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 equity options can have different implied volatilities across strike prices, while foreign exchange options may show a more uniform pattern away from at-the-money. It attributes the equity skew to a negative relationship between…
The document addresses why an option can have the same arbitrage-free price under real-world and risk-neutral probability descriptions even though the expected future stock price differs between them. It presents a one-period binomial setup with up and down…
The note describes how to infer a terminal risk-neutral distribution from option prices across strikes. For a given expiry, the call price as a function of strike encodes the distribution: its second strike derivative, adjusted for discounting, gives the…
The note explains why a probability calculated from a one-period up/down asset model can exceed one. The displayed equation uses the risk-free rate, rather than the stock’s expected return, to determine the risk-neutral probability when pricing under the…
The document asks whether machine learning needs to predict option gamma when a trader can rebalance delta hedges over time. It frames a portfolio containing a call, shares, and a second option, with target conditions for zero delta and zero gamma at a later…
The exchange explains why equity options can have higher implied volatilities away from the at-the-money strike. Black–Scholes translates option prices into implied volatility under a model with continuous, normally distributed price changes. If actual stock…
The document compares FFT and direct integration for evaluating the Carr–Madan option-pricing formula across strikes. The central computational tradeoff is that the characteristic function can be reused across strikes, but the direct method still evaluates…
The document explains how arbitrage arguments imply shape constraints on call option prices. For a fixed maturity, a call price must not rise as the strike increases. It must also be convex in strike: otherwise, a butterfly position can produce an upfront…
The document presents a proposed valuation expression for a swaption that gives its holder the right to enter an equity swap as a fixed-rate payer and equity receiver. It defines the swap rate from the discount factors for the swap's payment dates, then…
The document asks how to price a down-and-out call in a Black–Scholes setting. It specifies a strike and barrier at the same level, a spot price above them, zero interest rate, and a stated volatility. The request seeks a simple formula or proxy for this…
The document compares positive and negative bond convexity from an investor’s perspective. Positive convexity generally means price gains accelerate more as yields fall than price losses accelerate when yields rise. The answer therefore favors greater…
The document explains the meaning of requiring a super-replicating portfolio to dominate a payoff at a zero underlying price. In the exercise’s setup, the portfolio holds alpha shares and beta bonds. At zero spot, the shares contribute no value, and the…
The document explains why the diffusion coefficient in a backward Black–Scholes type partial differential equation is required to be positive. It generalizes the equation’s coefficients and focuses on the role of the second-derivative term: a negative…
The discussion outlines modeling choices for VIX options, including whether to model the VIX directly or build a structural model linking it with the S&P 500 options market. It notes that, although the VIX is calculated from SPX options, VIX derivatives may…
The note explains how to estimate relationships between option volatility buckets for a covariance-based view of portfolio vega. It proposes dividing maturities into intervals and using at-the-money implied volatility, with each later interval represented by…
The document considers ways to obtain historical S&P 500 options data when a researcher has WRDS access but lacks direct access to OptionMetrics. Suggested routes include checking whether faculty or other university researchers can share institutional…
The document gives an interview-style question about estimating a European option price without a calculator and points to a simple approximation for at-the-money calls and puts: about 0.4 times the underlying price, volatility, and the square root of time…
The discussion outlines how to estimate the probability that an option’s delta reaches a target at least once during a specified period. The probability depends on the underlying price model and parameters; under Black–Scholes, volatility and other inputs…
The document addresses how to identify the implied volatility of a 25-delta call or negative-25-delta put, and how to infer the corresponding strike when volatility is known. It presents the Black–Scholes call delta as a function of spot, strike, time to…
The document considers adapting Longstaff–Schwartz Monte Carlo to a put that can be exercised at several discrete dates. It describes simulating geometric Brownian motion paths, estimating continuation values by regressing discounted future payoffs on a…
The document clarifies the relationship between option prices and dynamic replication in the frictionless Black–Scholes framework. Frictionless trading means rebalancing a self-financing replicating portfolio has no transaction costs or imposed limits; it…
The document explains a step in deriving the Black–Scholes formula from a binomial tree. The number of upward moves is modeled as a random variable, and the derivation uses its binomial distribution, with mean equal to the number of trials times the up-move…
The document explains why butterfly spreads with different strike widths may not have prices proportional to their maximum payoffs. It offers an approximation that treats each butterfly’s payoff as a digital indicator for whether the underlying finishes…
The document explains that the second derivative of an option’s value with respect to the underlying price is gamma. It measures how quickly delta, the option’s price sensitivity to the underlying, changes as the underlying moves. This curvature matters…