Information-Based Trading

TL;DR

It is proved that the value of the first trader's position is strictly greater than that of the second, and the results are generalized to situations where (a) there is a hierarchy of traders, (b) there are multiple successive trades, and (c)there is inventory aversion.

Abstract

We consider a pair of traders in a market where the information available to the second trader is a strict subset of the information available to the first trader. The traders make prices based on the information available concerning a security that pays a random cash flow at a fixed time $T$ in the future. Market information is modelled in line with the scheme of Brody, Hughston & Macrina (2007, 2008) and Brody, Davis, Friedman & Hughston (2009). The risk-neutral distribution of the cash flow is known to the traders, who make prices with a fixed multiplicative bid-offer spread and report their prices to a game master who declares that a trade has been made when the bid price of one of the traders crosses the offer price of the other. We prove that the value of the first trader's position is strictly greater than that of the second. The results are analyzed by use of simulation studies and generalized to situations where (a) there is a hierarchy of traders, (b) there are multiple successive trades, and (c) there is inventory aversion.

Figures (8)

• Figure 1: Distribution of trading times under Scenario 3 based on 100,000 simulations, including both buys and sells. In this case $r=0$ and $p=0.8$.
• Figure 2: Heat chart of average profit as a function of the spread and the information flow rate. We look at per-trade profits on the left and per-session profits on the right. Here $r=0$ and $p=0.8$.
• Figure 3: The total profitability of Trader $A$ is illustrated in this example involving six trades, where $\epsilon_1=1$, $\epsilon_2=1$, $\epsilon_3=1$, $\epsilon_4=-1$, $\epsilon_5=-1$, $\epsilon_6=1$. A line leading up to a trading point indicates a buy at that point and a line leading down indicates a sell. The inventory rises to 3 at trade 3, then drops to 1 at trade 5, then rises to 2 at trade 6. Thus $\alpha_1=4$, $\alpha_2=5$, $\beta_1=3$, $\beta_2=2$. The value of Trader $A$'s position is given by the risk-neutral probability of the trade sequence multiplied by $(1-\phi^{-1}) + (1-\phi^{-2}\psi^{-1}) + (1-\phi^{-3}\psi^{-2}) - (1-\phi^{-2}\psi^{-3}) - (1-\phi^{-1}\psi^{-2}) + (1-\phi^{-2}\psi^{-1})$. We observe that since $\phi > \psi \geq 1$ the difference between the third trade and the fourth trade is positive, and likewise the difference between the second trade and the fifth trade is positive.
• Figure 4: Positivity of profitability for $\phi=1.02$ and $\psi_A=1.01$. The positivity of \ref{['eq: lemma ineq']} is shown for these parameters and for all combinations of buys and sells up to a total of ten trades ($m\leq10$). For $n\geq2$ we consider the binary representation of $n$ given by the binary number $1\upsilon_1\,\upsilon_2 \cdots \upsilon_m$ and we plot the value of \ref{['eq: lemma ineq']} for the sequence $(\epsilon_k)_{1\leq k\leq m}$ where $\epsilon_k=2\,\upsilon_k-1$ for $k\in\{1,\dots,\,m\}$.
• Figure 5: Trading dynamics when game master allows up to ten trades. The model parameters are $T=1$, $\sigma_B=1$, $\sigma_A=\sqrt{2}$, $\phi=1.02$, $\psi_A=1$, $\psi_B=1$ and $r=0$. A single outcome of chance is shown spanning the first one-tenth of the trading session. The top left panel shows the bid and offer quotes of Trader $A$ (red dashed line for offer, red solid line for bid) and for Trader $B$ (blue dash-dot line for offer, blue dotted line for bid). We indicate that a trade has taken place with an upward pointing arrow when Trader $A$ buys and a downward pointing arrow when Trader $A$ sells. The top right panel shows the inventories of Trader $A$ (dotted red line) and Trader $B$ (solid blue line). The bottom left panel shows the trajectories of $S^A_t$ (dotted red line) and $S_t^B$ (solid blue line). The bottom right panel shows the trajectory of the quotient of the quoted mid-prices and the boundaries $\phi^2$ and $\phi^{-2}$. A trade occurs whenever the quotient process hits a boundary.

Theorems & Definitions (24)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof