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.
12,303 documents
The document considers why the Black–Scholes stock-price expression includes volatility while its expected price growth appears independent of volatility. It asks whether this means investors should accept a risky asset whose expected return equals the…
The document asks whether the Poisson-mixture formula used to price European options under Merton’s jump-diffusion model can also price American options. For European contracts, the cited formula weights Black–Scholes prices adjusted for the number of jumps…
The document explains how to interpret a table reporting the risk of a dynamically delta-hedged option position. It describes simulating stock-price paths at the stated weekly frequency, computing the option delta at each step, and adjusting the share…
The document discusses how to compare the VIX with realized volatility calculated from daily S&P 500 index prices. The questioner computes rolling volatility from log returns, annualizes it using the square root of 252, and asks whether a 30-day or roughly…
The document proposes modeling a positive process by making its logarithm an Ornstein–Uhlenbeck process. In this construction, the log value mean-reverts toward a constant level, so the corresponding process is positive and its long-run central level is the…
The document explains how risk-neutral probabilities in a binomial valuation tree are tied to the asset's forward price. For a stock with continuous dividend yield, the expected next-step stock price under the risk-neutral measure should equal the forward…
The question asks how to simulate the next price in a geometric mean-reverting model from the previous price, elapsed time, drift, reversion speed, volatility, and a standard normal draw. The accepted answer gives a discrete transition: evolve the log price…
The document describes an attempt to price calls and puts with Black–Scholes using historical volatility. Prices near the underlying stock price appear close to observed prices, while estimates farther away diverge. The author calculates implied volatility…
The document explains the distinction between extreme price changes and jumps in a stochastic price path. A lognormal distribution can assign low probability to very large price changes over a time interval, but that does not mean the model’s path jumps…
The document examines why a single simulated geometric Brownian motion path with zero drift can produce a maximum likelihood drift estimate far from zero. It defines log returns and gives the likelihood setup used to estimate drift and volatility, then…
The document explains why realized volatility used to evaluate forecasts is not directly observable. Researchers must choose a proxy, such as one derived from returns, and recognize that the resulting measure depends on assumptions about prices, sampling,…
The document examines an Ornstein–Uhlenbeck process whose drift targets a level equal to the process’s limiting variance. It asks whether that apparent relationship is mathematically consistent, using a process with positive mean-reversion speed, constant…
The document describes an attempt to calibrate the two-factor G2++ interest-rate model to cap volatility quotes using QuantLib's Python interface. The example builds a yield curve, creates cap helpers from maturities and volatility inputs, sets an…
The document compares traditional asset-class risk premia with factor investing and alternative risk premia (ARPs). It frames factors as return sources within established markets, such as equity value, size, momentum, and quality, which may have different…
The document presents an APGARCH model output with a negative estimate for gamma and asks what its sign means and whether it indicates a leverage effect. The table also reports the estimate’s standard error, test statistic, and p-value, alongside estimates…
This note compares approaches for fitting European call prices across strike and recovering the risk-neutral distribution implied by options. One method fits call prices directly with arbitrage-constrained cubic smoothing splines; the second derivative of…
This note explains two assumptions desks use to update an implied volatility surface between full calibrations. Under sticky strike, implied volatility stays fixed for each strike as the underlying price changes. An option at a given strike is therefore…
The document asks why stochastic-volatility modeling often introduces a market price of risk when option pricing can begin directly under a risk-neutral measure. It describes a traditional setup in which variance has historical dynamics, its drift changes…
The document offers two practical ways to build intuition about whether an option’s volatility skew is unusually cheap or expensive. One approach is to consider how realized volatility might differ when the underlying is near a particular strike. If…
The document discusses constructing a model-free implied-volatility measure from a range of traded call and put prices. The described procedure first builds an implied-volatility skew by interpolating and extrapolating market observations with a cubic…
The document asks how to price a floorlet whose payoff depends on a linearly compounded rate derived from a bond price. It specifies a risk-neutral model in which the short rate and a second state variable, sigma, both evolve stochastically. The proposed…
The answer adapts the Heston stochastic volatility model to a forward by defining a zero-start log-return process whose exponential is a martingale under the pricing measure. It gives the characteristic function for that process, using the numerically stable…
The document distinguishes the sign of a diffusion coefficient in a stochastic differential equation from the standard deviation of modeled returns. A negative coefficient can be represented by reversing the sign of the Brownian motion, which remains a…
This overview lists stochastic processes used in quantitative finance and gives their representative stochastic differential equations. Brownian motion provides the random driver; geometric Brownian motion models an asset with proportional drift and constant…