Chapter 10.3-10.4: Market Making with Utility and Adverse...
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Chapter 10.3-10.4: Market Making with Utility andAdverse Selection
Joshua Novak
University of Calgary
August 24rd, 2016
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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[�±]
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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+ sup
δ+
{e−κ
+δ+ (δ+ − ε+ + hq−1 − hq)}
1q>q
+λ− supδ−
{e−κ
−δ−(δ− − ε− + hq+1 − hq)}
1q
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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 + �+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
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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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