Posts Tagged: Research

EC4004 — No Comments
23
Sep 08

EC4004 Lecture 5: Uncertainty

CHONGQING, CHINA - OCTOBER 9: A black swan re...

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This is the second part of the course, where we look at strategic behaviour under uncertainty. Lectures one, two, three, and four were all about establishing the theory of demand under certain conditions. We don’t live in a world of certainty, but its exact opposite, so we must change the theory to account for the presence of the unknown in economic life. We have not been too successful at this, but the theory remains, and we will learn it.

The mainstream theory of how we deal with choice under uncertainty and our interactions with others is built on probability theory. 

Definition (Probability). The probability of an event occurring is the relative frequency with which it occurs. 

Example. Toss a coin a large number of times, and the likelihood you get Heads will be 1/2. Roll a dice a large number of times, and you will roll a 6 on average 1/6 of the time.

Definition (Expected Value). The expected value of a game with several uncertain outcomes is the size of the prize the winner will get, on average. It is the average outcome from an uncertain gamble. 

Example. In a 2-player game of tossing a coin and predicting whether it comes up heads (X_{1}) or tails (X_{2}), with payoffs to each player being +1 and -1 respectively, then the expected value of the game is 

0.5 X_{1}+ 0.5X_{2} = 0.5 \times (1) + 0.5 \times (-1) = 0


So in this game, neither player would make out too well over time. 

Risk Aversion. People prefer less risk, especially when the stakes are quite high. The reason is the interaction of the potential payoff (the X above) and the probability of obtaining that payoff. We say there is diminishing marginal utility from the consumption of further and further amounts of risk. In terms of pricing risk, we can look at Figure 1 below:

Here we see the utility associated with a certain level of risk is concave to the origin—meaning when things get more risky, the person is less likely to derive a correspondingly high utility from that risky situation. If the person were truly risk-neutral, then they would pay D-E for a gamble, but they don’t, because for them, the bet simply is not worth it. As the HET site writes:

Notice by comparing points D and E in Figure 1 that the concavity of the elementary utility function implies that the utility of expected income, u[E(z)] is greater than expected utility E(u), i.e. u[pz1 + (1-p)z2] > pu(z1) + (1-p)u(z2). This represents the utility-decreasing aspects of pure risk-bearing. We can think of it this way. Suppose there are two lotteries, one that pays E(z) with certainty and another that pays z1 or z2 with probabilities (p, 1-p) respectively. Reverting to our von Neumann-Morgenstern notation, the utility of the first lottery is U(E(z)) = u(E(z)) as E(z) is received with certainty; the utility of the second lottery is U(z1, z2; p, 1-p) = pu(z1) + (1-p)u(z2). Now, the expected income in both lotteries is the same, yet it is obvious that if an agent is generally averse to risk he would prefer E(z) with certainty than E(z) with uncertainty, i.e. he would choose the first lottery over the second. This is what is captured in Figure 1 as u[E(z)] > E(u).


We have many ways to cope with this type of risk, but the three most common are:

  1. Insurance
  2. Diversification
  3. Options


We’ll go through each of these in class with a demonstration about the market for used cars. 

Not that there is always and everywhere a fundamental uncertainty—we can’t know what we don’t know. For more on this, see The Black Swan by Nassim Taleb, or my review of that book here. 

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Research — 1 Comment
1
Aug 08

Limit Order Only Markets

Levy distributionImage via Wikipedia

Attention conservation notice: apologies for all the \LaTeX code/typos/stupidity here, I’m working on some notes on Limit Orders for a paper I’m writing. Any comments most welcome.

A limit order is an instruction to trade a specific quantity of an asset at a specified price, or a better price. The order is an ex-ante pre-commitment (t, j, x, p) made on a date t to trade up to a given amount x of an asset j at a prespecified limit price p. The order is in force until filled or cancelled, so unfilled orders accumulate in an limit order book (Kyle, 1989).

Limit orders differ from standard market orders, which are requests to trade immediately at the best available price in the market. In a limit order setting the order execution is always filled at the limit price set by the buyer or seller. In limit orders, price priority holds, which means the limit orders offering better terms of trade (buyers buying higher, sellers selling lower) execute ahead of limit orders at worse prices. Time priority can also hold, where at each price p, older limit orders are executed ahead of newer ones in a queuing system which thus rewards first movers who give up liquid positions to allow the LOOM to work more effectively. The price and time priorities may be taken together to define a probability distribution over execution timing. The notion of equilibrium in limit order markets is different to other continuously clearing markets, because buyers arrive and trade asynchronously in a limit order market, so there is no unique market-wide `market-clearing‘ price, except in degenerate cases. Rather, there is a sequence of bilateral transaction prices at which endogenously matched pairs of investors choose to trade over time.

A limit order to buy 100 shares can be filled at \$47.50 or below. A limit order to sell at \$50.25 can be filled at \$50.25 and above. The existence of the limit order is independent of having anyone to actually trade to fulfil the order. As time passes, one might not be able to fill the order. There several advantages to limit order markets. The first is the ability to obtain a better price within tolerant risk boundaries as defined or perceived by the buyer or seller. The second is the discrete nature of each trade: a limit order to buy simply means you specify how much of something you want, and and how much you are willing to pay for it, and vice versa for the supplier. Both demand and supply are discrete functions of price. Any combination of these orders in a specific time and place gives rise to a limit order only interupted market where the orders are matched up for execution (Osborne, 1977, Chapter 2) is the classic exposition of LOOM). The third advantage of the LOOM structure is it’s simplicity in execution: one can easily define a greedy algorithm for such a market.

The execution of a simple market algorithm will always generate either a highest bid (if there are no sell orders), a lowest offer (no buy orders), a quote, (bid and offer but no transaction) or a transaction at a unique price. The only requirement is the orders have finite price limits on them. The market algorithm puts the orders to buy and sell in ranks, applies an ambiguity rule to ensure there is a unique price.

And this algorithm will look something like this:

Assigning transactions Algorithm

1. Begin. Sort buyers in descending order by the suggested price per unit.

2. Sort sellers in ascending order by the requested price per unit.

3. Traverse the two lists from top to bottom, and perform transactions while maintaining the total amount sold equal to the total amount bought. Repeat.

4. Stop when the offered price in the buyer’s list is lower than the requested price in the seller’s list.

Next bit of the paper—simulate this thing, and see what happens. Then apply it somewhere cool

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