Optimal stopping in L evy models for non-monotone discontinuous - - PowerPoint PPT Presentation

Outline

Optimal stopping in L´ evy models for non-monotone discontinuous payoffs

Svetlana Boyarchenko and Sergei Levendorski˘ i

University of Texas at Austin; University of Leicester

March 8, 2012

Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 1 / 35

Outline

Outline

1

Introduction

2

Bad and good news principles

3

EPV-operators and Wiener-Hopf factorization

4

Stochastic expressions and optimal stopping

5

No-remorse principle for exit problems

6

Simple sufficient conditions

7

Application I: entry with the embedded option to exit

8

Conclusion

Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 1 / 35

Introduction

Motivation

Abundance of optimal stopping problems in economics and finance If pay-off streams are monotone, several general methods are applicable and closed form solutions obtain For options with non-monotone payoff streams, in some cases of interest, the answer can be guessed For options with non-monotone payoff streams, there are no general efficient verification theorems

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Introduction

Important situations different from BM model and monotone payoff streams

Complicated processes

• 1. jump-diffusion models
• 2. regime-switching models
• 3. models with additional stochastic factors, e.g., stochastic interest rate

and/or stochastic volatility Complicated payoff functions

• 1. embedded options, e.g., entry with option to exit/default
• 2. policy interventions, e.g., caps and floors for output price
• 3. models with strategic interactions

Models with ambiguity

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Introduction

Aims of the talk

For options to acquire or abandon a stream of payoffs explain general principles, which allow one to guess the solution present several general simple verification theorems Apply these theorems to entry problem with an embedded option to exit preemption game (talk tomorrow) For simplicity, consider one-factor jump-diffusion models (L´ evy models) Generalizations for models with more than one factor can be done as in a series of papers Boyarchenko-Levendorskiˇ i 2006-2009 for options with monotone payoff streams

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Bad and good news principles

Notation:

q > 0 – the discount rate X = {Xt}t≥0 – a L´ evy process (jump-diffusion with i.i.d. increments), which models the underlying uncertainty X is defined on a filtered probability space (Ω, F, (Ft)0≤t<∞, P) satisfying the usual properties E – the expectation operator under a chosen equivalent martingale measure (EMM) Q Ψ(β) – the L´ evy exponent of Xt under Q L – infinitesimal generator of X Leβx = Ψ(β)eβx X t = sup0≤s≤t Xs – the supremum process X t = inf0≤s≤t Xs – the infimum process

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Bad and good news principles

L´ evy exponent

Ψ(β) is definable from E

• eβXt

= etΨ(β) If X is BM with drift b and variance σ2, Ψ(β) = σ2 2 β2 + bβ. If X is DEJD process, with the density of jumps F(dy) = c+λ+e−λ+y1(0,+∞)(y)dy + c−(−λ−)e−λ−y1(−∞,0)(y)dy, then Ψ(β) = σ2 2 β2 + bβ + c+β λ+ − β + c−β λ− − β , where c+ > 0 (resp., c− > 0) is the intensity of positive (resp., negative) jumps, 1/λ+ (resp., 1/λ−) is the average size of a positive (resp., negative) jump.

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Bad and good news principles

Bad and good news principles

It is optimal to acquire a non-decreasing stream f (Xt) the first time the expected present value (EPV) of the stream under the infimum process becomes non-negative: enter at X0 = h iff Eh ∞ qe−qtf (X t)dt

• ≥ 0.

(1) It is optimal to abandon a non-decreasing stream f (Xt) the first time the expected present value (EPV) of the stream under the supremum process becomes non-positive: exit at X0 = h iff Eh ∞ qe−qtf (X t)dt

• ≤ 0.

(2)

• S. Boyarchenko, 2004: g(x) = ex − K;

Boyarchenko and Levendorskiˇ i 2005-2007: monotone payoff functions, both in continuous and discrete time

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EPV-operators and Wiener-Hopf factorization

Realization via EPV-operators

Supremum and infimum processes X t = sup0≤s≤t Xs - the supremum process X t = inf0≤s≤t Xs - the infimum process Normalized EPV operators under X, X, and X: (Eqg)(x) := qEx +∞ e−qtg(Xt)dt

• (E+

q g)(x) := qEx

+∞ e−qtg(X t)dt

• ,

(E−

q g)(x) := qEx

+∞ e−qtg(X t)dt

• .

