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The document explains that option Greeks are sensitivities of the Black–Scholes–Merton price to model inputs. It defines delta, gamma, rho, vega, theta, and sensitivity to the dividend rate as partial derivatives, then works through the call delta…
The document discusses how to identify common risk factors in private markets when asset prices are infrequently observed or unavailable. One suggested route is to describe companies using observable characteristics such as industry, business relationships,…
Put–call parity links prices of European calls and puts with the same strike to the value of a forward contract. With the risk-free rate and time to expiry known, the parity relationship gives the forward price and can be rearranged to infer the asset’s…
The document asks how to choose a probability measure when pricing derivatives with Monte Carlo simulation. It starts from risk-neutral valuation, where price is the discounted expected payoff, and notes that simulations may require modeling state variables…
The document poses an inverse pricing problem: fitting a Black–Scholes model to binary option prices observed across strike prices when the time to maturity, volatility, risk-free rate, and underlying price are all unknown. The stated goal is to predict…
The document explains why a single expiration payoff diagram may not fully describe a portfolio containing short American-style options or options with different maturities. A short American option can be exercised by its holder before expiration, so the…
The document presents a derivation of the CAPM by treating the market portfolio as a portfolio on the capital allocation line that maximizes the Sharpe ratio. Under the assumption that investors choose mean–variance efficient portfolios, individual holdings…
The document discusses choosing the ARCH and GARCH lag orders for a volatility model. One proposed approach estimates a range of candidate orders, checks whether coefficients are statistically significant, inspects residual autocorrelation, and uses…
The document distinguishes the Samuelson effect’s maturity-related volatility pattern from volatility measured directly on futures price levels. The effect concerns returns volatility: nearby commodity contracts are often more volatile than deferred…
The document explains how to obtain the Radon–Nikodym derivative when changing from a probability measure associated with numeraire M to one associated with numeraire N. The derivation uses the martingale property of asset prices expressed in units of the…
This note addresses the European call formula for a dividend-paying stock and whether an expected stock growth rate should replace dividend yield. It explains that the dividend-adjusted Black–Scholes formulation and a cost-of-carry formulation are equivalent…
This note explains why a plain floating-rate bond can have interest-rate duration close to the time until its next coupon reset. Under a single-curve assumption, future coupons are projected from forward rates and discounted using the same curve.…
The note discusses how options traders describe exposure to changes in implied volatility skew and volatility convexity. Skew refers to differences in implied volatility across strikes, such as between out-of-the-money puts and calls. Traders may…
This note clarifies how to set up valuation equations for an asset whose dividends follow a Markov process. Its central distinction is between cum-dividend value, which includes a payment made today, and ex-dividend price, which excludes it. The investor’s…
The document explains how to generate multivariate Gaussian returns with a target covariance structure. Starting with a vector of independent standard normal draws, apply a matrix square root of the desired covariance matrix. A Cholesky factor is one such…
The discussion describes how to think about risk in a single-currency swap exchanging floating payments based on different tenors, such as six-month and three-month rates. It separates sensitivity to changes in the tenor basis from outright rate delta. A…
The discussion asks whether traders can profit from the tendency of equity markets to fall quickly and recover more gradually. It explains that this asymmetry is reflected in option prices: downside puts often carry a premium relative to comparable calls. A…
The discussion examines why a zero-fee Euro Stoxx 50 ETF appeared to outperform its benchmark. It identifies securities lending as a possible source of revenue, since an index itself holds no assets to lend. Holding an index portfolio can therefore add…
The document explores representing Brazilian DI1 interest rate futures with rateslib for curve calibration and sensitivity analysis. It proposes a zero-coupon swap representation and distinguishes the futures’ annualized compounded internal rate of return…
The document examines a proposed monthly regression of year-over-year stock index returns on lagged returns and macroeconomic variables, including credit spreads, inflation, and industrial production. It describes a forecasting setup that uses ARIMA…
The document explains why quasi-Monte Carlo methods use low-discrepancy sequences: these points fill a space more evenly than random samples, reducing large gaps. It defines discrepancy as a measure of how closely the fraction of points in a region matches…
The document explains why hedging a mortgage-backed security with a swap is complicated when the underlying mortgages can be prepaid. Because prepayments depend in part on future interest rates, the security’s outstanding principal and the matching swap…
The document addresses why a stock’s cumulative return can fall below the risk-free return, even when investors expect equities to earn a premium over time. Its explanation is that risky assets can have poor outcomes over particular periods. The equity…
The question seeks a long historical daily closing-price series for the individual companies that made up the Dow Jones Industrial Average, rather than a series for the index itself. The response points to a dataset of intraday one-minute bars covering the…