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 considers why a local volatility model for an exotic option may be calibrated to a broad implied volatility surface when hedging in practice uses only a limited set of options. It connects the local volatility construction to prices across…
The document asks whether a stock forecast based on real-world return probabilities can support decisions among owning shares, buying protective puts, or substituting calls, and what risk-neutral pricing adds. Its proposed real-world model extrapolates…
The document explains how to use a volatility surface quoted by delta to price an option on a futures contract. For a given option strike, first find the implied volatility associated with that strike: when the surface is quoted by delta, solve for the…
The document explains that VIX9D applies the VIX methodology to S&P 500 options whose expirations bracket a nine-day horizon. The calculation uses two option maturities, derives implied volatility for each using the standard VIX procedure, and interpolates…
The note offers a probability-based intuition for why options far from the strike can have little time or extrinsic value, including calls that are deeply in the money. It frames time value around the chance that the underlying crosses the strike before…
The note explains why two expressions for a European call’s delta agree when the cost of carry is defined as the risk-free rate minus the dividend yield. The risk-free rate is a cost of holding the asset because that capital could earn interest elsewhere,…
The document discusses why an implied volatility smile may reach its minimum at strikes above the current spot price. For equity indices, it identifies demand from call overwriting as a possible flow-based explanation: investors sell upside calls to earn…
The document examines the gamma-weighted P&L of a delta-hedged option. Its expression weights the difference between realized and implied variance by dollar gamma over time, raising the question of whether a lower average realized volatility than the initial…
The document asks whether a variance swap can be replicated explicitly with vanilla options under the Black–Scholes model when both the risk-free rate and dividend yield are nonzero. It contrasts this request with a familiar replication formula derived for…
The document asks whether stock and vanilla option positions can create a synthetic long single-stock future when short puts must be fully cash-secured. It considers buying the stock and writing a put as a possible substitute, while noting the familiar…
The discussion offers two rough ways to assess whether a convertible bond may be inexpensive. One is to compare its market price with the value of a comparable straight bond, which isolates the premium paid for the embedded conversion feature. The premium is…
This note explains why higher interest rates can raise call values and lower put values, even when rate increases may coincide with falling stock prices. It gives the Black–Scholes sensitivity to the risk-free rate: call rho is positive, while put rho is…
The document describes simulating jump increments from Kou’s asymmetric double-exponential model. Its proposed procedure draws a Poisson number of jumps, splits them into upward and downward counts with a binomial draw, then models the total jump size on…
The document addresses how to derive forward-price dynamics when both the underlying asset and interest rates are stochastic. One response applies Itô’s lemma to a function of time, spot price, and rate, including the rate’s variance and the covariance…
The document explains why option rho is generally positive for calls and negative for puts as interest rates rise. It connects call sensitivity to put-call parity: a call can be viewed as a synthetic combination that includes exposure to the underlying and a…
The document discusses why short-dated implied volatility often reacts more visibly to near-term events than long-dated implied volatility. Longer expiries average expected variance over more time, which can dilute the effect of a temporary volatility shock.…
The document compares two proposed ways to price a Margrabe option on two correlated assets in a Black–Scholes setting. The first changes numeraire to express the payoff using the asset used as numeraire; under that measure, the relative asset price has…
The document distinguishes realized volatility, which is estimated from observed underlying prices, from implied volatility, which is inferred by reversing an option-pricing model using a market option price. A model requires a volatility input to produce a…
The document derives the limiting early-exercise boundary for an American put as expiration approaches. It compares immediate exercise, which pays intrinsic value, with waiting for a short interval and then receiving the expected payoff under risk-neutral…
The document considers a call payoff based on the terminal underlying price minus the average price observed over an earlier period ending at time τ. This is described as an Asian strike call, with the averaging window allowed to end before the option…
The document addresses why converting smile data from implied volatilities into Black model option prices does not produce a curve with the same visual shape. Its answer explains that when puts are converted to equivalent calls using put-call parity, call…
The document asks whether cubic spline interpolation across swaption strikes guarantees an arbitrage-free implied volatility smile, and how that approach differs from a stochastic model. The answer says ordinary spline routines do not impose the necessary…
The document asks whether European options under GARCH volatility require simulating the full sequence of stock prices and variances, or whether forecast average volatility can be inserted into a geometric Brownian motion solution. The response explains that…
The document distinguishes the probability assumptions used in derivatives pricing from those used to forecast outcomes and manage risk. Under the risk-neutral measure, simulations support option valuation and the calculation of sensitivities used in…