37  Price Limits and Volatility

Note

In this chapter, we examine how Vietnam’s daily price limit regime distorts observed return distributions, biases volatility estimates, and affects the validity of standard asset pricing tests. We develop corrections that allow researchers to work with censored returns and present volatility estimation methods robust to price limits.

Vietnam is one of a handful of active equity markets that still enforce daily price limits on individual stocks. HOSE imposes a \(\pm\) 7% limit, HNX imposes \(\pm\) 10%, and UPCoM imposes \(\pm\) 15%, each measured relative to the prior day’s closing (or reference) price. When a stock’s equilibrium price change exceeds the limit, the observed return is censored at the boundary. The stock closes at the limit price, but the unobserved “true” return—the price change that would have occurred without the constraint—remains unknown.

This censoring has pervasive consequences for empirical finance. Return distributions are truncated, biasing mean and variance estimates. Volatility models that ignore censoring understate true risk. Factor betas are attenuated. Event study abnormal returns are compressed. Bid-ask spread estimators that rely on return serial correlation are distorted. Any researcher working with Vietnamese equity data must understand these effects and either correct for them or demonstrate that they do not materially affect conclusions.

Price limits exist for a stated policy purpose: to prevent panic selling and speculative excess, thereby “cooling” the market during periods of stress (Brennan 1986). Whether they achieve this objective—or merely delay price discovery and create magnet effects—is an empirical question with a large international literature and no consensus. We examine the Vietnamese evidence.

37.1 The Vietnamese Price Limit Regime

37.1.1 Institutional Details

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import statsmodels.api as sm
from scipy import stats, optimize
from arch import arch_model
import warnings
warnings.filterwarnings('ignore')

plt.rcParams.update({
    'figure.figsize': (12, 6),
    'figure.dpi': 150,
    'font.size': 11,
    'axes.spines.top': False,
    'axes.spines.right': False
})

The price limit structure has evolved over time. HOSE began trading in July 2000 with a \(\pm\) 2% limit, which was widened to \(\pm\) 5% in 2002 and to \(\pm\) 7% in 2013. HNX has operated at \(\pm\) 10% since its current form, and UPCoM at \(\pm\) 15% Table 37.1. The limits apply to the adjusted closing price relative to the reference price (typically the prior day’s close, adjusted for corporate actions).

Table 37.1: Vietnamese daily price limit regime by exchange.

Exchange	Current Limit	Effective Date	Prior Limits
HOSE	\(\pm\) 7%	June 2013	\(\pm\) 2% (2000), \(\pm\) 5% (2002)
HNX	\(\pm\) 10%	—	Various, stabilized at \(\pm\) 10%
UPCoM	\(\pm\) 15%	—	Wider limits reflecting OTC nature

Importantly, the limits are asymmetric in practice: they apply equally to up and down moves, but the economic consequences differ. A stock hitting the upper limit prevents buyers from bidding higher (excess demand persists), while hitting the lower limit prevents sellers from offering lower (excess supply persists). Both create unfilled orders that spill over to subsequent trading days.

from datacore import DataCoreClient

client = DataCoreClient()

# Daily data with high, low, open, close, volume, and limit indicators
daily = client.get_daily_prices(
    exchanges=['HOSE', 'HNX', 'UPCoM'],
    start_date='2008-01-01',
    end_date='2024-12-31',
    include_delisted=True,
    fields=[
        'ticker', 'date', 'exchange',
        'open', 'high', 'low', 'close', 'adjusted_close',
        'reference_price', 'ceiling_price', 'floor_price',
        'volume', 'turnover_value',
        'limit_up_hit', 'limit_down_hit'
    ]
)

daily['date'] = pd.to_datetime(daily['date'])
daily = daily.sort_values(['ticker', 'date'])

# Compute daily returns
daily['daily_return'] = daily.groupby('ticker')['adjusted_close'].pct_change()

# Flag limit hits from price data if not provided
if 'limit_up_hit' not in daily.columns or daily['limit_up_hit'].isna().all():
    daily['limit_up_hit'] = (daily['close'] >= daily['ceiling_price'])
    daily['limit_down_hit'] = (daily['close'] <= daily['floor_price'])

# Exchange-specific limits
exchange_limits = {'HOSE': 0.07, 'HNX': 0.10, 'UPCoM': 0.15}
daily['limit_pct'] = daily['exchange'].map(exchange_limits)

print(f"Daily observations: {len(daily):,}")
print(f"Date range: {daily['date'].min()} to {daily['date'].max()}")
print(f"Unique tickers: {daily['ticker'].nunique()}")

37.2 Prevalence of Limit Hits

37.2.1 Aggregate Frequency

How often do Vietnamese stocks hit their price limits? The answer varies dramatically by exchange, market capitalization, and market conditions.

# Overall frequencies
limit_stats = daily.groupby('exchange').agg(
    n_obs=('daily_return', 'count'),
    n_up=('limit_up_hit', 'sum'),
    n_down=('limit_down_hit', 'sum'),
).assign(
    pct_up=lambda x: x['n_up'] / x['n_obs'] * 100,
    pct_down=lambda x: x['n_down'] / x['n_obs'] * 100,
    pct_either=lambda x: (x['n_up'] + x['n_down']) / x['n_obs'] * 100
)

print("Limit Hit Frequencies by Exchange:")
print(limit_stats[['pct_up', 'pct_down', 'pct_either']].round(2).to_string())

# Monthly aggregate: fraction of stock-days hitting limits
daily['year_month'] = daily['date'].dt.to_period('M')
monthly_limit = (
    daily.groupby(['year_month', 'exchange'])
    .agg(
        n_obs=('daily_return', 'count'),
        n_up=('limit_up_hit', 'sum'),
        n_down=('limit_down_hit', 'sum')
    )
    .assign(
        pct_up=lambda x: x['n_up'] / x['n_obs'] * 100,
        pct_down=lambda x: x['n_down'] / x['n_obs'] * 100,
        pct_any=lambda x: (x['n_up'] + x['n_down']) / x['n_obs'] * 100
    )
    .reset_index()
)
monthly_limit['date'] = monthly_limit['year_month'].dt.to_timestamp()

fig, axes = plt.subplots(2, 1, figsize=(14, 8), sharex=True,
                          gridspec_kw={'height_ratios': [3, 1]})

hose_monthly = monthly_limit[monthly_limit['exchange'] == 'HOSE']

axes[0].fill_between(hose_monthly['date'], 0, hose_monthly['pct_up'],
                      color='#27AE60', alpha=0.6, label='Upper limit hits')
axes[0].fill_between(hose_monthly['date'], 0, -hose_monthly['pct_down'],
                      color='#C0392B', alpha=0.6, label='Lower limit hits')
axes[0].axhline(y=0, color='black', linewidth=0.5)
axes[0].set_ylabel('% of Stock-Days')
axes[0].set_title('Panel A: HOSE Daily Price Limit Hits')
axes[0].legend(loc='upper left')

# VN-Index for context
vnindex = client.get_index_returns(
    index='VNINDEX', start_date='2008-01-01', end_date='2024-12-31',
    frequency='monthly'
)
vnindex['date'] = pd.to_datetime(vnindex['date'])
axes[1].bar(vnindex['date'], vnindex['return'] * 100,
            color=np.where(vnindex['return'] > 0, '#27AE60', '#C0392B'),
            width=25, alpha=0.7)
axes[1].set_ylabel('VN-Index (%)')
axes[1].set_title('Panel B: VN-Index Monthly Returns')

plt.tight_layout()
plt.show()

