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 note answers a narrow options question: what happens to the Black–Scholes delta of an out-of-the-money call as implied volatility tends to infinity. It presents the call delta formula and observes that, for a remaining time to expiry, the formula’s…
The document asks for an intuitive explanation of the drift that appears when a forward rate in the LIBOR market model is expressed under a different forward measure. It presents the adjacent-rate drift adjustment, which depends on the correlation between…
The document examines how the index factor should enter the exercise price of an option on a credit default swap index. It contrasts the factor included in the live portfolio swap value at option expiry with the factor that might belong in the exercise…
The document examines how the risk-free rate affects risky debt yields and credit spreads in the Merton structural credit model. It presents the model’s zero-coupon debt yield formula and differentiates the spread with respect to the risk-free rate,…
The note asks whether an underlying’s risk-neutral implied distribution can be recovered from forward prices, as it can from a cross-section of option prices using the Breeden–Litzenberger relationship. The answer explains that a forward price alone does not…
The document examines whether Lévy’s characterization implies that every discounted traded asset price is Brownian motion. It distinguishes the cited result—that a continuous martingale with continuous paths and finite quadratic variation can be represented…
The document poses a martingale condition problem for a process formed from the square of standard Brownian motion, a linear Brownian term, a deterministic time term, and a constant. Its hint evaluates the conditional expectation between two times by…
The document addresses a QuantLib failure while stripping cap and floor volatilities into individual optionlet volatilities. The reported error says a parameter falls outside its allowed range during bootstrap, at a particular strike and expiry. The answer…
The document explains how an exponential change of measure shifts a Brownian motion’s drift. It first uses the Radon–Nikodym derivative and moment-generating functions to show that the original Brownian value at a fixed time has a normal distribution with…
The document distinguishes a bond futures contract’s expected return from the yield associated with its underlying bond. A futures price is set so the contract’s expected payoff at settlement is zero under the pricing measure; that does not make the bond’s…
The document describes numerical instability when using the Carr–Madan Fourier transform to price deep in-the-money and out-of-the-money options under the Heston model. Direct numerical integration may produce implausible results, such as a call price above…
The document distinguishes real options in corporate finance from exchange-traded puts and calls. A real option is a time-limited business opportunity: a company can spend money to pursue a project if conditions are favorable, or let the opportunity lapse.…
The document uses a one-period stock example to explain why an option hedge cannot be evaluated by treating physical outcome probabilities as pricing probabilities. A call is paired with a short stock position, and the stated up and down probabilities make…
The document examines whether the usual calendar spread arbitrage condition for European calls on a stock carries over to options on futures. The key complication is that options with different expirations can reference futures contracts with different…
The document presents a pricing question for vanilla quanto options on commodity forwards. It proposes adjusting the forward by an exponential term involving the correlation between the underlying and foreign exchange rate, their volatilities, and time to…
The document asks what mathematical assumptions underlie a continuous asset price model introduced heuristically in an option valuation text. It proposes that log returns over an interval are normally distributed, with a mean adjusted by half the variance…
The document distinguishes the overnight indexed swap from the overnight cash rate it references. An OIS is a derivative with a tenor whose value reflects expectations for the compounded overnight rate over that period. Banks may lend or borrow overnight in…
The document asks whether daily variation margin changes futures prices relative to forwards, using stock-index futures as its starting point. It notes that a forward’s fair value reflects financing and asks how the resulting sensitivity to the cash index…
The document examines how to obtain volatility exposure with options when trading a complete option strip is impractical. A single dynamically hedged option retains path dependence, while a dynamically hedged straddle or strangle is described as less…
The document explains how to generalize first-passage results for geometric Brownian motion when the starting value and threshold are not normalized to convenient constants. Taking logarithms turns the process into Brownian motion with drift, so the relevant…
The document values a perpetual claim that pays one unit when a non-dividend-paying stock first falls to a lower barrier. It presents two equivalent routes: risk-neutral expectation of the discount factor at the hitting time, or an ordinary differential…
The document defines a forward volatility agreement as exposure to the at-the-money implied volatility observed at a future date for an option expiring later. It asks how such a contract is hedged in practice. The response mentions forward-start options or…
The document asks for freely available, widely accepted papers on pricing options on fixed-income instruments, including Treasury and Bund options and options embedded in callable or putable bonds. The author distinguishes these products from more commonly…
The document explains why a model-free measure built from option prices ideally requires a continuum of strikes, and how finite market listings change the problem. With observable prices across strikes and negligible bid-ask spreads, vanilla calls and puts…