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Chapter 10.3-10.4: Market Making with Utility and Adverse Selection Joshua Novak University of Calgary August 24rd, 2016
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• Chapter 10.3-10.4: Market Making with Utility andAdverse Selection

Joshua Novak

University of Calgary

August 24rd, 2016

• 10.3- Utility Maximizing Market Maker

Consider an exponential utility function and corresponding performance criteria,

u(x) = e−γx G δ(t, x ,S , q) = Et,x,S,q[− exp {−γ(X δT + QδT (ST − αQαT ))}]

With the ansatz G (t, x ,S , q) = −e−γ(x+qS+g(t,q)), we get the HJB equation:

∂tg −1

2σ2γq2 + sup

δ+

λ+e−κ+δ+ 1− e−γ(δ

++g(t,q−1)−g(t,q))

γ

+λ−e−κ−δ− 1− e−γ(δ

−+g(t,q+1)−g(t,q))

γ= 0

g(t, 0) = 0 g(T , q) = −αq2

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• 10.3- Utility Maximizing Market Maker

Solving the HJB with calculus leads to the optimal posting depth thatmaximizes utility,

δ±,∗ =

{δ±,∗0 +

(γ[log(1 + γκ±

)]−1 − [κ±]−1) γ > 0δ±,∗0 γ = 0

Where the 0 subscript indicates the parameter of the base model in 10.2. Thisshows us that a model using an exponential utility function is just a constantshift of the model using an inventory penalty.

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• 10.4-Market Making with Adverse Selection

We suppose that the midprice follows the dynamics,

dSt = σdWt + �+dM+t − �−dM−t

Where M±t are Poisson processes with intensities λ+ and λ− respectively. Let

ε± = E[�±]

• 10.4.1- Impact of Market Orders on Midprice

We get the HJB, where q and q are the minimum and maximum inventorylevels respectively,

φq2 = ∂th + λ+ sup

δ+

{e−κ

+δ+ (δ+ − ε+ + hq−1 − hq)}

1q>q

+λ− supδ−

{e−κ

−δ−(δ− − ε− + hq+1 − hq)}

1q

• 10.4.1- Impact of Market Orders on Midprice

Define w(t) = eA(T−t)z ∈ Rq−q+1. We find the optimal control to be, assumingκ+ = κ− = κ,

δ+,∗(t, q) = ε+ + 1κ

(1 + log

(wq(t)

wq−1(t)

))q 6= q

δ−,∗(t, q) = ε− + 1κ

(1 + log

(wq(t)wq+1(t)

))q 6= q

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• Behaviour of the Strategy

We see that more risk aversion results in a larger fill rate and a highermagnitude of inventory drift.

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• Short-term alpha and Adverse Selection

Suppose our midprice includes the alpha of the stock,

dSt = (ν + αt)dt + σdWt

dαt = −ζαtdt + ηdW αt + �+1+M+t−dM+t − �−1+M−

t−dM−t

Where �±i , i ∈ N are i.i.d. random variables representing the size of marketimpact. Define `±t ∈ {0, 1} denote whether or not a limit order is posted on therespective side of the book at time t. The agent has the cash process,

dX `t =

(St +

2

)dN+,`t −

(St −

2

)dN−,`t

Where ∆ is the spread and N±t is the counting process for filled limit orders.

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• Short-term alpha and Adverse Selection

We use a running penalty performance criteria,

H`(t, x ,S , α, q) = E

X `T + Q`T (ST − (ST − (∆2 + ϕQ`T))− φ

T∫t

(Q`u)2du

The ansatz of choice is,

H(t, x ,S , α, q) = x + qS + h(t, α, q)

with the terminal condition h(T , α, q) = −q(

∆2 + ϕq

). An implicit optimal

control is found to be,

`+,∗(t, q) = 1{∆2 +E[h(t,α+�+,q−1)−h(t,α+�+,q)]>0}∩{q>q}

`−,∗(t, q) = 1{∆2 +E[h(t,α−�−,q+1)−h(t,α−�−,q)]>0}∩{q

• Features of the Strategy

We see that α < 0 encourages sell orders, and α > 0 encourages buy orders.

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• Features of the Strategy

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