Figure 37.1

37.2.2 By Market Capitalization

# Merge with lagged market cap
monthly_mcap = client.get_monthly_returns(
    exchanges=['HOSE'],
    start_date='2008-01-01',
    end_date='2024-12-31',
    fields=['ticker', 'month_end', 'market_cap']
)
monthly_mcap['month_end'] = pd.to_datetime(monthly_mcap['month_end'])

# Assign size quintiles each month
monthly_mcap['size_quintile'] = (
    monthly_mcap.groupby('month_end')['market_cap']
    .transform(lambda x: pd.qcut(x.rank(method='first'), 5,
                                   labels=['Q1\n(Small)', 'Q2', 'Q3', 'Q4',
                                           'Q5\n(Big)']))
)

# Map to daily
daily_hose = daily[daily['exchange'] == 'HOSE'].copy()
daily_hose['month_end'] = daily_hose['date'].dt.to_period('M').dt.to_timestamp('M')
daily_hose = daily_hose.merge(
    monthly_mcap[['ticker', 'month_end', 'size_quintile']],
    on=['ticker', 'month_end'], how='left'
)

size_limit = (
    daily_hose.dropna(subset=['size_quintile'])
    .groupby('size_quintile')
    .agg(
        pct_up=('limit_up_hit', 'mean'),
        pct_down=('limit_down_hit', 'mean'),
        n=('daily_return', 'count')
    )
)
size_limit[['pct_up', 'pct_down']] *= 100

fig, ax = plt.subplots(figsize=(10, 5))

x = np.arange(len(size_limit))
width = 0.35
ax.bar(x - width / 2, size_limit['pct_up'], width,
       color='#27AE60', alpha=0.85, label='Upper limit', edgecolor='white')
ax.bar(x + width / 2, size_limit['pct_down'], width,
       color='#C0392B', alpha=0.85, label='Lower limit', edgecolor='white')

ax.set_xticks(x)
ax.set_xticklabels(size_limit.index)
ax.set_ylabel('% of Stock-Days')
ax.set_title('Price Limit Hit Frequency by Size Quintile (HOSE)')
ax.legend()

plt.tight_layout()
plt.show()

Figure 37.2

37.2.3 Consecutive Limit Days

A single limit hit might simply reflect a large information event that is absorbed within one day. Consecutive limit hits in the same direction are more problematic because they indicate that the limit is actively preventing price discovery over multiple days.

def count_consecutive_limits(group):
    """Count consecutive limit-up and limit-down sequences."""
    up_runs = []
    down_runs = []
    
    up_count = 0
    down_count = 0
    
    for _, row in group.iterrows():
        if row['limit_up_hit']:
            up_count += 1
            if down_count > 0:
                down_runs.append(down_count)
                down_count = 0
        elif row['limit_down_hit']:
            down_count += 1
            if up_count > 0:
                up_runs.append(up_count)
                up_count = 0
        else:
            if up_count > 0:
                up_runs.append(up_count)
            if down_count > 0:
                down_runs.append(down_count)
            up_count = 0
            down_count = 0
    
    if up_count > 0:
        up_runs.append(up_count)
    if down_count > 0:
        down_runs.append(down_count)
    
    return up_runs, down_runs

# Sample: compute for HOSE stocks
hose_tickers = daily_hose['ticker'].unique()
all_up_runs = []
all_down_runs = []

for ticker in hose_tickers:
    group = daily_hose[daily_hose['ticker'] == ticker].sort_values('date')
    up_runs, down_runs = count_consecutive_limits(group)
    all_up_runs.extend(up_runs)
    all_down_runs.extend(down_runs)

print("Consecutive Limit Hit Distribution (HOSE):")
for direction, runs in [('Upper', all_up_runs), ('Lower', all_down_runs)]:
    if not runs:
        continue
    runs_series = pd.Series(runs)
    print(f"\n  {direction} limit sequences:")
    print(f"    Total sequences: {len(runs_series):,}")
    print(f"    1 day:  {(runs_series == 1).sum():,} ({(runs_series == 1).mean():.1%})")
    print(f"    2 days: {(runs_series == 2).sum():,} ({(runs_series == 2).mean():.1%})")
    print(f"    3 days: {(runs_series == 3).sum():,} ({(runs_series == 3).mean():.1%})")
    print(f"    4+ days: {(runs_series >= 4).sum():,} ({(runs_series >= 4).mean():.1%})")
    print(f"    Max consecutive: {runs_series.max()}")

37.3 Return Distribution Distortion

37.3.1 Censoring Mechanics

Price limits create Type I censoring (also called “truncation at a known point”): the latent (unobserved) return \(r^*\) is generated from some continuous distribution, but the observed return is:

\[ r^{\text{obs}} = \begin{cases} \bar{L} & \text{if } r^* \geq \bar{L} \quad \text{(upper limit hit)} \\ r^* & \text{if } \underline{L} < r^* < \bar{L} \quad \text{(interior)} \\ \underline{L} & \text{if } r^* \leq \underline{L} \quad \text{(lower limit hit)} \end{cases} \tag{37.1}\]

where \(\bar{L}\) and \(\underline{L}\) are the upper and lower limits. For HOSE, \(\bar{L} = +0.07\) and \(\underline{L} = -0.07\).

The censoring has predictable effects on the observed distribution:

1. Mean bias. If the uncensored distribution is symmetric, censoring from both sides preserves the mean approximately. But if the distribution is skewed (as stock returns are, with negative skewness), the bias can go either way.
2. Variance underestimation. Censoring always reduces the observed variance relative to the true variance, because extreme returns are compressed to the limit values.
3. Kurtosis distortion. Probability mass piles up at the limit values, creating spikes in the distribution.

hose_returns = daily_hose['daily_return'].dropna()
hose_returns = hose_returns[hose_returns.abs() < 0.15]  # Remove data errors

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Panel A: Full distribution
axes[0].hist(hose_returns, bins=200, density=True,
             color='#2C5F8A', alpha=0.7, edgecolor='none')
axes[0].axvline(x=0.07, color='#C0392B', linewidth=2, linestyle='--',
                label='$\pm$ 7% limit')
axes[0].axvline(x=-0.07, color='#C0392B', linewidth=2, linestyle='--')
axes[0].set_xlabel('Daily Return')
axes[0].set_ylabel('Density')
axes[0].set_title('Panel A: HOSE Daily Return Distribution')
axes[0].legend()

# Panel B: Zoom on tails
bins_tail = np.linspace(-0.09, -0.05, 40)
bins_tail_up = np.linspace(0.05, 0.09, 40)

axes[1].hist(hose_returns[hose_returns < -0.04], bins=80, density=True,
             color='#C0392B', alpha=0.6, label='Left tail')
axes[1].hist(hose_returns[hose_returns > 0.04], bins=80, density=True,
             color='#27AE60', alpha=0.6, label='Right tail')
axes[1].axvline(x=0.07, color='black', linewidth=2)
axes[1].axvline(x=-0.07, color='black', linewidth=2)
axes[1].set_xlabel('Daily Return')
axes[1].set_ylabel('Density')
axes[1].set_title('Panel B: Tail Behavior at Limits')
axes[1].legend()

plt.tight_layout()
plt.show()

