[PPT] - On Non-stability of some Inverse Problem in Inverse Problem in PowerPoint Presentation

SLIDE 1

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

On Non-stability of some Inverse Problem in Option Pricing

Peter Mathé

Weierstrass Institute, Berlin joint work with R. Krämer: Modulus of Continuity of Nemytski˘ ı operators with application to the problem of option pricing, J. Inv. Ill-Posed Problems (16):435–461, 2008 E-mail: mathe@wias-berlin.de Homepage:http://www.wias-berlin.de/people/mathe

Linz, October 30, 2008

SLIDE 2

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Outline

1

Introduction, main results

2

Modulus of continuity for classes of Nemytski˘ ı operators

3

Local analysis of forward and backward Black-Scholes kernels

4

Summary, prospective

SLIDE 3

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Problem formulation

We consider a time-dependent Black-Scholes model dPτ = µPτ dτ + σ(τ)Pτ dWτ , for a time-dependent volatility σ(t) > 0 on a finite time-horizon 0 ≤ τ ≤ T.

SLIDE 4

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Problem formulation

We consider a time-dependent Black-Scholes model dPτ = µPτ dτ + σ(τ)Pτ dWτ , for a time-dependent volatility σ(t) > 0 on a finite time-horizon 0 ≤ τ ≤ T. The price of a European call is then obtained as C(t) = ucall

BS (P, K, r, t, S(t)) where

P is the actual asset price K is the strike price r is the short interest rate t is the maturity, and S(t) := t

0 σ2(τ) dτ is the integrated volatility.

Remark This is the same model problem as considered in Bernd Hofmann’s talk.

SLIDE 5

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

The Black-Scholes function

Below we keep K, P, r fixed! The function ucall

BS (P, K, r, t, s) =

PΦ(d + √s) − Ke−rtΦ(d)

, if s > 0,

P − Ke−rt

+

, for s = 0 , with c(t) := log P Ke−rt , and d(t, s) := c(t) − s/2 √s , defines a kernel k(t, s), t ∈ [0, T], s > 0, C(t) = k(t, S(t)).

SLIDE 6

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

The Black-Scholes function

Below we keep K, P, r fixed! The function ucall

BS (P, K, r, t, s) =

PΦ(d + √s) − Ke−rtΦ(d)

, if s > 0,

P − Ke−rt

+

, for s = 0 , with c(t) := log P Ke−rt , and d(t, s) := c(t) − s/2 √s , defines a kernel k(t, s), t ∈ [0, T], s > 0, C(t) = k(t, S(t)). Remark The option price depends on the volatility only through the integrated volatility S(t)! The pricing is a composition of the linear operator S(t) and the non-linear Black-Scholes mapping.

SLIDE 7

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Nemytski˘ ı operators (superposition operators)

Let k(t, s), t ∈ I := [0, T], 0 < s < smax(t) be any kernel

function. We assign the non-linear mapping

[NS](t) := k(t, S(t)), 0 ≤ t ≤ T, with domain of definition D+(N) := {f ∈ C(I), f(0) = 0, 0 < f(t) < smax(t), t ∈ I} , and with range R(N). For the general theory we refer to [Appell-Zabrejko].

SLIDE 8

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Nemytski˘ ı operators (superposition operators)

Let k(t, s), t ∈ I := [0, T], 0 < s < smax(t) be any kernel

function. We assign the non-linear mapping

[NS](t) := k(t, S(t)), 0 ≤ t ≤ T, with domain of definition D+(N) := {f ∈ C(I), f(0) = 0, 0 < f(t) < smax(t), t ∈ I} , and with range R(N). For the general theory we refer to [Appell-Zabrejko]. Problem (Pricing) Determine C(t) := [NS](t), t ∈ I!

