Ruin Analysis in the Constant Elasticity of Variance Model (CEV - PowerPoint PPT Presentation

Klebaner, Liptser Ruin Analysis in the Constant Elasticity of Variance Model (CEV Model). Fima Klebaner 1 Robert Liptser 2 1 Monash University, Australia 2 Tel Aviv University, Israel 16 - 20 of July, 2007 / in Bressanone Outline

1. The model The Constant Elasticity of Variance Model (CEV) Klebaner, Liptser - was introduced by Cox (1996) - was applied in Option Pricing Models (Delbaen & Shirakawa (www.math.ethz.ch/ delbaen), Lu & Hsu (2005). CEV is defined by the Itô equation w.r.t. Brownian motion B t : dX t = µ X t dt + σ X γ t dB t , X 0 = K > 0 , where µ, σ � = 0 are arbitrary constants, � 1 � γ = 1 γ ∈ 2 , 1 ; 2 ( branching diffusion ) - existence of strong solution - Delbaen & Shirakawa; - uniqueness - Yamada & Watanabe.

2. Time to ruin (infinite horizon). Result Klebaner, Liptser In contrast to Black-Scholes: ( γ = 1), when X t is always positive, CEV process admits ruin at τ 0 = inf { t : X t = 0 } . � 1 , for µ ≤ 0 � � P K τ 0 < ∞ = 1 − S ( K ) S ( ∞ ) , for µ > 0 , � x � σ 2 ( 1 − γ ) y 2 ( 1 − γ ) � µ where S ( x ) = exp − dy . 0

3. Time to ruin (finite horizon, lower bound). Result Klebaner, Liptser �  � � K 1 − γ 2 µ σ √ Φ − , for µ � = 0  1 − e − 2 ( 1 − γ ) µ T 1 − γ P K ( τ 0 ≤ T ) ≥ � � K 1 − γ Φ − , for µ = 0 ,  √ σ ( 1 − γ ) T where Φ is the standard ( 0 , 1 ) -Gaussian distribution. Moreover, 1 lim K 2 ( 1 − γ ) log P K ( τ 0 ≤ T ) K →∞ � µ ( 1 − γ )[ 1 − e − 2 ( 1 − γ ) µ T ] , for µ � = 0 ≥ − 1 � σ 2 1 , for µ = 0 . 2 ( 1 − γ ) 2 T

4. Time to ruin (finite horizon, upper bound). Result Klebaner, Liptser Only logarithmic asymptotic in K → ∞ is computable: 1 lim K 2 ( 1 − γ ) log P K ( τ 0 ≤ T ) K →∞ � µ ( 1 − γ )[ 1 − e − 2 ( 1 − γ ) µ T ] , for µ � = 0 ≤ − 1 � σ 2 1 , for µ = 0 . 2 ( 1 − γ ) 2 T

5. Most likely path to ruin ( τ 0 ≤ T , K is large). Result Klebaner, The function Liptser  e µ t � � 1 / ( 1 − γ ) 1 − 1 − e − 2 µ ( 1 − γ ) t  , if µ � = 0  1 − e − 2 µ ( 1 − γ ) T   u ∗ t =  � � 1 / ( 1 − γ )   1 − t , if µ = 0 .  T is the estimate for the random process t ∧ τ 0 = X t ∧ τ 0 x K provided that K → ∞ . K This estimate is of the maximum likelihood type in the Large Deviation Scale.

6. Paths to extinction. Example 1 Klebaner, Liptser µ = 0, γ = 1 2 , T = 1 and any σ . 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 t

7. Paths to extinction. Example 2 Klebaner, µ = ± 10, γ = 1 2 , T = 1 and any σ . The same path for Liptser µ = 10 and µ = − 10 but different P ( τ 0 ≤ T ) . 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 t µ = − 10, σ 2 = 1, T = 1 ⇒ P ( τ 0 ≤ 1 ) ≈ e − 20 K µ = 10, σ 2 = 1, T = 1 ⇒ P ( τ 0 ≤ 1 ) ≈ e − 20 Ke ( − 10 ) .

