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 why the discounted price of an American option need not be a martingale under risk-neutral pricing. A European contract has a specified maturity payoff, so its discounted price can be represented as a conditional expectation of that…
The document discusses estimating option gamma with a central second difference and choosing the underlying price shock, h. The estimate is an approximation, so h should not be reverse-engineered to force agreement with a market gamma. A small step is a…
The document describes how to choose the forward input when extracting implied volatility from European SPX options. It treats the forward as the primary quantity for a given maturity, with the yield rate derived from the forward and spot index level.…
The document explores how an issuer might set the coupon on a barrier reverse convertible (BRC), a structured product with a comparatively high stated payment. It proposes that the coupon may be linked to the value of the embedded down-and-in put on the…
The note explains how to read volatility shocks in the CCAR 2018 severely adverse scenario. It clarifies that the listed shock sizes are additive percentage-point changes to implied volatility, rather than relative percentages or basis-point changes. For…
The document asks how to derive an approximation for the volatility of the profit and loss on a hedged short at-the-money option. It describes a simulation setup in which an underlying follows geometric Brownian motion, the option is priced using the process…
The document considers how to value a corporate rights issue, in which existing shareholders receive rights to buy additional shares at a stated price. Although an individual right resembles a call option, the issue grants rights broadly to shareholders, so…
The document explains why swaps do not share a single pricing formula: their fair terms depend on the underlying, cash flows, market conventions, and valuation framework. It distinguishes setting a contract’s fixed rate or strike at inception so its value is…
The document raises questions about how volatility smiles affect the strikes used to construct currency risk reversals. It asks why, under a flat smile, strikes are approximately symmetric around at-the-money in log space, and why that symmetry can place the…
The document explains why a local volatility model can imply a spot–implied-volatility correlation of either positive or negative one, or zero. In this model, the implied volatility of a European option is a deterministic function of time and spot. Applying…
The document shows how to transform the Black–Scholes partial differential equation into a simpler equation for the discounted option value as a function of the discounted underlying. It begins with the original equation and defines both discounted…
The discussion compares buying a zero-dividend stock with buying a hypothetical American call on one share that has a zero strike and no expiry. It explains that, without dividends and under no-arbitrage assumptions, the discounted stock price is a…
The document explains two ways to value rebates in barrier options under Black–Scholes assumptions. For a down-and-out call with a rebate paid at maturity, the payoff can be separated into the ordinary surviving call payoff and a rebate paid if the barrier…
The document discusses how to express the price of a European call as a discounted expectation under a risk-neutral measure, starting from a binomial stock-price model and considering its continuous-time limit. The key pricing correction is to discount the…
Local volatility models can be calibrated to European vanilla option prices across strikes and maturities, reproducing the current implied-volatility surface. Dupire’s relationship derives a risk-neutral diffusion consistent with a sufficiently smooth…
The question describes estimating a risk-neutral density by choosing distribution parameters that minimize squared differences between model and observed call and put prices. The answer says that when the assumed terminal distribution is lognormal, its…
The answer explains the square-root-of-time rule of thumb for the vega of at-the-money options. Under Black–Scholes, vega is proportional to the underlying price, the normal density evaluated at d1, and the square root of time to expiry. Near the money, for…
The discussion asks why a stock's expected return does not directly appear in the Black-Scholes option-pricing formula. Several replies explain the result through replication and hedging: a dynamically adjusted stock position can offset exposure to the…
The document concerns sourcing US credit default swap data for a study of the Corona pandemic period. A response suggests that Bloomberg's CMAN historical CDS spreads may be adequate when IHS Markit historical consensus data are unavailable, while noting a…
The document asks about simulating an integral driven by a Black–Scholes asset process, with the integrand switching according to whether the asset price is below a strike. The author describes this as a naive hedge for a European put and notes that such a…
The document poses an asymptotic option-pricing question. With the underlying price, interest rate, and time to expiry normalized, it defines a put price as the larger of a linear function of strike above a threshold and the put’s intrinsic value. Below or…
The document explains why a trader might neutralize an option portfolio’s delta: to reduce sensitivity to directional moves in the underlying while retaining exposure to volatility. It gives options market making as an example, where the objective is to…
The discussion explains how an FX option table organized by strike relates to a volatility chart organized by delta. Delta links a strike to the option’s moneyness: at a given strike and maturity, the call and put have opposite-side deltas and one is in the…
The document frames a calibration problem for a one-period stock-price binomial model. The two possible next-period prices are represented as multiplicative up and down moves from the current price, with continuously compounded risk-free growth over the…