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 derives the expected time integral of an Ornstein–Uhlenbeck process up to its first hit of a fixed threshold. It first notes that conditioning on the process being at the threshold at that stopping time adds no information, since the…
The document raises a statistical question about uncertainty in option Greeks estimated with Monte Carlo simulation. It starts from a simulated option value with a standard error, then applies central finite differences to estimate Delta from prices at…
The document explains why option pricing becomes less determinate when an underlying follows a jump-diffusion process. In the Black–Scholes framework, market completeness yields a unique risk-neutral probability measure and therefore a unique arbitrage-free…
The document explains how overnight indexed swap rates relate to daily overnight rates and how compounding builds the floating leg over a term. It distinguishes the compounded overnight payment for a three-month period from the quoted three-month OIS swap…
The document discusses estimating sensitivities for a European arithmetic basket call, whose payoff depends on the sum or average of several underlying assets. Because the basket has multiple underlyings, it has separate deltas, vegas, and other…
The document examines a Monte Carlo simulation of a European call under the Black–Scholes–Merton model. It contrasts compounding normally distributed simple returns with exponentiating normally distributed log returns, then gives the drift adjustment needed…
The document describes a question about calibrating a local volatility surface through an inverse problem. It outlines a representation that uses B-splines across one dimension and linear interpolation across time, then penalizes variation in time and…
The procedure estimates an at-the-money implied volatility for a target maturity by using two listed option expirations that bracket it. It first selects nearby call and put strikes, estimates a forward price for each expiration from put–call parity, and…
The explanation clarifies that the Black–Scholes stock process is written for the ordinary price level, with drift and volatility proportional to the current price. The logarithm enters when Itô’s lemma is applied to that process: the log price then has a…
The document raises a pricing question about a down-and-out call when the underlying follows a jump process, with the Kou double-exponential and normal inverse Gaussian models suggested to represent asymmetric jumps. It considers raw Monte Carlo simulation…
The document poses a question about constructing a risk-neutral density from out-of-the-money options. Since the density is related to the second derivative of option price with respect to strike, it asks whether convexity of option prices is needed to…
The document explains why an FX option’s delta convention must match the convention used to build a volatility smile or surface. It lists spot and forward delta, each with premium-adjusted and unadjusted forms, and notes that converting between conventions…
The note addresses whether Black–Scholes prices a call and a put equally when their strikes are equally far from the current underlying price in opposite directions. It distinguishes terminal payoffs from option values before expiry. At expiry, if the…
The document explains how a Monte Carlo simulation can align with Black 76 without first converting futures prices into spot prices. It starts from a lognormal spot-price process with drift and volatility, then substitutes the spot-forward relationship so…
The document explains that a numeraire used for pricing does not always need to be a traded asset. It presents the risk neutral pricing relation under a measure associated with the chosen numeraire, then describes expressing derivative and underlying prices…
The document asks whether the publication of the Black–Scholes option-pricing paper in 1973 was directly connected to the founding and launch of the Chicago Board Options Exchange that year. The cited historical account says the formula was not developed to…
The document explains how to price a claim paying the cosine of a terminal stock price under risk-neutral pricing when the stock follows geometric Brownian motion. It writes the discounted expected payoff as an integral over the risk-neutral lognormal…
The document explains the combined payoff of a long up-and-in put and a short up-and-in call with the same strike, maturity, and barrier. If the underlying reaches the barrier before expiry, both options activate and behave like vanilla options. Put-call…
The document considers the impact of replacing established overnight discount rates with new risk-free rates on uncleared swaptions. It asks whether valuation changes should be compensated, distinguishing temporary changes in a swap’s discounted value from…
The document explores how a forward interest rate follows from the prices of zero-coupon bonds with different maturities. It proposes holding a long position in the longer-maturity bond and a short position in a scaled amount of the shorter-maturity bond,…
The document explains replication as matching a derivative’s payoff across every possible model state using positions in the underlying stock and a money market account. In a one-period example, stock and bond holdings are chosen so their combined terminal…
The document considers whether linear hedges, such as delta hedges, can meet most clients' needs without other derivatives. It notes that Black–Scholes theory can use a rebalanced linear hedge to replicate a nonlinear option payoff. For linear exposures, or…
The document explains why option traders sometimes treat theta as a proxy for gamma, focusing on a delta-neutral portfolio. It uses the Black–Scholes differential equation to relate portfolio time decay, delta, gamma, interest rates, and the underlying…
The document examines the Black–Scholes risk-neutral drift relation and asks whether its negative volatility adjustment conflicts with the observation that higher interest rates can coincide with falling stock prices. The replies explain that the relation…