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 addresses fitting SABR to market volatility smiles quoted in normal and lognormal conventions. It distinguishes the volatility convention used to translate SABR prices into implied volatility: normal market quotes should be compared with normal…
The document concerns modeling the log returns of two stocks with a Gaussian copula and Student-t marginal distributions. The author has already fit a Gaussian copula with normal margins and asks how to simulate data when the margins instead follow t…
The document asks how to estimate the price change that would make a risk-adjusted trading system add one share. The proposed procedure estimates annualized return volatility at the latest price, calculates the current position, then holds volatility fixed…
This exchange explains why a time-series model needs assumptions about both residuals and their squares. Independent, identically distributed residuals support inference about the conditional mean, while stable squared residuals help estimate forecast…
The document raises a pricing question about autocallable products: why their value can depend on volatility of volatility and the forward smile, and why a local stochastic volatility model may be appropriate. The questioner offers a hedging intuition for…
The document discusses how to infer the perturbation factor in a discrete-time Ho–Lee interest-rate tree when the initial bond term structure, risk-neutral up probability, and short rates are known. It gives a relationship between the logarithm of the…
The discussion asks how to price options with several expiration dates when volatility figures are supplied for successive years. The replies distinguish two interpretations: separate vanilla options observed in 2015, or forward volatility periods used to…
The document works through a Ho-Lee short-rate model with a Brownian short rate and derives the zero-coupon bond price, its stochastic differential equation, and the discounted bond’s dynamics. It then constructs the forward-measure density from the bond…
The document asks whether a European call’s delta must stay at or below one in an equity model with zero rates and dividends. Its answers show that this is not a general model-free bound: delta depends on how the option price changes with the current…
The document introduces the uncertain volatility model (UVM), which represents volatility as unknown within specified lower and upper bounds, and asks how it compares with stochastic volatility models. Its appeal is a tractable way to incorporate volatility…
The document explains the typical effect of decreasing time to expiration on option delta, holding strike and other conditions comparable. As time value declines, an in-the-money option tends to behave more like its underlying, so its delta moves toward one…
The document explores how to generate artificial asset-price paths when constant-volatility geometric Brownian motion does not fit the desired behavior. Its example modifies a GBM-style price update by drawing a separate volatility value at each time step…
The note contrasts two ways to represent volatility when modeling option prices. A local volatility model makes volatility a deterministic function of the underlying asset level and time. A stochastic volatility model instead allows volatility to evolve…
The document concerns recovering implied volatility from option prices when options are out of the money, where a direct closed-form inversion of the Black–Scholes price is unavailable. It points readers toward numerical methods and published work, including…
The document addresses negative prices produced for low-priced, out-of-the-money calls while calibrating a stochastic-volatility model. Its central explanation is numerical: if option prices are computed by numerical integration, approximation error can…
The document compares option deltas under sticky-strike and sticky-delta assumptions. With sticky strike, implied volatility is held fixed at a strike as spot changes, making the option’s spot sensitivity the usual Black–Scholes delta in that setup. With…
The document gives a proposed way to estimate the aggregate vega exposure of volatility-linked exchange-traded products. For an individual product, the answer scales its shares outstanding by share price and leverage, then divides by the one-month…
The document asks how market makers handle heavy customer selling in short-dated, deep out-of-the-money options. Its example describes a dealer buying puts, delta-hedging, and accumulating long exposure to volatility skew and tail convexity. The question…
The discussion presents the Black formula for a caplet and asks how to differentiate its price with respect to the forward rate and volatility. It states the familiar forward delta as the discounting and accrual factors multiplied by the normal cumulative…
The discussion derives the generalized Fourier transform of a European put payoff, defined as the positive part of strike minus the underlying price, when the underlying is represented in log-price coordinates. Integrating only over log prices below the log…
The discussion asks whether ATM swaption volatilities indexed by expiry and tenor must satisfy no-arbitrage conditions. It explains that there are no direct no-arbitrage restrictions across such quotes because each swaption is written on a different…
Volatility timing adjusts exposure to risky assets according to recent or current volatility. The basic approach reduces exposure when measured volatility is high and increases it when volatility is low; the document notes that this can mean cutting risk…
The document examines whether a steeper downside volatility skew raises or lowers the price of an at-the-money binary call expiring the next day. It explains that a digital call’s value is tied to the risk-neutral probability that the underlying finishes…
The document presents code for simulating the asset component of a Heston model with Andersen’s scheme. The update uses the current and next variance values, model parameters, and a normal draw to produce each asset price. The author reports that option…