Game-Theoretical Model of Debt Management Mohmmad AL Ganber, Kevin - - PowerPoint PPT Presentation

Game-Theoretical Model of Debt Management

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen September 3, 2014

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Control System

Consider the control system of the borrower ˙ x = I(x)x − u(t), x(0) = ¯ x, where

◮ x: total debt measured as a fraction of the yearly income of

the borrower.

◮ I(x): interest rate payed on the debt at a given time. ◮ u(t) ∈ [0, 1]: is the control variable for the borrower which

represents the payment rate as a fraction of the income.

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Optimal control problem

The total cost J(u, ¯ x) = ∞

0 e−rtL(u(t))dt,

where

◮ r is the discounted rate. ◮ L(u) is the cost to the borrower for implementing the control

u. L(0) = 0, L′ > 0, L

′′ > 0,

lim

u→1− L(u) = +∞

Our goal is to minimize the total cost, i.e., min

u(.) J(u, ¯

x)

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Hamliton Jacobi Equation

The value function V (¯ x) = min

u(.) J(u, ¯

x) By using the principle of dynamic programming, V is a solution of the Hamilton-Jacobi equation rV (x) = H(x, V ′(x)), with H(x, ξ) = {L(ω) − ξω + ξI(x)x}. The optimal control u∗(x) = arg min

ω∈[0,1]{L(ω) − V ′(x)ω}

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Difference between an explicit and an implicit ODE

Figure : Difference between an explicit and an implicit ODE

Figure 2 Right plot show a standard ODE and the left plot shows the

ODE in our case.

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Differential Inclusion

rV (x) = H(x, V ′(x)) Differential inclusion V ′(x) ∈

• F −(x, V (x)) , F +(x, V (x))
• .

◮ If V ′ = F −(x, V ), the optimal control u∗(x) < I(x)x, so

˙ x > 0 and the debt increases.

◮ If V ′ = F +(x, V ), the optimal control u∗(x) > I(x)x, so

˙ x < 0 and the debt decreases.

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Initial Data

Figure : Choosing initial data

We should the point x0 such that: V ′(x0) = W ′(x0) and V (x0) = W (x0)

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Example 1

If ¯ x > 5, then ˙ x < 0, limt→∞ x(t) = 5 . If 0 < ¯ x < 5 then ˙ x > 0 limt→∞ x(t) = 5

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

The Bankruptcy Cost model

Let ρ : [0, XB) → [0, ∞) be instantaneous risk of bankruptcy which depends on the total debt such that ρ(x) is not decreasing and ρ(0) = 0, lim

x→xB− ρ(x) =

+ ∞. xB= max amount of debt the borrow will be able to pay back. The total cost function is now J(u, ¯ x) = ∞ exp{−rt − t ρ(x(s))ds} (ρ(x(t))B + L(u(t)))dt where: B=bankruptcy cost to the borrower.

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

With Similar arguments and by using the principle of dynamic programming, The equations are modified to (r + ρ(x))V (x) = H(x, V ′(x)), with H(x, ξ) = {L(ω) − ξω + ξI(x)x + ρ(x)B}. The optimal control u∗(x) = arg min

ω∈[0,1]{L(ω) − V ′(x)ω}

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Differential Inclusion

The implicit ODE leads to a differential inclusion: V ′(x) ∈

• F −(x, V (x)) , F +(x, V (x))
• .

◮ If we choose V ′ = F −(x, V ) ≤ ξ#(x), the optimal control

u∗(x) < I(x)x, so ˙ x > 0 and the debt increases.

◮ If we choose V ′ = F +(x, V ) ≥ ξ#(x), the optimal control

u∗(x) > I(x)x, so ˙ x < 0 and the debt decrease.

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Example 2

Solution=Lower envelope of blue and green curves. If ¯ x < 9.5, then ˙ x < 0, limt→∞ x(t) = 0. If ¯ x > 9.5, then ˙ x > 0, limt→∞ x(t) = ∞, and bankruptcy occurs.

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

The Game Theoretical Model

A pool of risk neutral lenders who charge an interest α(x) depending on the bankruptcy risk ρ(x). ODE: (r + ρ(x))V (x) = H(x, α(x), V ′(x)) Hamiltonian: H(x, ξ) = {L(ω) − ξω + ξα(x)x + ρ(x)B} α′(x) = [α(x) + λ][α(x) − r − ρ(x)] α(x)x − u(x) Optimal control: u∗(x) = arg min

ω∈[0,1]{L(ω) − V ′(x)ω}

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management

Example 3

If ¯ x < 7.5, then ˙ x < 0, limt→∞ x(t) = 0. If ¯ x > 7.5, then bankruptcy occurs.

Mohmmad AL Ganber, Kevin Dastalfo, Karim Shikh Khalil Faculty Mentor: Wen Shen. Postdoc: Khai Nguyen Game-Theoretical Model of Debt Management