Two-sided exit problem for general L evy processes Theory and - - PowerPoint PPT Presentation

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)] = et k(p), c) where the function k(p) is called the characteristic exponent

• f the process and is given by the formula (Re p =0)

k(p) = 1 2 p2σ2 − αp + ∞

−∞

• e−px − 1 +

px 1 + x2

• Π(dx),

(1)

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

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.5 0.5 1 1.5 2 time sample path level x level −y

Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes

Statement of the problem

Introduce events: Ax = { ξ(χ) ≥ x } the exit from the interval by the process through the upper boundary; Ay = { 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)IAx + (−ξ(χ) − y)IAy 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; τx = inf{ t : ξ(t) ≤ −x }, Tx = −ξ(τx) − x

Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes

Figure: First passage time of a certain level

5 10 15 20 25 −15 −10 −5 5 10 15

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 ] =

• E [e−pξ+(νs)]

−1 E [ e−p(ξ+(νs)−x); ξ+(νs) > x ], E [ e −sτx−pTx ] =

• E e pξ−(νs)−1

E [ ep(ξ−(νs)+x); −ξ−(νs) > x ], where ξ+(t) = sup

u≤t

ξ(u), ξ−(t) = inf

u≤t ξ(u),

νs is an exponential variable with parameter s > 0, independent of the process {ξ(t), t ≥ 0}, P[ νs > t ] = e−st, and E[e −p ξ±(νs) ] = ∞ s e−st E [ e −pξ±(t)] dt, ±Re p ≥ 0.

Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes

* Introduce the functions (v ≥ 0)

K+(v, du, s) = ∞ E[ e−sτv+B; Tv+B ∈ dl] E [e−sτ l+B; T l+B ∈ du], K−(v, du, s) = ∞ E[e−sτ v+B; T v+B ∈ dl] E[e−sτl+B; Tl+B ∈ du], * and also, by recurrence K(1)

± (v, du, s) = K±(v, du, s),

(2) K(n+1)

±

(v, du, s) = ∞ K(n)

± (v, dl, s) K±(l, du, s),

n ∈ N.

Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes

Main results

Theorem E [ e−sχ; X ∈ du, Ax] = f s

+(x, du) +

∞ f s

+(x, dv) Ks +(v, du),

E [ e−sχ; X ∈ du, Ay] = f s

−(y, du) +

∞ f s

−(y, dv) Ks −(v, du),

Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes

where f s

+(x, du) = E [ e−sτ x; T x ∈ du ] −

∞ E [ e−sτy; Ty ∈ dv ] E [ e−sτ v+B; T v+B ∈ du ]; f s

−(y, du) = E [ e−sτy; Ty ∈ du ] − ∞

• E [ e−sτ x; T x ∈ dv ] E [ e−sτv+B; Tv+B ∈ du ];

and Ks

±(v, du) = ∞

• n=1

K(n)

± (v, du, s),

v ≥ 0 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 u≤t

ξ(u) < x] We derive the exact formula for the function Qs(p) = E [ e−p ξ(νs); χ > νs ] = x

−y

e−up P [ −y < ξ−(νs), ξ(νs) ∈ du, ξ+(νs) < x ], where ξ+(t) = sup

u≤t

ξ(u), ξ−(t) = inf

u≤t ξ(u).

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−(νs), ξ(νs), ξ+(νs) } satisfies the formula Qs(p) = Ux(s, p)−eyp ∞ E[ e−sχ; X ∈ dv, Ay ] evp Uv+B(s, p), where Ux(s, p) = E[ e−p ξ(νs); ξ+(νs) ≤ x ] = E[e−pξ−(νs)] E[ e−p ξ+(νs); ξ+(νs) ≤ x ]. In particular, P[ χ > νs ] = P[ ξ+(νs) ≤ x ] − ∞ E[ e−sχ; X ∈ dv, Ay] P[ ξ+(νs) ≤ v + B ],

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) = S0 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

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 path of S(t) upper barrier U lower barrier L

Tetyana Kadankova, No¨ el Veraverbeke Two-sided exit problem for general L´ evy processes

Applications

Under some assumptions, the option price is given by CL,U(t) = e−r(T−t)EQ

• (S(T) − K)+Iχ>T /Ft
• 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, Ft 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.

• f 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