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.
10,263 documents
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 discussion concerns the unconditional covariance of a multidimensional Ornstein–Uhlenbeck process, expressed as an integral involving the drift matrix and the diffusion covariance. It asks how to evaluate that integral in a dynamic Nelson–Siegel or…
The document asks whether a single constant volatility can approximate prices produced from a curve or surface of Black implied volatilities for products such as floating-rate notes with caps or floors and constant-maturity swaps. It also asks whether a…
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 note explains how the relationship between the up and down factors determines whether a binomial price tree recombines. When the factors are reciprocals, an up move followed by a down move returns to the same value as the reverse sequence. After seven…
The document addresses why a euro LIBOR interest-rate swap may calculate floating and fixed interest accruals using different day-count conventions. It attributes the floating leg’s convention to the euro money-market standard, while the fixed leg uses an…
The document examines how to convert tenor-specific parameters in a multi-factor Cheyette, or quasi-Gaussian, interest-rate model into factor parameters while preserving swap-rate variance. It describes a procedure that first matches the variance and…
The document addresses how to discount a payment when the yield curve uses one day-count convention and the instrument uses another. Its central principle is that the discount factor for a given payment date must be the same regardless of how the rate is…
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 explains how a zero-coupon bond price can be derived when the short rate follows a stochastic process, and how the bond’s market price of risk relates to pricing. It first applies Itô’s lemma to a bond price that depends on the short rate, then…
The document explains why a yen-based investor’s cost of hedging a dollar asset is connected to interest rates and forward exchange pricing. It describes using a cross-currency swap to exchange dollar interest payments for yen interest payments and lock in 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 compares two ways to value an investment whose expected cash flows grow exponentially. Strict present value discounts each future flow at the risky project rate from its payment date back to today. The alternative first values the stream at the…
The document explains how to calculate an interest rate swap’s bucketed delta in rateslib when the available market inputs are zero rates rather than quoted calibration swaps. It treats zero-rate observations as pseudo-instruments that calibrate the curve…
The document asks how the equivalent martingale measure changes when an asset follows an arithmetic Brownian motion with an additive drift, rather than a geometric Brownian motion whose drift multiplies the price. It frames the issue using a bank-account…
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…
The document considers how to reconstruct historical forward rate agreement (FRA) levels after a Mexican market moved from a 28-day fixing to an overnight index. With the new OIS index published only from September 2024, a direct history for the new-index…
The document explains how yield moves can change which bond is cheapest to deliver (CTD) into a bond futures contract. It defines net basis as a bond’s forward price at delivery minus its futures invoice price, calculated using the futures price and the…
The document discusses representing a fixed-rate, amortizing loan in QuantLib when a built-in loan or mortgage class is not apparent. The response constructs a fixed-rate bond for coupon payments, adds an initial amortizing payment for the loan principal,…