Univariate Volatilty

Kevin Sheppard

Today

• Realized Variance
• Bid-Ask Bounce
• Volatility Signature Plots
• Alternative Estimators
• Modeling and Forecasting $RV$
• Implied Variance

Realized Variance

• Use UHF data to precisely estimate variance over a given period
  • Typically 1 day
  • Avoids diurnal effects

$$ RV_{t}^{(m)}=\sum_{i=1}^{m}\left(p_{i,t}-p_{i-1,t}\right)^{2}=\sum_{i=1}^{m}r_{i,t}^{2} $$

• Like variance estimator except no need to demean
• Assumption is that $m\rightarrow\infty$
• Estimated integrated variance

$$ \lim_{m\rightarrow\infty} RV_{t}^{(m)} = \int_{t}^{t+1}\sigma_{s}^{2}ds $$

Simulated Brownian Motion

• Assume $\mu$ and $\sigma$ are annualized $$r_{i,t} \stackrel{iid}{\sim} N\left(\frac{\mu}{252 m},\frac{\sigma^2}{252 m}\right) $$

$$ p_{i,t} = p_{i-1,t} + r_{i,t} $$

• $p_{0,t} = \ln(100)$

The price path

plot(p)

Implementing $RV$

rv = {}

for i in (13,39,78,390,1560,23400):
    step = m // i
    rets = p.iloc[::step].diff().dropna()
    _rv = (rets**2).sum()
    rv[step] = _rv
np.sqrt(252 * pd.Series(rv))

1800    0.210890
600     0.242003
300     0.209546
60      0.180562
15      0.197455
1       0.200257
dtype: float64

$RV$ estimation error

plot_rv()

Bid-Ask Bounce

• Primary challenge of $RV$ implementation
• Not one price, but two
  • Bid
  • Ask or Offer
• Transactions tend to bound between these two prices
• Creates many high-frequency noise returns
• Observed return is $$ r_{i,t} = r^\star_{i,t} + \epsilon_{i,t} $$
• $r^\star_{i,t}$ is the return we would like see
• $\epsilon_{i,t}$ follows an MA(1) $$RV_{t}^{(m)}\approx\widehat{RV}_{t}+m\tau^{2}$$

Realistic UHF Asset Price

plot_trans()

Transaction, Bid, and Ask Prices

plot_trans(zoom=True)