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 raises a validation question from a team developing pricing engines for market risk in a Solvency II solution. The instruments named include bonds, callable bonds, credit default swaps, options, and futures. The central issue is whether…
The document asks how to find the supremum of a digital option’s value when a call option price is known. It gives a call price of 5 at strike 100 and asks about a digital option with strike 105. The proposed setup equates the call price to the Black–Scholes…
The document asks whether degree-one homogeneity of call and put prices is preserved when changing from the risk-neutral measure to the share measure. It focuses on stochastic-volatility models and starts from the premise that, under the risk-neutral…
The document examines why an out-of-the-money put’s delta may first move in one direction and then reverse as implied volatility rises. It gives the Black–Scholes–Merton delta formula and differentiates delta with respect to volatility. The sign of that…
The document explains why a put’s theta in the Bachelier, or normal, model cannot generally be represented by the discounting term alone. For a discounted put price, differentiating with respect to the passage of time gives a contribution from interest rate…
The document asks how to characterize a position consisting of a long in-the-money July call and a short at-the-money August put, including its delta and other option Greeks after delta hedging. The replies focus only on directional exposure: both owning a…
The note asks why CME Eurodollar interest-rate options include strikes above 100 basis points, which might appear to make puts always in the money and calls always out of the money if rates cannot fall below zero. The answer points out that this reasoning…
The question asks whether apparent calendar arbitrage exists between December and January options on the same stock. The researcher compares implied volatilities calculated from top-of-book midquotes and notes that total implied variance should not decrease…
The document sets up a comparison between investing with one exponential-utility optimization over a full horizon and reoptimizing at an intermediate date before the final horizon. It presents the martingale-method terminal wealth expression for a complete…
The document asks why the Black–Scholes change of measure shifts Brownian motion by a particular amount when removing the excess drift from the discounted asset price. The answer starts from the desired risk-neutral dynamics: the discounted price should have…
The document explains how a quoted cap volatility is used to value a cap made up of multiple caplets. Under the market convention described, one cap volatility is applied across the cap’s caplets so that their combined valuation matches the market price of…
The document considers a stochastic volatility model with more sources of risk than traded assets, so the market is incomplete. It asks whether derivative prices depend on the additional derivative selected to complete the market. The answer says they do:…
The document examines importance sampling for Longstaff–Schwartz Monte Carlo pricing of American put options. The proposed approach simulates paths under a drift shifted to generate more in-the-money observations, then weights payoffs using a likelihood…
The document asks how local volatility, defined as a function of spot and time, can appear in Dupire’s equation as a function of strike and maturity. It also asks what functional forms are used to represent local volatility in practice and whether a…
The document examines whether the delta of a European call on a futures contract is the normal cumulative probability at the first Black–Scholes term or that probability multiplied by a discount factor. The questioner differentiates a discounted futures…
The document explains why a finite difference estimate of a vanilla call option’s gamma can fail when implemented path by path in a Monte Carlo simulation. The estimator compares call payoffs under upward-shifted, unshifted, and downward-shifted initial…
The discussion distinguishes implied volatility from realized volatility as displayed by trading platforms. Implied volatility is inferred from an option's market price using a pricing model, commonly Black–Scholes. Options across strikes and a shared…
The discussion explains that option prices do not come from a single forecast of an upcoming stock move. They reflect market trading and expectations about the size and likelihood of future price changes. A trader can compare the market-implied risk with…
The document presents a one-period binomial example for pricing a European call and constructing a replicating portfolio. With the underlying ending in one of two states, the option's state-contingent payoffs determine the stock holding that, together with…
The discussion compares buying in-the-money (ITM) and out-of-the-money (OTM) calls when anticipating a large upward stock move. ITM calls cost more because they include intrinsic value and generally have higher delta, so they can participate more if the…
This note asks when an options model, particularly a stochastic local volatility model, needs to reproduce the observed vanilla implied volatility skew exactly. It suggests that close calibration matters when trading vanilla options and European products…
The document analyzes a bond coupon that follows a reference rate within a band and is capped at either end. The questioner decomposes the payoff into a fixed lower amount, a long caplet at the lower boundary, and a short caplet at the upper boundary.…
The note identifies a payoff equal to the positive difference between two asset prices at maturity as an exchange option. This is equivalent to a call on one asset priced in units of the other, with a zero strike. For a maturity of three years in the stated…
The discussion addresses whether weekly options lose a predictable fraction of their value on each day before expiry. It distinguishes theoretical time decay from observed market prices: a model and assumed volatility can illustrate a decay path, but there…