commit9091c6cc8410904522b00c99a89fc59bbae238c4parent5171b30eb3ed865acd5ea3b2eadce0c615423cbeAuthor:Shimmy Xu <shimmy.xu@shimmy1996.com>Date:Tue, 5 May 2020 16:54:08 -0500 Update notes for Chapter 01 and 02Diffstat:

M | notes.org | | | 49 | +++++++++++++++++++++++-------------------------- |

1 file changed, 23 insertions(+), 26 deletions(-)diff --git a/notes.org b/notes.org@@ -7,15 +7,13 @@ #+INCLUDE: "~/.emacs.d/static/math_macros.sty" src latex-macros #+Options: toc:nil num:t -* Chapter 1 +* Chapter 01 Walrasian auction: traders express commitment to trade at certain prices (liquidity provision) to auctioneer and auctioneer periodically clears market at intersection of supply and demand curve. -Market-making: market-maker (designated and are privileged) provides public quotes themselves and clears at a price that minimizes unsatisfied orders. market-maker's quotes helps coordination between sellers and buyers, a mechanism that Walrasian auction lacks. +Priority rules is needed when price is not continuous and Walrasian auction \(p^{*}\) cannot not be achieved, as some people won't get to trade. Continuous-time double auction: use a public limit order book (LOB) and a designated market-makers are no longer necessary. -Fundamental analysis attempts to decide whether an asset is over-priced or under-priced. Quantitative analysis (or technical analysis) attempts to predict price movements by identifying price patterns. - Empirically, dispersion of future price changes is much larger than the mean predicted price change, which means informed trades are either very rare and very successful or more common but with low degree of individual success due to weak information content. Price changes are close to being martingales. This implies that the real information contained in the signals used by traders or investors is extremely weak. The overall nagging feeling is that speculative traders trade too much, probably as a result of overestimating the predictive power of their signals and underestimating the costs of doing. @@ -24,13 +22,20 @@ In LOB modern markets, a simple separation of market participants could be: spec Speculators faces risk from inaccurate predictions and market-maker faces adverse selection risk. market-maker adjust quotes to mitigate adverse selection costs and thus causing trades to have price impact. market-maker then need a spread twice as wide as price impact to profit. -Distribution of price changes after a trade is very broad, with a heavy tail in the direction of the trade. This means while most trades are noisy, a small percentage have high information content. Overall this causes market-maker's P&L to be negatively skewed. Market-making is akin to selling insurance: although profitable on average, this strategy may generate enormous losses in the presence of informed traders. +Simple model for market maker shows return variance decreases with informed trade probability, but skewness increases in magnitude. +Break even spread is given by (assume uninformed flow has equal chance of being from either direction) +\begin{align*} + 0 + &= \frac{1}{2}(1 - \phi)s - \phi J,\\ + s + &= 2\phi J. +\end{align*} -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. +Market operates in a regime of small revealed liquidity but large latent liquidity as market makers face adverse selection. -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. +Scarcity of revealed liquidity forces large trades to be fragmented, which means prices cannot be in equilibrium in the traditional sense: 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 +* Chapter 02 ** 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. @@ -84,17 +89,17 @@ where \(\tau_{\eta}\) is the error correction time. \(\eta_{t}\) is often called \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. +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. This similarity between short term and long term volatility, while also displayed by simple random walks, can also be extremely unintuitive given rather large fundamental uncertainty about asset prices. ** 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}}, + &\propto \frac{1}{(r^{2} + a^{2})^{\frac{1 + \mu}{2}}} \to_{\abs{r} \ll a} \frac{1}{\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 \(\mu \to \infty\). +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 \(\sigma^{2}\), 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 +To account for volatility clustering, we decompose \(r_{t} = \sigma_{t}\xi_{t}\), where \(\xi_{t}\) is a unit variance IID random variable accounting for directional component and \(\sigma_{t}\) is a long memory positive random variable accounting for volatility component. A common form is \begin{align*} \sigma_{t} &= \sigma_{0}\E^{\omega_{t}}, @@ -105,12 +110,9 @@ where \(\omega_{t}\) is approximately Gaussian with variogram &= \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. +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 characteristic time scale is fixed: this means in reality, volatility fluctuations are multi-time scale. + +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. However, past volatility doesn't help much in prediction future returns. 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 \(\sim t^{-0.2}\)). @@ -119,14 +121,9 @@ Market activity also clusters. Define average market activity \(\mean{\phi}\) as \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). +where \(\dd N_{t}\) is number of price changes occurred during \([t, t + \dd t]\). The covariance \(\cov(\der[N_{t}]{t}, \der[N_{t + \tau}]{t})\) 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. +Prices in financial markets tend to exhibit two types of volatilities: small frequent moves and rare extreme moves. 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. @@ -138,7 +135,7 @@ Are prices fundamentally efficient (in the sense that they are always close to s 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. +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, but rather, originates from the price changes themselves. Seeing how long term volatility doesn't differ too drastically to higher frequency ones, microstructure can indeed be relevant to lower frequency price dynamics. 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.