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 Liptser2

1Monash University, Australia 2Tel Aviv University, Israel

16 - 20 of July, 2007 / in Bressanone Outline

Klebaner, Liptser

• 1. The model

The Constant Elasticity of Variance Model (CEV)

• 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 Bt: dXt = µXtdt + σX γ

t dBt,

X0 = K > 0, where µ, σ = 0 are arbitrary constants, γ ∈ 1 2, 1

• ;

γ = 1 2 (branching diffusion)

• existence of strong solution - Delbaen & Shirakawa;
• uniqueness - Yamada & Watanabe.

Klebaner, Liptser

• 2. Time to ruin (infinite horizon). Result

In contrast to Black-Scholes: (γ = 1), when Xt is always positive, CEV process admits ruin at τ0 = inf{t : Xt = 0}. PK

• τ0 < ∞
• =
• 1,

for µ ≤ 0 1 − S(K)

S(∞),

for µ > 0, where S(x) = x exp

• −

µ σ2(1 − γ)y2(1−γ) dy.

Klebaner, Liptser

• 3. Time to ruin (finite horizon, lower bound).

Result

PK(τ0 ≤ T) ≥    Φ

• −

K 1−γ σ√ 1−γ

• 2µ

1−e−2(1−γ)µT

• ,

for µ = 0 Φ

• −

K 1−γ σ(1−γ) √ T

• ,

for µ = 0, where Φ is the standard (0, 1)-Gaussian distribution. Moreover, lim

K→∞

1 K 2(1−γ) log PK(τ0 ≤ T) ≥ − 1 σ2

• µ

(1−γ)[1−e−2(1−γ)µT ],

for µ = 0

1 2(1−γ)2T

• ,

for µ = 0.

Klebaner, Liptser

• 4. Time to ruin (finite horizon, upper bound).

Result

Only logarithmic asymptotic in K → ∞ is computable: lim

K→∞

1 K 2(1−γ) log PK(τ0 ≤ T) ≤ − 1 σ2

• µ

(1−γ)[1−e−2(1−γ)µT ],

for µ = 0

1 2(1−γ)2T

• ,

for µ = 0.

Klebaner, Liptser

• 5. Most likely path to ruin (τ0 ≤ T, K is large).

Result

The function u∗

t =

         eµt 1 − 1−e−2µ(1−γ)t

1−e−2µ(1−γ)T

1/(1−γ) , if µ = 0

• 1 − t

T

1/(1−γ) , if µ = 0. is the estimate for the random process xK

t∧τ0 = Xt∧τ0

K provided that K → ∞. This estimate is of the maximum likelihood type in the Large Deviation Scale.

Klebaner, Liptser

• 6. Paths to extinction. Example 1

µ = 0, γ = 1

2, T = 1 and any σ.

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 t

Klebaner, Liptser

• 7. Paths to extinction. Example 2

µ = ±10, γ = 1

2, T = 1 and any σ. The same path for

µ = 10 and µ = −10 but different P(τ0 ≤ T).

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 t

µ = −10, σ2 = 1, T = 1 ⇒ P(τ0 ≤ 1) ≈ e−20K µ = 10, σ2 = 1, T = 1 ⇒ P(τ0 ≤ 1) ≈ e−20Ke(−10).

Klebaner, Liptser

• 8. Proofs: Infinite Horizon

The proof uses the generator of Xt: L = 1 2σ2x2γ ∂2 ∂x2 + µx ∂ ∂x . The equation LS = 0 has a solution S(x) = x exp

• −

µ σ2(1 − γ)y2(1−γ) dy

S(b)−S(K) S(b)

gives the probability of hitting {0} before {b}. Finally the fact that sum of probabilities of hitting {0} before {b} and of hitting {b} before {0} equals 1 and limb→∞ S(b) completely describes the solution.

Klebaner, Liptser

• 9. Proofs. Finite horizon (lower bound)

For t < τ0, by Itô’s formula 0 ≤ X 1−γ

t

e−(1−γ)µt = K 1−γ − t σ2 2 γ(1 − γ)e(1−γ)µs X 1−γ

s

ds + t σ(1 − γ)e−(1−γ)µsdBs ≤ K 1−γ + t σ(1 − γ)e−(1−γ)µsdBs. Hence, {τ0 > T} ⊆

• − K 1−γ ≤

T σ(1 − γ)e−(1−γ)µsdBs

• .

