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 raises the problem of checking whether prices produced by models such as Black–Scholes or Heston are accurate. It distinguishes a priori validation, which includes choosing a model and fitting it to market data, from a posteriori validation…
The document discusses how to choose the spot-price bump when approximating call-option Greeks numerically and comparing them with Black–Scholes analytical values. It favors central differences, which reprice at spots on both sides of the current level, over…
This note considers how to model future spot when log returns are assumed to follow a Student-t distribution rather than a Gaussian one. It highlights a key theoretical problem: the Student-t distribution lacks a moment-generating function, so the…
This note derives the delta of a European cash-or-nothing call that pays when the terminal asset price exceeds a strike. Under the Black–Scholes framework, the option value is the discounted probability of finishing above the strike, and differentiating it…
This note explains why a delta-hedged position that is short options can lose when the underlying price moves. A zero net delta at the hedge point describes the position’s immediate sensitivity; it does not mean the option price falls when spot rises. For a…
This note examines when a floating rate note (FRN) prices at par and how its rate exposure depends on the relation between its coupon index and discount curve. With matching forecast and discount curves, the floating coupons telescope against the discounted…
The document answers whether a country-specific market risk premium and inflation rates belong in a currency forward-rate calculation. Under covered interest rate parity, the forward exchange rate is determined by the spot exchange rate and the two…
The document explains what a strip of Eurodollar futures can imply when its quarterly rates are compounded over a one-year period. The resulting rate represents a December-starting one-year compounded three-month LIBOR forward, and is closely related to a…
The document asks how to obtain a Green's function for zero-coupon bond pricing under the Vasicek short-rate model. It gives a pricing partial differential equation with a market price of risk adjustment and compares it with the forward Fokker–Planck and…
The document explains how to convert an annually compounded interest rate into the continuously compounded rate used in the Black–Scholes model. Equating the accumulated value under each convention over the same time horizon gives the conversion: take the…
This exchange challenges the claim that a Constant Maturity Swap keeps a company’s debt duration near the tenor of its CMS index throughout the life of the trade. CMS names the rate used to reset a leg, such as the prevailing five-year swap rate; it does not…
This discussion examines the commonly used SABR at-the-money volatility approximation that keeps only the leading term, alpha divided by the forward price raised to one minus beta. The fuller Hagan expression includes correction terms involving expiry, the…
This exchange explains how to bootstrap an EONIA curve in QuantLib-Python with a reference date that stays fixed when the evaluation date changes. The proposed solution is to use the `PiecewiseLogCubicDiscount` constructor that accepts an explicit reference…
The document explains why a stock modeled with geometric Brownian motion has drift equal to the risk-free rate under a risk-neutral probability measure. It contrasts the expected growth and discounted value of a riskless account with those of a risky stock…
The document asks whether extensions to the SABR stochastic-volatility model, often motivated by rates that can become negative, offer benefits for options on non-negative underlyings such as equities and foreign exchange. The response argues against using…
The document asks how to calibrate the one-factor Hull–White interest-rate model under a historical measure for counterparty credit risk calculations. It distinguishes this from risk-neutral calibration, which commonly fits market prices of swaptions or…
The document addresses a performance question about calculating option prices or Greeks across many inputs in R. Its answer explains that the fOptions package's GBSGreeks function already accepts vector inputs, so a separate compiled-loop package is not…
The document explains why risk-neutral probabilities can be useful even though they are not forecasts of real-world frequencies. It uses a currency example: a terminal probability of a specified euro decline against the dollar can be viewed as an implication…
Perpetual bonds have no contractual maturity date, so they provide a real-market example of instruments modeled with an infinite lifetime. The document explains that this legal feature does not necessarily mean investors should expect the bond to remain…
The document formulates a mean-variance approach to pricing a European call in an incomplete market. It seeks an equivalent martingale measure that minimizes the expected squared difference between the terminal value of a dynamically managed portfolio and…
The note clarifies what volatility implied by VIX options represents. VIX itself reflects an estimate of S&P 500 volatility over a forward 30-day period, but a VIX option does not simply measure the future volatility of today’s spot VIX. VIX options expire…
The document clarifies that implied volatility is a Black–Scholes-equivalent volatility inferred from a European vanilla option price for a given strike and maturity. A smile implied by a different pricing model is constructed in two steps: fix that model’s…
The document explains why an interest rate swap’s sensitivity to its forecast curve can change when its discount curve moves. Changing discounting alters the present value assigned to projected floating-leg cash flows; as those values change, the swap’s…
The document corrects the claim that a European option’s binomial price does not depend on probability. It distinguishes physical probabilities, which describe actual beliefs about market moves, from risk-neutral probabilities, which are implied by the…