Motivation Problem formulation Swing options in the jump diffusion model Viscosity solution Further research Optimal multiple stopping time problems of jumps diffusion processes Im` ene BEN LATIFA ENIT-LAMSIN New advances in Backward SDEs for financial engineering applications Tamerza (Tunisia) October 25 - 28, 2010 Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Swing options in the jump diffusion model Viscosity solution Further research Outline 1 Motivation 2 Problem formulation 3 Swing options with a jump diffusion model 4 Viscosity solution 5 Further research Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Swing options in the jump diffusion model Application to electricity market Viscosity solution Further research Outline Motivation 1 Application to electricity market Problem formulation 2 Some notations Existence of an optimal stopping strategy Swing options in the jump diffusion model 3 Formulation of the Multiple Stopping Problem Regularity Viscosity solution 4 uniqueness of viscosity solution Further research 5 Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Swing options in the jump diffusion model Application to electricity market Viscosity solution Further research In this talk we study the optimal multiple stopping time problem which consists on computing the essential supremum of the expectation of the pay-off over multiple stopping times, which is defined for each stopping time S by E [ ψ ( τ 1 , ..., τ d ) |F S ] . v ( S ) = ess sup τ 1 ,...,τ d ≥ S We will specify this set later. Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Swing options in the jump diffusion model Application to electricity market Viscosity solution Further research Example: Swing options A swing option is an option with many exercise rights of American type. Swing options are usually embedded in energy contracts. Pricing Swing options are important (the energy market will be deregulated and energy contacts will be priced according to their financial risk). In the energy market, the consumption is very complex. it depends on exogenous parameters (Weather, temperature). The consumption could increases sharply and prices follow. Such problems are studied by Davison and Anderson (2003) and also Carmona and Touzi (2008). In their paper Carmona and Touzi showed that pricing Swing option is related to optimal multiple stopping time problems. They studied the Black Scholes case. In this paper we extend their results in a market where jumps are permitted. Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research Outline Motivation 1 Application to electricity market Problem formulation 2 Some notations Existence of an optimal stopping strategy Swing options in the jump diffusion model 3 Formulation of the Multiple Stopping Problem Regularity Viscosity solution 4 uniqueness of viscosity solution Further research 5 Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research Let (Ω , F , F , P ) be a complete probability space, where F = {F t } 0 ≤ t ≤ T a filtration which satisfies the usual conditions. • Let T ∈ (0 , ∞ ) be the option’s maturity time i.e. the time of expiration of our right to stop the process or exercise. • S the set of F -stopping times with values in [0 , T ]. • S σ = { τ ∈ S ; τ ≥ σ } for every σ ∈ S . • δ > 0 the refracting period which separates two successive exercises. • We also fix ℓ ≥ 1 the number of rights we can exercise. • � � ( τ 1 , ..., τ ℓ ) ∈ S ℓ , τ 1 ∈ S σ , τ i − τ i − 1 ≥ δ on { τ i − 1 + δ ≤ T } a . s , S ( ℓ ) := σ τ i = T on { τ i − 1 + δ > T } a . s , ∀ i = 2 , ..., ℓ Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research Outline Motivation 1 Application to electricity market Problem formulation 2 Some notations Existence of an optimal stopping strategy Swing options in the jump diffusion model 3 Formulation of the Multiple Stopping Problem Regularity Viscosity solution 4 uniqueness of viscosity solution Further research 5 Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research Let X = { X t } t ≥ 0 be a non-negative c` adl` ag F -adapted process. We assume that X satisfies the integrability condition : � ¯ X p � ¯ E < ∞ for some p > 1 , where X = sup X t . (2) 0 ≤ t ≤ T We introduce the following optimal multiple stopping time problem : � ℓ � Z ( ℓ ) � := sup E X τ i . (3) 0 ( τ 1 ,...,τ ℓ ) ∈S ( ℓ ) i =1 0 The optimal multiple stopping time problem consist in computing the maximum expected reward Z ( ℓ ) and finding the optimal 0 exercise strategy ( τ 1 , ..., τ ℓ ) ∈ S ( ℓ ) at which the supremum in (3) is 0 attained, if such a strategy exist. Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research Carmona and Touzi (2008) have related the optimal multiple stopping time problems to a cascade of ordinary stopping time problems. Let us define Y ( i ) as the Snell envelop of the reward process X ( i ) which is defined as follows Y (0) = 0 � � Y ( i ) X ( i ) τ |F t ∀ t ≥ 0 , ∀ i = 1 , ..., ℓ, and = ess sup E , t τ ∈S t where the i-th exercise reward process X ( i ) is given by : � � X ( i ) Y ( i − 1) t + δ |F t for 0 ≤ t ≤ T − δ = X t + E (4) t and X ( i ) for t > T − δ = X t t Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research Now, we shall prove that Z ( ℓ ) can be computed by solving 0 inductively ℓ single optimal stopping time problems sequentially. This result is proved in the paper of Carmona and Touzi [1] under the assumption that the process X is continuous a.s. In this work we prove this result for c` adl` ag processes. Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research We recall the result of Nicole EL KAROUI (1981) to prove the existence of optimal stopping time for each ordinary stopping time problem. So we need to define inf { t ≥ 0 ; Y ( ℓ ) = X ( ℓ ) τ ∗ = } 1 t t and for i = 2 , ..., ℓ τ ∗ inf { t ≥ δ + τ ∗ i − 1 ; Y ( ℓ − i +1) = X ( ℓ − i +1) = } 1 { δ + τ ∗ i − 1 ≤ T } + T 1 { δ + τ ∗ i − 1 > T } . i t t Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research We recall then the result of Nicole EL KAROUI (1981) Theorem 1 (Existence of optimal stopping time) For all i = 1 , ..., ℓ let ( X ( ℓ − i +1) ) t , be a non negative c` adl` ag process t that satisfies the integrability condition E [sup t ∈ [0 , T ] X ( ℓ − i +1) ] < ∞ . t Then, for each t ≥ 0 there exists an optimal stopping time for Y ( ℓ − i +1) . Moreover τ ∗ i is the minimal optimal stopping time for t � � Y ( ℓ − i +1) i − 1 + δ , i.e. Y ( ℓ − i +1) X ( ℓ − i +1) |F τ ∗ = E a.s.. τ ∗ τ ∗ τ ∗ i − 1 + δ i − 1 + δ i Which show the existence of optimal stopping time for each ordinary stopping time problem. Im` ene BEN LATIFA Optimal multiple stopping time problems of jumps diffusion
Motivation Problem formulation Some notations Swing options in the jump diffusion model Existence of an optimal stopping strategy Viscosity solution Further research Our aim now is to prove the existence of optimal stopping time for each ordinary stopping time problem. So, let us prove that X ( i ) satisfies the assumptions of the last theorem. Lemma 2 We have that X is a c` adl` ag adapted process, then for i = 1 , ..., ℓ , X ( i ) is a c` adl` ag adapted process . Integrability condition: Lemma 3 Under our assumptions on the state process X we have the following integrability condition of the process X ( i ) when p > 1 X ( i ) � p � �� X ( i ) = X ( i ) ¯ ¯ E < ∞ where sup t , 0 ≤ t ≤ T where p is the moment order of ¯ Im` ene BEN LATIFA X. Optimal multiple stopping time problems of jumps diffusion
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