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 concerns pricing options under the Heston stochastic volatility model with an Euler Monte Carlo scheme. The question reports unstable convergence when using Sobol quasi-random numbers and asks whether a Brownian bridge can address apparent…
The document derives a formula for an asset-or-nothing European call when implied volatility varies with strike. It starts from the Black–Scholes prices for cash-or-nothing and asset-or-nothing payoffs, then applies a tight call-spread argument and the chain…
The document explains an apparent violation of the no-arbitrage lower bound for a European put. For a put with strike K and maturity T, the cited bound is the greater of zero and the present value of the strike minus the current underlying price. A plotted…
The document asks how to estimate the early exercise premium for an American option when the underlying pays discrete cash dividends. The proposed approach integrates a probability term over possible asset prices and times before the ex-dividend date, then…
The document addresses numerical breakdown in SABR normal implied volatility calculations when the forward price equals the strike. In the cited formulation, the intermediate function chi depends on zeta, which becomes zero at this at-the-money point; direct…
The document explains option Greeks in the Heston stochastic volatility model, focusing on delta as the sensitivity of theoretical option value to the underlying price. It also defines gamma, rate sensitivity, theta, sensitivity to variance, vanna, and…
The document asks how to express time-sensitive inputs when using a closed-form option pricing approximation, specifically the Bjerksund–Stensland model. The question lists the risk-free rate, volatility, time to expiration, and cost of carry, and asks…
The document examines why European call and put prices are equal when the strike equals the asset’s forward price. Put-call parity states that their price difference is the spot price minus the discounted strike; setting the strike to the no-arbitrage…
The document develops a definition of realized gamma from the second-order term in an option’s price change. It defines gamma dollars as one half of spot squared times option gamma, then uses the diffusion component of spot’s change to show that the option’s…
The post considers how to maintain a bearish view on Tesla while limiting the ongoing expense of a stock borrow over a potentially long holding period. The response explains that derivative prices generally reflect the costs associated with carrying the…
The document answers which exchange rate to enter as the spot parameter when pricing an FX option with the Garman–Kohlhagen model. Its central point is to use the current spot rate; the forward rate for the option’s expiry is already implied by the model’s…
The document explores an empirical approach to estimating fair implied volatilities for SPX options. It proposes using a proxy for future volatility, such as VIX, to select similar historical observations, fitting a distribution to those observations, and…
The document describes a discrepancy between a closed-form price for a geometric Asian call and a Monte Carlo estimate. The author uses a formula parameterized by spot, strike, maturity, volatility, interest rate, and a monitoring count, and reports that the…
The document reviews a Monte Carlo implementation for a payoff capped at an upper barrier and floored at a lower barrier at expiry. The accepted answer identifies errors in the simulation: each price update should multiply the current spot by the geometric…
The document explains how a European call behaves in the Black–Scholes setting when volatility is zero. With no dividends, the stock follows a deterministic path growing at the risk-free rate. The call’s payoff is therefore known at expiry; discounting that…
The document explains how to turn a Dupire local volatility model into a European option price. Its example starts with an implied volatility surface, constructs a Black–Scholes–Merton process that uses the surface, and sets up a vanilla call. It then…
The document explains why option pricing can use an equal-probability illustration for upside and downside outcomes even when historical returns are asymmetric. It distinguishes the risk-neutral measure used to price derivatives from historical return…
The document works through the time-zero value of a claim in a one-period, two-state market. A portfolio holding the risky asset and a bond is chosen so that its terminal value matches the claim in both the up and down states. The hedge ratios are obtained…
The document considers how to derive a pricing PDE for a derivative under the Heston–Hull–White model, where the stock, stochastic variance, and short rate are correlated. It identifies a problem with applying the usual discounting transformation directly:…
The document asks why Laguerre polynomials are used as regression basis functions in Longstaff–Schwartz Monte Carlo pricing of American options. It cites discussion of multicollinearity among basis functions and a separate numerical comparison that found…
The document asks how to implement Rebonato’s approximation for swaption volatility in C++. The formula combines swap weights, forward rates, correlations, and the integrated products of forward-rate volatilities. With piecewise-constant volatilities, each…
The document clarifies the maturity value of a money market holding that is worth K multiplied by a discount factor at time t. Because the amount is invested in an account earning the risk-free rate, its value grows from time t to T by the corresponding…
The discussion asks whether a binary option always has a Black–Scholes implied volatility and presents several competing answers. A binary call price can rise and then fall as volatility increases, so an observed price may correspond to two volatility…
The document examines valuing a corridor variance swap partway through its life, when realized variance is known for the elapsed period but future variance remains uncertain. It explains that a standard variance swap can be replicated in theory by weighting…