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 + Qδ

T (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 + γ

κ±

)]−1 − [κ±]−1)

γ > 0

δ±,∗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 + ε+dM+t − ε−dM−t

Where M±t are Poisson processes with intensities λ+ and λ− respectively. Letε± = E[ε±] <∞.We consider the running inventory penalty objective function,

Hδ(t, x ,S , q) = Et,x,S,q

X δT + Qδ

T (St − αQδT )− φ

T∫t

(Qδu )2du

We use the ansatz,

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

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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+ supδ+

{e−κ

+δ+

(δ+ − ε+ + hq−1 − hq)}

1q>q

+λ− supδ−

{e−κ

−δ−(δ− − ε− + hq+1 − hq)}

1q<q

+(ε+λ+ − ε−λ−)q

h(T , q) = −αq2

Define the vector z ∈ Rq−q+1 such that zj = e−ακj2

j = q . . . , q .Define the

matrix A ∈M(q−q+1)×(q−q+1) where,

Ai,q =

qκ(ε+λ+ − ε−λ−)− φκq2, i = q

λ+e−1−κε+

i = q − 1

λ−e−1−κε− i = q + 1

0 otherwise

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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 + ε+

1+M+

t−dM+

t − ε−1+M−t−dM−t

Where ε±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+ε+,q−1)−h(t,α+ε+,q)]>0}∩{q>q}

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

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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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