Since the market maker is adjusting his beliefs according to Bayes’ theorem, his beliefs and thus, subsequently, his prices will evolve over the course of the trading day. The
55 movements in these beliefs and prices correspond to his learning about the true population
distribution given the underlying observed information. In Figure 2, all trader actions are constant, except in the case in which there is a realisation of a high value compared to a low value. In this situation, the informed will buy the asset; as opposed to selling it had the asset’s value be low. Thus, the probability of a buy is only dependent upon a realised value outcome. Moreover, the traders are serially independent throughout the trading day, which means that these probabilities will remain constant (until the market maker’s prices converge to the true value). This is demonstrated in equations (3.13) and (3.14) which state that at any given point in time, i, the probability of a buy is equal to the probability of another buy at any other given point in time, j.
(
0) (
0) ( )
Pr Buyi SΩ∩V =Pr Buyj SΩ∩V = + −μ 1 μ γ ∀i j, (3.13)
(
0) (
0) ( )
Pr Buyi SΩ∩V =Pr Buyj SΩ∩V = −1 μ γ ∀i j, (3.14)
This result then suggests that the probability of observing b amount of buys in n total number of trades will, thus, form a binomial distribution condition upon the realised value of the asset, V, and any given signal outcome S0Ω:
(
0) ( ) ( (
1) )
Pr Buys , 1
1
b n b if V V
b n S V p p where p
if V V
μ μ γ
μ γ
Ω∩ = − − = ⎨⎧ + −⎪⎪⎩ − == (3.15)
56 Thus, the probability of observing b number of buys, conditional only upon n trades and the
signal outcome, is the weighted average of the two binomial distributions according to the probabilities associated with the realisation of V,
( ) ( ) ( ) ( ) ( )
Equation (3.16) is graphically demonstrated in Figure 3 below. The number of observed trades, n, is equal to 100, the probability of an uninformed trader buying the asset is fixed at γ= 0.5, the probability that the asset is high is θ= 0.5, whilst the proportion of informed trading varies, specifically μ∈{0, 0.5, 0.9}. The interpretation of the figure is that if
(
or)
V V= V V= , as the proportion of informed traders increases, then the market maker should notice an increasingly (decreasingly) disproportionate number of buys to sells out of all given observed trades, n. Interestingly, the distribution becomes bimodal as the proportion of informed trading in the market increases, which allows the market maker to revise his prior belief of the true asset’s value as he observes any given amount of buys and sells throughout the trading day.
57 Figure 3 Distribution of buys over 100 trades
Binomial distribution for the expected number of buys conditional on the true value. The proportion of uninformed trading is kept constant at γ =0.5, the probability of a high value of the asset occurring is θ =0.5. The binomial distributions are then graphed according to the proportion of informed traders in the market, which has been shown for μ∈{0, 0.5, 0.9}
The process through which the market maker adjusts his beliefs about the asset’s value is thus conditional upon the transaction history. Specifically, he will adjust his belief after each observed trade, conditional upon that most recent action and his initial information set.
The sequential revision process is thus described by the following process. The market maker sets an initial spread based on his prior belief, which will be conditional upon the receipt of a signal or not. A trader arrives at the market and transacts at one of the market maker’s quotes. The market maker revises his belief about the asset’s value, now conditional upon the direction of the recent trade, and subsequently revises his quotes, at which point another trader arrives and trades, and then the market maker will revise his beliefs again such that this process will be repeated.
