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 derives a first order approximation for the one year total return on a coupon bond when its yield changes. It distinguishes the bond's initial price from its price one year later, when it has one fewer year to maturity. The latter price is…
The document explains why a deterministic partial differential equation can describe the value of a claim whose payoff depends on a stochastic process. In a Markov setting, the conditional expected payoff is a function of the current state and time. Under…
The discussion concerns sensitivity-preserving cash-flow mapping for a fixed-versus-Euribor six-month interest rate swap under a multi-curve framework. The question asks how to assign cash flows to regulatory pillar dates when discounting uses an overnight…
The document considers geometric Brownian motion paths that remain above a lower price barrier throughout a fixed horizon. It starts from a supplied survival probability and frames the requested conditional mean and variance of the terminal price in terms of…
The document explains why pricing a plain vanilla foreign exchange option from spot or from the outright forward can lead to equivalent formulations. Garman–Kohlhagen uses spot and domestic and foreign interest rates, while Black–76 uses the forward price.…
The document explains how the standard Black–Scholes partial differential equation can be rearranged so that its second-order price term appears as the derivative of a product. Applying the product rule to the diffusion term creates an additional…
The document asks whether a multi-factor stochastic volatility model, especially a two-factor model, can be approximated by a faster single-factor model for Monte Carlo calibration. It defines the distinction in terms of the number of Brownian motions…
The document explains why early exercise does not add value to an American call when there are no dividends or repo costs and interest rates are nonnegative. Put-call parity implies that a European call is worth at least its intrinsic value under these…
The document explains why otherwise identical call options should not become more expensive as their strike prices rise. If a lower-strike call costs no more than a higher-strike call, an investor can buy the lower-strike option and sell the higher-strike…
The document addresses valuation of a cross-currency basis swap whose two currency legs are collateralized in a third currency. Its example uses Indian rupee and South African rand cash flows with US dollar collateral. The stated procedure is to convert each…
The document explains why the coupon required to price a callable bond at par can vary non-monotonically with the call date. The coupon spread over a comparable non-callable bond reflects the value of the issuer’s call option. That value depends on time to…
A self-financing portfolio changes value only through changes in the assets it holds, with no external deposits or withdrawals. For holdings phi_i in assets priced S_i, its infinitesimal value change is the sum of each holding multiplied by that asset’s…
This note traces the use of q for continuous dividend yield in derivative pricing. It contrasts Merton’s use of D in an early treatment of continuous dividends with later use of q in work by Hull and White and in Hull’s textbook. Other papers used d or the…
The answer addresses whether time can be divided by a Brownian motion differential and whether one can interpret a quotient such as dV/dS for Itô processes. It warns that manipulating formal differentials as ordinary algebra can produce misleading results.…
The discussion considers how an issuer might hedge an ETN linked to a VIX strategy, especially when replicating an index with VIX futures could involve trading costs and limited liquidity in deferred contracts. The answers emphasize that a hedge need not…
The VIX calculation uses out-of-the-money SPX options with nonzero bids, so its option count changes as volatility and the available strike range change. The document asks how many options might be needed to replicate the index, but provides no practical…
A par yield curve shows the coupon rate at which a bond would trade at par across maturities. The document explains why practitioners use this representation instead of plotting yields of existing coupon bonds: coupon differences and bond-specific pricing…
The document explains why the product of a pricing density and a self-financing portfolio value is a martingale in the Black–Scholes setting. Its central point is that martingales are defined relative to a probability measure. Under the physical measure, the…
The document explains the approximate conversion between volatility swap notional and variance swap notional. A volatility swap payoff changes with volatility, while a variance swap payoff changes with squared volatility. Applying a local linear…
The document explains why cross-currency basis swaps exchanging overnight risk-free rates may trade away from zero, despite a theoretical flat-pricing argument based on covered interest parity. It describes a proposed arbitrage: borrow in one currency at its…
The document derives a change-of-numeraire Radon–Nikodym derivative for two positive assets. It assumes the ratio of their prices is a martingale under the original measure, uses that ratio to define a density process with expectation one, and constructs the…
The discussion checks a far-field boundary condition for a continuously averaged Asian call under the risk-neutral Black–Scholes process. When the current asset price is very large, the option is expected to finish in the money, so its value can be…
The document asks how to interpret and scale parameters estimated for a Bates stochastic volatility model from daily log returns, using the mean reversion parameter κ as an example. It distinguishes estimates under the real-world probability measure from…
The document asks whether positive gamma is generally costly, negative gamma earns premium, and theta tends to have the opposite sign from gamma. It frames theta as the carrying cost of holding long gamma and asks whether these observations can be proved…