfolio is locally risk-free, so its return is tied to the financing rate. The stock’s real-world…
The document asks how to distinguish a market-price-based delta from a model-price-based delta for options on Treasury futures. It describes delta as the option price's sensitivity to the underlying futures price and proposes estimating that sensitivity by…
The discussion asks why equity index implied volatility surfaces often show a pronounced downside skew rather than a smile, and why smiles may have appeared historically. One explanation is that index volatility combines the volatilities of its constituents…
The document explains how to approximate the curvature of call prices around an at-the-money strike when option strikes are unevenly spaced. Since the risk-neutral density is related to the second derivative of call price with respect to strike, the method…
The document investigates why QuantLib’s finite-difference Black–Scholes engine produces American put prices that diverge from a reference spreadsheet when discrete cash dividends are scheduled near expiration. The question reports that call prices are close…
The document presents a slow nested-loop implementation for pricing European calls, Asian options, digital calls, and up-and-in barrier calls under the Merton jump-diffusion model. It motivates the problem as a comparison with the exponential variance-gamma…
The discussion considers how to estimate the expected termination time of an American put, counting maturity as the termination date when the holder never exercises early. One proposed approximation uses the time-zero price difference between otherwise…
The document examines delta hedging a quanto call when the underlying is priced in a foreign currency and the portfolio is accounted for in a domestic currency. A hedge based only on the foreign share count misses the currency conversion: the domestic value…
The document derives a lower bound for a European put on a non-dividend-paying stock by comparing two portfolios. One portfolio holds the put and one share; the other holds a zero-coupon bond that pays the strike at expiration. At maturity, the put-stock…
The document asks how a delta-hedged portfolio of two options leads to a Black–Scholes pricing equation, including a term for the underlying asset’s market price of risk. It writes the portfolio as one option minus a delta-scaled position in another, then…
The document considers fitting an implied-volatility curve to intraday option observations, where data become sparse and variable at extreme out-of-the-money strikes. The question is whether unusual observations can be discounted and whether option-market…
The document asks whether a closed-form valuation exists for an American down-and-out call whose principal amortizes linearly with elapsed time if the option is knocked out. It frames the payoff as a possible time-dependent rebate and compares the problem…
The document derives a self-financing replicating portfolio for a European put in a Black-Scholes market with a risky asset and a deterministic bank account. Applying Itô’s lemma to the option value and matching the diffusion terms identifies the risky-asset…
The document explains how to build and plot a Heston model volatility surface using Python QuantLib. The example creates a Heston process from spot, flat risk-free and dividend curves, and model parameters, then wraps it in a Heston model and passes it to a…
The document explains a Monte Carlo check for an implementation of Dupire local volatility. Simulated underlying paths use the local volatility evaluated at the current simulated spot and time, with the process initialized at the market spot. For each…
The note explains a market convention for describing the right side of an implied volatility smile as call skew, even though put-call parity links those prices to in-the-money puts. The answer suggests that traders often build the smile from at-the-money and…
The note compares delta choices for hedging short-dated at-the-money options in an index dispersion trade, which pairs long single-stock variance with short index variance. It explains that using Black–Scholes delta with the option’s initial implied…
The document explains why the Black–Scholes call price contains two different probabilities: N(d2), the risk-neutral probability that the option finishes in the money, and N(d1), the corresponding probability under a measure associated with using the stock…
The document explains that option traders can seek directional gains as well as profit from volatility. Buying calls can express a bullish view with less upfront capital than buying the underlying or futures, while limiting the buyer’s loss to the premium…
The document asks how to choose volatility when using Black–Scholes deltas to hedge an option, and whether updating the volatility input with each day’s implied volatility changes expected profit and loss or its variance. It contrasts implied volatility,…
The document frames a relative-value trade between American-style options on different futures, potentially quoted in different currencies. The proposed view is to sell volatility in the option with higher implied volatility and buy volatility in the other,…
The document explains why a process introduced as the stock price under the real-world measure may later be written with a new name and then relabeled as the stock price in a Feynman–Kac representation. In the Black–Scholes PDE, the drift appearing in the…