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 note asks whether an American put’s time derivative, or theta, is continuous where the optimal exercise boundary meets the continuation region. It distinguishes this question from the familiar smooth-pasting condition, which says the option’s delta…
The note explains the intuition behind Hull’s statement that a U-shaped implied volatility smile corresponds to a distribution with more probability in both small and extreme moves than a lognormal distribution. Higher implied volatility away from…
The document examines negative variance-equation intercept estimates in a GJR-GARCH model with a pandemic indicator as an exogenous input. The response writes a simplified model with a regime indicator and an asymmetric term for negative shocks, then argues…
The document asks how to generate observations from a four-factor t-copula when the dependence matrix changes over time. Its setting is a series of 2,000 correlation matrices from a copula-DCC model, with one matrix intended for each time step. The accepted…
Structured products can embed options that transfer specific risks between investors and issuing banks. For example, autocallable notes may pay higher coupons in exchange for investors selling downside protection, often through barrier puts. This can leave…
The document compares two discrete approximations for estimating the volatility parameter in a mean-reverting stochastic differential equation with a deterministic time-varying mean. One approach begins from the continuous-time solution and approximates the…
The document raises a practical question about estimating European option vega under the Heston stochastic volatility model. The author reports computing delta and gamma by bumping spot and recalculating prices or delta, with results matching Bloomberg…
The document explains why a standard variance swap payoff uses a scaled sum of squared log returns without subtracting the sample mean squared. One answer distinguishes realized variance in the market convention from the statistical sample variance: the…
The document poses a pricing puzzle under Black–Scholes assumptions: with spot at 100, zero interest, and volatility chosen so the one-year one-standard-deviation price range is 50 to 200, the call at 200 costs more than the put at 50. It asks why strikes…
The document questions whether daily leveraged ETFs can serve as long-term investments and whether switching between bullish and bearish funds could improve returns. It describes a simple simulation that compounds daily leveraged returns generated from…
The document asks whether the negative half-variance term in the logarithm of a geometric Brownian motion price process is related to volatility drag when converting arithmetic returns into geometric returns. It supplies the Black–Scholes stochastic…
The note explains that a product term in a regression represents an interaction: the association between one predictor and the outcome changes with the value of another predictor. For two predictors, including their product allows the fitted slope for one…
The note explains that delta 10 or delta 30 implied volatility quotes identify points on an options volatility surface, rather than a different definition of option delta. Each delta quote maps to a particular strike or moneyness under the relevant market…
The document derives the probability density of the minimum of a standard Wiener process over a fixed time interval using the reflection principle. It first observes that the minimum has the same distribution as the negative of the process maximum, since…
The document asks why the characteristic function used in Heston stochastic-volatility option pricing satisfies the same partial differential equation as the probability terms in a call-price representation. It presents the governing PDE for those terms,…
The position buys two deep out-of-the-money calls and puts, while selling one nearer out-of-the-money call and put. One response describes it as a ratioed strangle switch; another views it as an iron condor combined with a long strangle. These are…
The document compares a normal model for a forward swap rate, where volatility measures absolute rate changes, with a lognormal model, where volatility measures proportional changes or log returns. It derives the relationship by considering the lognormal…
The document presents QuantLib code that constructs a Black–Scholes process and an American-style vanilla option with specified dividend dates and amounts. The pricing function accepts a valuation date, expiry, call or put designation, strike, time-step…
The document surveys ways to express a view on how volatility itself changes. It distinguishes realized volatility of volatility from implied volatility of volatility and describes several candidate exposures: options on volatility products, variance-swap…
The document raises a modeling question about combining GARCH volatility filtering with peaks-over-threshold extreme value theory (POT) and Markov chain Monte Carlo estimation. The researcher fits several conditional volatility specifications, including…
The document explains why a stochastic local volatility model can combine features of local volatility and stochastic volatility. It writes the asset’s instantaneous volatility as a stochastic factor multiplied by a state- and time-dependent local factor.…
The document addresses whether CME options on one-month SOFR futures continue trading during the month when the underlying SOFR rate is measured. This matters for implied volatility modeling: if trading overlaps the averaging period, remaining uncertainty…
The document compares a rolling historical-volatility calculation with an ARCH(q) specification and asks how to describe each. It distinguishes a direct calculation from a model that assumes a return process and requires parameter fitting. It also mentions…
The document addresses how to simulate the variance process in a Heston-style model when its correlated Brownian driver includes a drift adjustment linked to the wealth process. The proposed approach is to substitute the adjusted Brownian differential into…