commit5171b30eb3ed865acd5ea3b2eadce0c615423cbeparent2550ab7d555499b37f15ea3092c532f1e0c5e626Author:Shimmy Xu <shimmy.xu@shimmy1996.com>Date:Sat, 20 Apr 2019 23:48:08 -0400 Add notes for Chapter 2Diffstat:

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1 file changed, 121 insertions(+), 0 deletions(-)diff --git a/notes.org b/notes.org@@ -29,3 +29,124 @@ Distribution of price changes after a trade is very broad, with a heavy tail in A typical US large cap stock's daily traded volume would be around \(0.5\%\) of total market capitalization. A small cap would be even lower at around \(0.1\%\). At any given time, the quantity available on the order book would be \(10^{-4}\) of market cap, which means even "highly liquid" markets might not be very liquid at all. This is because market-makers are reluctant to commit to large quantities due to adverse selection. As a result, markets operate in a regime of small revealed liquidity but large latent liquidity and large trades must be fragmented. Prices cannot be in equilibrium: since trades are fragmented, at any instant, the traded volume is much smaller than the latent supply and demand; at larger time scales, information start to display it's long term effects, causing price to change. + +* Chapter 2 +** The Random Walk Model +Bachelier argues future price changes are unpredictable and considers price series as Gaussian random walks. So Bachelier framework is basically a fancy name for Brownian motion model. + +Since price changes are typically proportional to stock price (multiplicative), geometric Brownian motion makes more sense for the price changes. On shorter time scales, however, additive price changes is closer to empirical data due to restrictions on minimum allowable price change. + +Bachelier's first law: price variogram under i.i.d. price increments is linear in \(\tau\): +\begin{align*} + \fV(\tau) + &= \ev((p_{t + \tau} - p_{t})^{2})\\ + &= D\tau. +\end{align*} +Definition of volatility follows that of multiplicative model (GBM) due to its popularity: +\begin{align*} + D + &= \sigma_{r}^{2}\mean{p}^{2}. +\end{align*} + +For \(\tau > 0\), a trending random walk has ACF \(C_{r}(\tau) > 0\) and a mean-reverting one has \(C_{r}(\tau) < 0\). This would cause observed volatility depend on sampling frequency \(\frac{1}{\tau}\) (we can view \(\tau\) as trade time, so \(\tau = 1\) recovers the original \(\sigma_{r}\)): +\begin{align*} + \sigma^{2}(\tau) + &= \frac{\fV(\tau)}{\tau \mean{p}^{2}}\\ + &= \frac{1}{\tau \mean{p}^{2}}\ev((p_{t + \tau} - p_{t})^{2})\\ + &= \frac{1}{\tau \mean{p}^{2}}\ev((\sum_{u = 1}^{\tau}\mean{p} r_{t + u})^{2})\\ + &= \frac{\sigma_{r}^{2}}{\tau}(\sum_{u = 1}^{\tau}\sum_{k = 1}^{\tau}C_{r}(k - u))\\ + &= \frac{\sigma_{r}^{2}}{\tau}(\tau + 2\sum_{u = 1}^{\tau}(\tau - u)C_{r}(u))\\ + &= \sigma_{r}^{2}(1 + 2\sum_{u = 1}^{\tau}(1 - \frac{u}{\tau})C_{r}(u))). +\end{align*} +Plot of \(\sigma(\tau)\) vs. \(\tau\) is a volatility signature plot. Under exponentially decaying ACF \(C_{r}(u) = \rho^{u}\), positive correlation (\(\rho\)) leads to increasing \(\sigma(\rho)\) (in \(\tau\)) and negative correlation the opposite: +\begin{align*} + \lim_{\tau \to \infty}\frac{\sigma^{2}(\tau)}{\sigma^{2}(1)} + &= \lim_{\tau \to \infty}\frac{\sigma_{r}^{2}(1 + 2\sum_{u = 1}^{\tau}(1 - \frac{u}{\tau})C_{r}(u)))}{\sigma_{r}^{2}}\\ + &= 1 + 2\lim_{\tau \to \infty}\sum_{u = 1}^{\tau}(1 - \frac{u}{\tau})\rho^{u}\\ + &= 1 + 2\lim_{\tau \to \infty}\sum_{u = 1}^{\tau}\rho^{u} - \frac{2}{\tau}\lim_{\tau \to \infty}\sum_{u = 1}^{\tau}u\rho^{u}\\ + &= 1 + 2\lim_{\tau \to \infty}\frac{\rho - \rho^{\tau + 