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The document asks whether the volatility smile and the volatility-of-volatility smile can both arise from dependence between spot and volatility. It proposes viewing the relationship in two directions: volatility moves affecting spot may help explain the…
The document explains how to express a view that policy rates will stay higher for longer than the market expects. In a fixed-for-floating swap, paying fixed and receiving floating benefits if the floating rate over the relevant period is higher than the…
The document clarifies an example about a forward-start variance swap and whether a statement about implied volatility contains a typo. It explains that the relevant forward-start swap strike represents today’s expectation of the variance-swap strike for a…
The document explains the roles of limit orders and marketable orders in exchange matching. A resting limit order supplies liquidity as the maker; an incoming order that executes against it takes liquidity, even if it was submitted as a limit order. Orders…
The document explains how Dupire’s formula relates local volatility to the time and strike derivatives of call prices, assuming no dividends. It walks through a small strike-and-maturity table to estimate those derivatives with finite differences and obtain…
The document introduces an approach for inferring a liquidity parameter from market bid and ask prices for a European call. The parameter, denoted by λ, represents a symmetric distortion around the option’s mid price. Rather than deriving liquidity from the…
The discussion addresses a common fixed-income desk intuition: a receiver swap entered at a rate below a later spot swap rate may still have positive total performance because earlier carry can offset later mark-to-market losses. In its simplified example,…
The document examines why an American call on a stock with one known dividend can be worth more than a European call with the same strike and expiry. It first proposes estimating the difference through European calls around the ex-dividend date, adjusting…
In the Basel market risk framework, the reduced set of risk factors is used to calibrate stressed expected shortfall when a sufficiently long history is unavailable for the full model. Banks select factors relevant to their portfolio and with adequate…
The document clarifies that geometric Brownian motion can be simulated by multiplying lognormal price relatives across time steps. In the direct approach, each step’s gross price ratio should itself be lognormally distributed, with log-mean and…
The document explains why digital options can be difficult to hedge near expiry: their payoff changes abruptly at the strike, creating steep or concentrated sensitivities. One practical approach is to approximate a digital with a call spread. A tighter…
The document discusses three probability and option-pricing questions: when to stop drawing from a balanced deck while betting on card color, how to value a call whose underlying payoff is the number of heads in a fixed sequence of flips, and whether equal…
The document considers how much historical data is appropriate for evaluating a trading strategy, especially one that trades frequently. It argues that the useful horizon depends partly on average holding period and market structure: longer-horizon…
The note proposes a behavioral explanation for differences in stock momentum: investors may underreact to firm-specific news, causing prices to adjust gradually. It asks whether this effect should be stronger or persist longer among stocks held more by…
The note presents a way to obtain the probability density of a basket formed by summing two correlated geometric Brownian motion assets. Although each asset is lognormally distributed, their sum is generally not lognormal. The proposed density follows from…
The note derives a forward price when an asset pays a deterministic proportional dividend before contract expiry. Its main method compares the forward payoff with a cash-and-carry position: borrow funds to buy the stock, account for the dividend cash flow,…
The note explains why an at-the-money short-dated straddle can be approximated using implied volatility. In a Black–Scholes setting, put-call parity reduces the value of an ATM-forward straddle to twice the call value. The ATM call value is approximated as a…
The discussion clarifies whether the VIX is leading or lagging. Because it is calculated from option prices and represents the market’s implied expectation of volatility over the coming period, it is forward-looking in that sense. It does not independently…
Treasury futures allow the short to choose which eligible bond to deliver, creating a quality option for the short and affecting the futures price. If the cheapest-to-deliver bond is already known, pricing can be based on that bond’s forward price adjusted…
The discussion explains why out-of-the-money VIX puts expiring on a given date can lose value even when the spot VIX falls sharply. The key distinction is that VIX options are tied to the VIX futures contract for their expiration, not directly to the spot…
The document explains the notation for a EUR trade combining three forward-starting interest rates with different start dates and tenors. It identifies the position as a curve fly, where the intermediate point is the belly and the shorter- and longer-term…
The document concerns how to compress a portfolio of long and short commodity forward or futures positions across delivery months into a single front-month-equivalent exposure. The goal is to approximate portfolio profit and loss using that exposure and the…
The document asks how to price a European claim that pays the natural logarithm of a stock price at maturity, including the possibility of a negative payoff when the stock finishes below one. It presents the general risk-neutral valuation rule: discount the…
The document surveys ways to build a user interface for controlling trading software whose strategy runs in C++. The suggested design is to expose a small service from the strategy host, then let a separate web or desktop client send commands and display…