SLIDE 9

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Nemytski˘ ı operators (superposition operators)

Let k(t, s), t ∈ I := [0, T], 0 < s < smax(t) be any kernel

function. We assign the non-linear mapping

[NS](t) := k(t, S(t)), 0 ≤ t ≤ T, with domain of definition D+(N) := {f ∈ C(I), f(0) = 0, 0 < f(t) < smax(t), t ∈ I} , and with range R(N). For the general theory we refer to [Appell-Zabrejko]. Problem (Pricing) Determine C(t) := [NS](t), t ∈ I! Problem (Calibration, subject of Bernd’s talk) Determine σ2(τ) = S−1 N−1(C(t))

from option

prices C(t), t ∈ I.

SLIDE 10

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Nature of ill-posedness

Both the pricing and calibration problems were studied in several papers, [Hein-diss, Hein/Hofmann, Hofmann/Kraemer]. The following facts appear important: The operator S : C(I) → C(I) is a compact linear

perator, and hence the inversion is ill-posed.

Both, the non-linear opeators

N : D(N) ⊂ C(I) → C(I), and its inverse N−1 : R(N) → C(I)

are continuous! The latter is remarkable and was first proved by R. Krämer, see [Kraemer/Richter].

SLIDE 11

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Modulus of continuity

To any continuous (non-linear) operator K : D(K) ⊂ X → Y

ne can assign its modulus of continuity at x† ∈ D(K) as

ω(K, x†, δ) := sup

K(x) − K(x†)Y, x ∈ D(K), x − x†X ≤ δ
.

SLIDE 12

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Modulus of continuity

To any continuous (non-linear) operator K : D(K) ⊂ X → Y

ne can assign its modulus of continuity at x† ∈ D(K) as

ω(K, x†, δ) := sup

K(x) − K(x†)Y, x ∈ D(K), x − x†X ≤ δ
.

Remark This resembles the modulus of continuity of a real valued continuous function ω(f, δ) := sup

f(x) − f(x′)
, x, x′ ∈ [a, b],
x − x′

≤ δ

,

and it is its local counterpart, we refer to approximation theory, see [Korne˘ ıchuk]. When further restricting the domain, then better bounds possible.

SLIDE 13

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Pictures from [Romy’s dissertation]

The following behavior was observed from simulation data:

0.01 0.02 0.03 0.04 0.05 1 2 3 4 5 6 7 x 10

−3

δk measurements

Figure: (left) Behavior of modulus of continuity ω(N−1, u†, δ) as δ → 0.

SLIDE 14

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Pictures from [Romy’s dissertation]

The following behavior was observed from simulation data:

0.01 0.02 0.03 0.04 0.05 1 2 3 4 5 6 7 x 10

−3

δk measurements

1 1.5 2 2.5 5 5.2 5.4 5.6 5.8 6 6.2 6.4

log log(1/δk) log(1/ω(N −1, Nsmax, δk)) measurements linear regression

Figure: (left) Behavior of modulus of continuity ω(N−1, u†, δ) as δ → 0. (right) Linear regression on the model ω(N−1, u†, δ) ≈ log−µ(1/δ) yields ˆ µ = 1.1124. (parameters P = 1, K = 0.85, r = 0.05, T = 0.2)

SLIDE 15

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Why the modulus of continuity?

Given x† ∈ D(K) let us introduce the set Kδ(x†) :=

x ∈ D(K), x − x†X ≤ δ
.

We want to approximate y† := K(x†) from noisy data x ∈ Kδ(x†). Let φ be any method of reconstruction. Its error is then e(φ, x†) := sup

K(x) − K(x†)Y,

x ∈ Kδ(x†)

SLIDE 16

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Why the modulus of continuity?

Given x† ∈ D(K) let us introduce the set Kδ(x†) :=

x ∈ D(K), x − x†X ≤ δ
.

We want to approximate y† := K(x†) from noisy data x ∈ Kδ(x†). Let φ be any method of reconstruction. Its error is then e(φ, x†) := sup

K(x) − K(x†)Y,

x ∈ Kδ(x†)

≥ inf

η∈Y

K(x) − ηY,

x ∈ Kδ(x†)

SLIDE 17

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Why the modulus of continuity?

Given x† ∈ D(K) let us introduce the set Kδ(x†) :=

x ∈ D(K), x − x†X ≤ δ
.

