Keywords: mean field game; optimal control; trading
execution strategy.
Content
1. The model
2. The solution
3. Numerical experiments (animation 1, animation 2, animation 3, interactive application, app)
4. References
The retail traders
model
Let:
·
R be the set of real numbers,
·
T_{1}ÎR be a positive number that represents the time
horizon of the mean field game model that describes the retail traders. This
means that the model that describes
the retail traders is solved in the time interval [0,T_{1}],
·
x_{t}, tÎ[0,T_{1}], be a real
stochastic process that represents the number of shares of the risky asset held
by the retail traders as a function of the time t, tÎ[0,T_{1}]. The variable
x_{t}_{ }is called trading
execution strategy or trading position of the retail traders at time t, tÎ[0,T_{1}]. Recall that the variable x_{t}, tÎ[0,T_{1}], describes the retail traders in the mean
field approximation. Positive values of x_{t}, tÎ[0,T_{1}], mean that the retail traders have a long
position on the asset shares, negative values of x_{t}, tÎ[0,T_{1}], mean that the retail traders have a
short position on the asset shares,
·
be a continuous function that represents the
trading rate of the retail traders,
·
m(t,x), xÎR, tÎ[0,T_{1}], be the probability density function of the random
variable x_{t}, tÎ[0,T_{1}],
·
σ
be a (non zero) real constant,
·
W_{t}, , be a standard Wiener process such
that W_{0}=0 and dW_{t}, , be its stochastic differential.
We assume that the trading execution strategy
of the retail traders x_{t}, tÎ[0,T_{1}], is
solution of the following stochastic differential equation:
(1)
with
initial condition:
(2)
where is a known random variable whose
probability density function is denoted with m_{0}(x), xÎR. The probability density function of x_{t}_{ }is denoted by m(t,x), xÎR, tÎ[0,T_{1}].
Equation
(1) is the mean field equation that describes the “mean dynamics”
of the trading position of the “mean retail trader”.
Note that in (1) the function is not a known coefficient. The function is the control variable that must be
determined solving the mean field game model.
Let E( denote the expected value of ,
A_{1} be
the set of the square integrable processes on [0,T_{1}]
, and λ, θ, a be parameters such that λ>0, θ≥0 and
aÎR.
The retail traders adopt the trading execution
strategy whose rate =_{t}=(t,x_{t}), tÎ[0,T_{1}],
solves the following problem:
(3)
where:
(4)
subject to
the constraints (1), (2).
The model
(3), (4), (1), (2) is the mean field game model used to describe the behaviour of the retail traders.
The big trader model and the asset share price
dynamic equation
Let:
·
T_{2}ÎR be a positive number such that T_{2}≤T_{1}, T_{2} is the time horizon of the
liquidation problem. This means that the selling order must be executed in the
time interval [0,T_{2}],
·
Y>0
be a positive integer that represents the number of asset shares that must be
sold in the time interval [0,T_{2}],
·
β:[0,T_{2}]
× R → R be a continuous function that
represents the trading rate of the big trader,
·
y_{t}, tÎ[0,T_{2}], be a real
function that represents the number of asset shares of the risky asset held by
the big trader as a function of the time t, tÎ[0,T_{2}].
The variable y_{t}, tÎ[0,T_{2}], is called trading execution strategy of the big
trader.
The liquidation problem of the big trader
consists in determining how to execute the order of selling Y asset shares in
the time interval [0,T_{2}] in order to maximize the expected (total)
revenue resulting from the sale.
We assume that y_{t}, tÎ[0,T_{2}], is solution of
the following differential equation:
(5)
with
initial condition:
= Y,
(6)
and final
condition:
(7)
The function β_{t}= β(t,y_{t}), tÎ[0,T_{2}], is the control
variable of the control problem that defines the behaviour
of the big trader.
Note that the differential equation (5) is the
equation used in similar circumstances by Almgren
[1].
The asset price dynamic equation of the model
presented in [2] is a simple generalization of the asset share price dynamic
equation used by Almgren in [1].
Let:
·
be a stochastic process that
represents the price of the asset share “in absence of trading”, we
assume that d= εdB_{t}, [0,T_{2}], where ε>0 is a real parameter
and B_{t}, [0,T_{2}], is a standard
Wiener process, such that B_{0}=0 and dB_{t}
is its stochastic differential. The Wiener processes W_{t},
Z_{t} and B_{t},
[0,T_{2}], are assumed to be independent.
·
ζ_{2
}be nonnegative
constants.
Let S_{t} be the asset share
price at time t, [0,T_{2}], we assume that S_{t}, [0,T_{2}], is a real
stochastic process defined as follows:
S_{t} = + M_{t} + β_{t} + ζ_{2}(Y-y_{t}), tÎ[0,T_{2}],
(8)
given,
(9)
where M_{t }= E(α(t,x_{t})) = E(α_{t}) = , t[0,T_{1}], β_{t} = β(t,y_{t}), t[0,T_{1}], and we assume that is a positive random variable
concentrated in a point.
Equation (9) is the asset price dynamics
equation.
Note that the
asset price dynamic equation (9) allows negative asset share prices. The same
is true for the asset share price dynamic equation used by Almgren
in [1].
Let A_{2} be the set of the square integrable processes on [0,T_{2}]. The problem of finding the optimal
trading execution strategy for the big trader consists in solving the following
optimal control problem:
(10)
where:
V(β) = , βA_{2},
(11)
subject to
the constraints (5), (6), (7).
The problem (10),
(11), (5), (6), (7), (8), (9) is a linear quadratic optimal control
problem. The function V(β),
βA_{2} is the expected final revenue resulting
from the liquidation order when the trading execution strategy y_{t}, tÎ[0,T_{2}], determined by β through (5), (6), (7) is adopted.