# Quantify the spike
n_at_upper = ((hose_returns >= 0.069) & (hose_returns <= 0.071)).sum()
n_at_lower = ((hose_returns >= -0.071) & (hose_returns <= -0.069)).sum()
n_total = len(hose_returns)
print(f"Observations at upper limit ($\pm$ 0.1% of 7%): {n_at_upper:,} "
      f"({n_at_upper/n_total:.2%})")
print(f"Observations at lower limit: {n_at_lower:,} "
      f"({n_at_lower/n_total:.2%})")

Figure 37.3

37.3.2 Comparing HOSE vs. HNX vs. UPCoM

The three Vietnamese exchanges have different limit widths, creating a natural experiment: if limits distort the distribution, wider limits should produce distributions closer to the uncensored benchmark.

fig, axes = plt.subplots(1, 3, figsize=(16, 4.5))

for i, (exchange, limit, color) in enumerate([
    ('HOSE', 0.07, '#2C5F8A'),
    ('HNX', 0.10, '#C0392B'),
    ('UPCoM', 0.15, '#27AE60')
]):
    rets = daily[daily['exchange'] == exchange]['daily_return'].dropna()
    rets = rets[rets.abs() < limit + 0.05]
    
    axes[i].hist(rets, bins=150, density=True,
                  color=color, alpha=0.7, edgecolor='none')
    axes[i].axvline(x=limit, color='black', linewidth=1.5, linestyle='--')
    axes[i].axvline(x=-limit, color='black', linewidth=1.5, linestyle='--')
    axes[i].set_title(f'{exchange} ($\pm$ {limit*100:.0f}%)')
    axes[i].set_xlabel('Daily Return')
    if i == 0:
        axes[i].set_ylabel('Density')
    
    # Stats
    pct_at_limit = ((rets.abs() >= limit - 0.001).sum() / len(rets) * 100)
    axes[i].text(0.95, 0.95, f'At limit: {pct_at_limit:.2f}%',
                  transform=axes[i].transAxes, ha='right', va='top',
                  fontsize=9, bbox=dict(boxstyle='round', facecolor='white',
                                         alpha=0.8))

plt.suptitle('Return Distributions by Exchange', fontsize=13)
plt.tight_layout()
plt.show()

Figure 37.4

37.4 Variance Bias from Censoring

37.4.1 Analytical Bias

If the true return follows \(r^* \sim N(\mu, \sigma^2)\), the variance of the censored return can be derived analytically. Let \(a = (\underline{L} - \mu) / \sigma\) and \(b = (\bar{L} - \mu) / \sigma\):

\[ \text{Var}(r^{\text{obs}}) = \sigma^2 \left[1 - \frac{b \phi(b) - a \phi(a)}{\Phi(b) - \Phi(a)} - \left(\frac{\phi(a) - \phi(b)}{\Phi(b) - \Phi(a)}\right)^2 \right] + \text{boundary terms} \tag{37.2}\]

where \(\phi\) and \(\Phi\) are the standard normal PDF and CDF. The key result is that \(\text{Var}(r^{\text{obs}}) < \sigma^2\) always—censoring systematically underestimates variance.

def simulate_censored_variance(true_sigma, limit, n_sim=100000, mu=0):
    """Simulate observed vs true variance under censoring."""
    rng = np.random.default_rng(42)
    r_star = rng.normal(mu, true_sigma, n_sim)
    r_obs = np.clip(r_star, -limit, limit)
    
    var_true = np.var(r_star)
    var_obs = np.var(r_obs)
    
    pct_censored = ((r_star >= limit) | (r_star <= -limit)).mean()
    
    return {
        'true_sigma': true_sigma,
        'true_var': var_true,
        'obs_var': var_obs,
        'var_ratio': var_obs / var_true,
        'bias_pct': (1 - var_obs / var_true) * 100,
        'pct_censored': pct_censored * 100
    }

# Sweep across volatility levels for each exchange limit
results_bias = []
sigmas = np.linspace(0.005, 0.08, 50)

for limit_name, limit in [('HOSE $\pm$ 7%', 0.07), ('HNX $\pm$ 10%', 0.10),
                            ('UPCoM $\pm$ 15%', 0.15)]:
    for sigma in sigmas:
        res = simulate_censored_variance(sigma, limit)
        res['exchange'] = limit_name
        results_bias.append(res)

bias_df = pd.DataFrame(results_bias)

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

colors_exch = {'HOSE $\pm$ 7%': '#2C5F8A', 'HNX $\pm$ 10%': '#C0392B',
                'UPCoM $\pm$ 15%': '#27AE60'}

for exch in colors_exch:
    subset = bias_df[bias_df['exchange'] == exch]
    axes[0].plot(subset['true_sigma'] * 100, subset['var_ratio'],
                  color=colors_exch[exch], linewidth=2, label=exch)

axes[0].axhline(y=1, color='gray', linewidth=0.5, linestyle='--')
axes[0].set_xlabel('True Daily Volatility (%)')
axes[0].set_ylabel('Observed / True Variance')
axes[0].set_title('Panel A: Variance Ratio')
axes[0].legend()
axes[0].set_ylim([0.5, 1.05])

for exch in colors_exch:
    subset = bias_df[bias_df['exchange'] == exch]
    axes[1].plot(subset['true_sigma'] * 100, subset['pct_censored'],
                  color=colors_exch[exch], linewidth=2, label=exch)

axes[1].set_xlabel('True Daily Volatility (%)')
axes[1].set_ylabel('% of Returns Censored')
axes[1].set_title('Panel B: Censoring Rate')
axes[1].legend()

plt.tight_layout()
plt.show()

Figure 37.5

37.4.2 Empirical Variance Bias by Size

# Cross-listed stocks or transfer events provide a natural experiment:
# Same stock, different limit regime
# Alternative: compare variance of HOSE returns to variance of the same
# stock's returns implied from intraday data (not censored by closing limit)

# Approach: Tobit-based variance estimation
# Model observed returns as censored normal
def tobit_variance(returns, limit):
    """
    Estimate true variance via Tobit MLE under censored normal.
    """
    r = returns.dropna().values
    upper = limit
    lower = -limit
    
    # Classify observations
    at_upper = r >= (upper - 1e-6)
    at_lower = r <= (lower + 1e-6)
    interior = ~at_upper & ~at_lower
    
    if interior.sum() < 20:
        return np.nan, np.nan
    
    def neg_loglik(params):
        mu, log_sigma = params
        sigma = np.exp(log_sigma)
        
        ll = 0
        # Interior observations
        if interior.sum() > 0:
            ll += np.sum(stats.norm.logpdf(r[interior], mu, sigma))
        # Upper censored
        if at_upper.sum() > 0:
            ll += np.sum(np.log(1 - stats.norm.cdf(upper, mu, sigma) + 1e-15))
        # Lower censored
        if at_lower.sum() > 0:
            ll += np.sum(np.log(stats.norm.cdf(lower, mu, sigma) + 1e-15))
        
        return -ll
    
    # Initial values
    mu0 = r[interior].mean() if interior.sum() > 0 else 0
    sigma0 = r[interior].std() if interior.sum() > 0 else r.std()
    
    try:
        result = optimize.minimize(
            neg_loglik, [mu0, np.log(max(sigma0, 1e-6))],
            method='Nelder-Mead', options={'maxiter': 5000}
        )
        mu_hat = result.x[0]
        sigma_hat = np.exp(result.x[1])
        return mu_hat, sigma_hat
    except Exception:
        return np.nan, np.nan

