definition of mid price in literature

Question

In literature, the mid-price is often used along with the terms "fair value", "true value" among others. I take it alot of the times it means the same thing because the mid-price doesnt necessarily have to between the best bid and best ask. Is this generally true or more or less at the realm of the author to define.

Answer 1

As someone who has contributed to literature, I am purposefully vague with the use of mid price. Not that I don't define it but that it is difficult to state which definition is the best in which context. Here are an example of a few definitions of mid price:

Last Trade: The physical price at which the most recent trade physically took place. This is therefore known to have been the most recent price at which buyers and sellers agreed. Often has large discrepancy as can only print a tradeable price, and depending upon time passed can not be representative of current market state.

Average Price: The current average of the best bid and best offer. Less discrepancy between above due to averaging and more reactive to current market but can continually mis-price trades which are consistently trading on one side of the bid-offer.

Weight Price: Considers the bid-size and offer-size of the best bids and offers. Usually is more accurate but is very sensitive to manipulation in less liquid markets (I switched away from this choice of mid market in some instruments I made markets for when customers began manipulating on screen exchange prices)

Depth Weighted Prices: you can use 2nd, 3rd, etc. depth of bids and offers to aggregate more weighted prices, by considering the average of the prices at which you would have to pay (or receive) for each quantity of instrument. This is a short description of a more mathematical technique. A study I performed for interest rates swaps on a millisecond trading exchange with some excellent data offered to me showed this to be one of the most reliable and accurate methods.

Instinct: Not all prices are exchange or electronically quoted so in this case you can rely on human pattern recognition and assimilation of numerous qualitative features to assess where you believe the next trade will occur, which is a valid assessment of mid market.

Multiple Loss Function: If you have numerous prices with relative prices, each with bid offer you can setup a modelling procedure to optimise the mid market values such that the difference to the implied weighted mid markets.

e.g.
price 1: 1 / 2
price 2: 3 / 4
price 3: 6 / 7
price 2 minus price 1 (a spread trade): 1.75 / 2
price 3 minus price 2 (a spread trade): 3 / 3.25
2*price 2 minus price 1 minus price 3: -1.25 / -1.0

Uncertainty: A big misconception is that the mid-market price is a fixed definite value. It is better represented as a probability distribution of next trading price with a mean and some uncertainty around that mean.

====== Edited for comment ======

Im sure there are different ways of considering depth weighted markets. This is one measure I performed:

As a mid price take the mid between the average buy price and average sell price measured over various different volumes

E.g. Suppose the state of the market is
Bid Size | Bid | Offer | Offer Size
1 | 10 | 11 | 3
4 | 9 | 12 | 3

So if I were to buy:
1: total cost is 11
2: total cost is 11
3: total cost is 11
4: total cost is 11.25
5: total cost is 11.40
Mean over all volumes: 11.13

And if I were to sell:
1: total cost is 10
2: total cost is 9.5
3: total cost is 9.333
4: total cost is 9.25
5: total cost is 9.2
Mean over all volumes: 9.457

And the mid of these is: 10.2935
Compared with the weighted mid using only first depth: 10.25

The sensitivity of these prices to the first depth is obviously different, and so their susceptibility to this form of manipulation is different. Note you can also use this method in combination with multiple loss function and have a very reliable set of mid prices.

You will notice that I use only 1-5 volumes since the bid only had total 5, this is a consideration (hyper parameter) of the model as to how much of the depth you are able utilise, continually at all hours of operation.

Answer 2

In a dealer market the public is always overcharged when they try to buy, and receives less than value when they try to sell (the dealer makes a living from the difference). It is reasonable to assume the midpoint between these prices, i.e. between the bid and the ask, represents the unobservable "fair value" for a small amount of the asset. It is just an assumption based on symmetry.