We want to approximate y† := K(x†) from noisy data x ∈ Kδ(x†). Let φ be any method of reconstruction. Its error is then e(φ, x†) := sup

K(x) − K(x†)Y,

x ∈ Kδ(x†)

≥ inf

η∈Y

K(x) − ηY,

x ∈ Kδ(x†)

=: rad(K(Kδ(x†))).

SLIDE 18

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Why the modulus of continuity? cont’d

The set K(Kδ(x†)) has both a radius and a diameter diam(K(Kδ(x†))) := sup

K(x) − K(x′), x, x′ ∈ K(Kδ(x†))
,

and it holds diam(K(Kδ(x†))) ≥ rad(K(Kδ(x†))) ≥ 1/2 diam(K(Kδ(x†))). This implies that e(φ, x†) ≥ 1/2ω(K, x†, δ)!

SLIDE 19

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Why the modulus of continuity? cont’d

The set K(Kδ(x†)) has both a radius and a diameter diam(K(Kδ(x†))) := sup

K(x) − K(x′), x, x′ ∈ K(Kδ(x†))
,

and it holds diam(K(Kδ(x†))) ≥ rad(K(Kδ(x†))) ≥ 1/2 diam(K(Kδ(x†))). This implies that e(φ, x†) ≥ 1/2ω(K, x†, δ)! Theorem Suppose we want to approximate y† = K(x†) from noisy data x ∈ Kδ(x†). Then for any reconstruction method it holds true that e(φ, x†) ≥ 1/2ω(K, x†, δ).

SLIDE 20

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Why the modulus of continuity? cont’d

The set K(Kδ(x†)) has both a radius and a diameter diam(K(Kδ(x†))) := sup

K(x) − K(x′), x, x′ ∈ K(Kδ(x†))
,

and it holds diam(K(Kδ(x†))) ≥ rad(K(Kδ(x†))) ≥ 1/2 diam(K(Kδ(x†))). This implies that e(φ, x†) ≥ 1/2ω(K, x†, δ)! Theorem Suppose we want to approximate y† = K(x†) from noisy data x ∈ Kδ(x†). Then for any reconstruction method it holds true that e(φ, x†) ≥ 1/2ω(K, x†, δ). Remark For linear operators this is well known, see [V/V].

SLIDE 21

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Results for the pricing problem

Theorem (not at the money: well-conditioned) Suppose that P = K. For each S† ∈ D+(N) there are δ0 > 0 and L < ∞ such that ω(N, S†, δ) ≤ Lδ, 0 < δ ≤ δ0.

SLIDE 22

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Results for the pricing problem

Theorem (not at the money: well-conditioned) Suppose that P = K. For each S† ∈ D+(N) there are δ0 > 0 and L < ∞ such that ω(N, S†, δ) ≤ Lδ, 0 < δ ≤ δ0. Theorem (at the money: ill-conditioned) Suppose that P = K and r = 0. For each ε > 0 there is δ(ε) > 0 such that ω(N, D(N), δ) ≤ (1 + ε) P √ 2π √ δ, 0 < δ ≤ δ(ε).

SLIDE 23

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Results for the pricing problem

Theorem (not at the money: well-conditioned) Suppose that P = K. For each S† ∈ D+(N) there are δ0 > 0 and L < ∞ such that ω(N, S†, δ) ≤ Lδ, 0 < δ ≤ δ0. Theorem (at the money: ill-conditioned) Suppose that P = K and r = 0. For each ε > 0 there is δ(ε) > 0 such that ω(N, D(N), δ) ≤ (1 + ε) P √ 2π √ δ, 0 < δ ≤ δ(ε). Remark The interesting case P = K and r > 0 is open!

SLIDE 24

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Results for calibration

Theorem (not at the money: ill-conditioned) Suppose that P = K and u† = N(S†) for some S† ∈ D+(N). For each ε > 0 there is δ(ε) such that ω(N−1, u†, δ) ≤ (1 + ε)log2(P/K) 2 log(1/δ) , 0 < δ ≤ δ(ε).

