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 describes how practitioners can approximate a digital option using a narrow call spread around its strike. Buying the lower-strike call and selling the upper-strike call creates a payoff concentrated between the strikes; as the gap narrows, the…
The document explains a model-based route for estimating a European option’s value when its American counterpart is quoted. First, specify assumptions for the underlying’s risk-neutral dynamics, such as a diffusion or jump-diffusion model. Then select a…
The document explains why higher volatility can increase the cost of delta-hedging a short option. Its central point is that the volatility used to price an option is an estimate of what the underlying will realize over the option’s life. If actual realized…
The document explains how forward delta for a vanilla foreign-exchange option relates to a hedge using a forward contract. Under the stated Black pricing setup, differentiating the call value with respect to the outright forward rate gives a discounted…
The Cornish-Fisher expansion approximates a target distribution’s quantiles using a reference distribution and information about the target’s cumulants. In modified VaR applications, it is commonly presented with the normal distribution as the reference and…
The document explains when early exercise changes the value of American options on a non-dividend-paying underlying. Under the no-arbitrage assumption and a nonnegative risk-free rate, early exercise of a call is not optimal because it gives up remaining…
The document explains why a short-rate model may be used to value bonds and swaps even when their cash flows have no embedded options. A correctly calibrated model must reproduce the present value obtained by discounting fixed cash flows against the spot…
The document explains how introducing stochastic interest rates changes the uniqueness of a Black–Scholes call price. The key issue is whether the interest rate risk can be traded and hedged. If no claims linked to the stochastic rate are available, the…
A constant maturity swap rate is the par swap rate for a chosen tenor, observed on each reset date. A CMS leg therefore references a swap rate such as the ten-year rate even when payments reset more frequently. In a CMS swap, that rate may be exchanged for a…
The post asks how a trader should adjust a volatility surface when pricing a product that includes a long option. The answer describes shifting the parameters of a chosen surface model in absolute terms, with adjustments reflecting the trader’s exposure to…
The post points readers seeking a C# implementation of the Barone-Adesi-Whaley option pricing model to QLNet, a C# port of QuantLib. It identifies the library’s pricing engine as a place to find a native implementation, offering a practical route for someone…
The document describes reducing the pricing of a fixed-strike Asian call to a partial differential equation in time and a transformed state variable. The state variable tracks the remaining strike after accounting for the average accumulated so far, scaled…
The document examines how an option’s theta changes as time passes, sometimes described as theta acceleration. It distinguishes the textbook Black–Scholes theta from a finite-difference estimate obtained by repricing after a one-day time shift. Comparing…
The document disentangles the delta of a forward contract’s value from the sensitivity of a quoted forward or futures price. The value of an existing forward contract has spot delta of one when other inputs are held fixed. The forward delivery price, by…
The document clarifies what is a martingale under a forward measure and whether clean or dirty bond prices qualify. A bond price alone is not generally a martingale under the T-forward measure; the relevant martingale is the asset price expressed relative to…
The document explains why a Monte Carlo estimate for an up-and-out call can be too high when simulated paths use discrete time steps. A path may cross the barrier between sampled points and return below it before the next observation, so checking only the…
The document gathers advice for a mathematician with statistics and programming experience who wants to enter quantitative finance without first joining a hedge fund. It recommends exploring quant roles at banks or in academia, building relationships with…
The document explains the net basis in a Treasury cash bond and futures trade, addressing why it may be negative even though it is associated with delivery options. A negative reading does not by itself establish an arbitrage opportunity: funding constraints…
The document examines whether a forward-starting swap can be statically replicated using spot-starting zero-coupon swaps. The proposed hedge combines paying fixed on a one-year-forward, one-year swap, paying fixed on a spot-starting one-year zero-coupon…
The document explains why the ratio of two zero-coupon bond prices, with a common valuation horizon, is a martingale under the forward measure associated with one bond. It focuses on the conditional expectation and the correct use of Bayes’ rule when…
The document examines how to price IBOR-linked caplets and floorlets when the underlying FRA uses different day-count conventions for its floating and fixed legs. The standard Black formula multiplies the option value by one accrual fraction, but real market…
The document considers calibrating an equity stochastic-volatility model alongside a stochastic short rate, with correlations among the driving processes. It explains that the stated combination may lack a tractable closed-form pricing formula, making direct…
The document investigates systematic profit-and-loss errors in a simulated dynamic hedge of an American put. It describes pricing and recalculating delta with a fine-step binomial tree while simulating the stock as geometric Brownian motion, financing the…
The document frames the model-selection problem for pricing vanilla swaptions on fixed-versus-floating or floating-versus-floating legs. Requirements include multiple floating-rate tenors, OIS and LIBOR legs across currencies, mid-curve structures, and both…