# Estimate for each HOSE stock
hose_stocks = daily_hose.groupby('ticker').filter(
    lambda x: len(x) >= 250
)['ticker'].unique()

tobit_results = []
for ticker in hose_stocks[:500]:  # Sample for speed
    rets = daily_hose[daily_hose['ticker'] == ticker]['daily_return'].dropna()
    if len(rets) < 250:
        continue
    
    naive_sigma = rets.std()
    mu_hat, sigma_hat = tobit_variance(rets, 0.07)
    
    if np.isfinite(sigma_hat) and sigma_hat > 0:
        tobit_results.append({
            'ticker': ticker,
            'naive_sigma': naive_sigma,
            'tobit_sigma': sigma_hat,
            'bias_pct': (sigma_hat - naive_sigma) / naive_sigma * 100
        })

tobit_df = pd.DataFrame(tobit_results)

print("Tobit vs Naive Volatility Estimation (HOSE):")
print(f"  Mean naive σ:  {tobit_df['naive_sigma'].mean():.4f}")
print(f"  Mean Tobit σ:  {tobit_df['tobit_sigma'].mean():.4f}")
print(f"  Mean bias:     {tobit_df['bias_pct'].mean():.1f}%")
print(f"  Median bias:   {tobit_df['bias_pct'].median():.1f}%")
print(f"  Max bias:      {tobit_df['bias_pct'].max():.1f}%")

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

axes[0].scatter(tobit_df['naive_sigma'] * 100,
                 tobit_df['tobit_sigma'] * 100,
                 s=15, alpha=0.5, color='#2C5F8A', edgecolors='none')
lim = max(tobit_df['tobit_sigma'].max(), tobit_df['naive_sigma'].max()) * 100 + 0.5
axes[0].plot([0, lim], [0, lim], 'k--', linewidth=1)
axes[0].set_xlabel('Naive σ (% daily)')
axes[0].set_ylabel('Tobit σ (% daily)')
axes[0].set_title('Panel A: Tobit vs Naive Volatility')

axes[1].hist(tobit_df['bias_pct'], bins=50, color='#C0392B',
             alpha=0.7, edgecolor='white', density=True)
axes[1].axvline(x=0, color='black', linewidth=1)
axes[1].axvline(x=tobit_df['bias_pct'].median(), color='#2C5F8A',
                linewidth=2, linestyle='--',
                label=f"Median: {tobit_df['bias_pct'].median():.1f}%")
axes[1].set_xlabel('Bias (%): (Tobit - Naive) / Naive')
axes[1].set_ylabel('Density')
axes[1].set_title('Panel B: Distribution of Correction')
axes[1].legend()

plt.tight_layout()
plt.show()

Figure 37.6

37.5 Volatility Estimation Under Price Limits

37.5.1 Range-Based Estimators

Range-based volatility estimators use the daily high and low prices rather than close-to-close returns, making them partially robust to closing-price censoring (since intraday prices may approach but not be censored at the same points). However, they are biased when the intraday price trajectory itself is constrained by the limits.

def parkinson_vol(high, low, n_periods=20):
    """
    Parkinson (1980) range-based volatility estimator.
    σ² = (1/4ln2) * E[(ln(H/L))²]
    """
    log_hl = np.log(high / low)
    var = (1 / (4 * np.log(2))) * (log_hl ** 2)
    return np.sqrt(var.rolling(n_periods).mean())

def garman_klass_vol(open_p, high, low, close, n_periods=20):
    """
    Garman-Klass (1980) OHLC volatility estimator.
    More efficient than Parkinson by using open and close.
    """
    log_hl = np.log(high / low)
    log_co = np.log(close / open_p)
    var = 0.5 * log_hl ** 2 - (2 * np.log(2) - 1) * log_co ** 2
    return np.sqrt(var.rolling(n_periods).mean())

def yang_zhang_vol(open_p, high, low, close, n_periods=20):
    """
    Yang-Zhang (2000) drift-independent estimator.
    Combines overnight, Rogers-Satchell, and open-to-close components.
    """
    log_oc = np.log(open_p / close.shift(1))  # Overnight
    log_co = np.log(close / open_p)
    log_ho = np.log(high / open_p)
    log_lo = np.log(low / open_p)
    
    # Rogers-Satchell component
    rs = log_ho * (log_ho - log_co) + log_lo * (log_lo - log_co)
    
    k = 0.34 / (1.34 + (n_periods + 1) / (n_periods - 1))
    
    var_overnight = log_oc.rolling(n_periods).var()
    var_open_close = log_co.rolling(n_periods).var()
    var_rs = rs.rolling(n_periods).mean()
    
    var = var_overnight + k * var_open_close + (1 - k) * var_rs
    return np.sqrt(var.clip(lower=0))

# Compute for HOSE sample
sample_ticker = 'VNM'  # Large, liquid stock
sample = daily_hose[daily_hose['ticker'] == sample_ticker].copy()
sample = sample.sort_values('date').set_index('date')

# Close-to-close realized vol
sample['cc_vol'] = sample['daily_return'].rolling(20).std() * np.sqrt(252)

# Range-based
sample['parkinson'] = parkinson_vol(
    sample['high'], sample['low'], 20
) * np.sqrt(252)

sample['garman_klass'] = garman_klass_vol(
    sample['open'], sample['high'], sample['low'], sample['close'], 20
) * np.sqrt(252)

sample['yang_zhang'] = yang_zhang_vol(
    sample['open'], sample['high'], sample['low'], sample['close'], 20
) * np.sqrt(252)

fig, ax = plt.subplots(figsize=(14, 5))

ax.plot(sample.index, sample['cc_vol'], color='#BDC3C7',
        linewidth=1, label='Close-to-Close', alpha=0.8)
ax.plot(sample.index, sample['parkinson'], color='#2C5F8A',
        linewidth=1.5, label='Parkinson')
ax.plot(sample.index, sample['yang_zhang'], color='#C0392B',
        linewidth=1.5, label='Yang-Zhang')

ax.set_ylabel('Annualized Volatility')
ax.set_title(f'Volatility Estimators: {sample_ticker}')
ax.legend(ncol=3)
ax.set_ylim([0, ax.get_ylim()[1]])

plt.tight_layout()
plt.show()

Figure 37.7

37.5.2 GARCH Models with Censored Returns

Standard GARCH models assume returns are fully observed. When returns are censored, the log-likelihood must account for the probability mass at the limit values. We implement a censored GARCH(1,1):

\[ r_t = \mu + \varepsilon_t, \quad \varepsilon_t = \sigma_t z_t, \quad z_t \sim N(0, 1) \tag{37.3}\]

\[ \sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \tag{37.4}\]

The censored log-likelihood replaces the standard normal density for limit-hit observations:

\[ \ell_t = \begin{cases} \log \phi\left(\frac{r_t - \mu}{\sigma_t}\right) - \log \sigma_t & \text{if interior} \\ \log \Phi\left(\frac{\underline{L} - \mu}{\sigma_t}\right) & \text{if lower limit} \\ \log\left[1 - \Phi\left(\frac{\bar{L} - \mu}{\sigma_t}\right)\right] & \text{if upper limit} \end{cases} \tag{37.5}\]

def censored_garch11(returns, limit, max_iter=500):
    """
    GARCH(1,1) with censored normal likelihood.
    
    Parameters
    ----------
    returns : array-like
        Observed daily returns (censored at $\pm$ limit).
    limit : float
        Price limit (e.g., 0.07 for HOSE).
    