Notice that Eq = q(q − L)−1

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EPV-operators and Wiener-Hopf factorization

EPV-operators

Action on exponentials κ±

q (β) =

• E±

q eβx

|x=0. For any L´ evy process, κ+

q (β) < ∞, ∀ β ≤ 0, and κ− q (β) < ∞, ∀ β ≥ 0. If

the no-bubble condition q − Ψ(1) > 0 holds, then κ+

q (β) < ∞, ∀ β ≤ 1.

Another interpretation E+

q f (x) = E[f (x + X Tq)],

E−

q f (x) = E[f (x + X Tq)],

where Tq is an exponential random variable of mean 1/q, independent of the process X.

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EPV-operators and Wiener-Hopf factorization

Examples

Brownian motion: X Tq is an exponentially distributed random variable on R+ of mean 1/β+, and X Tq is an exponentially distributed random variable on R− of mean 1/β−, where β− < 0 < β+ are the roots of the characteristic equation r − Ψ(β) = 0 EPV-operators are of the form E+

q u(x) = β+

+∞ e−β+yu(x + y)dy, E−

q u(x) = (−β−) −∞

e−β−yu(x + y)dy.

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EPV-operators and Wiener-Hopf factorization

Examples

Double-exponential jump-diffusion model: EPV-operators are of the form E+

q u(x) =

• j=1,2

a+

j β+ j

+∞ e−β+

j yu(x + y)dy,

E−

q u(x) =

• j=1,2

a−

j (−β− j ) −∞

e−β−

j yu(x + y)dy,

where β−

2 < λ− < β− 1 < 0 < β+ 1 < λ+ < β+ 2 are the roots of

q − Ψ(β) = 0, and a±

j > 0 are constants.

General case: E±

q are PDOs with the symbols φ± q (ξ), which are WH factors.

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EPV-operators and Wiener-Hopf factorization

Wiener-Hopf factorization formula

Three versions:

• 1. Let Tq ∼ Exp(q) be the exponential random variable of mean q−1,

independent of process X. For ξ ∈ R, E[eiξXTq ] = E[eiξX Tq ]E[eiξX Tq ];

• 2. For ξ ∈ R,

q q − Ψ(ξ) = φ+

q (ξ)φ− q (ξ),

where φ±

q (ξ) admits the analytic continuation into the corresponding

half-plane and does not vanish there

• 3. Eq = E−

q E+ q = E+ q E− q .

3 is valid in appropriate function spaces, and can be either proved as 1 or deduced from 2 because Eq = q(q + Ψ(D))−1, E±

q = φ± q (D).

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Stochastic expressions and optimal stopping

Stochastic expressions

Notation τ – stopping time For a stopping time τ and a measurable f , define Vex(τ; f ; x) = q−1E[1Tq<τf (x + XTq)] = E τ−0 e−qtf (x + Xt)dt

• (3)

Ven(τ; f ; x) = q−1E[1τ≤Tqf (x + XTq)] = E +∞

τ

e−qtf (x + Xt)dt

• (4)

To ensure finiteness, we assume that E[|f (x + XTq)|] < ∞ ∀ x; (5) in some cases, this condition can be relaxed.

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Stochastic expressions and optimal stopping

Stochastic expressions, cont’d

Connection between entry and exit problems Vex(τ; f ; x) = q−1Eqf (x) + Ven(τ; −f ; x) The exit problem with stream f is equivalent to the entry problem with stream −f , and optimality conditions for one problem can be easily reformulated in terms of optimality conditions for the other problem. Value of a swap f0 – current stream fn – new stream V (τ; f0, fn; x) = Vex(τ; f0; x) + Ven(τ; fn; x) This problem is equivalent to maximization of Vex(τ; f0 − fn; x), or, alternatively, to maximization of Ven(τ; fn − f0; x).

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Stochastic expressions and optimal stopping

Standing assumptions

Assumption 1. Function f is measurable and satisfies (5). Assumption 2. X is a L´ evy process satisfying (ACP)-property, with non-trivial supremum and infimum processes. (ACP)-property: For any f ∈ L∞(R), Eqf is continuous. A sufficient condition is: for some t > 0, the transition measure PXt is absolutely continuous.

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Stochastic expressions and optimal stopping

Optimal stopping lemma

Lemma 1 Let τB be the first entrance time into a Borel set B. Let B be a Borel set such that Wex(τB; f ; ·) = (q − L)Vex(τB; f ; ·) is universally measurable (6) Wex(τB; f ; x) = f (x), x ∈ R \ B, a.e. (7) Wex(τB; f ; x) ≥ f (x), x ∈ B, a.e. (8) Vex(τB; f ; x) ≥ 0, ∀ x. (9) Then τB maximizes Vex(τ; f ; ·) in the class of all stopping times.

