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 an investor holding long-dated calls might also buy deep out-of-the-money puts with shorter expirations. Such puts can provide convex protection in a sharp decline: their delta and gamma may increase as the underlying falls, while…
The document poses a question about the conditional expectation of stock returns in a model with a return driver, a volatility term, and an ARCH(1) variance process. The innovation is written as conditional volatility multiplied by a standard normal shock.…
The document discusses simulating returns without assuming a fixed unconditional distribution. It warns that financial returns have time-varying conditional moments, especially persistent volatility clustering, so resampling the unconditional return…
The document presents code intended to simulate a GARCH(1,1) process. It updates conditional variance using a constant plus a coefficient on the previous squared return and another on the previous variance, then multiplies the resulting volatility by a…
The answer argues that implied volatility need not be extracted only from the Black–Scholes lognormal model. It describes constructing option valuation formulas under alternative distributions, including log-hyperbolic-secant, log-Cauchy, log-t, and…
The discussion explains why a normal random-walk increment over a time step of length t/n is scaled by the square root of t/n. Each increment can be written as a standard normal random variable multiplied by that scale, so its variance is t/n. For…
The question concerns a chart that overlays exchange-rate log differences with a conditional volatility series and observes that the latter is much smaller, despite similar-looking peaks in other examples. The response emphasizes that log returns and…
The discussion distinguishes the role of two conditions used when building a trinomial pricing tree. The conditional expected stock price is set to grow at the risk-free rate under the tree’s transition probabilities. This is the no-arbitrage condition: it…
The discussion explains why Black–Scholes remains useful even though it cannot establish a uniquely fair option price. Its broader contribution is the dynamic hedging framework: under suitable assumptions, an option payoff can be replicated by adjusting…
The document explains a volatility innovation as the part of future volatility that was not captured by the forecast made with currently available information. If realized volatility differs from its prediction, that forecast error is the innovation. In…
The answer explains that the Black–Scholes formula follows from the terminal-price distribution of a geometric Brownian motion (GBM). A Monte Carlo method simulates terminal prices under that process, applies the option payoff, averages the payoffs, and…
The document explains why an option skew measure based on volatility differences divided by log-strike distance, with a square-root-of-maturity adjustment, can remain similar across listed expiries. The key comparison uses strikes set at a fixed number of…
The document asks why historical return quantiles can be used for Value-at-Risk when returns are dependent, as in ARCH-type models. The response clarifies that dependence among returns does not prevent modeling their marginal distribution: volatility models…
The document considers whether options can reveal volatility over a specific upcoming interval, such as a trading day containing a scheduled central-bank announcement. It notes that implied volatility is only directly observable for horizons with liquid…
Co-terminal, or diagonal, swaptions share a common final maturity while their expiries step forward and their underlying swap tenors shorten. The document illustrates the structure with a callable bond that can be redeemed at regular intervals before…
The question derives daily log returns under geometric Brownian motion and estimates their sample variance from a month of stock prices. Since the variance of a return over a daily interval is the annual variance scaled by the length of that interval, the…
The document asks how to choose a moving-average smoothing parameter without relying only on strategy optimization, then presents an example for an exponentially weighted moving average (EWMA) volatility estimate. In the described approach, returns are…
The discussion compares two ways to express the profit and loss of a long delta-hedged option when realized volatility exceeds implied volatility: marking the option using realized volatility, or retaining the implied-volatility mark and allowing…
The document asks how to obtain Black-Scholes implied volatility from a model’s characteristic function using an integral relation attributed to Gatheral’s treatment of Lewis inversion. It focuses on the contour used to value a covered call, why the payoff…
The note explains why a short position’s compounded return can differ in magnitude from a long position’s return, even though the profit and loss on a matched long and short exposure are equal and opposite. It distinguishes a simple short in a future from a…
This note compares sensitivity methods for discrete Asian options when the underlying follows geometric Brownian motion. A geometric-average Asian option has an analytic price under Black–Scholes assumptions, allowing Greeks to be obtained from the…
The document considers how to estimate an implied volatility surface for an asset that has spot data but no options market. It outlines several practical approaches: estimate realized volatility with a model such as GARCH and add a chosen risk premium; scale…
The document considers how to decompose the forward-looking yield of an index of agency fixed-rate residential mortgage-backed securities. The proposed framework adds a duration-matched Treasury yield, the option-adjusted spread, and an adjustment intended…
The document clarifies that negative variance values can arise from numerical discretization even though the continuous-time Heston variance process does not become negative. Flooring values, taking absolute values, or otherwise altering simulated variance…