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The document explores extending the Heston stochastic volatility model to represent mean reversion in the asset price or returns as well as in variance. The answer proposes a price-level adjustment in which the drift of log price pulls it toward a long-run…
The document considers whether individual quarterly returns can be recovered from a sequence of rolling one-year returns. Each rolling figure combines several quarterly returns, so the unknown quarterly values can be represented as a system of equations. The…
The document asks how wealth changes when each repeated bet either increases the current balance by 50% or reduces it by 50%, with equal probability. It highlights the difference between calculating expected wealth and describing the path followed by a…
The document examines how volatility affects the risk-neutral probability that an option finishes in the money under Black-Scholes. For a call, that probability is represented by the cumulative normal value at d2, so its response to volatility depends on…
The document explains why a curve is not defined by the fixed or floating legs of a swap. A curve instead supplies discount factors or forecast rates, and those functions must be distinguished when valuing a cross-currency swap. For a fixed-fixed USD-INR…
The document explains the two components of a bond’s carry and roll-down over a holding period. Carry is the net income earned, such as coupon income after financing costs, and can be related to the difference between forward and spot yield. Roll-down is the…
The document distinguishes market value at risk from credit value at risk. Market VaR estimates a portfolio loss threshold over a chosen horizon and confidence level using market risk factors such as rates, foreign exchange, spreads, and equity prices.…
The document explains risk-neutral pricing through a one-period binomial model with a stock and a risk-free account. When the risk-free growth factor lies between the stock's down and up factors, the model rules out simple arbitrage and defines risk-neutral…
The document considers whether a portfolio objective should combine several performance and risk measures, such as multiplying Sharpe and Sortino ratios or dividing multiple return measures by multiple risk measures. The response cautions that an objective…
The discussion asks where to find practice problems for market risk, with particular interest in vanilla and exotic options. Responses suggest using free FRM-style exercises and an exam with explanations, consulting Carol Alexander's Market Risk Analysis,…
The discussion distinguishes two sets of normative principles that apply to different objects. Von Neumann–Morgenstern axioms describe rational preferences over choices and gambles, including completeness, transitivity, continuity, and independence. Coherent…
The discussion explains what principal component analysis does when applied to asset returns: it forms orthogonal components as linear combinations of the original assets. Keeping fewer components can simplify covariance calculations and represent correlated…
This discussion collects risk management topics relevant to automated trading, with particular attention to high-frequency systems. It names drawdowns, runs of losses, empirical loss distributions, confidence intervals, value at risk, slippage, liquidity,…
The document asks how a martingale should appear in a simulation of a Cox–Ross–Rubinstein stock-price model. It proposes checking whether the expected ratio of successive prices is one, then wonders whether plotting prices through time would reveal the…
The document raises a question about how to express an FX option’s delta when the trading book uses an accounting currency different from either currency in the option. Its example is a one-year at-the-money EUR call and USD put, with option and rate inputs…
The document considers choosing weights for two stocks to minimize portfolio variance while targeting a beta of one. It supplies expected returns, volatilities, a correlation, and betas measured against an index, and shows that the covariance information can…
The document outlines a factor-based approach to estimating one-day value at risk for a currency portfolio containing forwards. A forward’s value is affected by its exchange rate and the interest rates of both currencies. For a USD/JPY forward, for example,…
This discussion considers how to measure a bond portfolio's sensitivity to a yield shift in one currency when the holdings are denominated in several currencies. A weighted average of bond modified durations describes sensitivity to a simultaneous parallel…
This note outlines the factors said to inform the currencies and weights in the International Monetary Fund's Special Drawing Rights basket. The answer describes the SDR primarily as a unit of account for the IMF and related institutions, rather than as a…
This discussion explains the CAPM intuition for why an asset with negative market beta may have a lower expected return than the risk-free rate, even if its standalone volatility matches that of a positive-beta asset. CAPM relates expected return to…
This note explains why a Bloomberg VCUB option matrix may show different values for strike and straddle views. The distinction is the displayed measure: the strike view presents volatility, while the straddle view presents a premium. A premium-based straddle…
The note proposes a set of indicators for monitoring foreign exchange risk at different horizons. For short-term market stress, it recommends tracking implied-volatility indices across equity, bond, and FX markets, alongside funding and credit signals such…
The note addresses whether days with zero observed stock returns should be removed before calculating excess returns in a CAPM analysis. Its main guidance is to first determine whether the zeros represent genuine market observations or data errors. Erroneous…
The note shows how to use the Gaussian shift theorem to evaluate an expectation involving two correlated standard normal variables. It first states the theorem: an exponential tilt inside an expectation can be converted into a shifted argument of the…