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Stochastic expressions and optimal stopping

Proof of the optimal stopping lemma

Proof. Let τ be a stopping time. Then, using Dynkin’s formula and (7)–(9), we

• btain

Vex(τB; f ; x) = E τ e−qt(q − L)Vex(τB; f ; x + Xt)dt

• +E
• e−qτVex(τB; f ; x + Xτ)
• ≥

E τ e−qtf (x + Xt)dt

• .

With τ = τB, we obtain the equality, which means that τB is optimal.

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Stochastic expressions and optimal stopping

Exit problem in more details: value of exit

Theorem 2 Let f be a measurable function , which is either semi-bounded or satisfies (5). Let τ −

h be the first entrance time into (−∞, h], For any h,

Vex(τ −

h ; f ; x) = q−1E− q 1(h,+∞)E+ q f (x).

(10)

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Stochastic expressions and optimal stopping

Proof of Theorem 2

Proof. Assuming that X0 = 0, we use the following facts (i) the obvious identity XTq = (XTq − X Tq) + X Tq (ii) the random variables X Tq and XTq − X Tq are identical in law, (iii) the random variables X Tq and XTq − X Tq are independent, and definitions of the EPV-operators: qVex(τ −

h ; f ; x)

= E[1τ −

h <Tqf (x + XTq)]

= E[1x+X Tq >hf (x + X Tq + (XTq − X Tq))] = E[1x+X Tq >hE+

q f (x + X Tq)]

= E−

q 1(h,+∞)E+ q f (x).

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Stochastic expressions and optimal stopping

Exit problem in more details: optimal timing

Theorem 3 a) Let f be a non-decreasing stream that changes sign. Then (i) there exists h such that E+

q f (x) ≤ 0,

x ≤ h, and E+

q f (x) ≥ 0,

x ≥ h. (11) (ii) τ −

h , the first entrance time into (−∞, h], is an optimal exit time.

b) If f is not monotone but (11) holds, then τ −

h is an optimal exit time in

the class of stopping times of the threshold type.

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Stochastic expressions and optimal stopping

Intuition behind Theorem 3

Set w(x) = E+

q f (x).

Since E−

q is a positive operator, Vex(τ − h ; f ; x) is maximized only if

1(h,+∞)w(x) is maximized. Condition (11) ensures this.

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Stochastic expressions and optimal stopping

Insufficiency of the result for non-monotone streams

It is possible that the optimal exercise region is disjoint, and then τ −

h is

not optimal; but condition (11) holds. How to verify optimality of τ −

h ?

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No-remorse principle for exit problems

No-remorse principle for exit problems

Notation: F(dy) –the density of jumps of the L´ evy process {Xt}. Theorem 4 Let the bad news principle (11) hold, and let, for all x < h, Uex(τ −

h ; f ; x) := f (x) +

+∞

h−x

Vex(τ −

h ; f , x + y)F(dy) ≤ 0.

(12) Then τ −

h is an optimal exit time.

LHS is called the remorse index If the remorse index is non-positive in the action region, then exit is optimal, and there is no reason to regret the decision to exit.

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No-remorse principle for exit problems

Interpretation

Rewrite condition (12) as ∞

h−x

Vex(τ −

h ; f ; x + y)F(dy) ≤ −f (x),

x < h, a.e.. (13) The RHS is the (absolute value of) instantaneous losses, which the firm would suffer at Xt = x should the management decide to continue

• perations and not exit, and the LHS is the instantaneous gains due to

jumps from x into the inaction region, where the firm value is positive. Therefore, the optimality condition can be formulated as follows: (−∞, h] is an optimal exit region if, at each point x ∈ (−∞, h], the absolute value

• f instantaneous losses is not smaller than the instantaneous gains due to

jumps from x into the inaction region.

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No-remorse principle for exit problems

Proof of Theorem 4

Verify conditions of the optimal stopping Lemma 1. Recall that Wex(τ −

h ; f ; x) = (q − L)Vex(τ − h ; f ; x).

Wex(τ −

h ; f ; x) = f (x), x > h follows from a generalization of the Black-

Scholes equation. For x ≤ h, we have Vex(τ −

h ; f ; x) = 0. From action

Lu(x) = σ2 2 u′′(x) + bu′(x) +

• R\0

(u(x + y) − u(x))F(dy), Wex(τ −

h ; f ; x) = −

+∞

h−x

Vex(τ −

h ; f , x + y)F(dy) x < h.