    Returns
    -------
    Dictionary with estimated parameters and conditional variances.
    """
    r = np.array(returns, dtype=float)
    T = len(r)
    upper = limit
    lower = -limit
    
    at_upper = r >= (upper - 1e-6)
    at_lower = r <= (lower + 1e-6)
    interior = ~at_upper & ~at_lower
    
    def neg_loglik(params):
        mu, omega, alpha, beta = params
        
        if omega <= 0 or alpha < 0 or beta < 0 or (alpha + beta) >= 1:
            return 1e10
        
        sigma2 = np.zeros(T)
        sigma2[0] = omega / (1 - alpha - beta) if (alpha + beta) < 1 else r.var()
        
        ll = 0
        for t in range(T):
            if t > 0:
                eps = r[t - 1] - mu
                sigma2[t] = omega + alpha * eps ** 2 + beta * sigma2[t - 1]
            
            sigma2[t] = max(sigma2[t], 1e-10)
            sigma = np.sqrt(sigma2[t])
            
            if interior[t]:
                ll += stats.norm.logpdf(r[t], mu, sigma)
            elif at_upper[t]:
                prob = 1 - stats.norm.cdf(upper, mu, sigma)
                ll += np.log(max(prob, 1e-15))
            elif at_lower[t]:
                prob = stats.norm.cdf(lower, mu, sigma)
                ll += np.log(max(prob, 1e-15))
        
        return -ll
    
    # Initial values from standard GARCH
    mu0 = r[interior].mean() if interior.any() else 0
    var0 = r[interior].var() if interior.any() else r.var()
    
    try:
        result = optimize.minimize(
            neg_loglik,
            [mu0, var0 * 0.05, 0.10, 0.85],
            method='Nelder-Mead',
            options={'maxiter': max_iter, 'xatol': 1e-8}
        )
        mu, omega, alpha, beta = result.x
        
        # Reconstruct conditional variance
        sigma2 = np.zeros(T)
        sigma2[0] = omega / max(1 - alpha - beta, 0.01)
        for t in range(1, T):
            eps = r[t - 1] - mu
            sigma2[t] = omega + alpha * eps ** 2 + beta * sigma2[t - 1]
        
        return {
            'mu': mu, 'omega': omega, 'alpha': alpha, 'beta': beta,
            'persistence': alpha + beta,
            'uncond_var': omega / max(1 - alpha - beta, 0.01),
            'sigma2': sigma2,
            'loglik': -result.fun,
            'converged': result.success,
            'n_censored': at_upper.sum() + at_lower.sum(),
            'pct_censored': (at_upper.sum() + at_lower.sum()) / T * 100
        }
    except Exception as e:
        return None

# Compare standard vs censored GARCH for a volatile stock
volatile_stock = daily_hose.groupby('ticker')['limit_up_hit'].mean()
volatile_stock = volatile_stock.sort_values(ascending=False).head(20)
test_ticker = volatile_stock.index[0]

test_returns = (
    daily_hose[daily_hose['ticker'] == test_ticker]
    .sort_values('date')['daily_return']
    .dropna()
    .values
)

# Standard GARCH (arch library)
std_garch = arch_model(test_returns * 100, vol='GARCH', p=1, q=1,
                         mean='Constant', dist='normal')
std_result = std_garch.fit(disp='off')

# Censored GARCH
cens_result = censored_garch11(test_returns, limit=0.07)

print(f"Stock: {test_ticker}")
print(f"Observations: {len(test_returns)}, "
      f"Censored: {cens_result['pct_censored']:.1f}%\n")

print(f"{'Parameter':<12} {'Standard':>12} {'Censored':>12}")
print("-" * 36)
print(f"{'μ':<12} {std_result.params['mu']/100:>12.6f} "
      f"{cens_result['mu']:>12.6f}")
print(f"{'ω':<12} {std_result.params['omega']/10000:>12.8f} "
      f"{cens_result['omega']:>12.8f}")
print(f"{'α':<12} {std_result.params['alpha[1]']:>12.4f} "
      f"{cens_result['alpha']:>12.4f}")
print(f"{'β':<12} {std_result.params['beta[1]']:>12.4f} "
      f"{cens_result['beta']:>12.4f}")
print(f"{'α+β':<12} "
      f"{std_result.params['alpha[1]']+std_result.params['beta[1]']:>12.4f} "
      f"{cens_result['persistence']:>12.4f}")
print(f"{'Uncond σ':<12} "
      f"{np.sqrt(std_result.params['omega']/(1-std_result.params['alpha[1]']-std_result.params['beta[1]'])/10000):>12.4f} "
      f"{np.sqrt(cens_result['uncond_var']):>12.4f}")

test_data = daily_hose[daily_hose['ticker'] == test_ticker].sort_values('date')
dates = test_data['date'].values[-len(test_returns):]

fig, axes = plt.subplots(2, 1, figsize=(14, 8), sharex=True,
                          gridspec_kw={'height_ratios': [1, 2]})

# Panel A: Returns with limit hits highlighted
axes[0].plot(dates, test_returns, color='#2C5F8A', linewidth=0.5, alpha=0.7)
limit_up_mask = test_returns >= 0.069
limit_down_mask = test_returns <= -0.069
axes[0].scatter(dates[limit_up_mask], test_returns[limit_up_mask],
                 color='#27AE60', s=10, zorder=3, label='Upper limit')
axes[0].scatter(dates[limit_down_mask], test_returns[limit_down_mask],
                 color='#C0392B', s=10, zorder=3, label='Lower limit')
axes[0].axhline(y=0.07, color='gray', linewidth=0.5, linestyle='--')
axes[0].axhline(y=-0.07, color='gray', linewidth=0.5, linestyle='--')
axes[0].set_ylabel('Return')
axes[0].set_title(f'Panel A: Daily Returns ({test_ticker})')
axes[0].legend(fontsize=8)

# Panel B: Conditional volatility
std_sigma = std_result.conditional_volatility / 100  # Convert from % to decimal
cens_sigma = np.sqrt(cens_result['sigma2'])

axes[1].plot(dates, std_sigma * np.sqrt(252), color='#BDC3C7',
             linewidth=1, label='Standard GARCH')
axes[1].plot(dates, cens_sigma * np.sqrt(252), color='#C0392B',
             linewidth=1.5, label='Censored GARCH')
axes[1].set_ylabel('Annualized Conditional σ')
axes[1].set_title('Panel B: Conditional Volatility')
axes[1].legend()

plt.tight_layout()
plt.show()

Figure 37.8

37.6 Effects on Asset Pricing Tests

37.6.1 Beta Attenuation

Price limits attenuate the covariance between stock returns and factor returns, biasing beta estimates toward zero. The intuition is simple: on days when the market moves 3% but a stock’s true return would have been 6%, the observed return is capped at 7%, understating the stock’s sensitivity.

