# In the Kyle Market Maker Model, an Insider sets the quantity of trades to maximize her profit.

## Question:

In the "Kyle Market Maker Model", an Insider sets the quantity of trades to maximize her profit. Let's assume the last trade price of a stock, oh say, TWTR is $33.03, and was the true price before the insider enters the market. An insider believes (based on perhaps credible knowledge of a pending acquisition announcement), that the true price would be $45.00 after the announcement (acquisition price contemplated were $54.20, with a probability of success 57%. So $45.00 is the probability weighted expected price post announcement.) Let's assume TWTR's price volatility is $0.05 between trades, and the observed trade volume is 5,000 shares, and volatility is 500 shares.

(Inspired by: https://en.wikipedia.org/wiki/Proposed_acquisition_of_Twitter_by_Elon_Musk)

What will be the optimal amount of shares for the insider to attempt to buy?

What is the price the insider can expect to buy at and the maximum profit expected by the insider after the merger announcement?

One of the most important assumptions in the Kyle model is MMs (Market Makers) cannot tell between the IT (insider) and LTs (liquidity traders). However, MMs as well as all participants can estimate a distribution of uninformed volume.

We suggested that the trading volume from uninformed LT's is a distributed with a mean of 5,000 shares and standard deviation of 500 shares. It is interesting and potentially important for IT and MMs to have a sense of how sensitive price impact is to the distribution assumptions of uninformed trades.

The optimal bid price that MM will post to make the market.

Please calculate optimal bid prices for mean volume ranging from 0 to 10,000 in steps of 1,000 shares, and volatility of volume ranging from 100 to 1,000 in steps of 100.

Is the optimal bid price more sensitive to the mean or volatility of volume of uninformed trades?

Insider realizes that this is a huge opportunity to profit, but she is budget constrained and not able to place the full volume of order (to buy shares). She, perhaps foolishly, decides to mortgage her house and buy 50,000 shares immediately, and then she would whisper the "rumor" about the pending acquisition announcements to some friends and family. She believes (again foolishly), she is sharing profit with her friends and family. Now, one of the MM got word of this. Since it is a rumor, he is assessing that the probability of the acquisition to go through is only 30%. Again the whisper acquisition price is $54.20. All other parameters stay the same.

What will this lucky MM's strategy be to take advantage of this whisper? That is what is the bid price and volume (bit size) he is going to place.

What is the total expected profit for the original insider, and what is the total profit for the lucky MM?

**Related Book For**

## Microeconomics An Intuitive Approach with Calculus

ISBN: 978-0538453257

1st edition

Authors: Thomas Nechyba