Prof. Guhr corrections

paper/financial_response_spread_year_paper/sections/02_abstract.tex

@@ -1,19 +1,20 @@
 \abstract{
 Recent research on the response of stock prices to trading activity revealed
-long lasting effects, even across stocks of different companies.
-These results imply non-markovian effects in price formation and when trading
-many stocks at the same time, in particular trading costs and price
-correlations.
+long lasting effects, even across stocks of different companies. These results
+imply non-markovian effects in price formation when trading many stocks at the
+same time, in particular trading costs and price correlations.
 How the price response is measured depends on data set and research focus.
 However, it is important to clarify, how the details of the price response
 definition modify the results. Here, we evaluate different price response
 implementations for the Trades and Quotes (TAQ) data set from the NASDAQ stock
 market and find that the results are qualitatevily the same for two different
 definitions of time scale, but the response can vary by up to a factor of two.
-Further, we confirm the dominating contribution of immediate price response
-directly after a trade, as we find that delayed responses are suppresed.
-Finally, we test the impact of the spread in the price response, detecting that
-large spreads have stronger impact.
+Further, we show the key importance of the order between trade signs and
+returns, displaying the changes in the signal strenght. Moreover, we confirm
+the dominating contribution of immediate price response directly after a trade,
+as we find that delayed responses are suppresed. Finally, we test the impact of
+the spread in the price response, detecting that large spreads have stronger
+impact.
 \PACS{
       {89.65.Gh}{Econophysics} \and
       {89.75.-k}{Complex systems} \and

paper/financial_response_spread_year_paper/sections/03_introduction.tex

@@ -2,8 +2,8 @@ \section{Introduction}\label{sec:introduction}
 
 Financial markets use order books to list the number of shares bid or asked at
 each price. An order book is an electronic list of buy and sell orders for a
-specific security or financial instrument organized by price level, agents can
-place different types of instructions (orders).
+specific security or financial instrument organized by price levels, where
+agents can place different types of instructions (orders).
 
 In general, the dynamics of the prices follow a random walk. There are two
 extreme models that can describe this behavior: the Efficient Market Hypothesis
@@ -20,7 +20,7 @@ \section{Introduction}\label{sec:introduction}
 effects due to stategies or liquidity costs are not contained either.
 
 There are diverse studies focused on the price response
-\cite{prop_order_book,dissecting_cross,r_walks_liquidity,subtle_nature,Bouchaud_2004,large_prices_changes,pow_law_dist,theory_market_impact,spread_changes_affect,master_curve,EMH_lillo,quant_stock_price_response,ori_pow_law,Wang_2018_b,Wang_2018_a,Wang_2016_avg,Wang_2016_cross},
+\cite{prop_order_book,dissecting_cross,r_walks_liquidity,subtle_nature,Bouchaud_2004,large_prices_changes,pow_law_dist,theory_market_impact,spread_changes_affect,master_curve,EMH_lillo,quant_stock_price_response,ori_pow_law,Wang_2018_b,Wang_2018_a,Wang_2016_avg,Wang_2016_cross}.
 In our opinion, a critical investigation of definitions and methods and how
 they affect the results is called for.
 

paper/financial_response_spread_year_paper/sections/05_response_definitions.tex

@@ -29,7 +29,7 @@ \subsection{Key concepts}\label{subsec:key_concepts}
 \cite{prop_order_book,subtle_nature,account_spread,limit_ord_spread,stat_theory}.
 The price gap between them is called the spread
 $s\left(t\right) = a\left(t\right)-b\left(t\right)$
-\cite{subtle_nature,market_digest,Bouchaud_2004,account_spread,em_stylized_facts,large_prices_changes,stat_theory}.
+\cite{subtle_nature,market_digest,Bouchaud_2004,account_spread,large_prices_changes,em_stylized_facts,stat_theory}.
 Spreads are significantly positively related to price and significantly
 negatively related to trading volume. Companies with more liquidity tend to
 have lower spreads
@@ -61,7 +61,7 @@ \subsection{Key concepts}\label{subsec:key_concepts}
     r^{\left(l\right)}\left(t,\tau\right) = \ln S\left(t + \tau\right)
     - \ln S\left(t\right) = \ln \frac{S\left(t + \tau\right)}{S\left(t\right)}
 \end{equation}
-Equation \ref{eq:return_general} and Eq. \ref{eq:log_return_general} agree if
+Equations (\ref{eq:return_general}) and (\ref{eq:log_return_general}) agree if
 $\tau$ is small enough \cite{subtle_nature,empirical_facts}.
 
