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 Introduction 1 Bad and good news principles 2 EPV-operators and Wiener-Hopf factorization 3 Stochastic expressions and optimal stopping 4 No-remorse principle for exit problems 5 Simple sufficient conditions 6 Application I: entry with the embedded option to exit 7 Conclusion 8 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 Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 2 / 35
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 Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 3 / 35
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 Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 4 / 35
Bad and good news principles Notation: q > 0 – the discount rate X = { X t } 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 , ( F t ) 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 X t under Q L – infinitesimal generator of X Le β x = Ψ( β ) e β x X t = sup 0 ≤ s ≤ t X s – the supremum process X t = inf 0 ≤ s ≤ t X s – the infimum process Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 5 / 35
Bad and good news principles L´ evy exponent � e β X t � = e t Ψ( β ) Ψ( β ) is definable from E If X is BM with drift b and variance σ 2 , Ψ( β ) = σ 2 2 β 2 + b β. If X is DEJD process, with the density of jumps F ( dy ) = c + λ + e − λ + y 1 (0 , + ∞ ) ( y ) dy + c − ( − λ − ) e − λ − y 1 ( −∞ , 0) ( y ) dy , then Ψ( β ) = σ 2 c + β c − β 2 β 2 + b β + λ + − 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. Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 6 / 35
Bad and good news principles Bad and good news principles It is optimal to acquire a non-decreasing stream f ( X t ) the first time the expected present value (EPV) of the stream under the infimum process becomes non-negative: enter at X 0 = h iff �� ∞ � E h qe − qt f ( X t ) dt ≥ 0 . (1) 0 It is optimal to abandon a non-decreasing stream f ( X t ) the first time the expected present value (EPV) of the stream under the supremum process becomes non-positive: exit at X 0 = h iff �� ∞ � E h qe − qt f ( X t ) dt ≤ 0 . (2) 0 S. Boyarchenko, 2004: g ( x ) = e x − K ; Boyarchenko and Levendorskiˇ i 2005-2007: monotone payoff functions, both in continuous and discrete time Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 7 / 35
EPV-operators and Wiener-Hopf factorization Realization via EPV-operators Supremum and infimum processes X t = sup 0 ≤ s ≤ t X s - the supremum process X t = inf 0 ≤ s ≤ t X s - the infimum process Normalized EPV operators under X , X , and X : �� + ∞ � ( E q g )( x ) := q E x e − qt g ( X t ) dt 0 �� + ∞ � ( E + q g )( x ) := q E x e − qt g ( X t ) dt , 0 �� + ∞ � ( E − e − qt g ( X t ) dt q g )( x ) := q E x . 0 Notice that E q = q ( q − L ) − 1 Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 8 / 35
EPV-operators and Wiener-Hopf factorization EPV-operators Action on exponentials � q e β x � κ ± E ± q ( β ) = | x =0 . evy process, κ + q ( β ) < ∞ , ∀ β ≤ 0, and κ − For any L´ q ( β ) < ∞ , ∀ β ≥ 0. If the no-bubble condition q − Ψ(1) > 0 holds, then κ + q ( β ) < ∞ , ∀ β ≤ 1. Another interpretation E + E − q f ( x ) = E [ f ( x + X T q )] , q f ( x ) = E [ f ( x + X T q )] , where T q is an exponential random variable of mean 1 / q , independent of the process X . Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 9 / 35
EPV-operators and Wiener-Hopf factorization Examples Brownian motion: X T q is an exponentially distributed random variable on R + of mean 1 /β + , and X T q 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 − β + y u ( x + y ) dy , E + q u ( x ) = β + 0 � 0 E − q u ( x ) = ( − β − ) e − β − y u ( x + y ) dy . −∞ Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 10 / 35
EPV-operators and Wiener-Hopf factorization Examples Double-exponential jump-diffusion model: EPV-operators are of the form � + ∞ e − β + E + � a + j β + j y u ( x + y ) dy , q u ( x ) = j 0 j =1 , 2 � 0 e − β − E − � a − j ( − β − j y u ( x + y ) dy , q u ( x ) = j ) −∞ j =1 , 2 2 < λ − < β − 1 < λ + < β + where β − 1 < 0 < 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. Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 11 / 35
EPV-operators and Wiener-Hopf factorization Wiener-Hopf factorization formula Three versions: 1. Let T q ∼ Exp ( q ) be the exponential random variable of mean q − 1 , independent of process X . For ξ ∈ R , E [ e i ξ X Tq ] = E [ e i ξ X Tq ] E [ e i ξ X 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. E q = 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 E q = q ( q + Ψ( D )) − 1 , E ± q = φ ± q ( D ). Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 12 / 35
Stochastic expressions and optimal stopping Stochastic expressions Notation τ – stopping time For a stopping time τ and a measurable f , define �� τ − 0 � q − 1 E [ 1 T q <τ f ( x + X T q )] = E e − qt f ( x + X t ) dt V ex ( τ ; f ; x ) = (3) 0 �� + ∞ � q − 1 E [ 1 τ ≤ T q f ( x + X T q )] = E e − qt f ( x + X t ) dt V en ( τ ; f ; x ) = (4) τ To ensure finiteness, we assume that E [ | f ( x + X T q ) | ] < ∞ ∀ x ; (5) in some cases, this condition can be relaxed. Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 13 / 35
Stochastic expressions and optimal stopping Stochastic expressions, cont’d Connection between entry and exit problems V ex ( τ ; f ; x ) = q − 1 E q f ( x ) + V en ( 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 f 0 – current stream f n – new stream V ( τ ; f 0 , f n ; x ) = V ex ( τ ; f 0 ; x ) + V en ( τ ; f n ; x ) This problem is equivalent to maximization of V ex ( τ ; f 0 − f n ; x ), or, alternatively, to maximization of V en ( τ ; f n − f 0 ; x ). Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 14 / 35
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 ), E q f is continuous. A sufficient condition is: for some t > 0, the transition measure P X t is absolutely continuous. Boyarchenko and Levendorski˘ i (UT ) No-remorse principles 03/08/12 15 / 35
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