# Compare betas estimated with all days vs excluding limit-hit days
# Also compare betas from HOSE stocks vs same stocks if they were on HNX

daily_hose_merged = daily_hose.merge(
    client.get_index_returns('VNINDEX', frequency='daily',
                              start_date='2008-01-01',
                              end_date='2024-12-31')[['date', 'return']],
    on='date', how='left'
).rename(columns={'return': 'mkt_return'})

# For each stock, estimate beta:
# (a) Using all days
# (b) Excluding limit-hit days
# (c) Tobit-corrected (censored regression)
beta_comparison = []

for ticker in hose_stocks[:300]:
    stock = daily_hose_merged[daily_hose_merged['ticker'] == ticker].dropna(
        subset=['daily_return', 'mkt_return']
    )
    if len(stock) < 250:
        continue
    
    # (a) All days
    X_all = sm.add_constant(stock['mkt_return'])
    model_all = sm.OLS(stock['daily_return'], X_all).fit()
    beta_all = model_all.params['mkt_return']
    
    # (b) Exclude limit-hit days
    interior = stock[~stock['limit_up_hit'] & ~stock['limit_down_hit']]
    if len(interior) < 200:
        continue
    X_int = sm.add_constant(interior['mkt_return'])
    model_int = sm.OLS(interior['daily_return'], X_int).fit()
    beta_interior = model_int.params['mkt_return']
    
    # Limit hit frequency for this stock
    pct_limit = (stock['limit_up_hit'].sum() + stock['limit_down_hit'].sum()) / len(stock) * 100
    
    beta_comparison.append({
        'ticker': ticker,
        'beta_all': beta_all,
        'beta_interior': beta_interior,
        'beta_diff_pct': (beta_interior - beta_all) / abs(beta_all) * 100,
        'pct_limit_hits': pct_limit
    })

beta_df = pd.DataFrame(beta_comparison)

print("Beta Attenuation from Price Limits:")
print(f"  Mean β (all days):      {beta_df['beta_all'].mean():.3f}")
print(f"  Mean β (interior only): {beta_df['beta_interior'].mean():.3f}")
print(f"  Mean difference:        {beta_df['beta_diff_pct'].mean():.1f}%")
print(f"  Correlation(diff, limit_freq): "
      f"{beta_df['beta_diff_pct'].corr(beta_df['pct_limit_hits']):.3f}")

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

axes[0].scatter(beta_df['beta_all'], beta_df['beta_interior'],
                 s=15, alpha=0.5, color='#2C5F8A', edgecolors='none')
lim = max(beta_df['beta_all'].abs().max(),
           beta_df['beta_interior'].abs().max()) + 0.2
axes[0].plot([-0.5, lim], [-0.5, lim], 'k--', linewidth=1)
axes[0].set_xlabel('β (all days)')
axes[0].set_ylabel('β (interior only)')
axes[0].set_title('Panel A: Beta with vs without Limit Days')

axes[1].scatter(beta_df['pct_limit_hits'], beta_df['beta_diff_pct'],
                 s=15, alpha=0.5, color='#C0392B', edgecolors='none')
# Add regression line
z = np.polyfit(beta_df['pct_limit_hits'], beta_df['beta_diff_pct'], 1)
x_line = np.linspace(0, beta_df['pct_limit_hits'].max(), 100)
axes[1].plot(x_line, np.polyval(z, x_line), 'k-', linewidth=1.5)
axes[1].axhline(y=0, color='gray', linewidth=0.5)
axes[1].set_xlabel('Limit Hit Frequency (%)')
axes[1].set_ylabel('Beta Increase When Excluding Limit Days (%)')
axes[1].set_title('Panel B: Attenuation vs Limit Frequency')

plt.tight_layout()
plt.show()

Figure 37.9

37.6.2 Effect on Factor Premia

If betas are attenuated by censoring, then cross-sectional Fama-MacBeth risk premia estimates are biased upward (because the denominator of the slope coefficient is too small). We quantify this effect:

# Monthly returns: compute with all days vs excluding limit-hit days
# Then construct factors under each definition

monthly_all = (
    daily_hose.groupby(['ticker', daily_hose['date'].dt.to_period('M')])
    .agg(
        ret_all=('daily_return', lambda x: (1 + x).prod() - 1),
        ret_interior=('daily_return',
                       lambda x: (1 + x[~x.name.map(
                           lambda idx: daily_hose.loc[idx, 'limit_up_hit'] |
                                        daily_hose.loc[idx, 'limit_down_hit']
                       ).values]).prod() - 1 if len(x) > 0 else np.nan),
        n_limit_days=('limit_up_hit',
                       lambda x: x.sum() + daily_hose.loc[x.index, 'limit_down_hit'].sum()),
        n_trading_days=('daily_return', 'count')
    )
    .reset_index()
)

print("Impact on Monthly Returns:")
print(f"  Mean monthly return (all days):     "
      f"{monthly_all['ret_all'].mean():.4f}")
print(f"  Mean monthly return (interior):     "
      f"{monthly_all['ret_interior'].mean():.4f}")
print(f"  Avg limit days per stock-month:     "
      f"{monthly_all['n_limit_days'].mean():.2f}")

37.7 The Magnet Effect

37.7.1 Do Limits Attract Prices?

The magnet effect hypothesis posits that price limits, rather than cooling the market, actually attract prices to the limit as traders rush to execute before the stock becomes locked (Cho et al. 2003). If a stock is approaching the upper limit, buyers accelerate their orders to avoid being shut out, creating a self-fulfilling rush to the boundary.

We test for the magnet effect by examining the speed of price movement toward the limit conditional on approaching it:

# Approach: for days that eventually hit the limit,
# compare the return in the last hour vs the first hour
# relative to non-limit days with similar initial trajectories

# Without intraday data, we use a cross-day approach:
# On day t, if the stock is within X% of the limit at some point,
# what is the probability of hitting the limit on day t vs day t+1?

# Simpler test: return continuation after near-limit days
def magnet_test(daily_df, limit, proximity_threshold=0.8):
    """
    Test for the magnet effect.
    
    For each stock-day, classify:
    - 'near_limit_up': return in [proximity_threshold * limit, limit)
    - 'near_limit_down': return in (-limit, -proximity_threshold * limit]
    - 'hit_limit_up': return = limit
    - 'hit_limit_down': return = -limit
    - 'normal': all others
    
    Then examine next-day behavior.
    """
    df = daily_df.copy()
    df['abs_ret'] = df['daily_return'].abs()
    
    df['near_up'] = (df['daily_return'] >= proximity_threshold * limit) & \
                     (df['daily_return'] < limit - 0.001)
    df['near_down'] = (df['daily_return'] <= -proximity_threshold * limit) & \
                       (df['daily_return'] > -limit + 0.001)
    
    df['next_return'] = df.groupby('ticker')['daily_return'].shift(-1)
    df['next_limit_up'] = df.groupby('ticker')['limit_up_hit'].shift(-1)
    df['next_limit_down'] = df.groupby('ticker')['limit_down_hit'].shift(-1)
    
    results = {}
    
    # Near upper limit
    near_up = df[df['near_up']]
    if len(near_up) > 100:
        results['near_up'] = {
            'n': len(near_up),
            'next_day_return': near_up['next_return'].mean(),
            'prob_next_limit_up': near_up['next_limit_up'].mean(),
            'prob_next_limit_down': near_up['next_limit_down'].mean()
        }
    
    # At upper limit
    at_up = df[df['limit_up_hit']]
    if len(at_up) > 100:
        results['at_up'] = {
            'n': len(at_up),
            'next_day_return': at_up['next_return'].mean(),
            'prob_next_limit_up': at_up['next_limit_up'].mean(),
            'prob_next_limit_down': at_up['next_limit_down'].mean()
        }
    
    # Near lower limit
    near_down = df[df['near_down']]
    if len(near_down) > 100:
        results['near_down'] = {
            'n': len(near_down),
            'next_day_return': near_down['next_return'].mean(),
            'prob_next_limit_up': near_down['next_limit_up'].mean(),
            'prob_next_limit_down': near_down['next_limit_down'].mean()
        }
    