1}}{1 - \rho} - \lim_{\tau \to \infty}\frac{2}{\tau}\frac{\rho - (\tau + 1)\rho^{\tau + 1} + \tau \rho^{\tau + 2}}{(1 - \rho)^{2}}\\ + &= 1 + 2\frac{\rho}{1 - \rho} - 0\\ + &= \frac{1 + \rho}{1 - \rho}. +\end{align*} + +To account for microstructure noise caused by Bid-ask bounce and temporarily price deviations, we can include high frequency noise term \(\eta_{t}\), which is zero mean, independent from \(r_{t}\), and has ACF +\begin{align*} + C_{\eta}(\tau) + &= \E^{-\tau / \tau_{\eta}}, +\end{align*} +where \(\tau_{\eta}\) is the error correction time. \(\eta_{t}\) is often called an Ornstein-Uhlenbeck process. If we include this, then our measured volatility becomes +\begin{align*} + \sigma_{\eta}^{2}(\tau) + &= \frac{1}{\tau \mean{p}^{2}}\ev((p_{t + \tau} - p_{t})^{2})\\ + &= \frac{1}{\tau \mean{p}^{2}}\ev(\mean{p}^{2}(\sum_{u = 1}^{\tau} r_{t + u} + \eta_{t + \tau} - \eta_{t})^{2})\\ + &= \sigma^{2}(\tau) + \frac{1}{\tau}\ev((\eta_{t + \tau} - \eta_{t})^{2})\\ + &= \sigma^{2}(\tau) + \frac{2\sigma_{\eta}^{2}}{\tau}(1 - \E^{-\tau / \tau_{\eta}}). +\end{align*} +The new term is similar to mean-reversion: creates a higher short-term volatility than long-term volatility. + +Empirically, volatility signature plots are mostly flat with a very weak mean reversion effect (decreasing) over a wide range of time scales, from seconds to months. + +** Jumps and Intermittency in Financial Markets +Unconditional distribution of returns has fat tails that decay as power law. On short time scales, Student's \(t\) distribution is a reasonable approximation: +\begin{align*} + f_{r}(r) + &\propto_{\abs{r} \ll a} \frac{a^{\mu}}{\abs{r}^{1 + \mu}}, +\end{align*} +where \(\mu\) is the tail parameter (close to \(3\)) and \(a\) is related to variance through \(\sigma^{2} = a^{2} / (\mu - 2)\). For fixed variance, Gaussian distribution is recovered through \(\mu \to \infty\). + +To account for volatility clustering, we decompose \(r_{t} = \sigma_{t}\xi_{t}\), where \(\xi_{t}\) is a unit variance IID random variable and \(\sigma_{t}\) is a long memory positive random variable. A common form is +\begin{align*} + \sigma_{t} + &= \sigma_{0}\E^{\omega_{t}}, +\end{align*} +where \(\omega_{t}\) is approximately Gaussian with variogram +\begin{align*} + \fV_{\omega}(t) + &= \ev((\omega_{t} - \omega_{t + \tau})^{2})\\ + &= \chi_{0}^{2}\ln(1 + \min(\tau, T)), +\end{align*} +where \(\chi_{0}\) is "vol of vol" and \(T\) is cut-off time of covariance. \(T\) is typically on order of years and \(\chi_{0}^{2} \approx 0.05\). This is similar but fundamentally different from an Ornstein-Uhlenbeck log-volatility process, where we would have +\begin{align*} + \fV_{\omega}(t) + &= \chi_{0}^{2}\ln(1 + \E^{-\tau / \tau_{\omega}}), +\end{align*} +in which \(\tau_{\omega}\) is relaxation time. We can see that this gives exponential ACF functions for \(\omega_{t}\). Empirically, \(\sigma\) and \(\xi\) are not independent: positive past returns tend to decrease future volatilities while negative ones tend to increase them. This is called the leverage effect. + +Why do returns remain substantially non-Gaussian on time scales up to weeks or even months? Because correlation between random variables causes convergence to CLT to be much slower than the IID case (IID \(\sim t^{-1}\) and long memory \(\sim t^{-0.2}\)). + +Market activity also clusters. Define average market activity \(\mean{\phi}\) as +\begin{align*} + \ev(\dd N_{t}) + &= \mean{\phi} \dd t, +\end{align*} +where \(\dd N_{t}\) is number of price changes occurred during \([t, t + \dd t]\). The ACF of \(\phi = \der[N_{t}]{t}\) describes the temporal