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 works through the expiry payoff of a position that buys a put, sells a call at a higher strike, and buys another call at a still higher strike. Ignoring premiums, the long put produces gains below its strike, the position is flat between the put…
The document describes implied volatility as a way to represent option prices on a more interpretable and comparable scale. A pricing model maps a market option price to the volatility input that would reproduce that price, allowing traders to discuss an…
The document explains the distinction between an option’s notional exposure and the price paid for the contract. Using the SPX example, the response applies a contract multiplier of 100 to the index level to calculate notional value, and applies the same…
The document concerns hybrid models that combine Heston stochastic volatility with Hull–White interest rates. The question is how to approximate European option prices for model calibration, particularly when the pricing approach uses Monte Carlo simulation.…
The document poses a modeling question about whether local volatility derived from an observed implied volatility surface reproduces that surface or changes its skew. It outlines a workflow: obtain market implied volatilities, fit a model such as SVI, use…
The document examines why a derivative’s price should not depend on which other derivative is chosen to hedge volatility risk in a stochastic-volatility model. The proposed replication argument uses the underlying and a second derivative to span two Brownian…
The document explains a linear approximation for implied volatility in Emanuel Derman’s sticky implied tree model. The answer starts with local volatility modeled as a linear function of the underlying price, with slope determined by a parameter describing…
The document considers how to recover the risk-neutral density of an underlying at a given expiry from call option implied volatilities across strikes. It describes converting each implied volatility into a Black–Scholes call price, then applying the…
The document asks why a project option valued with real-world probabilities and the project’s required return differs from a risk-neutral valuation. Its binomial example has a successful project value of $10 million, an unsuccessful value of $2.7 million, an…
The note explains how a conditional expectation at a specific asset level can be expressed as a ratio of expectations weighted by the Dirac delta function. For a terminal asset value and a random variance, the numerator weights variance by the event density…
The document works through an Itô-calculus exercise involving two independent Brownian motions. It defines a process as their product minus one half of the time integral of the sum of their squared values, then differentiates that process using the product…
The document asks how to interpret specialness in bond futures when holding a long-only cash bond portfolio. It contrasts a cash bond’s implied forward price, determined by its carry, with the futures-implied forward price. When the futures-implied carry is…
The document describes a market practice for estimating cross-currency basis between quoted maturity dates. Rather than interpolating the quoted basis or outright FX forwards directly, it presents a representation of the FX forward using spot, domestic and…
The document examines a security paying one dollar if IBM reaches a specified price, with the stock initially below that level and no dividends or transaction costs. It contrasts a risk-neutral probability argument, which would imply a payoff value of one…
The document asks how to discount the legs of a fixed-to-fixed cross-currency swap when its collateral currency changes at a mandatory break date. Without a break, its proposed setup discounts euro cash flows using the euro collateral curve and uses a…
The document contrasts a funded equity collar with a margin loan used to build a stock position. In the collar, the client buys a put and sells a call to a bank; the bank may hedge its resulting exposure by borrowing and selling shares. A margin loan instead…
The document discusses a binomial no-arbitrage argument for pricing an option. A portfolio holds one option and shorts a quantity of the underlying chosen to offset the difference between the option values in the up and down states. If that hedge makes the…
The example considers a position short August puts, long August calls at a higher strike, and short futures. Its initial net delta, gamma, and vega are reported as slightly negative or close to zero. The question asks why the answer key says all three become…
The question asks how to sketch second-order Greek profiles for a European vanilla payoff, such as a butterfly, without computing a full profile. It is concerned with how Greeks vary as spot or volatility changes, including vanna, vomma, and the changes in…
The document asks why a study of a two-factor Double Heston model calibrates against option-market observations from many Wednesdays across a year, rather than conducting a separate calibration for each date. It contrasts that setup with research using…
The document examines a static hedge for two short calls with the same strike but different expirations, using one call whose expiration falls between theirs. The example shows that matching total notional does not keep the hedge balanced over time: the…
The document describes how to estimate a foreign-exchange risk-neutral density from vanilla option data, in the context of comparing the distribution for GBP/EUR around two historical dates. The required inputs include option prices across strikes and…
The document considers an arithmetic Brownian motion with constant volatility and asks how often it is expected to move a fixed distance from its current level, resetting the reference band after each crossing. The question is framed in terms of repeated…
The document connects the fundamental theorem of asset pricing to a one-period binomial tree. Choose the probability of an up move so that the stock price discounted by the bond is a martingale under the resulting measure. Solving this condition gives the…