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.
10,263 documents
The document explains why delta hedging a cash-or-nothing binary option becomes impractical near expiry. Under the stated lognormal pricing model, the option value is discounted and proportional to the cumulative normal probability of the terminal price…
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…
The document asks how to build a USD zero-rate curve from short LIBOR tenors, forward rate agreements, and interest-rate swaps. The response recommends bootstrapping: first infer discount factors or zero rates at the short maturities, then use those results…
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 discussion explains an algebraic step in proving that the one-period binomial model has a unique risk-neutral probability. At each step, the discounted asset price is a martingale, while the next asset value can move to an up or down state. Since the two…
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…
This explanation clarifies the role of conversion factors in bond futures and how they relate to the cheapest-to-deliver (CTD) bond. A conversion factor approximates the bond's price per unit of face value at a standardized 6% yield, using the bond's…
The document asks whether the two legs of an interest rate swap can use different payment calendars. The answer says that swap legs may differ in payment frequency and day-count convention, and distinguishes payment calendars from floating-rate reset…
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 explains how the expected product of two Brownian motions relates to their instantaneous correlation. Their quadratic covariation accumulates correlation over time, so when correlation is deterministic, the product expectation is the same under…
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 discusses how to fill missing maturities in constant-maturity Treasury par-yield data before bootstrapping a zero-coupon curve. It describes the published CMT yields as par yields based on a cubic-spline model, where each quoted yield serves as…
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 shows how to derive the dynamics of a transformed stochastic process from the dynamics of its underlying process. For a geometric diffusion and a power transformation, Itô’s lemma produces a new equation with both drift and stochastic terms; 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…