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Structured products can embed options that transfer specific risks between investors and issuing banks. For example, autocallable notes may pay higher coupons in exchange for investors selling downside protection, often through barrier puts. This can leave…
The document compares two discrete approximations for estimating the volatility parameter in a mean-reverting stochastic differential equation with a deterministic time-varying mean. One approach begins from the continuous-time solution and approximates the…
The note explains why a structural credit model’s implied spread may differ from the observed corporate bond spread. Observed spreads include compensation beyond expected default loss, such as liquidity effects, so a model focused on default risk can produce…
The note addresses why variance minimization can help find cointegrating combinations among nonstationary series. It explains that when a cointegrating relationship exists, the coefficients can be estimated super-consistently, giving more flexibility in how…
The document raises a practical question about estimating European option vega under the Heston stochastic volatility model. The author reports computing delta and gamma by bumping spot and recalculating prices or delta, with results matching Bloomberg…
The document asks whether a portfolio can be hedged simultaneously for value, delta, gamma, and vega, after considering hedges that match pairs of sensitivities. The answer frames the problem in terms of degrees of freedom: a hedge needs enough independent…
The document shows how a bond price process can define a change of probability measure under which a shifted Brownian motion is Brownian. Starting from the bond’s risk-neutral dynamics, it forms a density process from the discounted bond price divided by its…
The document explains how to value an existing forward before maturity and how that value relates to a transfer between counterparties. Under zero dividends and a constant interest rate, it derives the forward’s value as the spot price less the discounted…
The document examines the lower bound for a European call on a non-dividend-paying stock, stated as spot less the present value of the strike. It walks through an arbitrage argument for a call trading below that bound: short the stock, buy the call, and…
The document explains why a standard variance swap payoff uses a scaled sum of squared log returns without subtracting the sample mean squared. One answer distinguishes realized variance in the market convention from the statistical sample variance: the…
The document asks how to determine whether a strategy’s performance edge has weakened across time periods. It mentions comparing period means with a t-test as an approach encountered elsewhere, while expressing concern that this test may be unreliable for…
The document poses a pricing puzzle under Black–Scholes assumptions: with spot at 100, zero interest, and volatility chosen so the one-year one-standard-deviation price range is 50 to 200, the call at 200 costs more than the put at 50. It asks why strikes…
The document distinguishes flow trading from market making in a bank trading-desk context. It describes flow trading as taking speculative positions based on a view about future market direction. It describes market making as earning the spread on trades…
The document explains how to infer a discount curve from fixed-rate bond prices using Rateslib. It contrasts sequential bootstrapping, which solves curve values in maturity order and relies on suitable instrument and curve conditions, with Rateslib’s global…
The document clarifies two different rate queries for a vanilla interest-rate swap built in QuantLib. The fixed rate is the coupon already assigned to the swap when it is created. The fair rate is calculated from the swap’s market value and represents the…
The document explains how financial schedules handle a tenor addition when the target month lacks the same numbered day. Its example, adding six months to August 30, leads to February’s last day rather than carrying the excess days into March. It identifies…
The document questions whether daily leveraged ETFs can serve as long-term investments and whether switching between bullish and bearish funds could improve returns. It describes a simple simulation that compounds daily leveraged returns generated from…
The document examines how a power-law exponent affects the share of aggregate income received by the highest earners. It challenges an illustrative example by showing that merely doubling salaries while halving the number of earners at each level does not,…
The document asks whether the negative half-variance term in the logarithm of a geometric Brownian motion price process is related to volatility drag when converting arithmetic returns into geometric returns. It supplies the Black–Scholes stochastic…
The document considers a three-day, 99% value-at-risk estimate for a portfolio split equally between two stocks, given an annualized covariance matrix and joint Student t returns with five degrees of freedom. It asks whether portfolio returns can be combined…
The note outlines how a CDS upfront settlement value can be related to the par traded spread. It gives a simplified valuation relationship in which the difference between the contractual coupon and par spread is multiplied by the risky present value of one…
The note explains that a product term in a regression represents an interaction: the association between one predictor and the outcome changes with the value of another predictor. For two predictors, including their product allows the fitted slope for one…
This note frames trading under an assumed known stochastic price model as a finite-horizon stochastic control problem. The trader’s state includes capital, current position, price, and model parameters; the model evolves price and parameters using noise. The…
The note explains that delta 10 or delta 30 implied volatility quotes identify points on an options volatility surface, rather than a different definition of option delta. Each delta quote maps to a particular strike or moneyness under the relevant market…