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 derivative values are calculated using risk-neutral probabilities rather than real-world outcome probabilities. In a one-period binomial model, a replicating portfolio determines a contingent claim’s no-arbitrage price without…
The document explains why a bond’s implied spot rate can differ from a rate curve built with a SciPy univariate spline. The bond has two remaining cash flows, and its price and yield-to-maturity equation imply a final-cash-flow spot rate of about 2.8%, while…
Bond prices reflect more than scheduled cash flows. Even bonds with very low default risk can move as investors change their demand for safe assets or revise expectations for monetary policy. For riskier government and corporate debt, yield spreads over a…
The document asks whether a European option can be priced in a Black–Scholes setting when its payoff depends on a general transformation of the underlying asset. It illustrates the question with a call payoff equal to the positive part of a transformed…
The document presents practical ways to estimate the parameters of the Vasicek short-rate model without relying only on an unconstrained fit to bond prices. One approach discretizes the model into an approximate AR(1) process and matches theoretical and…
The document explains how TIPS can produce a positive nominal return despite a negative real yield, while still failing to preserve purchasing power. At maturity, inflation adjusts principal, but the adjusted payment may not offset the purchase price; the…
The document explains how to calculate an FX outright forward rate from spot and broker-quoted forward points. The stated calculation is to add the forward points to spot: for the one-month EUR/PLN example, 3.4550 plus 0.0079 gives 3.4629. The question also…
The document asks whether refinancing should be triggered by a simple comparison between lower interest payments and refinancing fees. It presents consumer heuristics for fixed-rate mortgages, including refinancing when market rates fall by a threshold or…
The document considers how to speed up repricing American equity options after constructing implied volatility surfaces. It describes the existing workflow of smoothing end-of-day surfaces and applying finite-difference pricing with discrete dividends, then…
The document discusses how to choose regression basis functions in the Longstaff–Schwartz least squares Monte Carlo method for valuing American-style options, including real options with multiple risk factors. The question is whether the few Laguerre…
The document asks how to define option vega when volatility varies with the underlying asset price, using the CEV form in which local volatility is proportional to a power of the asset price. It contrasts this setting with Black–Scholes, where volatility is…
A unilateral extinguisher is a swap portfolio that may be canceled after a credit event affecting one counterparty. Whether it ends depends on whether its mark-to-market value is above a contractually set threshold; a recovery payment may also apply. This…
The document considers a perpetual monthly payment obligation issued in exchange for an initial cash amount, with the issuer able to terminate it after a waiting period by repaying that original amount. It identifies the arrangement with a perpetual bond or…
The document derives a Black–Scholes–Merton value for a call-like payoff that is positive when the terminal asset price exceeds the strike and scales with the reciprocal of that price. It rewrites the payoff using an indicator and inverse asset, relating the…
The discussion compares possible ways to track changes in option volatility skew relative to at-the-money volatility. One answer proposes measuring the implied-volatility difference between the strikes where Black–Scholes d2 equals zero and d1 equals zero.…
The question sets up a discretized forward-rate model in which each forward is lognormal under its associated zero-coupon bond measure. It derives the resulting normal distribution for the log forward at its fixing date and uses that distribution to motivate…
The answer models an interest rate swap as an asset leg and a liability leg, then expresses generalized duration as the asset value times its duration minus the liability value times its duration. The sign reflects the liability’s negative contribution to…
The note contrasts pricing in complete and incomplete financial models. In a complete model, it describes a unique martingale measure and a derivative price given by the discounted expectation under that measure. In an incomplete model, multiple martingale…
The note asks how to differentiate a Carr–Madan Fourier integral for option prices with respect to a model parameter. Since the integration limits do not depend on that parameter, the proposed application of differentiation under the integral sign leaves the…
The note explains why option prices generated by a stochastic volatility model can imply a volatility smile when those prices are converted using the Black–Scholes formula. Under Black–Scholes pricing with one fixed volatility, the prices of calls across…
The document derives the futures price for an asset whose price process and short rate are driven partly by the same Brownian motion. Since the futures price is the expected future spot price, the solution first expresses the integrated short rate as a…
The document considers the probability that one of two correlated geometric Brownian motion prices finishes above the other, while the lower price also exceeds a threshold. It rewrites the event using shifted Brownian variables as two simultaneous…
The document derives the variance of a weighted sum of two correlated Brownian increments. For coefficients applied to each increment, the variance includes both individual variance terms and a covariance term determined by their correlation. Since each…
The document explains how derivatives can change a borrower’s practical liquidity and recovery profile even when derivative claims rank equally with senior debt in bankruptcy. Under collateral agreements, a firm may have to post cash or securities as market…