 At longer timescales, midpoint prices and transaction prices rarely differ by
@@ -102,8 +102,8 @@ \subsection{Key concepts}\label{subsec:key_concepts}
 large autocorrelation. A very plausible explanation of this phenomenon relies
 on the execution strategies of some major brokers on given markets. These
 brokers have large transactions to execute on the account of some clients. In
-order to avoid market making move because of an inconsiderable large order,
-they tend to split large orders into small ones \cite{empirical_facts}.
+order to avoid market making movements because of an inconsiderable large
+order, they tend to split large orders into small ones \cite{empirical_facts}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Time definition}\label{subsec:time_definition}
@@ -169,7 +169,7 @@ \subsubsection*{Trade time scale}\label{subsubsec:trade_time}
     \begin{array}{cc}
     \text{sgn}\left(S\left(t,n\right)-S\left(t,n-1\right)\right),
     & \text{if }\\ S\left(t,n\right) \ne S\left(t,n-1\right)\\
-    \varepsilon^{t}\left(t,n-1\right),
+    \varepsilon^{\left(t\right)}\left(t,n-1\right),
     & \text{otherwise}
     \end{array}\right.
 \end{equation}
@@ -227,8 +227,8 @@ \subsubsection*{Trade time scale}\label{subsubsec:trade_time}
 trade time scale. A drawback in the computation could come from the fact that
 the return of a given second is composed by the contribution of small returns
 corresponding to each change in the midpoint price during a second. As we are
-assuming only one value for the returns in each second, we are supposing  all
-the returns in one second interval to be positive or negative with the same
+assuming only one value for the returns in each second, we consider all the
+returns in one second interval to be positive or negative with the same
 magnitude, which could not be the case. This could increase or decrease the
 response signal at the end of the computation.
 
@@ -249,7 +249,7 @@ \subsubsection*{Trade time scale}\label{subsubsec:trade_time}
 \end{figure}
 
 Figure \ref{fig:return_contributions} illustrate this point. Suppose there are
-three different midpoint prices in a one second interval and thus, three
+three different midpoint prices in one second interval and thus, three
 different returns for these three midpoint price values. Furthermore, suppose
 that the volume of limit orders with the corresponding midpoint prices are the
 same in the bid and in the ask (the returns have the same magnitude). In the
@@ -279,23 +279,23 @@ \subsubsection*{Physical time scale}\label{subsubsec:physical_time}
 We use the trade sign definition in physical time scale proposed in Ref.
 \cite{Wang_2016_cross} and used in Refs.
 \cite{Wang_2017,Wang_2016_avg}, that depends on the classification in
-Eq. \ref{eq:trade_signs_trade} and reads
+Eq. (\ref{eq:trade_signs_trade}) and reads
 \begin{equation}\label{eq:trade_signs_physical}
     \varepsilon^{\left(p\right)}\left(t\right)=\left\{
     \begin{array}{cc}
-    \text{sgn}\left(\sum_{n=1}^{N\left(t\right)}\varepsilon^{t}
+    \text{sgn}\left(\sum_{n=1}^{N\left(t\right)}\varepsilon^{\left(t\right)}
     \left(t,n\right)\right),
     & \text{If }N \left(t\right)>0\\
     0, & \text{If }N\left(t\right)=0
     \end{array}\right.
 \end{equation}
-Where $N \left(t \right)$ is the number of trades in a second interval.
+where $N \left(t \right)$ is the number of trades in a second interval.
 $\varepsilon^{\left(p\right)}\left( t \right) = +1$ implies that the majority
 of trades in second $t$ were triggered by a market order to buy, and a value
 $\varepsilon^{\left(p\right)}\left( t \right) = -1$ indicates a majority of
 sell market orders. In this definition, they are two ways to obtain
 $\varepsilon^{\left(p\right)}\left( t \right) = 0$. One way is that in a
-particular second there is not trades, and then no trade sign. The other way is
+particular second there are no trades, and then no trade sign. The other way is
 that the addition of the trade signs ($+1$ and $-1$) in a second be equal to
 zero. In this case, there is a balance of buy and sell market orders.
 

