HFT: Approaching continuous time finance

• Does it help to use data sampled at a high frequency to estimate \(\mu_t\) and \(\sigma_t^2\)?
• Continuous-time finance helps
• Well-known example: Geometric Brownian motion (stock price movements are additive on the log scale) \[ \log(P_t) = X_t = X_0 + \mu t + \sigma W_t \]

where \(W_t\) is a Brownian motion

Brownian motion

The process \((W_t)_{0\leq t\leq T}\) is a Brownian motion provided that

1. \(W_0 = 0\)
2. \(t \rightarrow W_t\) is a continuous function of \(t\)
3. \(W\) has independent increments and for \(t>s\), \(W_t - W_s\) is normal with mean zero and variance \(t-s\)

Brownian motion: discrete sampling

• Suppose \(t = 0\) is the start of the trading day and \(t=1\) is the end of the day
• Assume there are \(n\) equidistant observations (transactions) of the log price
• One observation every \(\Delta_{t_n} = 1/n\) units of time
• We observe \(X_{t_{n,i}}\) with \(t_{n,i} = i\Delta t_n\) and get \[\Delta X_{t_{n,i}} = X_{t_{n,i}} - X_{t_{n,i -1}} \]
• By definition of the BM, the \(\Delta X_{t_{n,i}}\) are iid normal \(N(\mu\Delta t_n, \sigma^2 \Delta t_n)\)

Simulation of a Brownian motion

price = cumsum(c(
  start,
  rnorm(steps, mean = mu / steps, sd = sigma / sqrt(steps))
))

Estimating \(\mu\) and \(\sigma^2\) in the GBM model

• The natural estimator for \(\hat\mu\) is \[\hat\mu_n = \frac{1}{n\Delta t_n}\sum\limits_{i = 0}^{n-1}\Delta X_{t_n, i+1} = (X_1 - X_0)\]
• (Maybe) surprising: \(\hat\mu_n\) does not depend on the sampling frequency. Consistent estimation requires \(t\rightarrow \infty\)
• (Maybe) more surprising: \(\hat\sigma^2\) can be estimated consistently as \(n\rightarrow \infty\)!
• Set \(U_{n,i+1} = \Delta X_{t_n,i+1}/(\sigma\Delta t_n^{1/2}) \sim N\left(\frac{\mu}{\sigma}\Delta t_n^{1/2}, 1\right)\) and define the sample mean \(\bar U_n = \frac{1}{n}\sum\limits_{i = 0}^{n-1} U_{n,i+1}\)
• Then \[\begin{aligned}\hat\sigma_n^2 &= \frac{1}{(n-1)\Delta t_n}\sum\limits_{i = 0}^{n-1}\left(\Delta X_{t_n, i+1} - \bar{\Delta X_{t_n}}\right)^2 \\&= \frac{\sigma^2\Delta t_n}{(n-1)\Delta t_n}\sum\limits_{i = 0}^{n-1}\left(U_{n, i+1} - \bar U_n\right)^2 \stackrel{\mathcal{L}}{=} \sigma^2\frac{\chi^2_{n-1}}{n-1}\end{aligned}\]
• It follows that \(E\left(\hat\sigma_n^2\right) = \sigma^2\) and \(\text{Var}\left(\hat\sigma_n^2\right) = \frac{2\sigma^4}{n-1}\)

Non-centered estimator

• For high-frequency data, the mean \(\bar{\Delta X_{t_n}}\) is often not removed in estimation
• Instead, consider \[\hat\sigma^2_{n,\text{nocenter}} = \frac{1}{n\Delta t_n}\sum\limits_{i=0}^{n-1}\left(\Delta X_{t_{n, i +1}}\right)^2 = \frac{n-1}{n}\hat\sigma_n^2 + \Delta t_n\hat\mu_n^2 = \frac{n-1}{n}\hat\sigma_n^2 + \frac{1}{n}\hat\mu_n^2 \]
• Since \(\hat\mu_n\) does not depend on \(n\), it follows that \(\hat\sigma^2_{n,\text{nocenter}}\) is also consistent

