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 asks whether binary options can produce a position that doubles on success and halves on failure when the event is close to equally likely. Its algebra shows that the desired payoffs imply a particular option quantity and purchase price, but…
The document clarifies the meaning of delta when the underlying position is the stock itself. Delta is the derivative of a position’s value with respect to the stock price. For a long position whose value changes one-for-one with the stock price, that…
The document answers how to select the at-the-money strike when calibrating the volatility level in a shifted SABR model. The shift is added to both the forward and strike in the model’s implied-volatility calculation. Consequently, an option that is at the…
The document explains how market rates can be used to form measures of inflation expectations and expected real interest rates. It identifies inflation swap rates as the likely source of the cited medium-term inflation expectation measure, including the…
The document clarifies two uses of “forward interest rate”: a rate fixed today for borrowing or investing over a future interval, and a contract whose payoff depends on a short rate observed at a future date. It explains that these are distinct contract…
The document considers a borrower’s option to choose between a fixed-rate loan, a floating-rate loan, or cancellation after a decision window. It frames the choices as embedded options with value linked to the borrower’s credit spread, interest rates, and…
The document asks how to interpret quoted running and upfront market premiums at different tenors when working with preliminary data for estimating probability of default. It gives examples of five-year and eight-year quotes, each with a nonzero running…
The document presents a question about implementing the pathwise derivative method for estimating a call option’s delta under a geometric Brownian motion model. The proposed simulation code evolves prices through multiple time steps and then sums…
The answer explains how to derive the time-dependent bond price function in the Hull–White extended Vasicek model. Instead of trying to obtain the target result by manipulating a later equation alone, it solves the first-order equation for the bond-price…
The document considers a simplified model for valuing inflation options embedded in inflation-indexed contracts with guarantees. The author already has a stochastic nominal interest-rate model, G2++, and wants inflation scenarios linked to it. One proposed…
This discussion examines why longer-dated options can cost more even when implied volatility appears to fall with maturity. It cautions that implied volatility term structures vary: they may rise, fall, or fluctuate, including at the money. In the…
The document explains that continuous compounding is often a modeling choice rather than an assumption that interest is literally reinvested at every instant. Continuous-time asset-pricing theories describe evolving prices and discounting with continuous…
The document compares a one-touch option, which pays when the underlying crosses a threshold at any time through maturity, with a binary option that pays when the underlying is above the threshold at maturity. It reports that the one-touch price is…
The document presents a QuantLib setup for calibrating a one-factor Hull–White short-rate model to a EUR overnight-indexed discount curve and a normal-volatility surface for three-month Euribor swaptions. It describes fitting the model’s mean-reversion and…
The document considers why a riskless portfolio formed from an option and a hedge in the underlying must earn the risk-free rate in a binomial pricing model. The answer frames the argument as a contradiction: assume an arbitrage exists, then show that…
The document explains how to calculate the expectation of an exponential formed from several correlated Brownian motions. Its main method treats the Brownian vector at a fixed time as a multivariate normal random variable, with a covariance matrix determined…
The document distinguishes calculating implied volatility from choosing coordinates for interpolation and display. To calculate implied volatility from market prices, supply the actual spot, strike, maturity, rates, and dividend yield required by the pricing…
The document addresses whether the vanilla option portfolio used to replicate a variance swap still applies when the underlying pays dividends and the risk-free rate is nonzero. Its answer says the replication formula remains valid, provided the forward…
The document explains why a long position in a standard European vanilla option generally has positive vega, even when the option is already in the money. Vega measures how an option’s value responds to a change in volatility. The key idea is that a vanilla…
The document clarifies the distinction among a Libor fixing date, the start of the accrual period, and its end. In many markets, the rate is fixed shortly before the accrual period begins; it is a forward rate for the period from start to end. This means the…
The document addresses a common gap in the delta-hedging derivation of the Black–Scholes PDE: specifying a stock holding does not by itself explain how the portfolio remains self-financing as that holding changes. A trading strategy is described through its…
The document explains why the squared differential of an Itô process is represented by its diffusion variance times time. For a process with drift and a Brownian component, the quadratic variation accumulates at the rate given by the square of the diffusion…
The document explains how to derive state prices from asset payoffs by constructing portfolios that replicate Arrow–Debreu securities. For each state, solve a linear system in which the payoff matrix multiplied by portfolio weights equals a payoff of one in…
The document outlines a state-space formulation for an arbitrage-free affine extension of the Nelson–Siegel term-structure model. Yield observations are represented as a maturity-dependent adjustment plus a linear combination of three latent factors and…