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1 Markets I: The Ecosystem and Exchange-Traded Marketsالأسواق عبر الإنترنت 2 Markets II: Rates, FX and Creditالأسواق عبر الإنترنت 3 Markets III: Commodities, Energy and Cryptoالأسواق عبر الإنترنت 4 Quantitative Methodsالأساليب عبر الإنترنت 5 Derivatives and Volatilityالمشتقات عبر الإنترنت 6 Rates, Credit, XVA and Riskالفائدة والائتمان والمخاطر عبر الإنترنت 7 Research Craft: Predictors, Backtests, Measurement, Portfoliosالبحث عبر الإنترنت 8 Strategies I: Equities and Futuresالاستراتيجيات عبر الإنترنت 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desksالاستراتيجيات عبر الإنترنت 10 Microstructure and Executionالتنفيذ عبر الإنترنت 11 Market Making and High-Frequency Tradingصناعة السوق عبر الإنترنت 12 Machine Learning for Marketsتعلم الآلة عبر الإنترنت 13 Low-Latency Softwareالتكنولوجيا عبر الإنترنت 14 Networks, Hardware and Trading Infrastructureالتكنولوجيا عبر الإنترنت 15 Research, Data and Risk Platformsالتكنولوجيا عبر الإنترنت 16 The Desk and the Firmالشركة عبر الإنترنت 17 The Industry: Firms, Roles and Careersالمسارات المهنية عبر الإنترنت 18 The Interview Bookالمسارات المهنية عبر الإنترنت
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Quantitative Finance · المسرد

ما معنى Square-root process, Feller condition؟

يُعرف أيضًا باسم: square-root process · Feller condition

Definition 4.7 Quantitative Methods · الفصل 4 — Stochastic Differential Equations

The square-root process is dv=κ(vˉ−v) dt+ηv dWdv = \kappa(\bar v - v)\,dt + \eta\sqrt v\,dW with κ,vˉ,η>0\kappa, \bar v, \eta > 0 and v0>0v_0 > 0. The Feller condition is 2κvˉ≥η22\kappa\bar v \ge \eta^2; the ratio 2κvˉ/η22\kappa\bar v/\eta^2 is the Feller ratio.

Left: three Ornstein–Uhlenbeck paths from 0.5 with = 2 (half-life 0.35 years), x = 0, = 0.2, and their expected path 0.5e-2t (dashed). Right: three paths of the desk’s square-root process (= 2, v = 0.04, = 0.6), sampled exactly each day; with a Feller ratio of 0.44 they spend long spells near zero. Data: the chapter’s tutorial, seeded.
Figure 4.1. Left: three Ornstein–Uhlenbeck paths from 0.5 with κ=2\kappa = 2 (half-life 0.35 years), xˉ=0\bar x = 0, σ=0.2\sigma = 0.2, and their expected path 0.5e−2t0.5e^{-2t} (dashed). Right: three paths of the desk’s square-root process (κ=2\kappa = 2, vˉ=0.04\bar v = 0.04, η=0.6\eta = 0.6), sampled exactly each day; with a Feller ratio of 0.44 they spend long spells near zero. Data: the chapter’s tutorial, seeded.
Share of plain-Euler paths of the square-root process that produce a negative variance within one year, against the Feller ratio (vol-of-vol from 0.2 to 1.0). Refining the step does not remove the failures; exact and full-truncation schemes have none. Data: 20 000 and 10 000 seeded paths per point.
Figure 4.2. Share of plain-Euler paths of the square-root process that produce a negative variance within one year, against the Feller ratio (vol-of-vol η\eta from 0.2 to 1.0). Refining the step does not remove the failures; exact and full-truncation schemes have none. Data: 20 000 and 10 000 seeded paths per point.
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