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.
12,303 documents
The document examines parameter estimation for a simulated asset process whose log-price increments combine geometric Brownian motion with exponentially distributed time intervals. The author proposes estimating the mean and standard deviation of log…
The document describes a troubleshooting example for estimating conditional expected shortfall, also called CVaR, from a fitted EGARCH model with a Normal Inverse Gaussian return distribution. The author computes rolling distribution parameters, integrates…
The question compares broker-displayed theta for SPX index options and E-mini futures options with similar strikes and nearby expiries. Their quoted option prices are close, yet the reported theta differs, prompting a question about whether European versus…
This discussion outlines how short-rate models can support pricing for interest-rate derivatives. A modeled short rate produces zero-coupon bond prices and, in some models, closed-form prices for bond options that can be used for products such as caps,…
The document examines how a one-factor mean-reverting stochastic volatility model for instantaneous forward variance might extend to several factors. It describes a lognormal variance process driven by a mean-reverting Ornstein–Uhlenbeck factor, with the…
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 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 document examines whether significant autocorrelation in squared returns conflicts with an Augmented Dickey–Fuller test that indicates stationarity, and whether daily volatility can then be scaled to a longer horizon using the square-root-of-time rule.…
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 asks how several definitions of volatility relate: the standard deviation of log returns, the standard deviation of continuously compounded rates, and the standard deviation of percentage price changes. It notes that log returns and continuously…
The document asks how to simulate the Cox–Ingersoll–Ross (CIR) variance process efficiently on a GPU in MATLAB. It presents a nonnegative Euler discretization, where each step uses the prior variance, mean reversion, and a Brownian shock, then clips negative…
Returns and squared returns describe different features of a time series. Returns measure direction and can have little serial correlation even when their magnitudes are dependent over time. Thus, the absence of correlation in returns does not imply that…
The document asks how to estimate Expected Shortfall (ES) from a return series using the block maxima approach in Extreme Value Theory. It contrasts that question with a peaks-over-threshold formula based on Generalized Pareto Distribution parameters and…
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 clarifies how to interpret the statistic returned by an automatic variance ratio test applied to financial returns. The null hypothesis is that returns are serially uncorrelated, corresponding to a variance ratio of one; the alternative is…
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 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 discussion gives an economic interpretation of the Merton model relation between equity volatility and asset volatility: equity’s typical value change reflects the asset value change scaled by the sensitivity of equity to assets. Since the model treats…
The discussion considers how to obtain accurate quantiles for the Normal Inverse Gaussian (NIG) distribution, used here to model drivers of interest rate implied volatility risk. The questioner values the distribution’s parsimony and closure under…
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 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…