paper/financial_response_spread_year_paper/sections/06_response_implementations.tex

@@ -31,7 +31,7 @@ \section{Price response function implementations}
     \varepsilon_{j,d}^{\left(p\right)}\left(t\right)&=
     \text{sgn}\left(E_{j,d}\left(t\right)\right)\label{eq:sign_sum_trades}
 \end{align}
-Where the subscript $d$ refers to the days used in the response computation.
+where the subscript $d$ refers to the days used in the response computation.
 We use Eq. (\ref{eq:sum_trades}) to make easier the comparison between the
 results, as all the defined responses use the trade sign term.
 
@@ -46,16 +46,16 @@ \subsection{Response functions on trade time scale}
 \label{subsec:response_function_trade}
 
 We define the self- and cross-response functions in trade time scale, using the
-trade signs in trade time scale and for the returns, we select the last
-midpoint price of every second and compute them. We use this strategy with the
-TAQ data set considering that the price response in trade time scale can not
-be directly compared with the price response in physical time scale. In this
-case we relate each trade sign in one second with the midpoint price of the
-previous second. Then, to compute the returns, instead of using trades as the
-time lag (it would make no sense as all the midpoint price are the same in one
-second) we use seconds. Thus, we force the response in trade time scale to have
-a physical time lag, and then, be able to compare with the physical time scale
-response. This approximation is feasible considering the discussion in Sect.
+trade signs in trade time scale. For the returns, we select the last midpoint
+price of every second and compute them. We use this strategy with the TAQ data
+set considering that the price response in trade time scale can not be directly
+compared with the price response in physical time scale. In this case we relate
+each trade sign in one second with the midpoint price of the previous second.
+Then, to compute the returns, instead of using trades as the time lag (it would
+make no sense as all the midpoint price are the same in one second) we use
+seconds. Thus, we force the response in trade time scale to have a physical
+time lag, and then, be able to compare with the physical time scale response.
+This approximation is feasible considering the discussion in Sect.
 \ref{sec:response_functions_def}. The price response function in trade time
 scaled is defined as
 \begin{equation}\label{eq:response_functions_trade_scale_general}
@@ -286,12 +286,16 @@ \subsection{Activity response functions on physical time scale}
 As predicted by the weights, the event response is weaker than the physical
 response, and the activity response is the strongest response.
 
-In the three curves in the figure can be seen the increase-decrease behavior of
-the response functions.
+We propose a methodology to directly compare price response functions in trade
+time scale and physical time scale. Additionally, we suggest a new definition
+to measure the impact of the number of trades in physical time scale. In the
+three curves in the figure can be seen the increase-decrease behavior of the
+response functions.
 
 Our results are consistent with the current literature, where the results
 differ about a factor of two depending on the time scale. We note that the
 activity response function implementation is only a test and was never defined
-in previous works. That is why the factor of two difference can not be seen in
-Fig. \ref{fig:relation_responses} for
+in previous works. That is why the difference of a factor of two can not be
+seen in Fig. \ref{fig:relation_responses} for
 $R_{ij}^{\left(p,a\right)} \left( \tau \right)$.
+

paper/financial_response_spread_year_paper/sections/07_time_shift.tex

@@ -96,7 +96,7 @@ \subsection{Trade time scale shift response functions}
 obtained in Fig. \ref{fig:response_function_trade_scale}, it looks like the
 response function values with large time shift are stronger. However, this is
 an effect of the averaging of the functions. As the returns and trade signs are
-shifted, they are less values to average, and then the signals are stronger.
+shifted, there are less values to average, and then the signals are stronger.
 Anyway, the figure shows the importance of the position order between the trade
 signs and returns to compute the response function.
 

paper/financial_response_spread_year_paper/sections/08_short_long.tex

@@ -103,7 +103,11 @@ \section{Time lag analysis}\label{sec:short_long}
 the time lag $\tau = 10^{3}$, the peak is usually between $\tau = 10^{1}$ and
 $\tau = 10^{2}$. In these cases the long response are always negative after the
 $\tau'$ value and is comparable in magnitude with the random signal.
-
 On the other hand, the response functions that requires a bigger time lag to
 show the increase-decrease behavior, have non negative long responses, but
-still they are comparable in magnitude with the random signal.
+still they are comparable in magnitude with the random signal.
+
+According to our results, price response functions show a large impact on the
+first instants of the time lag. We proposed a new methodology to measure this
+effect and evaluate their consequences. The short response dominates the
+signal, and the long response vanishes.

