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 how to express a view that an option volatility skew is too flat or too steep while limiting exposure to the overall volatility level. Since option prices move with implied volatility, a trader can combine options at different strikes…
The document presents a bilateral equity option contract with American exercise, linked exercise between parties, post-exercise VWAP averaging, dividend-related strike reductions, and an additional premium tied to rates and the remaining call notional. It…
The discussion addresses why an at-the-money call can appear to have a delta near one when volatility is close to zero. One reply offers an intuition based on positive carry: with a risk-free rate above the dividend yield, a spot-at-the-money option can be…
A European put with strike K pays no more than K at maturity, since its payoff is the greater of K minus the underlying price and zero. The document applies the no-arbitrage principle: the put cannot be worth more before maturity than a zero-coupon bond…
The document presents a Python implementation of Longstaff–Schwartz least-squares Monte Carlo for an American-style Asian call with exercise available at discrete dates. The payoff depends on the average underlying price observed up to exercise. The reported…
The document explains why option volatility surfaces may use forward moneyness instead of raw strike. Moneyness measures a strike’s distance from a reference price; forward moneyness compares the strike with the forward price for that option’s maturity. This…
The document asks how a risk surface can turn sensitivities from a model’s implied volatility function into sensitivities to market quoted volatilities. The described method adds an interpolated surface of risk bumps to the model volatility. At the base…
The document raises a question about delta for American-style exotic options whose holders can make choices at specified times during the option’s life. The holder is assumed to follow an optimal policy that maximizes the option’s value, described in terms…
The document asks how to size a four leg iron butterfly and warns that expiration payoff calculations alone do not capture risks while the position is open. The response highlights execution costs, liquidity, assignment, margin, and the possibility of forced…
The document explains how an observer could estimate the gross premium associated with a reported SPY call trade using option-chain data. The key inputs are the option’s trading volume, the contract multiplier, and the option price. In the example, the cited…
The document examines whether combining a long butterfly, which benefits from a price near its center at expiration, with a long straddle, which benefits from a large move, can produce a profit across all terminal prices. The accepted answer explains the…
The document asks whether negative short-term risk-free rates require changes to the Black-Scholes option pricing formula. Its answer says the standard model does not require the rate to be positive, so a negative rate can be used directly. Under the stated…
The note addresses whether a rates Bermudan callable or swaption remains convex as its strike changes, even though possible exercise dates correspond to swaps with different remaining maturities. It represents the exercise payoff at a given date as the…
The document discusses how to express the Heston stochastic volatility pricing equation using a forward price, log-moneyness, time to expiry, and an undiscounted option value. It gives the spot and variance processes, their correlated Brownian shocks, and a…
The document explains why the Black–Scholes delta of an in-the-money call can first decline and later increase as volatility rises. Interpreting delta as the risk-neutral probability of exercise, it separates two effects: greater volatility makes the strike…
The document examines how a Black–Scholes call price approaches its terminal payoff as time to maturity shrinks to zero. It rewrites the price into a term involving the discounted intrinsic-value difference and a second term expressed as an integral of the…
The Snell envelope is explained through a sequence of choices where each decision may be taken only when its opportunity arises. In the highway example, each exit represents a stopping time, and the value at a given point is the best outcome still achievable…
The document explains how to find the probability mass under a normal distribution between a lower and upper bound. It uses the cumulative distribution function (CDF), which gives the accumulated probability from negative infinity up to a chosen value.…
The document introduces Fischer Black’s idea that the nominal short rate can be viewed as an option: it is the greater of zero and the sum of a shadow real interest rate and inflation. The shadow rate is described as tied to the investment opportunity set,…
The note explains how delta-hedging error arises when an option is valued using an assumed volatility that differs from the underlying asset's true volatility. The option's Black–Scholes pricing equation links its theta to its price, delta, gamma, interest…
The note asks why options on leveraged exchange-traded funds can appear cheaper than options on an unleveraged fund tracking the same index and expiry, despite an expectation that leverage should multiply implied volatility. It explains that a daily…
The note addresses why a call spread built from two outer strikes does not create arbitrage simply because its price exceeds the price of an intermediate-strike call. The intermediate call is cheaper than the weighted combination, so buying it and selling…
The document compares two numerical approaches for pricing European calls under the Heston–Nandi GARCH model using an inversion integral. One implementation evaluates the integrand through adaptive quadrature, while another precomputes…
The document explains how interest enters the overnight profit of a delta-hedged option position. It represents the portfolio as a short option combined with a position in the underlying sized by the option’s delta. The net portfolio value determines whether…