Realized variance

• Assume \(\mu_t\) and \(\sigma_t>0\) are predictable processes of finite variation driving the diffusion \[dX_t = \mu_t dt + \sigma_t dW_t.\]
• Then the continuously compounded return over \([t-k, t]\) is \[r(t,k) = X_t - X_{t-k} = \int\limits_{t-k}^t \mu_s ds + \int\limits_{t-k}^t \sigma_s dW_s\]
• The diffusive sample path variation is called the integrated variance \[IV(t-k, k) = \int\limits_{t-k}^t \sigma_s^2 ds \]
• Define the realized variance as \(RV^{(n)} := \sum\limits_{i=0}^{n-1}r_{i,n}^2\)
• As long as there are no jumps, \(RV^{(n)} \rightarrow IV(0, 1)\) as \(n\rightarrow\infty\)
• ‘Infill’ asymptotics: sampling at the highest possible frequency is crucial!

Infill asymptotics

Realized variance of the S&P 500

read_csv("../data-r/rv-data.csv", show_col_types = FALSE) |>
  ggplot(aes(x = Date, y = `Realized Variance (1-min)`)) +
  geom_line() +
  theme_bw()

Market Microstructure Noise

• Problem in practice: market microstructure frictions
• E.g., bid-ask spreads, price discreteness, asymmetric information, strategic order placement
• We can only observe \[Y_{i\Delta n} = X_{i\Delta n} + U _{i\Delta n}\] where \(Y_{i\Delta n}\) is the observed (log) transaction price or quote, \(X_{i\Delta n}\) is the efficient latent (log) price
• \(U_{i\Delta n}\) is white noise which captures microstructure frictions

Microstructure noise: the intuition

• So far: the highest possible sampling frequency is optimal

Microstructure noise: the problem

• Now: what is the RV estimator’s optimal sampling frequency/adjustment in the presence of microstructure noise?

Microstructure noise: formal setup

• For now, consider the simplest case with \(X_t = \sigma W_t\) and \(\tilde X_t = X_t + U_t\) where the \(U_t\) are iid noise with zero mean and variance \(a^2\)
• Then \[\begin{aligned}Y_{\tau_i} &= \tilde X_{\tau_i} - \tilde X_{\tau_{i-1}} \\&= \left(X_{\tau_i} - X_{\tau_{i-1}}\right) + U_{\tau_i} - U_{\tau_{i-1}} \\&= \sigma \left(W_{\tau_i} - W_{\tau_{i-1}}\right) + U_{\tau_i} - U_{\tau_{i-1}}\end{aligned}\]
• We get \(\text{Var}\left(Y_{\tau_i}\right) = \sigma^2\Delta_n + 2a^2\) and \(\text{Cov}\left(Y_{\tau_i}, Y_{\tau_{i-1}}\right) = -a^2\)
• The proportion of the total return variance that is market microstructure-induced is \[\pi = \frac{2a^2}{2a^2 + \sigma^2\Delta_n}\]
• As \(\Delta_n\) gets small, a larger fraction of the observed return variance is noise-driven
• Noise bias adjustment possible (Ait-Sahalia et al., 2005)

Bid-ask bounce and the volatility signature plot

Volatility signature plot: S&P 500

Optimal sampling frequency

• Optimal sampling frequency depends on the signal-to-noise ratio \(\pi\)

Two-scale realized variance

• Sparse sampling reduces noise bias but lowers efficiency: we throw away data
• Zhang, Mykland, and Ait-Sahalia (2005) propose a two-scale realized variance estimator based on
  1. sub-sampling
  2. averaging
  3. bias correction
• Start from observed noisy prices \(Y_0,\ldots,Y_n\)
• Split the data into \(K\) sparse sub-samples: \[G^{(k)} = \left\{Y_k, Y_{k+K}, Y_{k+2K}, \ldots\right\}, \quad k=1,\ldots,K\]
• Compute realized variance on each sparse grid and average: \[RV^{\text{avg}}_K = \frac{1}{K}\sum_{k=1}^K RV^{(k)}\]
• Correct the remaining noise bias using the all-data realized variance: \[TSRV = RV^{\text{avg}}_K - \frac{\bar n_K}{n}RV^{\text{all}}\]

where \(\bar n_K\) is the average number of returns per sparse grid.