8. Proofs: Infinite Horizon Klebaner, The proof uses the generator of X t : Liptser 2 σ 2 x 2 γ ∂ 2 L = 1 ∂ x 2 + µ x ∂ ∂ x . The equation L S = 0 has a solution � x � σ 2 ( 1 − γ ) y 2 ( 1 − γ ) � µ S ( x ) = exp − dy 0 S ( b ) − S ( K ) gives the probability of hitting { 0 } before { b } . S ( b ) Finally the fact that sum of probabilities of hitting { 0 } before { b } and of hitting { b } before { 0 } equals 1 and lim b →∞ S ( b ) completely describes the solution.

9. Proofs. Finite horizon (lower bound) Klebaner, For t < τ 0 , by Itô’s formula Liptser � t σ 2 2 γ ( 1 − γ ) e ( 1 − γ ) µ s e − ( 1 − γ ) µ t = K 1 − γ − 0 ≤ X 1 − γ ds t X 1 − γ 0 s � t σ ( 1 − γ ) e − ( 1 − γ ) µ s dB s + 0 � t ≤ K 1 − γ + σ ( 1 − γ ) e − ( 1 − γ ) µ s dB s . 0 Hence, � T � � − K 1 − γ ≤ σ ( 1 − γ ) e − ( 1 − γ ) µ s dB s { τ 0 > T } ⊆ . 0

10. ... continuation The previous inclusion is equivalent to Klebaner, Liptser � T � � K 1 − γ > − σ ( 1 − γ ) e − ( 1 − γ ) µ s dB s { τ 0 ≤ T } ⊇ . 0 � T 0 σ ( 1 − γ ) e − ( 1 − γ ) µ s dB s is zero mean Gaussian Since − random variable with the variance � T σ 2 ( 1 − γ ) 2 e − 2 ( 1 − γ ) µ s ds 0 � σ 2 ( 1 − γ ) ( 1 − e − 2 ( 1 − γ ) µ T , µ � = 0 2 µ = σ 2 ( 1 − γ ) T , µ = 0 and the lower bound follows.

11. Proofs. Finite horizon (lower bound, large K ) Klebaner, Liptser  � � � K 1 − γ 2 µ σ √ Φ − , for µ � = 0  1 − e − 2 ( 1 − γ ) µ T 1 − γ P K ( τ 0 ≤ T ) ≥ � � K 1 − γ Φ − , for µ = 0 ,  √ σ ( 1 − γ ) T where Φ is ( 0 , 1 ) -Gaussian distribution. Then, the lower bound is nothing but � � ξ ≥ K 1 − γ p , ξ is (0,1)-Gaussian r.v. , P where � 1 2 µ σ √ 1 − γ p = 1 − e − 2 ( 1 − γ ) µ T .

12. Continuation. .... (lower bound, large K ) A lower estimate for zero mean Gaussian tail Klebaner, Liptser � ∞ e − y 2 / 2 dy > 1 − a − 2 e − a 2 / 2 a a provides 1 � � τ 0 ≤ T lim K 2 ( 1 − γ ) log P K →∞ 1 � � ξ ≥ K 1 − γ p ≥ lim K 2 ( 1 − γ ) log P K →∞ K 2 ( 1 − γ ) log 1 − K − 2 ( 1 − γ ) p − 2 1 > − p 2 + lim K 1 − γ p K →∞ � �� � = 0 1 µ = − 1 − e − 2 ( 1 − γ ) µ T . σ 2 ( 1 − γ )

13. Proofs. Finite horizon (upper bound, large K ) We desire to have Klebaner, Liptser 1 1 µ � � lim K 2 ( 1 − γ ) log P τ 0 ≤ T ≤ − 1 − e − 2 ( 1 − γ ) µ T . σ 2 ( 1 − γ ) K →∞ Unfortunately, it is impossible without = X t “Large Deviations technique” for x K K . t Obviously, τ 0 = inf { t : X t = 0 } ≡ inf { t : x K = 0 } , t where σ dx K = µ x K K ( 1 − γ ) ( x K t ) γ dB t , x K t dt + 0 = 1 . t

14. Proofs. Large Deviation Setting x K is the diffusion process with small diffusion parameter: Klebaner, t Liptser σ 2 K 2 ( 1 − γ ) ( x ) 2 γ . Informally, the family { x K t ) t ≤ T } K →∞ is in the framework of Freidlin-Wentzell’s Large Deviation Principle (LDP) with the rate of speed 1 K 2 ( 1 − γ ) and the rate function  � T u t − µ u t ) 2 ( ˙ u 0 = 1 1 dt ,  du t ≪ dt i.e. du t = ˙ 2 σ 2 0 u 2 γ u ( t ) dt J T ( u ) = t ∞ , otherwise . 