SLIDE 25

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Results for calibration

Theorem (not at the money: ill-conditioned) Suppose that P = K and u† = N(S†) for some S† ∈ D+(N). For each ε > 0 there is δ(ε) such that ω(N−1, u†, δ) ≤ (1 + ε)log2(P/K) 2 log(1/δ) , 0 < δ ≤ δ(ε). Theorem (at the money: well-conditioned) If P = K and r = 0 then there is a constant L < ∞ such that ω(N−1, D¯

s(N−1), δ) ≤ Lδ.

SLIDE 26

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Results for calibration

Theorem (not at the money: ill-conditioned) Suppose that P = K and u† = N(S†) for some S† ∈ D+(N). For each ε > 0 there is δ(ε) such that ω(N−1, u†, δ) ≤ (1 + ε)log2(P/K) 2 log(1/δ) , 0 < δ ≤ δ(ε). Theorem (at the money: well-conditioned) If P = K and r = 0 then there is a constant L < ∞ such that ω(N−1, D¯

s(N−1), δ) ≤ Lδ.

Remark The last result was proven in Bernd’s talk. The interesting case P = K and r > 0 is open for general domain of definition!

SLIDE 27

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Results for calibration

Theorem (not at the money: ill-conditioned) Suppose that P = K and u† = N(S†) for some S† ∈ D+(N). For each ε > 0 there is δ(ε) such that ω(N−1, u†, δ) ≤ (1 + ε)log2(P/K) 2 log(1/δ) , 0 < δ ≤ δ(ε). Theorem (at the money: well-conditioned) If P = K and r = 0 then there is a constant L < ∞ such that ω(N−1, D¯

s(N−1), δ) ≤ Lδ.

Remark The last result was proven in Bernd’s talk. The interesting case P = K and r > 0 is open for general domain of definition!

SLIDE 28

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Lower bound

Proposition Suppose that r > 0. There are c > 0 and δ0 > 0 such that ω(N−1, D+,¯

s(N−1), δ) ≥

c log(1/δ), 0 < δ ≤ δ0.

SLIDE 29

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Lower bound

Proposition Suppose that r > 0. There are c > 0 and δ0 > 0 such that ω(N−1, D+,¯

s(N−1), δ) ≥

c log(1/δ), 0 < δ ≤ δ0. Problem We conjecture that ω(N, D+(N), δ) ≥ (1 − ε) P √ 2π √ δ, 0 < δ ≤ δ(ε) for P = K and r > 0.

SLIDE 30

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Lower bound

Proposition Suppose that r > 0. There are c > 0 and δ0 > 0 such that ω(N−1, D+,¯

s(N−1), δ) ≥

c log(1/δ), 0 < δ ≤ δ0. Problem We conjecture that ω(N, D+(N), δ) ≥ (1 − ε) P √ 2π √ δ, 0 < δ ≤ δ(ε) for P = K and r > 0.

SLIDE 31

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Well- and ill-conditioning of Nemytski˘ ı operators

Definition We shall call the operator K well-conditioned if ω(K, D, δ)/δ ≤ L is uniformly bounded as δ → 0, or it is ill-conditioned if the quotient is unbounded (along a sub-sequence) as δ → 0. The first result is immediate. Proposition If the family (kt)t∈I is uniformly Lipschitz then the corresponding operator is well-conditioned.

SLIDE 32

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Well- and ill-conditioning of Nemytski˘ ı operators

Definition We shall call the operator K well-conditioned if ω(K, D, δ)/δ ≤ L is uniformly bounded as δ → 0, or it is ill-conditioned if the quotient is unbounded (along a sub-sequence) as δ → 0. The first result is immediate. Proposition If the family (kt)t∈I is uniformly Lipschitz then the corresponding operator is well-conditioned. Remark Lipschitz continuity for Nemytski˘ ı operators was studied, see [Appell-Zabrejko].

SLIDE 33

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Classes of kernels

Assumption The kernel k is

1

continuously differentiable with respect to s and t for 0 < t ≤ T and 0 < s < smax(t),

2

for each t ∈ I the function kt is an index function.

