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The document proposes estimating portfolio profit distributions by simulating monthly generation for each energy asset and electricity forward prices, multiplying generation by price to estimate asset revenues, summing across assets, and taking lower-tail…
The document examines whether Lévy’s characterization implies that every discounted traded asset price is Brownian motion. It distinguishes the cited result—that a continuous martingale with continuous paths and finite quadratic variation can be represented…
The document poses a martingale condition problem for a process formed from the square of standard Brownian motion, a linear Brownian term, a deterministic time term, and a constant. Its hint evaluates the conditional expectation between two times by…
The document explains how a market maker can earn a spread while managing inventory risk with a related futures contract. In its simplified example, a dealer quotes both sides of an illiquid S&P 500 ETF, buys shares from a seller, and shorts index futures to…
The document describes bootstrapping spot rates from US Treasury par yields. It explains that Treasury par yields are bond-equivalent yields based on securities with semiannual coupon payments, then presents a Python example that solves sequentially for spot…
The document addresses a QuantLib failure while stripping cap and floor volatilities into individual optionlet volatilities. The reported error says a parameter falls outside its allowed range during bootstrap, at a particular strike and expiry. The answer…
The document explains how an exponential change of measure shifts a Brownian motion’s drift. It first uses the Radon–Nikodym derivative and moment-generating functions to show that the original Brownian value at a fixed time has a normal distribution with…
The document distinguishes a bond futures contract’s expected return from the yield associated with its underlying bond. A futures price is set so the contract’s expected payoff at settlement is zero under the pricing measure; that does not make the bond’s…
The document sets up a mean-reverting jump-diffusion with deterministic, time-varying mean-reversion, level, volatility, and jump-arrival parameters. It gives an integral representation of the process and derives expressions for its conditional mean and…
The document describes numerical instability when using the Carr–Madan Fourier transform to price deep in-the-money and out-of-the-money options under the Heston model. Direct numerical integration may produce implausible results, such as a call price above…
The document distinguishes real options in corporate finance from exchange-traded puts and calls. A real option is a time-limited business opportunity: a company can spend money to pursue a project if conditions are favorable, or let the opportunity lapse.…
The document uses a one-period stock example to explain why an option hedge cannot be evaluated by treating physical outcome probabilities as pricing probabilities. A call is paired with a short stock position, and the stated up and down probabilities make…
The document examines whether the usual calendar spread arbitrage condition for European calls on a stock carries over to options on futures. The key complication is that options with different expirations can reference futures contracts with different…
The document explains how to calculate a bond’s discrete-period return from its beginning and ending clean prices, accrued interest at both dates, and coupon payments received during the period. The calculation includes bond income as well as the change in…
The document presents a pricing question for vanilla quanto options on commodity forwards. It proposes adjusting the forward by an exponential term involving the correlation between the underlying and foreign exchange rate, their volatilities, and time to…
The document asks what mathematical assumptions underlie a continuous asset price model introduced heuristically in an option valuation text. It proposes that log returns over an interval are normally distributed, with a mean adjusted by half the variance…
The document distinguishes the overnight indexed swap from the overnight cash rate it references. An OIS is a derivative with a tenor whose value reflects expectations for the compounded overnight rate over that period. Banks may lend or borrow overnight in…
The document asks whether daily variation margin changes futures prices relative to forwards, using stock-index futures as its starting point. It notes that a forward’s fair value reflects financing and asks how the resulting sensitivity to the cash index…
The document examines how to obtain volatility exposure with options when trading a complete option strip is impractical. A single dynamically hedged option retains path dependence, while a dynamically hedged straddle or strangle is described as less…
The document raises a practical question about using estimated stock beta to hedge a concentrated portfolio. With a short sample or a volatile stock, an estimated beta can be large by chance even when the true relationship to the index is zero. A…
The document explains how to generalize first-passage results for geometric Brownian motion when the starting value and threshold are not normalized to convenient constants. Taking logarithms turns the process into Brownian motion with drift, so the relevant…
The document addresses how to calculate pivot points and first support and resistance levels from a stream of trade data. The key preprocessing step is to aggregate trades into five-minute bars using calendar time. Each bar should contain its open, high,…
The document explains how to estimate profit on a short Eurodollar futures position from the change in the quoted futures price, and asks how expiration timing and unchanged spot LIBOR affect a long position. The answer says the initial short-profit…
The document values a perpetual claim that pays one unit when a non-dividend-paying stock first falls to a lower barrier. It presents two equivalent routes: risk-neutral expectation of the discount factor at the hitting time, or an ordinary differential…