TSRV example with second-level data

Realized Covariances

• Similar in spirit – the efficient log-price process follows an \(N\)-dimensional Brownian semimartingale \[X_t = \int\limits_{0}^t a_u du + \int\limits_{0}^t \sigma_u dW_u\] where \(a\) is a predictable locally bounded drift process, \(\sigma_u\) is a volatility matrix process, and \(W\) is a vector of independent Brownian motions
• If \(X_t\) is discretely observed, a natural estimator for the quadratic covariation is \[RC_n = \sum\limits_{i=1}^n\left(X_{i/n} - X_{(i-1)/n}\right)\left(X_{i/n} - X_{(i-1)/n}\right)'\]
• Similar concept for realized betas \[R\beta_i = \frac{RC_{ip,n}}{RC_{pp,n}}\] where \(p\) corresponds to the market portfolio

Challenges in covariation estimation

• Positive definiteness: can we invert the covariance matrix?
• Conditioning: are matrix inversions numerically stable?
• Efficiency: are we throwing away too much data due to sparse sampling?
• Market microstructure effects
• Asynchronicity of observations in time: Epps (1979, JASA)

Multivariate Realized Kernels

• How to handle asynchronous trading?

• Definition: Refresh time for \(t\in[0,1]\). Define the first refresh time as \(\tau_1 = \max\left(t_1^{(1)}, \ldots, t_1^{(N)}\right)\). For each asset \(i\), let \(N_j^{(i)} = \min\left\{k: t_k^{(i)} > \tau_j\right\}\) be the next observation after \(\tau_j\); then define iteratively \[\tau_{j+1} = \max\left(t_{N_j^{(1)}}^{(1)}, \ldots, t_{N_j^{(N)}}^{(N)}\right) \]
• Compute synchronized returns based on \(\left\{\tau\right\}\)

High-frequency-based covariance matrix estimation

• Overcome asynchronicity by refresh time sampling: \[\begin{aligned}r\tau _{t,1}:=\max_{i}\left\{\tau_{t,1} ^{(i)}\right\}, \quad r\tau _{t,l+1}:=\max_{i}\left\{\tau_{t,N^{(i)}(r\tau _{t,l})} ^{(i)}\right\}\end{aligned}\]

• Realized kernel (Barndorff-Nielsen et al., 2011): \[\begin{aligned} K_t := \sum\limits_{h=-H_t } ^{H_t } k\left(\frac{h}{H_t +1 }\right) \Gamma_t ^{h}, \end{aligned}\] where \[\begin{aligned} k(x) = \left\lbrace \begin{array}{ll} 1 - 6x^{2} + 6x^{3} & 0 \leq x \leq 1/2\\ 2(1-x)^{3} & 1/2 \leq x \leq 1\\ 0 & x > 1 \end{array}\right.\end{aligned}\] \(\Gamma_t ^{h}\) is the \(h\)-lag auto-covariance matrix based on refresh-time synchronized returns, and \(H_t\) is a block-specific bandwidth.

Blocked Realized Kernels

• Blocked and regularized realized kernel (Hautsch et al., 2012)
• Idea: gaining efficiency and tractability in high dimensions by blocking
• Stocks ordered according to the average of daily mid-quotes
• 4 equal-sized groups result in \(b=10\) covariance blocks
• Compute realized kernel for each covariance block: \(K_t^b\)

Blocked Realized Kernels

• Ensure scale invariance by computing correlation blocks, \[\hat{H}_t^\text{b} = \left(V_t ^\text{b}\right)^{-1} K_t ^\text{b} \left(V_t ^\text{b}\right)^{-1},\hspace{1 cm}V_t ^\text{b} = \text{diag }\left[K_t ^{b}\right]^{1/2}\]

• Yields covariance estimator: \[\hat{\Sigma}_t^\text{BRK} = \text{diag} \left(\hat\sigma^2 _{t,1},\ldots,\hat\sigma^2 _{t,N}\right)^{1/2} \hat{H}_t \text{diag} \left(\hat\sigma^2 _{t,1},\ldots,\hat\sigma^2 _{t,N}\right)^{1/2}\]

• Averaging over the last five days: \[\hat{\Sigma}_{S,t} ^{BRK} := (1/5)\sum_{s=1}^5 \hat{\Sigma}_{t-s+1} ^{BRK}\]