structure of fluctuations in market activity, which displays long memory and periodicities (daily and weekly). + +A simple model to relate market activity and price volatility is to assume that price changes have size \(\vartheta\) and we would have +\begin{align*} + \sigma^{2} + &= \phi \vartheta^{2}. +\end{align*} +Prices in financial markets tend to exhibit two types of volatilities: small frequent moves and rare extreme moves, the latter of which is not captured by this simple model. Assuming returns follow \(t\) distribution, jumps defined as events greater than \(4\sigma\) contribute \(30\%\) of the total variance, a substantial component. More generally, some kind of self-excitation seems to be present in financial markets, i.e. increased volatility could trigger more activity in the future or in a different asset. + +** Why Do Prices Move? +Excessive volatility puzzle: the actual volatility of prices appears to be much higher than the one warranted by fluctuations of the underlying fundamental value, contrary to efficient-market theory. In reality, the number of large price jumps is much higher than the number of relevant news arrivals and occurs at a much higher frequency. This suggests that market activity is not only driven by news. + +Flat volatility puzzle: financial time series show very little under-reaction (which would create trend, a likely product of reaction to news arrivals) or over-reaction (which would lead to mean-reversion, a likely product of large latent supply and demand), leading to almost flat empirical signature plots. An explanation from microstructural view point is that, higher-frequency strategies feed on the inefficiencies generated by slower strategies (trends and correlations), finally resulting in white-noise returns on all time scales. + +Are prices fundamentally efficient (in the sense that they are always close to some fundamental value) or merely statistically efficient (in the sense that all predictable patterns are exploited and removed by technical trading)? + +Black argues prices are correct to within a factor of \(2\). If this is the case, the anchor to fundamental values can only be felt on a time scale \(T\) such that purely random fluctuations \(\sigma\sqrt{T}\) become of the order of say \(50\%\) of the fundamental price, leading to \(T = 6\) years for the stock market with a typical annual volatility of \(\sigma = 20\%\). Such long time scales suggest that the notion of a fundamental price is secondary to understanding the dynamics of prices at the scale of a few seconds to a few days. So we can mostly focus on statistical efficiency. + +** Summary and Outlook +A lot of quantitative volatility/activity feedback models (like ARCH-type models and Hawkes processes) attribute the excess volatility to self-referential effects, i.e. from the past price movements. This adds credence to the idea that majority of the short- to medium-term activity of financial markets is unrelated to any fundamental information or economic effects. + +The decoupling between price and fundamental value allows a theory of price moves that is mostly based on the endogenous, self-exciting dynamics of markets and not on long-term fundamental effects which are hard to model. One particularly important question is to understand the origin and mechanisms leading to price jumps, which seem to have a similar structure on all liquid markets. + +Flat volatility signature plot is a sign of statistical efficiency. + +** Errata +Formula (2.11) should be +\begin{align*} + p_{t} + &= p_{0} + \mean{p}(\sum_{t' = 0}^{t - 1}r_{t'} + \eta_{t}). +\end{align*}