# notes-TQR

Notes for /Trades, Quotes, and Prices/

git clone git://git.shimmy1996.com/notes-TQR.git
commit d6e413a49fca0789606f404e274a2ac654930a39
parent 9091c6cc8410904522b00c99a89fc59bbae238c4
Author: Shimmy Xu <shimmy.xu@shimmy1996.com>
Date:   Thu,  7 May 2020 09:34:59 -0500

Add notes for Chapter 03

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1 file changed, 47 insertions(+), 2 deletions(-)
diff --git a/notes.org b/notes.org
@@ -141,8 +141,53 @@ The decoupling between price and fundamental value allows a theory of price move

Flat volatility signature plot is a sign of statistical efficiency.

-** Errata
-Formula (2.11) should be
+* Chapter 03
+LOB is constrained to a predefined price and volume grid, resolutions of which are tick size and lot size. Orders in LOB are firm commitments organized in a queue (until canceled), and quantifies the visible liquidity.
+
+Each order is described by sign, price, volume, and submission time:
+\begin{align*}
+  x
+  &\defeq (\varepsilon_{x}, p_{x}, v_{x}, t_{x}).
+\end{align*}
+
+Evolution of orderbook is considered to be caglad (left continuous with right limits), so are new orders
+\begin{align*}
+  x
+  &\notin \cL(t_{x}),\\
+  x
+  &\in \overline{\cL}(t_{x}) = \lim_{t' \downarrow t_{x}}\cL(t'),
+\end{align*}
+and all relevant measures like bid $$b(t)$$, ask $$a(t)$$, mid $$m(t)$$, and spread $$s(t)$$.
+
+Think of market orders as orders that got matched upon submission, instead of orders with out a price specified (exchanges can fill in a price such that the entire order is matched), and those that don't match immediately are limit orders. Matching depends on the algorithm used.
+
+Limit orders in LOB is similar to an option contract written to the whole market, and are subject to the same adverse selection risk that market makers face. Limit orders typically earn profits when prices mean-revert but suffer losses when prices trend and drift beyond.
+
+When picking $$p_{x}$$ and $$v_{x}$$, we are limited to the LOB's predefined grid, which can be expressed as the resolution parameters lot size $$v_{0}$$ and tick size $$\vartheta$$.
+
+The tick size indicates how much more expensive it is for a trader to gain the priority associated with choosing a better price. Smaller tick sizes can lead to more volatile top layers of books. Relative tick size $$\vartheta_{r}$$ also comes into play as asset prices differ.
+
+Most LOB studies aggregates orders using same-side quote-relative coordinates, which expresses prices relative to the best quote on the same side:
+\begin{align*}
+  d(p_{x}, t)
+  &= \begin{cases}
+    b(t) - p_{x}, &\If x \text{ is a buy limit order},\\
+    p_{x} - b(t), &\If x \text{ is a sell limit order}.
+  \end{cases}
+\end{align*}
+Some times opposite-side quote-relative coordinates are used to avoid unwanted artifacts to appear (say division by 0):
+\begin{align*}
+  d^{\dagger}(p_{x}, t)
+  &= d(p_{x}, t) + s(t).
+\end{align*}
+
+Whenever such a matching occurs, it does so at the price of the limit orders, which can lead to price improvements.
+
+Most common priority rules include price-time and pro-rata. Priority rules do affect trader behaviors. Time priority rewards trades for displaying liquidity early, while pro-rata rewards traders for displaying more liquidity.
+
+The actions of traders in an LOB can be expressed solely in terms of the submission or cancellation of orders of lot size $$\v_{0}$$. Common order types include stop orders, iceberg orders, IOC/FOK orders, and market-on-close (MOC) orders. A MOC order is a market order that is submitted to execute as close to the closing price as possible, mostly to take advantage of the larger liquidity at close.
+
+For exchanges that are closed overnight, beginning and end of trading day may see so much volume that LOB trading would be prone to instabilities, so exchanges prefer to use auctions at those times. Auction clearing price is visible throughout, and traders can use this information to adjust their orders.
\begin{align*}
p_{t}
&= p_{0} + \mean{p}(\sum_{t' = 0}^{t - 1}r_{t'} + \eta_{t}).