Two-sided exit problem for general L´ evy processes Theory and application. Tetyana Kadankova, No¨ el Veraverbeke Center for Statistics 14th European Young Statisticians Meeting Debrecen, August 2005 Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Outline Definitions Main results Examples Applications Further research Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Introduction Definition A real-valued Markov process ξ = ( ξ ( t ) ∈ R , t ≥ 0) is called a L´ evy process if it has independent and stationary increments and its paths are right-continuous with left limits. Examples. Wiener process ( its increments are independent normally distributed) Poisson process (”the lack of memory property”). Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Properties a) Independence and stationarity = ⇒ Distribution of increments ξ ( t + s ) − ξ ( s ) is infinitely divisible. b) Therefore, by the L´ evy - Khinchine representation, in assumption that ξ (0) = 0 the transform E [ e − pξ ( t ) ] , Rep = 0 has the form E [ e − pξ ( t ) ] = e t k ( p ) , c) where the function k ( p ) is called the characteristic exponent of the process and is given by the formula (Re p =0) � ∞ k ( p ) = 1 � px � 2 p 2 σ 2 − αp + e − px − 1 + Π( dx ) , (1) 1 + x 2 −∞ Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Statement of the problem Consider a real valued L´ evy process { ξ ( t ); t ≥ 0 } , ξ (0) = 0 with the characteristic exponent given by (1) and the interval ( − y, x ) , x, y > 0 , x + y = B. Introduce χ = inf { t > 0 : ξ ( t ) / ∈ ( − y, x ) } the instant of the first exit from the interval ( − y, x ) by the process ξ ( t ) . Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Figure: Sample path of the process 2 1.5 sample path level x level −y 1 0.5 0 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Statement of the problem Introduce events: A x = { ξ ( χ ) ≥ x } the exit from the interval by the process through the upper boundary; A y = { ξ ( χ ) ≤ − y } the exit from the interval by the process through the lower boundary. We are interested in the distribution of χ and X = ( ξ ( χ ) − x ) I A x + ( − ξ ( χ ) − y ) I A y the value of the overshoot through the boundary at the epoch of the exit from the interval. Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Auxiliary functionals For x > 0 we define τ x = inf { t : ξ ( t ) ≥ x } , T x = ξ ( τ x ) − x ; τ x = inf { t : ξ ( t ) ≤ − x } , T x = − ξ ( τ x ) − x Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Figure: First passage time of a certain level 15 10 5 0 −5 −10 −15 0 5 10 15 20 25 Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Integral transforms of the joint distributions { τ x , T x } , { τ x , T x } , satisfy the following equalities E [ e − sτ x − pT x ] = � − 1 � E [ e − pξ + ( ν s ) ] E [ e − p ( ξ + ( ν s ) − x ) ; ξ + ( ν s ) > x ] , E e pξ − ( ν s ) � − 1 � E [ e − sτ x − pT x ] = E [ e p ( ξ − ( ν s )+ x ) ; − ξ − ( ν s ) > x ] , where ξ + ( t ) = sup ξ ( u ) , ξ − ( t ) = inf u ≤ t ξ ( u ) , u ≤ t ν s is an exponential variable with parameter s > 0 , independent of the process { ξ ( t ) , t ≥ 0 } , P [ ν s > t ] = e − st , and � ∞ E [ e − p ξ ± ( ν s ) ] = s e − st E [ e − pξ ± ( t ) ] dt, ± Re p ≥ 0 . 0 Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
* Introduce the functions ( v ≥ 0) � ∞ E [ e − sτ v + B ; T v + B ∈ dl ] E [ e − sτ l + B ; T l + B ∈ du ] , K + ( v, du, s ) = 0 � ∞ E [ e − sτ v + B ; T v + B ∈ dl ] E [ e − sτ l + B ; T l + B ∈ du ] , K − ( v, du, s ) = 0 * and also, by recurrence K (1) ± ( v, du, s ) = K ± ( v, du, s ) , (2) � ∞ K ( n +1) K ( n ) ( v, du, s ) = ± ( v, dl, s ) K ± ( l, du, s ) , n ∈ N . ± 0 Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Main results Theorem � ∞ E [ e − sχ ; X ∈ du, A x ] = f s f s + ( x, dv ) K s + ( x, du ) + + ( v, du ) , 0 � ∞ E [ e − sχ ; X ∈ du, A y ] = f s f s − ( y, dv ) K s − ( y, du ) + − ( v, du ) , 0 Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
