The Liquidity Risk Pricing

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The Liquidity Risk Pricing

• Andrey Marchenko
• GGY Inc., (AndreyMarchenko_at_GGY.com)

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Liquidity definition

• A seller wants to sell a volume V0 of common
  stock (in stock units)

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Liquidity model

• The more asset units he attempts to sell at the
  moment, the lower will be the price per unit,
  therefore ?p/?a lt 0 (for sales).
• Liquidity measure can be defined as l -?p(t,
  a)/?a the greater is l, the lower is the
  liquidity.
• If l 0 the price does not depend on the asset
  flow no friction market with absolute liquidity
  case.

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Some experimental data
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The problem

• Selling strategy the flow of sells u(t)
  generates cash flow p(t,
  a(t)) a(t)dt
• Its present value equals
• (1)
• Total volume sold equals
• (2)
• Find
• Maxa sa over strategies a(t) or its
  expectation for random p(t, a(t))
• The optimal strategy a(t)
• under restriction (2)
• Liquidity price V0 p(0, 0) - Maxa sa

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Different problems

•   Deterministic problem
• non-random stock price p(t, a) known in advance.
•    Stochastic problem p(t, a) ? p(? t, a)
• Static strategy a prefixed selling strategy in
  stochastic environment. It can be used by
  regulators in order to evaluate and limit
  possible losses.
• Dynamic strategy - optimal control approach
  assumes dynamic strategy depending on a current
  state of the random stock price.

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Deterministic problem analysis

• If
• p(t, a) p(a),
• d 0,
• Borrow V0 p(0) dollars at t 0 and sell the
  asset infinitely slow at a price p(0) max
  ap(a).
• So the liquidity price 0.
• If -?p(t, a)/? a 0 (absolute liquidity) p (t,
  a(t)) p(t) the optimal strategy is selling all
  the stock V0 at a moment t0 when p(t0) max t
  p(t)
• In both cases gain V0 max p.
• Therefore assume l, d gt 0.

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Generic solution formalism

• Standard variation calculus yields
• with is the (unknown) Lagrange coefficient z .
• Solving with respect to a we get a family in z of
  functions a(z , t).
• Condition
• adds an equation. Eliminate z and substitute to
  (3) to get s s (T).
• Find max Ts by simply differentiating.

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High liquidity approximation

• Assume for simplicity
• p(0, 0) 1 V0 1 d 1
• High liquidity p(t, a) p(t) - La/2
• Stationary liquidity L const

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Generic solution

• Assume L const then
• a(t) (p(t) - zet) / L.
• L z (eT - 1),
• Solve for ds/dT 0 in T and substitute to
  obtain s(L)

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Example linear asset price

• Assume p(t) 1bt . Then
• and

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Graphs of s(L, b)
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Comparison with simplest optimization
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Stochastic problem

• p(t, a) p(? t, a), ? being random parameter.
• Three approaches 3 problems
• Static strategy
• The seller is risk-neutral
• The seller is risk-sensitive
• Dynamic strategy

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Stochastic problem static a risk-neutral
seller

• Interested in maximization of Esa
• Since a(t) is not random, the Fubini theorem
  yields
• and reduces the problem to the deterministic one
  with Ep(t, a) substituting p(t, a)

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Stochastic problem static a risk-sensitive
seller

• Choose the risk metric and plot all admissible
  points in the Risk-Return plane
• Find the optimal point on the effective boundary
  maximizing utility function
• Analogy to the classic Markowitz theory
• portfolio choice ? sales allocation to different
  times a(t)
• return Esa
• risk metric standard deviation of sa

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Static a risk-sensitive seller 2

• Assume high liquidity approximation
• p(? t, a) p(? t) - La/2 with m (t)
  Ep(? t)
• return x
• risk
• measure y
• where C(t1, t2) Ep(t1) p(t2) - E p(t1)
  Ep(t2)

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Static a risk-sensitive seller 3

• Effective boundary equation
• dReturn/da 0.5kdRisk measure/da z

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Static a risk-sensitive seller 4

• These equations have unique solution for fixed T
  gt 0 and
• min0, T C(t, t)e-2t lt D lt max0, T C(t,
  t)e-2t
• The practical way to is to apply Galiorkin
  approximations and get a finite dimensional
  problem.

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Example

• Let dp p(ndt s dW) in risk-neutral world
• m(t) expnt,
• C(t1, t2) exps2(t1 t2)/2( exps2min(t1,
  t2) 1)
• Galiorkin approximations
• Order 0 a(t) ? 0, T (t) /T
• Order 1 a(t) ? 0, T (t)a0 a1 t /(a0 T
  a1 T 2/ 2)

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Example - results
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Stochastic problem - Dynamic strategy optimal
control

• Price process dp p(ndt s dW)
• Control dx -a(t) dt
• t time x (t) unsold stock a(t) ? 0.
• Optimized functional
• Boundary conditions
• p(0) 1 x(0) 1.

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Hamilton-Jacobi-Bellman equation

• Assume high liquidity approximation
• p(? t, a) p(? t) - La/2
• Hamilton-Jacobi-Bellman equation for V
• With t, ? ? 0 and 0 ? z ? 1
• max a- ?V/? z a (? - La /2)a e-t (et ?V/? z
  - ?)2e-t /2L
• Boundary conditions are V 0 for z 0 and for
  ? 0

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Equation for V

• Final ultra-parabolic equation
• For stationary process (µ and s do not depend on
  t) a substitution V(?, ?, t) U(?, ?) ? e-t
  leads to nonlinear parabolic equation

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Some problems to investigate

• Change price process
• Add random risk free rate
• Modify the liquidity model
• Change p(t, u(t))
• Consider random liquidity
• Complex liquidity models, modeling trade patterns
  and behavior
• Investigate time-discrete sales