SLIDE 34

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Classes of kernels

Assumption The kernel k is

1

continuously differentiable with respect to s and t for 0 < t ≤ T and 0 < s < smax(t),

2

for each t ∈ I the function kt is an index function. Case 1 For each t ∈ I there is 0 < s(t) ≤ smax(t) such that the function k(t, s) is convex on [0, s(t)). Case 2 For each t ∈ I there is 0 < s(t) ≤ smax(t) such that the function k(t, s) is concave on [0, s(t)). In addition, for each 0 < t0 ≤ T there is m := m(t0) ≤ T, such that for each 0 ≤ t ≤ t0 and 0 ≤ s < smax(t) it holds that k(t, s) ≤ k(m(t0), s).

SLIDE 35

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Pictures for the cases

s(t) kt(s(t)) kt(s) s s 00 00 00 s(t) s(t) s(t) kt(s(t)) kt(s(t)) kt(s(t)) kt(s) kt(s) kt(s) s

Figure: Kernels corresponding to Case 1 (left) and Case 2 (right),

respectively. The point s(t) is indicated.

Remark Case 1 corresponds to well-conditioning of the operator, whereas in Case 2 we shall provide a concave bound in δ, which reflects ill-conditioning.

SLIDE 36

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Pictures for the cases

s(t) kt(s(t)) kt(s) s s 00 00 00 s(t) s(t) s(t) kt(s(t)) kt(s(t)) kt(s(t)) kt(s) kt(s) kt(s) s

Figure: Kernels corresponding to Case 1 (left) and Case 2 (right),

respectively. The point s(t) is indicated.

Remark Case 1 corresponds to well-conditioning of the operator, whereas in Case 2 we shall provide a concave bound in δ, which reflects ill-conditioning.

SLIDE 37

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Pictures for the cases

s(t) kt(s(t)) kt(s) s s 00 00 00 s(t) s(t) s(t) kt(s(t)) kt(s(t)) kt(s(t)) kt(s) kt(s) kt(s) s

Figure: Kernels corresponding to Case 1 (left) and Case 2 (right),

respectively. The point s(t) is indicated.

Remark Case 1 corresponds to well-conditioning of the operator, whereas in Case 2 we shall provide a concave bound in δ, which reflects ill-conditioning.

SLIDE 38

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

General results

For each 0 < t0 ≤ T there are L(t0), C(t0) < ∞ such that sup

0≤t≤t0

∂k ∂s (t, s), s(t) ≤ s < smax(t)

≤ L(t0)

(1) sup

t∈[t0,T]

∂k ∂s (t, s), s ∈ [s0, S0]

≤ C(t0).

(2)

SLIDE 39

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

General results

For each 0 < t0 ≤ T there are L(t0), C(t0) < ∞ such that sup

0≤t≤t0

∂k ∂s (t, s), s(t) ≤ s < smax(t)

≤ L(t0)

(1) sup

t∈[t0,T]

∂k ∂s (t, s), s ∈ [s0, S0]

≤ C(t0).

(2) Proposition Suppose that there is 0 < t0 ≤ T such that (1) and (2) hold

true. Then, for each S† ∈ D+(N) there is δ0 > 0 such that

Case 1 ω(N, S†, δ) ≤ (L(t0) + C(t0))δ Case 2 ω(N, S†, δ) ≤ k(m(t0), δ) + (2L(t0) + C(t0))δ for 0 < δ ≤ δ0.

SLIDE 40

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

General results

For each 0 < t0 ≤ T there are L(t0), C(t0) < ∞ such that sup

0≤t≤t0

∂k ∂s (t, s), s(t) ≤ s < smax(t)

≤ L(t0)

(1) sup

t∈[t0,T]

∂k ∂s (t, s), s ∈ [s0, S0]

≤ C(t0).

(2) Proposition Suppose that there is 0 < t0 ≤ T such that (1) and (2) hold

true. Then, for each S† ∈ D+(N) there is δ0 > 0 such that

Case 1 ω(N, S†, δ) ≤ (L(t0) + C(t0))δ Case 2 ω(N, S†, δ) ≤ k(m(t0), δ) + (2L(t0) + C(t0))δ for 0 < δ ≤ δ0.