58 This recursion process is clear in the expression for 0
1
S
θAΩ from equation (3.1) as it maps the prior probability, θ0S0Ω, into a posterior probability conditional upon the direction of the
trader action. To illustrate this recursive pattern in general form, let θASk0Ωdenote the market maker’s posterior probability of a high value occurring given all past information until time
1 transacted trade direction, A ; where the market maker’s initial information set is given as k his prior belief conditional upon the receipt of a signal, θ0S0Ω =Pr
(
V S0Ω)
. Thus,Equation (3.17) can be expressed as such because all probabilities in the decision tree, except for θASiΩ, are constant across time. This equation defines the stochastic process of beliefs and demonstrates, based on observed trades, how the market maker will adjust the probability that he places on a high value of the asset occurring, conditional upon his initial signal outcome. Observing more buy actions than sell actions, for example, could possibly mean that the informed participants are trading on the realisation of a high value, which suggests that the market maker needs to increase θASk0Ω. However, there is a possibility that
59 this increased number of buys could be the result of a random draw of uninformed trading,
which would then suggest that 0
k
S
θAΩ should remain the same. Hence, the market maker will need to adjust θASk0Ωdependent upon the likelihood of each possibility occurring.29
Another implication of this learning process is that it is first influenced by the receipt of a signal; if the market maker initially receives a high signal (low signal), then his belief that a high value has occurred will be higher (lower) given the observed trade history, than otherwise. The revision in the market maker’s beliefs will subsequently affect the direction of the market maker’s prices. Thus, for the market maker’s belief of a high value occurring as defined by (3.17), his bid and ask prices at any time t k= − will now be defined as: 1
29 Specifically, the reason why this is so is because the probability of observing a buy is higher when the asset’s value is high, and, conversely, lower if the asset’s value is low. Thus, an observation of a buy at any one point in time will always cause an upward revision in his belief that the value of the asset is high.
Similarly, if he observes a sell then this will cause a downward revision in that same belief.
60
Note that given the parameter distributions, the spread as defined in (3.20) will form a convex function with maximum value defined as the unconditional expectation of the value of the asset, V*, and minimum values at V and V . The reason this is so is because the market maker becomes more confident about the true value. In other words, a buy or a sell will add little to the market maker’s belief about the true value of the asset as he becomes increasingly confident about the realisation of the true value of the asset.
As the market maker’s belief changes, so will his prices.30 The expected change in his prices across time will follow the notional theory of semi-strong form market efficiency as proved in Proposition 1 below.
Proposition 1. Transaction prices form a martingale relative to the market maker’s information set.
Proof: Denote P as the most recently transacted price. From (3.17), t Pt = ⎣E V⎡ θASt0Ω⎤⎦
30 These changes in prices can be thought of as market impact from trades. From an empirical standpoint, this implies that the observed price impact in the market can be related back to this model and hence be used as a proxy to measure information asymmetries across trades.
61
Hence, 0 0 0 0
1 t t1 t t
S S S S
t A A A A t
E P⎡⎣ + θ Ω⎤⎦=E E V⎡⎢⎣ ⎡⎣ θ +Ω⎤⎦ θ Ω⎥⎦⎤=E V⎡⎣ θ Ω⎤⎦=P
This proposition demonstrates an important implication for market efficiency, which states that in competitive markets with rational, risk-neutral participants, returns are unpredictable. This follows from what has been found in existing literature such as previously shown by Mandelbrot (1966).
Reverting to the revision process of the market maker’s beliefs, several questions follow.
Firstly, does the price process converge to a specific value in the limit? Secondly, if prices do converge, then what do they converge to? Finally, regarding the speed of convergence, how quickly do prices converge to that value? To answer this, the following two propositions are put forward: 31
Proposition 2. The posterior belief converges almost surely to the true realisation of the asset.
Proof: As shown in Appendix A.6, the log of the odds ratio of the market maker’s beliefs on the outcome of the asset’s value converges almost surely to the Kullback-Leibler (KL) Divergence between the probability of a buy conditional upon a high value occurring, p, and the probability of a buy conditional upon a low value occurring, q. That is,
31 The proofs put forward have been adapted from O’Hara (1995) to suit the nature of Model 1. See Appendix A.6 and Appendix A.7 for full proofs of Proposition 2 and Proposition 3 respectively. Appendix A.5 also contains an introduction to the concept of almost surely.
62
negative. This can only occur when
( )
(
0)
→∞ ∩ → which suggests that the market maker’s belief about a low value occurring given b buys and s sells converges to zero, which then implies that the true value V must be high. The implications of this proposition demonstrate that the market maker’s beliefs eventually converge and will actually converge to their full informational value; and as such, market prices will eventually become strong-form efficient. The next proposition demonstrates the speed at which this occurs.
Proposition 3. A Bayesian’s posterior who has observed an independent and identically distributed (i.i.d.) process over time converges exponentially.
Proof: Similarly, in Appendix A.7, it is shown that,
( )
63 This demonstrates that the market maker’s revision of beliefs converge to the true value at
an exponential rate, with respect to the number of trades in the market and the (KL) Divergence between the probabilities of a buy in either state of nature for V.