Klebaner, Liptser

• 10. ... continuation

The previous inclusion is equivalent to {τ0 ≤ T} ⊇

• K 1−γ > −

T σ(1 − γ)e−(1−γ)µsdBs

• .

Since − T

0 σ(1 − γ)e−(1−γ)µsdBs is zero mean Gaussian

random variable with the variance T σ2(1 − γ)2e−2(1−γ)µsds = σ2(1−γ)

2µ

(1 − e−2(1−γ)µT, µ = 0 σ2(1 − γ)T, µ = 0 and the lower bound follows.

Klebaner, Liptser

• 11. Proofs.

Finite horizon (lower bound, large K)

PK(τ0 ≤ T) ≥    Φ

• −

K 1−γ σ√ 1−γ

• 2µ

1−e−2(1−γ)µT

• ,

for µ = 0 Φ

• −

K 1−γ σ(1−γ) √ T

• ,

for µ = 0, where Φ is (0, 1)-Gaussian distribution. Then, the lower bound is nothing but P

• ξ ≥ K 1−γp
• , ξ is (0,1)-Gaussian r.v.,

where p = 1 σ√1 − γ

• 2µ

1 − e−2(1−γ)µT .

Klebaner, Liptser

• 12. Continuation. .... (lower bound, large K)

A lower estimate for zero mean Gaussian tail ∞

a

e−y2/2dy > 1 − a−2 a e−a2/2 provides lim

K→∞

1 K 2(1−γ) log P

• τ0 ≤ T
• ≥ lim

K→∞

1 K 2(1−γ) log P

• ξ ≥ K 1−γp
• > −p2 + lim

K→∞

1 K 2(1−γ) log 1 − K −2(1−γ)p−2 K 1−γp

• =0

= − 1 σ2(1 − γ) µ 1 − e−2(1−γ)µT .

Klebaner, Liptser

• 13. Proofs.

Finite horizon (upper bound, large K)

We desire to have lim

K→∞

1 K 2(1−γ) log P

• τ0 ≤ T
• ≤ −

1 σ2(1 − γ) µ 1 − e−2(1−γ)µT . Unfortunately, it is impossible without “Large Deviations technique” for xK

t

= Xt

K .

Obviously, τ0 = inf{t : Xt = 0} ≡ inf{t : xK

t

= 0}, where dxK

t

= µxK

t dt +

σ K (1−γ) (xK

t )γdBt,

xK

0 = 1.

Klebaner, Liptser

• 14. Proofs. Large Deviation Setting

xK

t

is the diffusion process with small diffusion parameter: σ2 K 2(1−γ) (x)2γ. Informally, the family {xK

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 JT(u) =   

1 2σ2

T

( ˙ ut−µut)2 u2γ

t

dt,

u0=1 dut≪dt i.e. dut= ˙ u(t)dt

∞,

• therwise.

Klebaner, Liptser

• 15. Proofs. The LDP setting (continuation)

However, formally, Freidlin-Wentzell’s LDP is not compatible with the CEV model: 1) unbounded “drift” 2) Hölder continuous and singular “diffusion” 3) xK

t

has paths in C[0,∞)(R+) ⊂ C[0,∞)(R) the subspace of nonnegative continuous functions

• n [0, ∞) absorbing at value zero if attaining zero.

C[0,∞)(R+) is the Polish relative to the local uniform metric: ̺(x′, x′′) =

• n≥1

2−n 1 ∧ sup

t≤n

|x′

t − x′′ t |

• .

Klebaner, Liptser

• 16. Proofs. LDP Theorem
• Theorem. The family {(xK

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 JT(u) =   

1 2σ2

T∧Θ(u)

( ˙ ut−µut)2 u2γ

t

dt,

u0=1 dUt≪dt, i.e. dut= ˙ utdt

∞,

• therwise,

where 0/0 = 0 by convention and Θ(u) = inf{t : ut = 0}, inf ∅ = ∞.

Klebaner, Liptser

• 17. Method of Theorem proof

Exponential Tightness: for stopping time ϑ ϑ ϑ and any positive constant η η η , lim

C→∞ lim K→∞

1 K 2(1−γ) log P

• sup

t≤T

xK

t

≥ C

• = −∞,

lim

∆→0 lim K→∞ sup ϑ ϑ ϑ≤T

1 K 2(1−γ) log P

• sup

t≤∆

|xK

ϑ ϑ ϑ+t − xK ϑ ϑ ϑ | ≥ η

η η

• = −∞.