Let N(∙,+)
be the normal probability distribution of mean ∙ and standard deviation +
and let ξ be a random
variable. The notation ξ N(∙,+) means that ξ is a
normal random variable distributed
N(∙,+). Using the ideas of Kalman [3]
under some hypotheses we reduce the solution of the mean field game model (3),
(4), (1), (2) and of the optimal control problem (10), (11), (5), (6), (7),
(8), (9) to the solution of two systems of Riccati
ordinary differential equations (see [2] for more details).
In particular in [2] under the hypotheses that: there
exist μ_{0},
ψ_{0}, ψ_{0}>0, such that N(μ_{0}, ψ_{0})
we show that N(μ_{t},
ψ_{t}), tÎ[0,T_{1}], where , tÎ[0,T_{1}],
is the solution of the mean field game model (4), (3), (2), (1) and μ_{t}, ψ_{t},
tÎ[0,T_{1}], are determined from the
solution of a system of six Riccati ordinary
differential equations.
Moreover
in [2] when tÎ[0,T_{1}],
we derive explicit
formulae for M_{t}, μ_{t}, t[0,T_{1}], these formulae
are_{ }substituted in the asset share price dynamic equation (8) in
order to solve the optimal control problem (10), (11), (5), (6), (7), (8), (9).
In [2] we show that the corresponding optimal trading execution strategy of the
big trader y_{t}= , tÎ[0,T_{2}], solution of the
problem (10), (11), (5), (6), (7), (8), (9) is the solution of the following
differential equation:
(12)
with
initial condition:
(13)
Recall
that since the formulae for M_{t}, μ_{t},
tÎ[0,T_{2}], deduced in [2] have been
obtained under the assumption N(μ_{0}, ψ_{0})
also the formula for t[0,T_{2}], deduced in [2]
from (12), (13) holds under the assumption N(μ_{0}, ψ_{0}).
Let us study the trading execution model
presented in Sections 1 and 2 via numerical simulation. In particular we consider
a liquidation order and in a test case we compare the value of the expected
total revenue of the big trader (11) associated to the optimal trading
execution strategy (12), (13) that we have found as solution of our model with
the expected total revenue of two alternative trading execution strategies.
We assume that:
·
the
number of asset shares held by the big trader at time t=0 is Y=2,
·
the
final time within which the sale must be completed is T_{2}=0.6,
·
the
remaining parameters of the models of Section 1 and 2 have the following
values: T_{1}=1, =10, =1, ζ_{2}=0, λ=0.5, σ=0.8, μ_{0}=10,
ψ_{0}=0.5 and a=-50, θ=1.
Note that the choice:
·
μ_{0}=10
implies that the retail traders are buyers at time t=0,
·
a=-50
implies that the retail traders are sellers at time T_{1}.
With the
previous choices the optimal trading execution strategy of the big trader
determined by our model from being a concave function of t when t is close to
zero becomes a convex function of t when t is close to T_{2}.
To fix the ideas we consider three trading
execution strategies (Strategy 1, Strategy 2, Strategy 3) that are shown in the
three animations that follow.
1.
In
animation 1 we show the optimal trading execution strategy of the big trader y_{t}=, tÎ[0,T_{2}],
obtained as solution of the model presented in Sections 1 and 2 defined by
(12), (13) and in [2]. This is Strategy 1.
2.
In
animation 2 we show the trading execution strategy of the big trader defined as
follows:
(14)
where , tÎ[0,T_{2}],
is the optimal trading execution strategy shown in animation 1. This is
Strategy 2.
3.
In
animation 3 we show the
optimal trading execution strategy of the big trader found in the “same
circumstances” using Almgren’s model [1]
that consists in selling with constant rate in the time interval [0,T_{2}],
that is:
(15)
This is Strategy
3.
Note that
Strategy 1 (see animation 1) makes use of
short selling (i.e.: y_{t} becomes
negative for some values of tÎ[0,T_{2}]). Strategy 2 and Strategy 3 do not use short selling.
In Figure 1 we compare the values of the
expected total revenue (11) associated to the trading execution Strategies 1,
2, 3. As it should be the expected total revenue associated to the optimal trading
execution strategy, Strategy 1, is greater than the expected total revenues
associated to Strategy 2 and Strategy 3.
In Figure 2 we show the difference between the
expected total revenues associated to Strategy 1 and Strategy 2 and the
difference between the expected total revenues associated to Strategy 1 and
Strategy 3.
Finally an interactive application and an app
are presented. The interactive application computes and draws on a graph the
optimal trading execution strategy of the big trader, the mean value of the
trading execution strategy of the retail traders and the asset price dynamics.
In the interactive application the user must assign the values of some of the model
parameters. To facilitate the use of the interactive
application a set of default values of these parameters is given. The
remaining parameters of the models involved in the numerical simulation are
chosen as follows: =10, =1, ζ_{2}=0, λ=0.5, σ=0.8, ψ_{0}=0.5
and θ=1. Recall that the random variable that appears in (9) is chosen to be a
random variable concentrated in the point (assigned by the user) with probability
one.
[1] R. Almgren (2003). Optimal execution with nonlinear impact functions and trading enhanced risk. Applied Mathematical Finance 10(1), 1-18.
[2] L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli (2014). A trading execution model based on
mean field games and optimal control. Applied
Mathematics 5(19), 3091-3116.
[3] R. E. Kalman
(1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering 82 (1),
35-45.
[4] J. M. Lasry,
P. L. Lions (2007). Mean field games. Japanese
Journal of Mathematics 2(1), 239-260.
[5] D. Revuz, M. Yor (1999). Continuous
Martingales and Brownian Motion. Springer-Verlag,
New York.
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