where + ( x, du ) = E [ e − sτ x ; T x ∈ du ] − f s � ∞ E [ e − sτ y ; T y ∈ dv ] E [ e − sτ v + B ; T v + B ∈ du ]; 0 f s − ( y, du ) = E [ e − sτ y ; T y ∈ du ] − ∞ � E [ e − sτ x ; T x ∈ dv ] E [ e − sτ v + B ; T v + B ∈ du ]; 0 and ∞ K ( n ) � K s ± ( v, du ) = ± ( v, du, s ) , v ≥ 0 n =1 is the series of the iterations of the K ( n ) ± ( v, du, s ) defined by (2). Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Applications Joint distribution of the supremum, the infimum and the value of the L´ evy process. Observe that, P [ χ > t ] = P [ − y < inf u ≤ t ξ ( u ) , sup ξ ( u ) < x ] u ≤ t We derive the exact formula for the function Q s ( p ) = E [ e − p ξ ( ν s ) ; χ > ν s ] = � x e − up P [ − y < ξ − ( ν s ) , ξ ( ν s ) ∈ du, ξ + ( ν s ) < x ] , − y where ξ + ( t ) = sup ξ − ( t ) = inf ξ ( u ) , u ≤ t ξ ( u ) . u ≤ t Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Applications Theorem The integral transform of the joint distribution of { ξ − ( ν s ) , ξ ( ν s ) , ξ + ( ν s ) } satisfies the formula � ∞ E [ e − sχ ; X ∈ dv, A y ] e vp U v + B ( s, p ) , Q s ( p ) = U x ( s, p ) − e yp 0 where U x ( s, p ) = E [ e − p ξ ( ν s ) ; ξ + ( ν s ) ≤ x ] = E [ e − pξ − ( ν s ) ] E [ e − p ξ + ( ν s ) ; ξ + ( ν s ) ≤ x ] . In particular, P [ χ > ν s ] = P [ ξ + ( ν s ) ≤ x ] − � ∞ E [ e − sχ ; X ∈ dv, A y ] P [ ξ + ( ν s ) ≤ v + B ] , 0 Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Applications Pricing double-barrier options Insurance risks Dam theory Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Applications We consider a financial model of type S ( t ) = S 0 exp { ξ ( t ) } where � S ( t ) is an asset price (price of a stock, an index, an exchange rate); � ξ ( t ) a general L´ evy process. Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Applications � Variance Gamma process � Generalized Hyperbolic model � Normal inverse Gaussian motion � Meixner process Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Double-knock option ⇓ lower barrier L and upper barrier U ⇓ knocks out if ⇓ either barrier is touched. ⇓ Otherwise, ⇓ pay-off max(0 , S ( T ) − K ) , at the maturity T ⇓ L < K < U is the strike price of the option. Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Figure: Double-knock option path of S(t) 80 upper barrier U lower barrier L 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Applications Under some assumptions, the option price is given by C L,U ( t ) = e − r ( T − t ) E Q ( S ( T ) − K ) + I χ>T / F t � � where � χ = inf { t > 0 : S ( t ) / ∈ ( L, U ) } , � r > 0 is the risk-free interest rate, � Q is the risk-neutral pricing measure, measure, � F t is information available at time t. Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Future research Asymptotic theory Numerical methods for inverting the Laplace transforms Simulation: Monte Carlo techniques Modeling of stock prices Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Questions ??? Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
References Kadankov V.F., Kadankova T.V. ”Intersections of an interval by a process with independent increments”. // Theor. of Stoch. Proc., Vol 11(27),54-68, 2005. Kadankov V.F., Kadankova T.V. ”Two-boundary problems for processes with independent increments”. //Ukr. Math.J., Vol 10, 2005. Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
Address Work Address: Tetyana Kadankova Mathematical Statistics Center for Statistics Universiteit Hasselt Universitaire Campus Agoralaan, Building D BE-3590 Diepenbeek Belgium Tel: +32 (11) 26.82.97 Email: tetyana.kadankova@uhasselt.be Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes
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