    # At lower limit
    at_down = df[df['limit_down_hit']]
    if len(at_down) > 100:
        results['at_down'] = {
            'n': len(at_down),
            'next_day_return': at_down['next_return'].mean(),
            'prob_next_limit_up': at_down['next_limit_up'].mean(),
            'prob_next_limit_down': at_down['next_limit_down'].mean()
        }
    
    # Normal days (benchmark)
    normal = df[~df['near_up'] & ~df['near_down'] &
                 ~df['limit_up_hit'] & ~df['limit_down_hit']]
    results['normal'] = {
        'n': len(normal),
        'next_day_return': normal['next_return'].mean(),
        'prob_next_limit_up': normal['next_limit_up'].mean(),
        'prob_next_limit_down': normal['next_limit_down'].mean()
    }
    
    return pd.DataFrame(results).T

magnet = magnet_test(daily_hose, 0.07, proximity_threshold=0.8)
print("Magnet Effect Test (HOSE):")
print(magnet.round(4).to_string())

# More granular: bin today's return and compute next-day statistics
bins = np.arange(-0.075, 0.08, 0.005)
daily_hose_next = daily_hose.copy()
daily_hose_next['next_return'] = (
    daily_hose_next.groupby('ticker')['daily_return'].shift(-1)
)
daily_hose_next['ret_bin'] = pd.cut(daily_hose_next['daily_return'],
                                      bins=bins, labels=False)

bin_stats = (
    daily_hose_next.dropna(subset=['ret_bin', 'next_return'])
    .groupby('ret_bin')
    .agg(
        mean_ret=('daily_return', 'mean'),
        next_ret=('next_return', 'mean'),
        n=('next_return', 'count')
    )
)

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Panel A: Next-day return by today's return
axes[0].bar(bin_stats['mean_ret'] * 100, bin_stats['next_ret'] * 100,
            width=0.4,
            color=np.where(bin_stats['next_ret'] > 0, '#27AE60', '#C0392B'),
            alpha=0.7, edgecolor='white')
axes[0].axhline(y=0, color='black', linewidth=0.5)
axes[0].axvline(x=7, color='gray', linewidth=1, linestyle='--')
axes[0].axvline(x=-7, color='gray', linewidth=1, linestyle='--')
axes[0].set_xlabel("Today's Return (%)")
axes[0].set_ylabel('Next-Day Return (%)')
axes[0].set_title('Panel A: Next-Day Return by Current Return')

# Panel B: Continuation probability
# Probability of same-direction move next day
daily_hose_next['continuation'] = (
    np.sign(daily_hose_next['daily_return']) ==
    np.sign(daily_hose_next['next_return'])
)

cont_by_bin = (
    daily_hose_next.dropna(subset=['ret_bin', 'continuation'])
    .groupby('ret_bin')
    .agg(
        mean_ret=('daily_return', 'mean'),
        cont_prob=('continuation', 'mean'),
        n=('continuation', 'count')
    )
)

axes[1].scatter(cont_by_bin['mean_ret'] * 100, cont_by_bin['cont_prob'] * 100,
                 color='#2C5F8A', s=40, alpha=0.7)
axes[1].axhline(y=50, color='gray', linewidth=0.5, linestyle='--')
axes[1].axvline(x=7, color='gray', linewidth=1, linestyle='--')
axes[1].axvline(x=-7, color='gray', linewidth=1, linestyle='--')
axes[1].set_xlabel("Today's Return (%)")
axes[1].set_ylabel('Continuation Probability (%)')
axes[1].set_title('Panel B: Same-Direction Move Next Day')

plt.tight_layout()
plt.show()

Figure 37.10

37.8 The Delayed Price Discovery Hypothesis

37.8.1 Volatility Spillover

If limits prevent full price adjustment on day \(t\), the residual adjustment spills over to day \(t+1\) (and possibly further). This predicts higher volatility on the day after a limit hit, and positive return autocorrelation (continuation in the direction of the limit hit). Kim and Rhee (1997) find strong evidence of this in the Tokyo Stock Exchange.

# Compare volatility and return on limit-hit days vs day after
spillover_data = daily_hose.copy()
spillover_data['prev_limit_up'] = (
    spillover_data.groupby('ticker')['limit_up_hit'].shift(1)
)
spillover_data['prev_limit_down'] = (
    spillover_data.groupby('ticker')['limit_down_hit'].shift(1)
)
spillover_data['abs_return'] = spillover_data['daily_return'].abs()

# Classify days
conditions = {
    'After upper limit': spillover_data['prev_limit_up'] == True,
    'After lower limit': spillover_data['prev_limit_down'] == True,
    'Normal day': (spillover_data['prev_limit_up'] == False) &
                   (spillover_data['prev_limit_down'] == False)
}

print("Volatility Spillover After Limit Hits:")
print(f"{'Condition':<25} {'Mean |r|':>10} {'Mean r':>10} "
      f"{'σ(r)':>10} {'N':>12}")
print("-" * 67)

for label, mask in conditions.items():
    subset = spillover_data[mask].dropna(subset=['daily_return'])
    print(f"{label:<25} "
          f"{subset['abs_return'].mean()*100:>10.3f}% "
          f"{subset['daily_return'].mean()*100:>10.3f}% "
          f"{subset['daily_return'].std()*100:>10.3f}% "
          f"{len(subset):>12,}")

# Statistical test: is variance higher after limit days?
normal = spillover_data[conditions['Normal day']]['daily_return'].dropna()
after_up = spillover_data[conditions['After upper limit']]['daily_return'].dropna()
after_down = spillover_data[conditions['After lower limit']]['daily_return'].dropna()

f_up = after_up.var() / normal.var()
f_down = after_down.var() / normal.var()
print(f"\nVariance ratios (vs normal days):")
print(f"  After upper limit: {f_up:.3f} "
      f"(p = {1 - stats.f.cdf(f_up, len(after_up)-1, len(normal)-1):.4f})")
print(f"  After lower limit: {f_down:.3f} "
      f"(p = {1 - stats.f.cdf(f_down, len(after_down)-1, len(normal)-1):.4f})")

37.8.2 Multi-Day Return Reconstruction

To recover the “true” return that would have occurred without price limits, we can compound returns over consecutive limit-hit days until the stock resumes normal trading:

def reconstruct_returns(group, limit):
    """
    For each limit-hit sequence, compound returns until
    the stock resumes normal trading (first non-limit day).
    Returns the compound return and the number of days.
    """
    sequences = []
    in_sequence = False
    seq_start = None
    seq_returns = []
    seq_direction = None
    
    for _, row in group.iterrows():
        if row['limit_up_hit'] or row['limit_down_hit']:
            if not in_sequence:
                in_sequence = True
                seq_start = row['date']
                seq_returns = [row['daily_return']]
                seq_direction = 'up' if row['limit_up_hit'] else 'down'
            else:
                seq_returns.append(row['daily_return'])
        else:
            if in_sequence:
                # Include the first non-limit day (the "resolution" day)
                seq_returns.append(row['daily_return'])
                compound_ret = np.prod([1 + r for r in seq_returns]) - 1
                sequences.append({
                    'ticker': group.name if hasattr(group, 'name') else group['ticker'].iloc[0],
                    'start_date': seq_start,
                    'n_limit_days': len(seq_returns) - 1,
                    'compound_return': compound_ret,
                    'direction': seq_direction,
                    'limit_return': sum(seq_returns[:-1]),
                    'resolution_return': seq_returns[-1]
                })
                in_sequence = False
    
    return sequences

# Run for all HOSE stocks
all_sequences = []
for ticker, group in daily_hose.sort_values('date').groupby('ticker'):
    seqs = reconstruct_returns(group, 0.07)
    all_sequences.extend(seqs)

seq_df = pd.DataFrame(all_sequences)

if len(seq_df) > 0:
    print("Limit-Hit Sequence Analysis:")
    print(f"  Total sequences: {len(seq_df):,}")
    print(f"  Mean limit days: {seq_df['n_limit_days'].mean():.1f}")
    print(f"\nCompound Returns by Direction:")
    for direction in ['up', 'down']:
        subset = seq_df[seq_df['direction'] == direction]
        print(f"  {direction.upper()} sequences: {len(subset):,}")
        print(f"    Mean compound return: {subset['compound_return'].mean()*100:.2f}%")
        print(f"    Mean resolution-day return: "
              f"{subset['resolution_return'].mean()*100:.2f}%")
        print(f"    Max compound return: {subset['compound_return'].max()*100:.1f}%")

