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 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…
The document examines the normal implied volatility formula for SABR when beta is zero, corresponding to a Bachelier model for the forward. It contrasts a formula using the forward-minus-strike difference with a further approximation that substitutes a…
The document explains why a volatility view alone does not settle whether an option position should be delta hedged. A short straddle held to expiry retains directional exposure as the underlying moves, while hedging repeatedly removes much of that local…
The document considers a payoff at a future date based on the positive difference between a strike and an average built from the compounded product of several forward rates. Under lognormal forward-rate dynamics, it changes to the payment-date forward…
This discussion considers European call pricing when the risk-free rate varies deterministically over time. It starts from a stock following geometric Brownian motion under the risk-neutral measure and distinguishes the bank-account-discounted stock from the…
The document asks why some derivations of the Black–Scholes equation use a stock move of one standard deviation when describing a self-financing delta hedge. The replies distinguish continuous-time distributional assumptions from a particular discrete…
The document questions a proof that an attainable option payoff has a unique initial value and hedging strategy. The proof equates two candidate strategies and concludes that their stochastic integral being a constant forces their integrands to match. The…
The document examines whether a European call with zero strike should trade like its underlying stock. It distinguishes an exact zero strike from a small positive strike and explains that a zero-strike call on a positive stock has no exercise uncertainty:…
The document explains why a delta-hedged option position is not necessarily a source of trading profit. In an idealized model, a perfectly hedged portfolio earns only the risk-free rate. A dealer may earn revenue by selling an option above its theoretical…
The document addresses an apparent contradiction in a Black–Scholes delta-hedging calculation when one simulated stock-price increment is zero. A literal reading of the differential notation suggests that the gamma term vanishes at that step, while the time…
The document explains how to estimate an implied volatility for an option maturity that is not directly quoted, using options with nearby expiries as market anchors. It frames the task as building a volatility surface with strike and maturity dimensions,…
The document explains a limitation in valuing a European option on a coupon-bearing bond as the sum of options on its individual cash flows. In a one-factor interest-rate model, the coupons and principal are driven by the same underlying factor and are…
The document considers how to limit the risk of a short GBP/JPY position after the exchange rate rises. The question proposes taking a long position through a binary option as a hedge, apparently in response to the adverse move. The answer redirects the…
The document explains how to express historical spread volatility when using the Bachelier single-factor spread option formula. Since the option’s underlying is the price difference between two futures contracts, the volatility input is measured in…
The document poses a question about deriving the delta of a standard call option when the underlying follows a local volatility process, with volatility specified as a function of both price and time. It asks how the option value responds to changes in the…
The document considers how to price options when the underlying can fall below zero, a situation that conflicts with the standard lognormal price assumption in Black–Scholes–Merton. It presents two alternatives: the Bachelier model, which uses a normal price…
The document explains the shape of a down-and-in call's value as spot moves around a lower barrier. At high spot levels, the chance of touching the barrier is small, so the option is worth little. As spot approaches the barrier, the probability of activation…
The answer explains stochastic local volatility (SLV) as a hybrid that combines a local volatility surface calibrated to vanilla options with a stochastic process for volatility. The local component helps fit vanilla prices, while the stochastic component…
The document asks why a call option’s theta in the Black–Scholes model does not appear to include a term involving the time change in the underlying share price. It presents the familiar call pricing expression and a standard theta formula, then questions…
The document sets up a call option on a defaultable bond under a risk-neutral measure. Both the risk-free interest rate and the default hazard rate follow mean-reverting stochastic processes. The option payoff is conditional on the issuer surviving to the…
The response derives a pricing relationship between a call option on an exchange rate and a put option on the inverse exchange rate. It defines the rate as domestic currency per unit of foreign currency, then rewrites the call payoff using the reciprocal…
The document discusses whether the perpetual American put, a standard Black–Scholes example with an optimal lower exercise barrier, conflicts with the model’s assumptions because hedging may require a stock short held indefinitely. One answer focuses on…
The document asks for approachable introductions to non-constant volatility models, including CEV, Heston, and SABR, after finding a more advanced volatility-surface text too demanding. The replies recommend resources at different levels and scopes rather…
The document poses an interview problem for estimating the expectation of a payoff at a future time when the underlying follows a diffusion whose volatility depends on its current value. It asks for two approximate algorithms: a stochastic simulation and a…