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 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 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…
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 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…
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 examines how to construct option positions when an investor knows bounds on one or more future absolute stock price changes. It discusses using the first observed move to update the remaining bound for the next interval, then divides possible…
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 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…
The post considers how to decide whether an implied volatility smile has unusually large skew. It defines local skew as the slope of implied volatility against log moneyness, while noting that the raw slope depends on the volatility level and may need…
The post asks for readable resources that build intuition for the Black–Scholes Greeks, especially through visualizations of how measures such as delta and vega change with time and moneyness. It also seeks guidance on using Greeks to structure options…
The question considers insuring a portfolio of large- and small-cap US equity ETFs against a loss beyond a portfolio-wide threshold. It contrasts a single payoff based on the combined basket with separate options on each holding, which could pay when one…
The document explains how put–call parity and strike-spread bounds can determine whether a one-year put with a strike of 100 is mispriced, given a call with a strike of 90, the stock price, and the risk-free rate. It derives the value of the 90-strike put…
The document explains why using the same numerical volatility input in the Bachelier and Black–Scholes models can produce very different call prices. Although both models describe uncertainty in an underlying price, their volatility parameters use different…
The document answers a request for mathematically rigorous foreign-exchange modeling references that also reflect a practitioner quant perspective. Its recommendations focus on books about pricing FX options, modeling smile risk, and valuing structured…
The document explains how to extend the familiar Heston characteristic function for log price to the joint distribution of log price and variance at a future time. It first writes the model as a two-dimensional diffusion, using independent Brownian motions…
The document examines the claim that an option’s delta approximates its probability of expiring in the money. In the Black–Scholes framework, the answer connects delta to the risk-neutral probability of an in-the-money expiry, with an adjustment related to…
The document explains how open interest relates to long and short positions in options and other derivatives. Because each contract has both a long side and a short side, open interest can be counted from either side: the number of open longs equals the…
The document explains how to price a European call under Merton’s jump-diffusion model. A Black-Scholes calculation conditioned on a fixed number of jumps is only one component of the price; the full value averages those conditional prices across all…
The document outlines a simple Monte Carlo approach to valuing the wildcard option in Treasury futures. It proposes sampling late-day changes in the cheapest-to-deliver bond during the delivery period, then evaluating each day whether exercising the wildcard…
The document considers how to estimate an option’s implied volatility at a target maturity using options observed at nearby expiries. Its response recommends estimating an at-the-money volatility term structure rather than holding an arbitrary strike fixed…
This discussion considers whether quasi-Monte Carlo (QMC) can help price path-dependent derivatives and how to speed up Monte Carlo simulations. It explains that an option price is an expectation over risk-neutral paths, so QMC can in principle be used to…
The discussion contrasts European call and put prices as maturity changes. It gives an arbitrage argument that, under a positive interest rate, a call with a longer maturity cannot be cheaper than an otherwise identical shorter-dated call: if it were, buying…