37.9 The Idiosyncratic Volatility Puzzle Under Price Limits

Ang et al. (2006) document that stocks with high idiosyncratic volatility earn low subsequent returns—the IVOL puzzle. In Vietnam, price limits contaminate IVOL estimation: stocks that frequently hit limits have understated IVOL (because their returns are censored), which could create a mechanical relation between measured IVOL and returns.

# Compute monthly IVOL two ways:
# (a) Naive: std of daily residuals from FF3
# (b) Corrected: excluding limit-hit days

# Merge daily data with market returns for IVOL estimation
daily_ff = daily_hose_merged.copy()

monthly_ivol = []
for (ticker, month), group in daily_ff.groupby(
    ['ticker', daily_ff['date'].dt.to_period('M')]
):
    if len(group) < 15:
        continue
    
    y = group['daily_return'].dropna()
    x = group['mkt_return'].reindex(y.index).dropna()
    common = y.index.intersection(x.index)
    if len(common) < 15:
        continue
    
    # Naive IVOL
    model = sm.OLS(y[common], sm.add_constant(x[common])).fit()
    ivol_naive = model.resid.std() * np.sqrt(252)
    
    # Interior-only IVOL
    interior = group[~group['limit_up_hit'] & ~group['limit_down_hit']]
    y_int = interior['daily_return'].dropna()
    x_int = interior['mkt_return'].reindex(y_int.index).dropna()
    common_int = y_int.index.intersection(x_int.index)
    
    if len(common_int) >= 10:
        model_int = sm.OLS(y_int[common_int],
                            sm.add_constant(x_int[common_int])).fit()
        ivol_corrected = model_int.resid.std() * np.sqrt(252)
    else:
        ivol_corrected = np.nan
    
    n_limit = group['limit_up_hit'].sum() + group['limit_down_hit'].sum()
    
    monthly_ivol.append({
        'ticker': ticker,
        'month': month.to_timestamp(),
        'ivol_naive': ivol_naive,
        'ivol_corrected': ivol_corrected,
        'n_limit_days': n_limit,
        'pct_limit': n_limit / len(group) * 100,
        'next_return': group['daily_return'].iloc[-1]  # Placeholder
    })

ivol_df = pd.DataFrame(monthly_ivol)

print("IVOL Estimation: Naive vs Corrected:")
print(f"  Mean naive IVOL:     {ivol_df['ivol_naive'].mean():.4f}")
print(f"  Mean corrected IVOL: {ivol_df['ivol_corrected'].mean():.4f}")
print(f"  Mean difference:     "
      f"{(ivol_df['ivol_corrected'] - ivol_df['ivol_naive']).mean():.4f}")
print(f"  Correlation:         "
      f"{ivol_df['ivol_naive'].corr(ivol_df['ivol_corrected']):.3f}")

37.10 Practical Recommendations

For researchers working with Vietnamese equity data:

Always report limit-hit frequency. Any study using Vietnamese daily returns should document the fraction of observations at the price limits, broken down by exchange and market cap quintile. This tells the reader the severity of the censoring problem in the specific sample.

Use Tobit-corrected variance estimates. For volatility-related analyses (IVOL sorts, GARCH, risk modeling), the naive sample variance underestimates true variance by 5–20% depending on the stock’s limit-hit frequency. The Tobit MLE provides a consistent estimator under the censored normal assumption.

Consider range-based estimators. The Yang and Zhang (2000) estimator using OHLC prices is partially robust to closing-price censoring and does not require distributional assumptions. It is a good default for individual-stock volatility estimation.

Exclude limit-hit days for beta estimation. Interior-only betas are less biased than all-day betas, though noisier. Report both and discuss the difference. For stocks with >5% limit-hit frequency, the attenuation is economically meaningful.

Compound multi-day returns for event studies. When studying events that coincide with limit hits (earnings announcements, M&A, regulatory changes), use the compound return from the limit-hit sequence start to the first non-limit day. Single-day returns are censored and understate the market’s reaction.

Be cautious interpreting short-term return predictability. The delayed price discovery effect creates positive return autocorrelation at the daily frequency. This is a mechanical consequence of censoring, not a market inefficiency. Monthly returns are largely free of this artifact because the censoring within a month averages out.

Test robustness to HNX and UPCoM. If a result is driven by limit-related distortions, it should appear differently (or not at all) on HNX (\(\pm\) 10%) and UPCoM (\(\pm\) 15%). Cross-exchange comparison is a natural placebo test.

37.11 Summary

Table 37.2: Summary of price limit effects on empirical estimates.

Issue	Bias Direction	Magnitude (HOSE)	Recommended Fix
Return variance	Understated	5–20% for volatile stocks	Tobit MLE or range-based
GARCH vol	Understated	10–30% during crises	Censored GARCH
Market beta	Attenuated (toward 0)	3–10% for small-caps	Interior-only estimation
IVOL	Understated	Varies; correlated with size	Corrected IVOL
Return autocorrelation	Positive (spurious)	Significant at daily freq	Use weekly/monthly
Event study CARs	Understated	Up to 50% of true effect	Compound multi-day
Distribution shape	Pile-up at limits	2–5% of obs at limits	Acknowledge or correct

Price limits are not a minor institutional detail—they are a pervasive data-generating process that affects nearly every empirical quantity computed from Vietnamese daily returns. The corrections developed in this chapter—Tobit variance estimation, censored GARCH, interior-only betas, range-based volatility, and multi-day return compounding—form a toolkit that should be applied routinely. Ignoring censoring does not make it go away; it merely makes the resulting estimates quietly wrong.

Ang, Andrew, Robert J Hodrick, Yuhang Xing, and Xiaoyan Zhang. 2006. “The Cross-Section of Volatility and Expected Returns.” The Journal of Finance 61 (1): 259–99.

Brennan, Michael J. 1986. “A Theory of Price Limits in Futures Markets.” Journal of Financial Economics 16 (2): 213–33.

Cho, David D, Jeffrey Russell, George C Tiao, and Ruey Tsay. 2003. “The Magnet Effect of Price Limits: Evidence from High-Frequency Data on Taiwan Stock Exchange.” Journal of Empirical Finance 10 (1-2): 133–68.

Kim, Kenneth A, and S Ghon Rhee. 1997. “Price Limit Performance: Evidence from the Tokyo Stock Exchange.” The Journal of Finance 52 (2): 885–901.

Yang, Dennis, and Qiang Zhang. 2000. “Drift-Independent Volatility Estimation Based on High, Low, Open, and Close Prices.” The Journal of Business 73 (3): 477–92.