From Kyle to Glosten & Milgrom and Back Umut etin (based on a - - PowerPoint PPT Presentation

From Kyle to Glosten & Milgrom and Back

Umut Çetin (based on a joint work with Hao Xing)

London School of Economics

Paris, 18 June 2014

• U. Çetin

From Kyle to G-M and Back

Kyle’s model of insider trading

Back (1992) formalised the continuous time version of the model first introduced by Kyle (1985):

1 Noise traders: The noise traders have no information about

the future value of the risky asset and can only observe their own cumulative demands. The cumulative demand is modelled by a standard Brownian motion denoted with B.

2 Informed trader: The insider “knows” the value of the risky

asset, f(V), at time 1, where V is a standard normal random variable independent of B. Being risk-neutral, her

• bjective is to maximize her expected profit.

3 Market makers: The market makers observe the total order

and compete in a Bertrand fashion to clear the market. They are assumed to be risk-neutral.

• U. Çetin

From Kyle to G-M and Back

The pricing mechanism of the market

The market makers decide the price looking at the total

• rder,Y, which is given by

Yt = Bt + θt, where θt is the position of the insider in the risky asset at time t. Thus, the common filtration of the market makers is the

• ne generated by Y and is denoted with FY. Note that θ is

not necessarily adapted to FY, i.e. the insider’s trade is not

• bserved directly by the market makers.

A market maker has a pricing rule, H : [0, 1] × R → R, to assign the price in the following form: Pt = H(t, Yt), where Pt denotes the price quoted by the market maker for the net demand at time t.

• U. Çetin

From Kyle to G-M and Back

Objectives of the insider and the market maker

The pricing rule is assumed to be strictly increasing in demand leading to the conclusion that the insider has the full information.

• U. Çetin

From Kyle to G-M and Back

Objectives of the insider and the market maker

The pricing rule is assumed to be strictly increasing in demand leading to the conclusion that the insider has the full information. The market maker chooses a rational pricing rule, i.e. he chooses a pricing rule so that H(t, Yt) = E[f(V)|FY

t ],

for every t ∈ [0, 1]. The insider aims to maximize her expected profit out of

• trading. If X is continuous and of FV, the profit at time 1

equals

1

XtdPt + X1(f(V) − P1) =

1

{f(V) − Pt} dXt.

• U. Çetin

From Kyle to G-M and Back

Avenues for a robust approach if one must!

V is the private signal of the informed trader. There is no a priori reason that everyone knows the mean and the variance of this normal random variable. Also the noise traders trade for reasons exogenous to the

• model. Thus, their volatility is not necessarily known.

Both extensions will require some theory of robust filtering (possibly beyond what is already known to the statisticians), which is interesting in its own. In this talk I will ignore the impacts of above kinds of ambiguities.

• U. Çetin

From Kyle to G-M and Back

Equilibrium

Definition 1 A pair (H∗, θ∗) is said to form an equilibrium if H∗ is a pricing rule, θ∗ ∈ A, and the following conditions are satisfied:

1 Market efficiency condition: Given θ∗, H∗ is a rational

pricing rule.

2 The optimality condition: Given H∗, θ∗ maximizes the

expected profit of the insider.

• U. Çetin

From Kyle to G-M and Back

Optimality conditions for the insider’s problem

Working out the formal HJB computations yields that in an equilibrium H should satisfy Ht + 1

2Hyy = 0.

Following Back the trading strategies that are not absolutely continuous are necessarily suboptimal.

• U. Çetin

From Kyle to G-M and Back

Optimality conditions for the insider’s problem

Working out the formal HJB computations yields that in an equilibrium H should satisfy Ht + 1

2Hyy = 0.

Following Back the trading strategies that are not absolutely continuous are necessarily suboptimal. Therefore, it is enough to consider X ∈ A of the form dXt = αtdt for some measurable α. With an abuse of notation, the insider’s problem becomes: sup

α∈A

Ev

1

{V − H(t, Yt)} αtdt

• ,

(1)

• U. Çetin

From Kyle to G-M and Back

Optimality conditions for the insider’s problem (cont’d)

The value function can be computed explicitly: J(t, y) :=

y

ξ(t)

{H(t, u) − V} du + 1 2

1

t

Hy(s, ξ(s))ds where ξ(t) is the unique solution of H(t, ξ(t)) = V. Direct calculations lead to Jt + 1

2Jyy = 0 so that

sup

X∈A(H)

Ev

1

(V − H(t, Yt)) dXt

• = Ev [J(0, 0)]−

inf

X∈A(H) Ev [J(1, Y1)] .