SLIDE 41

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

General results

For each 0 < t0 ≤ T there are L(t0), C(t0) < ∞ such that sup

0≤t≤t0

∂k ∂s (t, s), s(t) ≤ s < smax(t)

≤ L(t0)

(1) sup

t∈[t0,T]

∂k ∂s (t, s), s ∈ [s0, S0]

≤ C(t0).

(2) Proposition Suppose that there is 0 < t0 ≤ T such that (1) and (2) hold

true. Then, for each S† ∈ D+(N) there is δ0 > 0 such that

Case 1 ω(N, S†, δ) ≤ (L(t0) + C(t0))δ Case 2 ω(N, S†, δ) ≤ k(m(t0), δ) + (2L(t0) + C(t0))δ for 0 < δ ≤ δ0.

SLIDE 42

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Reducing the problem

In the previous proposition we established bounds in terms of the kernel function k(t0, ·), and this is supposed to be an index function! To understand the behavior as δ → 0 we need to understand the asymptotics near zero. This is even more complicated for the backward kernel, since this is given implicitly!

SLIDE 43

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Reducing the problem

In the previous proposition we established bounds in terms of the kernel function k(t0, ·), and this is supposed to be an index function! To understand the behavior as δ → 0 we need to understand the asymptotics near zero. This is even more complicated for the backward kernel, since this is given implicitly! Problem Given an index function ϕ.

1

How to determine its asymptotics as δ → 0?

2

How to determine the asymptotics for the inverse index function?

SLIDE 44

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Local analysis for the forward kernel

Definition (see [Bourbaki: Functions of a real variable]) Two index functions ϕ, ψ are called equivalent at zero, and we write ϕ ∼ ψ, if lims→0+ ϕ(s)/ψ(s) = 1.

SLIDE 45

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Local analysis for the forward kernel

Definition (see [Bourbaki: Functions of a real variable]) Two index functions ϕ, ψ are called equivalent at zero, and we write ϕ ∼ ψ, if lims→0+ ϕ(s)/ψ(s) = 1. We introduce the auxiliary kernel ht(s) = h(t, s) :=

PKe−rt

2π 1 c(t)2 s3/2e− c(t)2

2s

s > 0, t > 0, where c(t) := log(K/P) + rt. Lemma For any t ∈ I the following asymptotics holds true. ˜ kt(s) ∼

ht(s)

, if c(t) = 0, P

s

2π

, if c(t) = 0, as s → 0.

SLIDE 46

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Pictures

We indicate the kernels and their asymptotics.

00 s0 u0 functions ˜ kt and ht ˜ kt ht s functions ˜ kt and P

s/(2π)

˜ kt P

s/(2π)

s

Figure: The forward kernels and their asymptotics for c(t) = 0 (left) and c(t) = 0 (right).

SLIDE 47

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Local analysis of the backward kernel

We have established the local behavior for the forward

kernel. How to obtain the asymptotics for the implicitly given

backward kernel? Problem Suppose that ϕ, ψ are equivalent index functions. Are the inverse functions equivalent, i.e. that ϕ−1 ∼ ψ−1?

SLIDE 48

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Local analysis of the backward kernel

We have established the local behavior for the forward

kernel. How to obtain the asymptotics for the implicitly given

backward kernel? Problem Suppose that ϕ, ψ are equivalent index functions. Are the inverse functions equivalent, i.e. that ϕ−1 ∼ ψ−1? Example It holds ϕ(s) = log−1(1/s) ∼ ψ(s) =

1 + 2 log−1(1/s) − 1

However, ϕ−1(s) = e−1/s ∼ ψ−1(s) = e−2/(2s+s2).

SLIDE 49

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Asymptotics of the backward kernel

One can ensure preservation of equivalence under Property (R) For each 0 < ε < 1 there are 0 < δ < 1, 0 < s0 ≤ a/2 with ̺((1 + δ)s) ̺(s) ≤ 1 + ε, 0 < s ≤ s0.

