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.
8,116 documents
The document presents an R routine that estimates SPX option implied volatility by repeatedly applying the Black–Scholes formula and narrowing a volatility bracket. Its failure occurs when the while-loop condition evaluates to a missing value rather than…
The question compares a short-dated SPY put’s premium with its quoted implied volatility and asks why the annualized figure seems much larger than the option’s break-even move. The central explanation is that implied volatility is conventionally reported as…
The document asks how to value early exercise for a contract paying the running arithmetic average of an asset price minus a fixed strike. The response says there is no closed-form value for the American exercise problem in the stated setting. It recommends…
The document derives a call option value under a Bachelier-style arithmetic diffusion with constant interest rate and volatility. It first solves for the conditional distribution of the terminal asset price, which is normal, then discounts the expected…
The document considers a Black–Scholes equation with deterministic, time-varying volatility and interest rates, and describes transformations that turn it into a heat equation. The discussion then points to a risk-neutral alternative: model the underlying…
The document demonstrates a common SVI implementation mistake: feeding strike prices directly into the smile formula when the plotted horizontal axis uses log moneyness. In the example, the strike grid is converted to log strike over forward, and that…
The document outlines a least-squares Monte Carlo setup for valuing a Bermudan swaption. At the final exercise date, the payoff is the positive part of the underlying swap value. At earlier exercise dates, the method regresses discounted future swaption…
The document examines using discounted terminal spot as a control variate to reduce Monte Carlo error when pricing an option. A control variate adjusts simulated discounted payoffs by their covariance with another variable whose expectation is known; the…
The document asks how to calibrate a stochastic discount factor so that a physical or risk-adjusted density derived from option-implied risk-neutral densities resembles a realized return distribution. It raises a practical alignment issue: realized densities…
The document explains that interest rate derivatives can be valued or hedged by matching their cash flows or market sensitivities with traded instruments. Possible building blocks include money market accounts, swaps, forward rate agreements, and zero-coupon…
The document explains that options let traders express views beyond whether an underlying asset will rise or fall. Long options can benefit when realized volatility exceeds the level implied by current prices, or when implied volatility itself increases.…
The document examines whether CMS options should be replicated with cash-settled or physically settled swaptions. The key distinction is the annuity used in settlement. Physical settlement uses a delivery annuity based on discount factors, while cash…
The document asks how to compare the risk and return of options portfolios, noting that standard performance measures, market-risk measures, and option sensitivities capture different aspects of risk. It highlights a difficulty with comparing options by…
The document addresses the theoretical basis for using the implied volatility of a short-maturity, at-the-money option as a proxy for the instantaneous volatility of a stock’s log price. It states the key limiting idea: as option maturity tends to zero, this…
The document derives a conditional expectation for a mean-reverting stochastic volatility process. It applies Itô’s lemma to an exponentially rescaled process, integrates the resulting stochastic differential equation over the forecast horizon, and takes the…
The document considers delta, vega, and gamma for a European call whose underlying is a weighted sum of two stock prices. It questions whether the basket’s vega can be obtained by weighting the individual asset vegas, and proposes shifting both volatilities…
A portfolio that matches a claim’s terminal payoff does not necessarily determine the claim’s price today. For replication to support pricing, the initial portfolio value must be enough to reach the payoff across possible outcomes, without later cash…
The document explains how the Black–Scholes quantity N(d2) can approximate the probability that an option finishes in the money at expiration. N(d1), by contrast, corresponds to the option’s delta. This gives a practical starting point for estimating an…
Implied volatility is quoted as an annualized rate, while total implied variance scales squared volatility by time to maturity. Under constant volatility, this scaling follows the diffusion model: variance accumulates with elapsed time, as it does for sums…
The document describes a proposed dimension-reduction approach for valuing options on the minimum of two equity prices. The author starts with implied-volatility surfaces for each asset, computes option prices on the minimum across strikes and maturities,…
This discussion examines why vanilla option deltas do not reliably estimate double no-touch option prices. For a single upper barrier, the answer gives a one-touch pricing formula under zero interest rates and compares it with the delta of a vanilla call at…
The document asks whether strike-dependent implied volatility can be explained without relying only on trader preferences. It sketches a model-based intuition using Black–Scholes sensitivities. In that model, options away from at-the-money can have positive…
The document explains how implied volatility can affect the Greeks of a put backspread, which holds one short put and two lower-strike long puts. It separates the effect on an individual option’s gamma from the net position’s vega. For gamma, the formula has…
The document asks whether Turnbull and Wakeman’s approximation for continuously monitored arithmetic Asian options uses an Edgeworth expansion, and whether the method originated with them. The question describes the approach as matching the first two…