Thus, the optimal strategy for the insider is to find an absolutely continuous X so that H(1, B1 + X1) = V.

• U. Çetin

From Kyle to G-M and Back

Equilibrium

The equilibrium level of the total demand, Y ∗, satisfies dY ∗

t = dBt + V − Y ∗ t

1 − t dt, so that it is a Brownian bridge. The price is given by Pt = H(t, Y ∗

t ), where Y solves

Ht + 1 2Hyy = 0, H(1, y) = f(y). Y ∗ is a Brownian motion in its own filtration. Related literature: Wu (1999), Cho (2003), Laserre (2004), Biagini and Øksendal (2005).

• U. Çetin

From Kyle to G-M and Back

An arbitrage strategy

In the original Kyle model f(V) = V so the profit of the insider at time 1 equals:

1

(V − Yt)2 1 − t dt > 0. The above is an arbitrage but due to feedback effects, it cannot be scaled, hence an equilibrium, in which the insider has a finite wealth, exists. In the original Kyle model the insider buys when the asset is undervalued, i.e. V − Yt ≥ 0, and sells otherwise leading to an ‘arbitrage’ in every infinitesimal trade.

• U. Çetin

From Kyle to G-M and Back

This is not always the case for any f since in general the final wealth is given by

1

(V − Yt)(f(V) − H(t, Yt)) 1 − t dt, and V − y ≥ 0 is not equivalent to f(V) ≥ H(t, y). The above considerations show that the insider does not trade based on whether the market price is higher than her

• wn valuation but rather on whether her own signal, V, is

more optimistic, i.e. ≥, than the signal of the market makers, Y.

• U. Çetin

From Kyle to G-M and Back

Glosten-Milgrom Model

Traders submit a fixed size of δ buy or sell orders for the asset and f(V) takes values from the set {0, 1}. The total demand of the noise traders, Z, is given by the difference of two pure jump processes NB and NS. NB/δ and NS/δ are assumed to be independent Poisson processes with constant intensity β. Thus, the noise trades still follow a martingale. The net order of the insider is denoted by θ := θB − θS where θB (resp. θS) denotes the cumulative buy (resp. sell)

• rders.

The market makers choose a Markovian pricing rule to set the price as before.

• U. Çetin

From Kyle to G-M and Back

Unlike the Kyle equilibrium, in this model the insider uses a mixed strategy. The characterisation of equilibrium is similar to that of Kyle’s in the sense that the insider trades without changing the distribution of the total demand while making sure that the market price matches to the announcement value at time 1.

• U. Çetin

From Kyle to G-M and Back

Unlike the Kyle equilibrium, in this model the insider uses a mixed strategy. The characterisation of equilibrium is similar to that of Kyle’s in the sense that the insider trades without changing the distribution of the total demand while making sure that the market price matches to the announcement value at time 1. In this special case this amounts to the insider adding a ‘drift’ to the noise demand which will vanish when projected on the filtration of the market makers but at the same time ensure that [Y1 ≥ yδ] = [f(V) = 1] where yδ is such that P(Z1 ≥ yδ) = P(f(V) = 1). In Kyle’s framework the way to guess the optimal strategy is to apply the initial enlargement of filtration formula to Brownian motion with its time-1 value and the so-called ‘information drift’ will be the optimal choice for the insider.

• U. Çetin

From Kyle to G-M and Back

If we apply the same recipe here, taking δ = 1 we obtain that the intensity of Z B, when the natural filtration of Z is enlarged with the set [Z1 ≥ y1], becomes 1[Z1≥y1]β h(Zt− + 1, t) h(Zt−, t) + 1[Z1<y1]β 1 − h(Zt− + 1, t) 1 − h(Zt−, t) , (2) where h(z, t) := P[Z1 ≥ y1 | Zt = z]. A similar formula for the intensity of Z S yields that the information drift is again absolutely continuous. This is problematic since a continuous trading will reveal the identity of the insider at once and the market makers will immediately adjust the market price to the true price causing the insider loosing her informational advantage.

• U. Çetin

From Kyle to G-M and Back

The optimal strategy for the insider

The optimal strategy for the insider is to find two point process θB and θS so that the intensity, in her own filtration,

• f Z B + θB (resp. Z S + θS) equals to that of Z B (resp. Z S)

when the natural filtration of Z is enlarged with the set [Z1 ≥ y1]. This is how the mixed strategy arises.

