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 considers how to estimate the probability that a forward swap rate finishes above or below a swaption strike. A simple approximation uses the cumulative normal distribution of a standardized forward-minus-strike distance under a Bachelier model,…
The discussion explains how to calculate the probability that a call finishes in the money in a discrete-time binomial model. First determine the minimum number of upward moves needed for the terminal stock price to exceed the strike. For a recombining tree,…
The document addresses whether QuantLib can model option value changes at finer than daily time intervals on the final trading day before expiry. It says intraday date resolution is possible, but requires rebuilding both the C++ library and its Python…
The document examines why a simplified Treynor–Black weighting expression appears to favor a call option over its underlying stock. In the example, the option’s alpha is scaled by delta while its variance is scaled by delta squared, which can make the…
The discussion explains why geometric Brownian motion (GBM) remains common despite unrealistic assumptions about returns. Its appeal comes from simplicity, intuitive results, and closed-form Black–Scholes pricing. Practitioners often use Black–Scholes as a…
An amortising swap’s declining payment notionals can be decomposed into a stack of constant-notional swaps with different maturities. Each component starts with part of the total notional and remains active through its own maturity; adding their notionals…
The document contrasts maximum-likelihood fitting of a Student-t model for future log returns with fitting the same model to realized option payoffs. In the first approach, parameters determine the conditional mean and volatility from current variance and…
The document asks how to price options on stocks when the premium is settled daily rather than paid upfront. It proposes adapting standard Black–Scholes pricing by multiplying the ordinary option value by the growth factor for the risk-free rate over the…
The document considers an option whose terminal payoff is the positive part of a constant multiple of the underlying asset price minus a strike. It shows that when the multiplier is positive, the multiplier can be factored outside the positive-part…
The document asks how to estimate the price impact on a convertible bond when its underlying stock falls sharply. It describes a second-order approximation using the bond’s delta for the linear component and gamma for the curvature component, then questions…
The document raises a model-risk question: when a trading desk uses the Heston stochastic-volatility model to generate a volatility surface, why might risk management use other models to produce adjustments? It names interest-rate mean reversion and other…
The document frames a derivatives-pricing question about which traded instruments are hardest or infeasible to value. Its concrete example is a best-of or other rainbow option with early exercise. A rainbow payoff depends on multiple underlying assets,…
The document asks whether a European swaption can be priced equivalently under the risk-neutral measure and the forward swap measure, using the swap annuity as numeraire. The key correction is that the annuity at expiry is stochastic because it depends on…
The document explains the link between gamma and realized volatility for a delta-hedged option position. An option’s value is curved with respect to the underlying price, so a hedge that offsets its current delta removes the local linear exposure but leaves…
The document explains why observed prices and implied volatilities need not match a pricing model perfectly, using SABR and interest rate swaptions as context. Its central point is that market prices arise from trading among participants; models are…
The document considers how to value American calls with five-to-ten-year maturities on an accumulation fund that has no readily available exchange price or option quotes. It notes that such long-tenor options may be available over the counter, but…
The document explains two ways to measure an option’s one-day theta when implied volatility varies across expiries. Fixed-expiry theta holds the option’s implied volatility constant during the time bump, matching the Black–Scholes convention. Fixed-tenor…
The document examines a lower bound for the variance swap strike, expressed as a risk-neutral average of Black-Scholes implied variance over a standard normal coordinate. It uses a Taylor expansion around the center of that coordinate. The linear, or skew,…
The document asks how to compare the predictive pricing errors of three option valuation models against Black–Scholes prices, using both in-sample and out-of-sample results. It raises the Diebold–Mariano test as a possible method and questions whether its…
The document illustrates a common failure in Monte Carlo control variates: the target and control estimates were simulated with separate random draws, so their covariance did not provide the intended variance reduction. In the displayed put and call example,…
The document asks how control variates might improve Monte Carlo pricing for European and American down-and-in basket puts under a Heston model simulated with the QE scheme. It considers using a vanilla put as a control for a barrier payoff, while noting…
The document explores whether a trader can neutralize an options portfolio’s sensitivity to implied volatility, or vega, while retaining directional exposure. Combining long and short options can offset vega because options carry vega whereas linear…
The document weighs the practical value of studying complex analysis for quantitative finance. Its clearest application is derivative pricing: some models have conditional characteristic functions that enable quasi-analytical European option prices through…
The document discusses why an implied volatility smile may be absent or not observable in some markets. Where options are not actively quoted, or only at-the-money contracts trade infrequently, there may be too little market information to construct a…