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 explains two Brownian motion calculations relevant to barrier option pricing: the midpoint distribution conditional on the terminal value, and the probability of crossing a barrier between fixed endpoints. For the midpoint, it uses the joint…
The document presents a canonical semimartingale decomposition into a bounded variation component, a continuous martingale part, a compensated jump integral, and a residual jump term. It clarifies that the process differential is the object used as an…
The document addresses why extending an option’s time to expiry does not always increase a call’s value when the underlying has a positive dividend yield. Its explanation invokes the Black–Scholes call formula and put–call parity: as the horizon grows,…
The document explains two ways a long option position can benefit from volatility: a rise in implied volatility can increase the option’s resale value, while large underlying price moves can benefit a long gamma position. A long straddle combines exposure to…
The document presents a Carr and Wu option-pricing identity for a Markov model in which the underlying asset price and time describe the state. A call expiring at a later date can be represented as an integral of calls expiring at an intermediate date,…
The document distinguishes the market price of a currency forward from the cost of hedging measured relative to spot. Covered interest rate parity links the two: under parity, the forward-to-spot ratio equals the ratio of the two currencies’ gross interest…
The document considers a bond whose coupon is determined randomly on the payment date and asks whether its value can be found by averaging the prices of bonds with fixed coupons under real-world probabilities. The central pricing distinction is whether the…
The document explains how to compare a quoted currency forward rate with a no-arbitrage rate derived from borrowing and lending rates. When the quoted rate differs, the arbitrageur buys the currency through the cheaper route and sells it through the more…
The document investigates why a QuantLib fixed-rate bond gives coupon amounts that differ slightly from the stated rate when configured with ACT/ACT, while a 30/360 Bond Basis setting produces the expected coupon amounts but a different accrued-interest…
The document addresses how to obtain an initial short-maturity zero-curve point from a tradable overnight index swap quote. Its answer focuses on day-count conventions: quoted OIS par rates may use an ACT/360 basis, while the zero rates delivered to users…
The document explains why credit default swap index contracts can use standardized fixed coupons even when the market spread differs from that coupon. Its central point is that the difference between the contractual coupon and the market spread is settled…
The discussion examines a Monte Carlo pricing expression that represents VIX through the risk-neutral expectation of the log of the S&P 500’s future level relative to its current level. The question is why simulated values might make the expression under the…
The document examines why discounted traded asset prices are martingales under a risk-neutral pricing measure. It raises a potential confusion in an argument that compares holding an asset until a future date with selling it now and investing the proceeds at…
The document discusses a proposed solution to a modified Black–Scholes equation for a European put when volatility depends on the underlying asset price. The proposed expression uses associated Laguerre polynomials and is evaluated as a series. A numerical…
The document clarifies that overnight (ON) and tomorrow-next (TN) FX quotes refer to swaps with two settlement dates, rather than ordinary forward tenors. ON exchanges the base currency between today and tomorrow; TN exchanges it between tomorrow and the…
The document discusses ways to generate returns that reflect skewness and kurtosis, which a normal model does not capture. One approach uses a truncated Gram–Charlier Type A density: its coefficients are linked to cumulants, and the described setup gives a…
This explanation shows how FX forwards and cross-currency swaps determine discount factors for cash flows collateralized in a currency different from the payment currency. It first identifies local-currency discount curves from interbank instruments, then…
This discussion concerns calibrating the default probability input to a collateralized debt obligation model using a one-factor Gaussian copula and the large homogeneous portfolio approximation. In that setup, expected tranche loss depends on the reference…
This note checks the stochastic differential for a discounted geometric Brownian motion and explains why a zero-drift process is a martingale under a changed probability measure. Applying Itô’s product rule to the discounted asset gives a factor of the…
The document investigates why a short put’s hedge delta increased even as the underlying price rose slightly. It reports that volatility also declined and two days elapsed, with a long time remaining to expiration. The response identifies charm, or delta…
The document raises practical questions about modeling American options with Black–Scholes–Merton, including early exercise, changing bid–ask spreads, intraday time to maturity, volatility estimation, historical risk-free rates, strike listings, and…
The document shows how to value an annuity using the cash flows of a fixed-rate bond in QuantLib. The method starts with a bond built from a payment schedule and a discounting engine tied to a spot yield curve. To represent an annuity with no principal…
This discussion recommends ways to learn PDE techniques used in derivatives pricing. It suggests starting with an options text that introduces Itô’s lemma, binomial pricing, and explicit and implicit finite-difference methods, then moving to more specialized…
The document explains how put-call parity for year-on-year inflation caps and floors can be used to infer the market rate of the underlying inflation swap. In present-value terms, the cap value minus the floor value equals the value of a swap that pays fixed…