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Quantitative Finance · Glosario

¿Qué es Moving-block bootstrap, stationary bootstrap?

También llamado: moving-block bootstrap · stationary bootstrap

Definition 13.3 Quantitative Methods · Capítulo 13 — Resampling

The moving-block bootstrap (Künsch, 1989) builds a resample by concatenating blocks of $b$ consecutive observations, $X_{j+1}, \dots, X_{j+b}$ , with starting points $j$ drawn uniformly (indices taken modulo $n$ in the circular version). The stationary bootstrap (Politis and Romano, 1994) draws the block lengths independently from a geometric law of mean $b$ : each resampled day continues the current block with probability $1 - 1/b$ and starts a new block at a uniform position with probability $1/b$ .

Bootstrap distributions of the volatility seller’s five-year Sharpe ratio (sample value 1.11, dashed): resampling days independently (shaded, standard error 0.53) and resampling random blocks of mean length 18.6 days (standard error 0.83). Data: the chapter’s tutorial, 4 000 resamples, seeded. — Figure 13.1. Bootstrap distributions of the volatility seller’s five-year Sharpe ratio (sample value 1.11, dashed): resampling days independently (shaded, standard error 0.53) and resampling random blocks of mean length 18.6 days (standard error 0.83). Data: the chapter’s tutorial, 4 000 resamples, seeded.

The stationary bootstrap’s standard error of the five-year Sharpe ratio against the mean block length, averaged over sixty simulated histories of the volatility seller, and the true standard deviation (1.10, from 4 000 histories). One-day blocks are the iid bootstrap (0.59); the plateau near 0.97 is the most any block length recovers. Data: the chapter’s tutorial, seeded. — Figure 13.2. The stationary bootstrap’s standard error of the five-year Sharpe ratio against the mean block length, averaged over sixty simulated histories of the volatility seller, and the true standard deviation (1.10, from 4 000 histories). One-day blocks are the iid bootstrap (0.59); the plateau near 0.97 is the most any block length recovers. Data: the chapter’s tutorial, seeded.

Ejemplos

Example 13.5 (The volatility seller)

Each day the strategy earns a premium and pays the day’s squared return, $x_t = \mu + 1 - r_t^2$ , with $r_t = \sigma_t\varepsilon_t$ and $\ln\sigma_t$ an AR(1) with coefficient 0.9 and stationary standard deviation 0.4, scaled so that $\E[\sigma_t^2] = 1$ . The market returns $r_t$ are uncorrelated, but the squared returns are not: $\Cov(r_t^2, r_{t+k}^2) = e^{0.64 \cdot 0.9^k} - 1$ , and the P&L’s long-run variance is 3.9 times its variance. The premium $\mu$ is set for a population Sharpe ratio of 1.1. The seeded five-year history has a Sharpe ratio of 1.11, a skewness of $-4.7$ , a kurtosis of 34, and autocorrelations of 0.13 at one day, 0.10 at five, $-0.02$ at twenty.

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