• U. Çetin

From Kyle to G-M and Back

The optimal strategy for the insider

The optimal strategy for the insider is to find two point process θB and θS so that the intensity, in her own filtration,

• f Z B + θB (resp. Z S + θS) equals to that of Z B (resp. Z S)

when the natural filtration of Z is enlarged with the set [Z1 ≥ y1]. This is how the mixed strategy arises. Observe from (2) that on the set [Z1 ≥ y1] the desired intensity for cumulative buy orders is greater than β. Similarly, on the same set the intensity for the cumulative sell orders is less than β. Let’s call the insider who knows that f(V) = 1 (resp. f(V) = 0) high-type (resp. low-type).

• U. Çetin

From Kyle to G-M and Back

The optimal strategy for the insider

The above considerations yield that a high-type insider will never sell but purchase the asset by placing orders at the arrival times of her optimal point process. She will do this by meeting the sell orders of the noise traders as they arrive with some probability that depends on the current level of the demand, and placing also some extra buy

• rders to match the distributional requirements of the
• equilibrium. A low-type insider does exactly the opposite.

It is possible to construct such point processes by using a countable family of independent uniform random variables.

• U. Çetin

From Kyle to G-M and Back

Existence of a pure-strategy equilibrium

It would be interesting to know if there exists a pure strategy equilibrium, i.e. an equilibrium without using auxiliary random

• variables. The answer to this question is equivalent to that of

the following question: Let Z = N1 − N2, where Nis are independent Poisson processes with intensity λ. If G is the the initial enlargement of the natural filtration of Z with the set [Z1 ≥ z], does Z admit a decomposition Z = ˜ Z + A, where ˜ Z is the difference of two independent G-Poisson processes with intensity λ?

• U. Çetin

From Kyle to G-M and Back

Convergence to Kyle’s model

Recall that β was the intensity of the noise orders. If we set β =

1 2δ2 and let δ → 0, then NB − NS converges weakly to

a standard Brownian motion. Also we are able to show that the laws of equilibrium demands are tight and the insider’s optimal strategies converge to her optimal strategy in Kyle’s framework. Similarly, the value functions and the market depths converge to their respective counterparts in the Kyle model. This convergence was observed in a different version of Glosten-Milgrom model by Back and Baruch (2004).

• U. Çetin

From Kyle to G-M and Back

Generalising Glosten-Milgrom

Suppose now that one can trade any integer multiple of δ. Consequently, the noise trades evolve as δ

∞

• i=1

iMδ,i

t ,

where Mδ,i is the difference of independent Poisson processes with intensity

αi 2i2δ2 such that α := ∞ i=1 αi < ∞.

As δ → 0 the noise trades converge to a Brownian motion with variance α. Similarly, the equilibrium pricing rule Pδ(t, y) converges to some P0(t, y) in the limit.

• U. Çetin

From Kyle to G-M and Back

Liquidity costs in the limit

In every δ-market, the average cost of a trade of size x is given by Sδ(t, y, x) :=

    

δ x

x/δ

i=1 Pδ(t, y + iδ),

if x ≥ δ; Pδ(t, y), if x = 0; − δ

x

−x/δ

i=1 Pδ(t, y − iδ),

if x ≤ −δ, given that the cumulative demand before the placement of the

• rder is y. To every trade we associate the liquidity cost

ℓδ(t, y, x) =

    

δ x/δ

i=1 Pδ(t, y + iδ) − Pδ(t, y),

if x ≥ δ; 0, if x = 0; −δ −x/δ

i=1 Pδ(t, y) − Pδ(t, y − iδ),

if x ≤ −δ.

• U. Çetin

From Kyle to G-M and Back

Thus, if θδ is a trading strategy in a δ-market, the associated cumulative liquidity cost until time equals Lδ

t =

• s≤t

ℓδ(s, Y δ

s−, ∆θδ s).

The total wealth of the investor at time 1 is then given by Vθδ

1 −

1

Pδ(s, Y δ

s−)dθδ s − Lδ 1.

• U. Çetin

From Kyle to G-M and Back

Let θδ be a sequence of trading strategies converging in law to a semi-martingale θ. By letting δ → 0 we can show that the liquidity cost converges weakly to 1 2

t

P0

y (s, Ys)d[θ, θ]c s

+

• s≤t

∆θs

• P0(s, Ys− + z) − P0(s, Ys−)
• dz.

This suggests a modification to the liquidity model of Ç., Jarrow and Protter (2004). In particular, the marginal stock price, P0(t, Yt), now depends on the position of the trader.

• U. Çetin

From Kyle to G-M and Back