15. Proofs. The LDP setting (continuation) Klebaner, However, formally, Freidlin-Wentzell’s LDP is not compatible Liptser with the CEV model: 1) unbounded “drift” 2) Hölder continuous and singular “diffusion” 3) x K has paths in C [ 0 , ∞ ) ( R + ) ⊂ C [ 0 , ∞ ) ( R ) t the subspace of nonnegative continuous functions on [ 0 , ∞ ) absorbing at value zero if attaining zero. C [ 0 , ∞ ) ( R + ) is the Polish relative to the local uniform metric: 2 − n � � � ̺ ( x ′ , x ′′ ) = | x ′ t − x ′′ 1 ∧ sup t | . t ≤ n n ≥ 1

16. Proofs. LDP Theorem Klebaner, Liptser Theorem. The family { ( x K t ) t ≤ T } K →∞ obeys the LDP in ( C [ 0 , ∞ ) ( R + ) , ̺ ) with the rate of speed 1 K 2 ( 1 − γ ) and the rate function  � T ∧ Θ ( u ) u t − µ u t ) 2 ( ˙ 1 u 0 = 1 dt ,  2 σ 2 u 2 γ dU t ≪ dt , i.e. du t = ˙ 0 u t dt J T ( u ) = t  ∞ , otherwise , where 0 / 0 = 0 by convention and Θ ( u ) = inf { t : u t = 0 } , inf ∅ = ∞ .

17. Method of Theorem proof Exponential Tightness: Klebaner, Liptser ϑ η for stopping time ϑ ϑ and any positive constant η η , 1 � � x K C →∞ lim lim K 2 ( 1 − γ ) log P sup ≥ C = −∞ , t K →∞ t ≤ T 1 � � | x K ϑ + t − x K η ∆ → 0 lim lim K →∞ sup K 2 ( 1 − γ ) log P sup ϑ | ≥ η η = −∞ . ϑ ϑ ϑ ϑ ϑ ϑ ϑ ≤ T t ≤ ∆ Local Large Deviation Principle: for any u ∈ C [ 0 , T ] ( R + ) 1 � � � � � x K � ≤ δ δ → 0 lim lim K 2 ( 1 − γ ) log P sup t − u t = − J T ( u · ) . K →∞ t ≤ T

18. Proofs. Upper bound for P ( τ 0 ≤ T ) via LDP Klebaner, Liptser Notice that { τ 0 ≤ T } = { ( x K t ) t ≤ T ∈ D } with � � D = u · ∈ C [ 0 , T ] ( R + ) : u 0 = 1 , Θ ( u ) ≤ T . the closed set in the uniform metric. Then, due to the LDP , 1 � � lim K 2 ( 1 − γ ) log P τ 0 ≤ T ≤ − inf u ∈ D J T ( u ) . K →∞

19. Proofs. Computation of inf u ∈ D J T ( u ) inf u ∈ D J T ( u ) inf u ∈ D J T ( u ) The minimization is related to the optimal control problem Klebaner, Liptser with the control action ( a s , r ) s ≤ r ≤ T and controlled process u s , s ≤ r : µ u s + u γ ˙ u s = s a s , s ≤ r , u 0 = 1 for the cost functional J T ( u ) . Thus, � Θ ( u ) ∧ T u t − u t ) 2 ( ˙ u · ∈ D J T ( u · ) = inf min dt u 2 γ u : J T ( u ) < ∞ 0 t � r a 2 = inf s ds . ( w , r ): u s > 0 , s < r ≤ T 0

20. Proofs. The optimal control Change of variables A control ( a ∗ s , r ∗ ) is optimal provided Klebaner, Liptser that for any control ( a s , r ) , � r ∗ � r ( a ∗ s ) 2 ds ≤ a 2 s ds . 0 0 A role of γ < 1 is crucial in the finding of the optimal control. Set v t = u 1 − γ . t It is obvious that v 0 = u 0 = 1, v r = u r = 0 and v t solves the linear differential equation ˙ v s = µ ( 1 − γ ) v s + ( 1 − γ ) a s .