paper/financial_response_spread_year_paper/sections/09_spread_impact.tex

@@ -42,7 +42,7 @@ \section{Spread impact in price response functions}\label{sec:spread_impact}
 trade signs. As long as the stock is liquid, the number of trade signs grow.
 Thus, at the moment of the averaging, the large amount of trades, reduces the
 response function signal. Therefore, the response function decrease as long as
-the liquidity grows. And as stated in the introduction the spread is negatively
+the liquidity grows, and as stated in the introduction the spread is negatively
 related to trading volume, hence, firms with more liquidity tend to have lower
 spreads.
 

paper/financial_response_spread_year_paper/sections/10_conclusion.tex

@@ -10,21 +10,22 @@ \section{Conclusion}\label{sec:conclusion}
 The classification and sampling of the data had impact on the results, making
 them smoother or stronger, but always keeping their shape and behavior.
 
-The response functions were analyzed according to the time scales. To compare
-price response functions from different scales, we used the same midpoint
-prices in physical time scale with the corresponding trade signs in trade time
-scale or physical time scale. This assumption allowed us to get an idea of how
+The response functions were analyzed according to the time scales. We proposed
+a new approach to compare price response functions from different scales. We
+used the same midpoint prices in physical time scale with the corresponding
+trade signs in trade time scale or physical time scale. This assumption allowed
+us to compare both price response functions and get an idea of how
 representative was the behavior obtained in both cases. For trade time scale,
 the signal is weaker due to the large averaging values from all the trades in a
 year. In the physical time scale, the response functions had less noise and
-their signal were stronger. The activity in every second highly impact the
-responses. As the response functions can not grow indefinitely with the time
-lag, they increase to a peak, to then decrease. It can be seen that the market
-needs time to react and revert the growing. In both time scale cases depending
-on the stocks, two characteristics behavior were shown. In one, the time lag
-was large enough to show the complete increase-decrease behavior. In the other
-case, the time lag was not enough, so some stocks only showed the growing
-behavior.
+their signal were stronger. We proposed an activity response to measure how the
+number of trades in every second highly impact the responses. As the response
+functions can not grow indefinitely with the time lag, they increase to a peak,
+to then decrease. It can be seen that the market needs time to react and revert
+the growing. In both time scale cases depending on the stocks, two
+characteristics behavior were shown. In one, the time lag was large enough to
+show the complete increase-decrease behavior. In the other case, the time lag
+was not enough, so some stocks only showed the growing behavior.
 
 We modify the response function to add a time shift parameter. With this
 parameter we wanted to analyze the importance in the order of the relation
@@ -39,14 +40,16 @@ \section{Conclusion}\label{sec:conclusion}
 time shift should be a value between $t_{s} = \left(0,2\right]$ time steps.
 
 We analyzed the impact of the time lag in the response functions. We divided
-the time lag in a short and long time lag. The response function that depended
-on the short time lag, showed a stronger response. The long response function
-depending on the stock could take negative and non-negative values. However, in
-general the influence were not intense.
+the time lag in a short and long time lag. With this division we adapted the
+price response function in physical time scale. The response function that
+depended on the short time lag, showed a stronger response. The long response
+function vanish, and depending on the stock could take negative and
+non-negative values comparable to a random signal.
 
-Finally, we checked the spread impact in self-response functions. We divided
-524 stocks from the NASDAQ stock market in three groups depending on the year
-average spread of every stock. The response functions signal were stronger for
-the group of stocks with the larger spreads and weaker for the group of stocks
-with the smaller spreads. A general average price response behavior was spotted
-for the three groups, suggesting a market effect on the stocks.
+Finally, we checked the spread impact in price self-response functions. We
+divided 524 stocks from the NASDAQ stock market in three groups depending on
+the year average spread of every stock. The response functions signal were
+stronger for the group of stocks with the larger spreads and weaker for the
+group of stocks with the smaller spreads. A general average price response
+behavior was spotted for the three groups, suggesting a market effect on the
+stocks.