Practical Math : Auctions - Introduction

stempede (57)in #steemstem • 7 years ago (edited)

Hi, this is I hope the beginning of Maths Friday (today a little late). I'll try to pick your interest, enrich your life on the subject and give you the tools to advance on your own easily. Feel free to ask any questions.

^{[Wiki: bulle und bär, Frankfurt]}

Price discovery is a signaling system where different actors decide on a way to valuate an asset whose intrinsic value is ignored due to limited supply. Whoever meets the price with an offer gets the item.

Auctions are systems that inherently increase prices, as the price is not static its a horrible system to valuate everyday items (bread, toilet paper, shampoo) but great for collectibles. They are games of incomplete information (Bayesian games), where whoever has the most and best information wins. "Although, big corporations competing with each other engage in this auctions and the consumer benefits in some markets, see perfect competition"

This often means assuming buyers and sellers as competitive price takers. Knows as the Competitive Market Hypothesis

Auctions are the natural way economies work, in fact, they must be forbidden by law otherwise business would speculate during natural disasters.

Auctions can be clasified as primary (single item) or secondary (multiple items that can be equal or different).

Out of the primary there are 4 types that are the most common. English auction, Sealed first price auction, Ductch auction and Vickrey's auction.

1. Oral ascending auction:

^{[Southeby's auction]}^A^, ^B

Depending on the available information, bidders might collude or compete to give the highest price, until noone else is willing to outbid the offer. All participants know the higest bid.

When extended in a double auction, the Market Price is set when offer and price asking match. The group of pending orders and its difference is known as Bid-ask spread and it shows how liquid is a market. This is a really simple market theory and can be easily disrupted to make it more complex when participants become more sophisticated or speed of transactions increase.

In the case of low liquidity markets price increase and fall quickly. This figure also depends on the relative leverage of the position.

$\frac{X\rho }{X-\rho } = [+P] \; \; \; \; \; \; ; \; \; \; \; \; \; \frac{100P}{x+P} = [-\rho ]$

Where X is the price at any given moment, p is the percentage a stock falls, +P the price it would need to increase to arrive at the same level after the fall.

Example: You have a stock at $10. If it goes down by -20% you would need an increase of +25% to arrive to the same place. This means less liquidity in is necesary to increase to percentage and more liquidity to decrease the percentage on the way down. $\frac{100*0.2}{100-0.2} = 0.25$

This obeys, normally, a sigmoid function $S(t) = \frac{1}{1+e^{-t}}$ in a logistic way. It has price level buffers, and once the price excedes a certain rate the changes become dramaticaly fast.

^{[People lose conciousness bellow that lower star]}

One example of this is the Saturation curve of hemoglobing. Normally if your oxigen saturation falls bellows 80% it will go in free fall! is a scary number for specialists in emergency rooms.

Those buffers can also be calculated borrowing from principles of chemistry like the limitant reactant although more seasoned trading practices use more sofisticated pricing models using physics like cosidering the price as part of wave functions.

This is the reason stop loss and stop-gain orders exist. Normally with heuristics like -5% stop loss and +10% stop gain based on risk/reward ratios and support (buffer) levels.

Now, if the highest bid is $9.9 and the minimum ask is $10.1, this means there's a $0.2 spread.

But if someone immediately after makes a lower ask, say $10.05, this decreases the spread by 50% making the spread smaller.

One must take into account not only the prize but also the size of the orders. In that case, you would have to account for more than just time. Such model S(t,x). Such that the price of the asset over time t if x shares are bought (x>0) or sold (x<0). So as the price of x increases S(t,x) should be equal or higher.

2. Sealed First bid auction (pay as bid)

In this system all the bids are given normally a single chance to bid. They normally can't share information and the highest bid wins. Normally governments use this system to diminish price collusion.

3. Sealed Second bid Auction (vikrey's auction)

In this case, the highest bidder wins the auction but pays the price offered by the second bidder. It works to a stop loss. It leaves the sensation that the seller is missing out on that highest bid but in reality is not, most of the time.

4. Dutch Auction

It starts high and diminishes in price. It was used during the tulips auctions and recently was used in the ICO of the Lighting Network for the Ethereum protocol. This type of auction is designed to make people overpay. While it's equal mathematically to the sealed bid auctions, the information of the price decreasing affects the bidders making them jump in early.