SLIDE 50

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Asymptotics of the backward kernel

One can ensure preservation of equivalence under Property (R) For each 0 < ε < 1 there are 0 < δ < 1, 0 < s0 ≤ a/2 with ̺((1 + δ)s) ̺(s) ≤ 1 + ε, 0 < s ≤ s0. Lemma Suppose that ϕ ∼ ψ. If the inverse function ϕ−1 has Property (R) then ϕ−1 ∼ ψ−1.

SLIDE 51

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Asymptotics of the backward kernel

One can ensure preservation of equivalence under Property (R) For each 0 < ε < 1 there are 0 < δ < 1, 0 < s0 ≤ a/2 with ̺((1 + δ)s) ̺(s) ≤ 1 + ε, 0 < s ≤ s0. Lemma Suppose that ϕ ∼ ψ. If the inverse function ϕ−1 has Property (R) then ϕ−1 ∼ ψ−1. Lemma (Asymptotics of backwards kernel) ˜ gt(u) ∼

c(t)2

2 log(1/u)

, if c(t) = 0, 2π (u/P)2 , if c(t) = 0.

SLIDE 52

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Summary

We established the asymptotics of the modulus of continuity in a time-dependent Black-Scholes model. For the calibration problem we proved ill-conditioning of the modulus for P = K with a logarithmic rate! We established a lower bound in this case, such that ill-conditioning extends to all models which are more involved! In contrast calibration is well-conditioned if P = K provided that r = 0.

SLIDE 53

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Summary

We established the asymptotics of the modulus of continuity in a time-dependent Black-Scholes model. For the calibration problem we proved ill-conditioning of the modulus for P = K with a logarithmic rate! We established a lower bound in this case, such that ill-conditioning extends to all models which are more involved! In contrast calibration is well-conditioned if P = K provided that r = 0. Problem Prove asymptotics at the money (P = K) under r > 0!

SLIDE 54

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

Summary

We established the asymptotics of the modulus of continuity in a time-dependent Black-Scholes model. For the calibration problem we proved ill-conditioning of the modulus for P = K with a logarithmic rate! We established a lower bound in this case, such that ill-conditioning extends to all models which are more involved! In contrast calibration is well-conditioned if P = K provided that r = 0. Problem Prove asymptotics at the money (P = K) under r > 0!

Thank’s!

SLIDE 55

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

◮ Jürgen Appell and Petr P

. Zabrejko. Nonlinear superposition operators, volume 95 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990.

◮ Nicolas Bourbaki.

Functions of a real variable. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004.

◮ Torsten Hein.

Analytic and numeric studies concerning an inverse problem of option pricing (in German). PhD thesis, TU Chemnitz, 2003. http: //archiv.tu-chemnitz.de/pub/2003/0160.

◮ Torsten Hein and Bernd Hofmann.

SLIDE 56

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

On the nature of ill-posedness of an inverse problem arising in option pricing. Inverse Problems, 19(6):1319–1338, 2003.

◮ Bernd Hofmann and Romy Krämer.

On maximum entropy regularization for a specific inverse problem of option pricing.

J. Inverse Ill-Posed Probl., 13(1):41–63, 2005.

◮ N. Korne˘

ıchuk. Exact constants in approximation theory, volume 38 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1991. Translated from the Russian by K. Ivanov.

◮ Romy Krämer.

Identification in Financial Models with Time-Dependent Volatility and Stochastic Drift Components.

SLIDE 57

On Non-stability

f some

Inverse Problem in Option Pricing Peter Mathé Introduction, main results Modulus of continuity for classes of Nemytski˘ ı

perators

Local analysis

f forward and

backward Black-Scholes kernels Summary, prospective

PhD thesis, TU Chemnitz, 2007. http: //archiv.tu-chemnitz.de/pub/2007/0080.

◮ Romy Krämer and Matthias Richter.

Ill-posedness vs. ill-conditioning - an example from inverse option pricing.

Applic. Analysis, 87(4): 465–477, 2008.