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 discussion examines whether a short callable bond can be hedged with a receiver Bermudan swaption when their exercise dates align. It explains that the bond issuer’s call feature can often be hedged by selling a receiver swaption, but the instruments are…
The discussion addresses how to construct a delta-hedged portfolio using S&P 500 index and futures data when a data source provides a continuous E-mini series without an explicit maturity. It suggests inferring a listed contract’s expiration from its ticker…
The document explains when a European call price from a Heston model can be represented by a Black–Scholes implied volatility. It states that the Heston asset-price and variance processes do not generally have explicit pathwise solutions, though Fourier…
The document asks why investment banks need quantitative models when exchange-traded options already have market prices. It distinguishes liquid calls and puts, whose prices are set through trading, from the role of Black–Scholes in organizing those prices…
The document derives an implied risk-free rate from European put–call parity. Starting from the relation between a call, a put, spot, and the discounted strike, it rearranges the equation to solve for the annualized rate using the option prices, strike, spot…
The document considers how to calculate a mid implied volatility from bid and ask implied volatilities without choosing a pricing model. It concludes that averaging the two volatilities is the only model-free approach. The explanation maps volatility to a…
The document proposes a way to hedge the convexity of long-dated fixed cash flows, using a long-maturity swap as the example. The theoretical construction is to sell at-the-money payer and receiver swaptions with very short expiries, sized according to the…
The document investigates why an American call’s market ask price cannot be reconciled with an implied volatility calculation using RQuantLib, while Yahoo Finance reports a volatility estimate. The author compares the market quote with model prices and…
The document explains swimming delta as an option’s total sensitivity to the underlying when implied volatility changes with the underlying price. Standard Black–Scholes delta is a partial derivative that holds volatility constant. Under a skew-aware view, a…
The explanation connects currency basis swap (CBS) rates to FX forward pricing in the Garman–Kohlhagen currency option model. A forward rate can be taken directly from the FX forward market or derived from spot and the two currencies’ interest rates using…
The document defines a European barrier call whose payoff depends on the asset finishing above the strike and staying above a lower barrier at specified observation times. It asks how to estimate its discounted risk-neutral value by Monte Carlo simulation…
The response offers an intuition for why at-the-money implied volatility can act like an average volatility, particularly when used in quanto option pricing. It describes a result in which squared implied volatility is represented as a time average of…
The author compares a personal implementation of the Kou double-exponential jump-diffusion model with published European option prices. The code aims to price vanilla calls by combining Poisson-weighted jump contributions, normal distribution terms, and…
The document sets up a perpetual option whose payoff is the larger of two put-like payoffs, one with strike K1 and slope −1 and the other with strike K2 and slope −2. It begins with the standard perpetual-option ordinary differential equation under geometric…
The note addresses why a set of SABR-calibrated volatilities may appear to show only a downward skew rather than a volatility smile. It explains that the referenced paper shifts all observed strikes so they become positive before fitting the SABR model, and…
The document compares the adoption of the MOVE and TYVIX interest-rate volatility indices with the popularity of VIX. It distinguishes MOVE, which measures implied yield volatility using rates-market quotes, from TYVIX, which measured Treasury price…
The document asks whether dynamically hedging an option exposure with futures can reproduce its payoff without paying an option premium. In an idealized Black–Scholes setting with zero interest rates, continuous rebalancing, and no transaction costs or basis…
The document distinguishes implied volatility from market implied volatility, concluding that the terms generally refer to the same quantity: volatility inferred from observed option prices. It then explains why traders quote options in volatility terms…
The document studies super-replication of a call option in a two-period trinomial model with a risky asset and a zero-rate bond. Each period, the asset can move up, stay unchanged, or move down. Because the market has more possible outcomes than traded…
The document asks how to detect arbitrage in a series of call option prices with a common maturity and different strikes. It proposes checking familiar static relationships, including butterfly arbitrage and monotonicity of call prices with respect to…
The document considers a Black–Scholes asset that pays known proportional dividends at discrete dates. Between payment dates, the asset follows the standard diffusion, while each dividend causes an immediate proportional price reduction. The author gives a…
The document considers a payoff equal to one stock’s terminal price when a second stock finishes above a strike. The stocks follow correlated geometric Brownian dynamics, with their correlation represented by a shared Brownian motion, and the risk-free rate…
The document examines an apparent conflict between two presentations of European put-call parity. One writes the call-minus-put value as the discounted difference between forward price and strike; another presents it as the undiscounted difference. The…
The document asks how to estimate implied volatility for an American option on a deferred commodity futures contract when the option expires on the date associated with a nearer futures contract. It distinguishes volatility by both option expiration and…