Local Large Deviation Principle: for any u ∈ C[0,T](R+) lim

δ→0 lim K→∞

1 K 2(1−γ) log P

• sup

t≤T

• xK

t − ut

• ≤ δ
• = −JT(u·).

Klebaner, Liptser

• 18. Proofs. Upper bound for P(τ0 ≤ T) via LDP

Notice that {τ0 ≤ T} = {(xK

t )t≤T ∈ D}

with D =

• u· ∈ C[0,T](R+) : u0 = 1, Θ(u) ≤ T
• .

the closed set in the uniform metric. Then, due to the LDP , lim

K→∞

1 K 2(1−γ) log P

• τ0 ≤ T
• ≤ − inf

u∈D JT(u).

Klebaner, Liptser

• 19. Proofs. Computation of infu∈D JT(u)

infu∈D JT(u) infu∈D JT(u)

The minimization is related to the optimal control problem with the control action (as, r)s≤r≤T and controlled process us, s ≤ r: ˙ us = µus + uγ

s as, s ≤ r,

u0 = 1 for the cost functional JT(u). Thus, inf

u·∈D JT(u·) =

min

u:JT (u)<∞

Θ(u)∧T ( ˙ ut − ut)2 u2γ

t

dt = inf

(w,r):us>0,s<r≤T

r a2

sds.

Klebaner, Liptser

• 20. Proofs. The optimal control

Change of variables A control (a∗

s, r ∗) is optimal provided

that for any control (as, r), r∗ (a∗

s)2ds ≤

r a2

sds.

A role of γ < 1 is crucial in the finding of the optimal

• control. Set

vt = u1−γ

t

. It is obvious that v0 = u0 = 1, vr = ur = 0 and vt solves the linear differential equation ˙ vs = µ(1 − γ)vs + (1 − γ)as.

Klebaner, Liptser

• 21. Proofs. The optimal control (continuation)

Hence, for any admissible control, 0 = vr = eµ(1−γ)r + r eµ(1−γ)(r−s)(1 − γ)asds implies − 1 1 − γ = r e−µ(1−γ)sasds. By Cauchy-Schwarz’s inequality, r a2

sds ≥

1 (1 − γ)2 r e−2µ(1−γ)sds ≥ 2µ (1 − γ)[1 − e−2µ(1−γ)T].

Klebaner, Liptser

• 22. Proofs. The optimal control (continuation)

The control (a∗

s, r), with fixed r and a∗ s, transforming the

Cauchy-Schwarz inequality to the equality, is optimal with r replaced by T. Thus, r ∗ = T a∗

s = −

2µ 1 − e−2µ(1−γ)T e−µ(1−γ)s, 0 ≤ s ≤ T.

Klebaner, Liptser

• 23. Proofs. Upper bound for P(τ0 ≤ T)

The optimal control (a∗

s, T)s≤T defines v∗ s and, in turn,

u∗

s = (v∗ s )1/(1−γ),

u∗

s = eµt

1 − 1 − e−2µ(1−γ)s 1 − e−2µ(1−γ)T 1/(1−γ) In particular, JT(u∗) = − 1 σ2 µ (1 − γ)[1 − e−2µ(1−γ)T] Thus, lim

K→∞

1 K 2(1−γ) log P

• τ0 ≤ T
• ≤ − 1

σ2 µ (1 − γ)[1 − e−2µ(1−γ)T].

Klebaner, Liptser

• 24. Most likely path to ruin

By the definition of u∗

s, we have

JT(u∗) ≤ JT(u), ∀ u ∈ D. On the other hand, by the local LDP , lim

δ→0 lim K→∞

1 K 2(1−γ) log P

• sup

t≤T

|xK

t − ut| ≤ δ

• = −JT(u)

lim

δ→0 lim K→∞

1 K 2(1−γ) log P

• sup

t≤T

|xK

t − u∗ t | ≤ δ

• = −JT(u∗).

Hence, for sufficiently large K P

• sup

t≤T

|xK

t − u∗ t | ≤ δ

• sup

t≤T

|xK

t − ut| ≤ δ

• .

(⋆)

Klebaner, Liptser

• 25. Most likely path to ruin (continuation)

u∗ maximizes P

• supt≤T |xK

t − u∗ t | ≤ δ

• in the LDP scale

and so, it u∗

s = eµt

1 − 1 − e−2µ(1−γ)s 1 − e−2µ(1−γ)T 1/(1−γ) .