Hence Wex(τ −

h ; f ; x) ≥ f (x) on (−∞, h) is equivalent to (12).

Since f and Vex(τ −

h ; f ; ·) are measurable, Wex(τ − h ; f ; ·) is measurable .

Vex(τ −

h ; f ; x) ≥ 0 is immediate from (10) and (11).

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Simple sufficient conditions

A general sufficient condition

The no-remorse condition (12) is close to a necessary condition but it is not very easy to verify. A general sufficient condition: the remorse index Uex(τ −

h ; f ; x) is non-decreasing.

Next slide: simpler sufficient conditions

• n f in the action region, and either on Vex(τ −

h ; f ; ·) in the inaction region

• r on the L´

evy density F+(dy) of positive jumps These conditions hold for some non-monotone and discontinuous f .

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Simple sufficient conditions

Sufficient conditions

(1) f is non-decreasing in the action region, and Vex(τ −

h ; f ; ·) is

non-decreasing in the inaction region; (2) f is non-decreasing in the action region, and F+(dy) is non-increasing; (3) there exists γ ∈ R such that eγxf (x) is non-decreasing in the action region, and eγxVex(τ −

h ; f ; ·) is non-decreasing in the inaction region;

(4) there exists γ ∈ R such that eγxf (x) is non-decreasing in the action region and e−γyF+(dy) is non-increasing.

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Application I: entry with the embedded option to exit

Application I: entry with the embedded option to exit

Let f (Xt) = eXt − c be the profit stream of an active firm, where c > 0 is

• perating cost, and let I be the sunk investment cost.
• I. Solving the exit problem, find the optimal exit threshold and the value of

the active firm, Vp(Xt), that has the option to exit.

• II. The investment problem is equivalent to the option to acquire a stream
• f payoffs g(Xt) = (q − L)V (Xt) − qI, which is non-monotone.

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Application I: entry with the embedded option to exit

Theorem 5 Let the following two conditions hold (i) k−(x), the pdf of X Tq, satisfies k−(x) =

• µ−(dβ)(−β)e−βx,

x < 0, (14) where µ−(dβ) is a non-negative measure supported at a subset of (−∞, 0) (by Bernstein’s theorem, (14) is equivalent to absolute monotonicity of k−(x)); (ii) the inverse (E+

q )−1 admits the representation

(E+

q )−1 = c+ q0 − c+ q1∂x − K +−,

(15) where c+

q0, c+ q1 > 0, and K +− q

is the convolution operator with the non-negative kernel k+−, which is monotone on (0, +∞). Then the optimal investment timing is given by the bad news principle. Conditions (i)–(ii) hold in BM and DEJD model, and for many other classes of L´ evy processes.

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Application I: entry with the embedded option to exit

Application II: preemption games under L´ evy uncertainty

Will be presented tomorrow.

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Conclusion

Conclusion

We presented a general method that allows to solve optimal entry and exit problems in L´ evy models when when payoff streams may have discontinuities and be non-monotone. As applications, we considered entry problem with an embedded

• ption to exit (non-monotone payoff) and preemption game

(non-monotone and discontinuous payoff). The preemption game will be presented tomorrow. The corner stone of our approach is the Wiener-Hopf factorization

• method. In our previous work we applied this method to

perpetual options with non-standard payoffs American options with finite time horizon Bermudan options efficient numerical procedures for wide classes of L´ evy processes models of endogenous default

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Conclusion

Earlier results: non-switching L´ evy models

Applications to real options capital expansion technology adoption partially reversible situations two-point boundary problems menus of options dynamic industry equilibrium models with heterogeneous firms and endogenous regime shifts

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Conclusion

Earlier results: regime-switching models

exit and entry problems perpetual American options American options with finite time horizon approximations of models with stochastic interest rates approximations of stochastic volatility models comparative statics: fast and accurate procedure of calculation derivatives of value functions and early exercise boundaries w.r.t. exogenous parameters

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Conclusion

Earlier results: regime-switching models

Iteration procedure for thresholds and value functions iteration method produces monotone sequences of investment boundaries and option values for each state no a priori assumption about the relative location of the boundaries in different states is needed Efficiency of the iteration procedure monotonicity ⇒ certain self-correcting features ⇒ fairly accurate CPU time: of order m2, where m is the number of states

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Conclusion

Advantages of our method in 2 or 3 factor models

very fast if values at many points are needed (fitting); jumps in all factors can be easily incorporated correlation between all factors can be easily modeled; an extension with the simultaneous calculation of Greeks is possible

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