J.P.Zubelli(IMPA) and O.Scherzer(Univ.Vienna) AdrianoDeCezaroe-mail:decezaro@impa.brJointworkwith ConvexRegularizationofLocalVolatilityModelsfromOptionPrices:ConvergenceAnalysisandRates

(1)

Outline Asset Models and Option Pricing The Formulation of the Inverse Problem Properties ofFand ill-posedness of the inverse problem State of the art from the literature Non-quadratic regularization of calibration problem Exponential Families Conclusions

Convex Regularization of Local Volatility Models from Option Prices: Convergence Analysis and

Rates

Adriano De Cezaro e-mail: decezaro@impa.br

Joint work withJ. P. Zubelli (IMPA)andO. Scherzer (Univ.

Vienna)

(2)

Plan

1 Asset Models and Option Pricing 2 The Formulation of the Inverse Problem

3 Properties ofF and ill-posedness of the inverse problem 4 State of the art from the literature

5 Non-quadratic regularization of calibration problem Convergence

Convergence rates 6 Exponential Families 7 Conclusions

(3)

Derivative Contracts

European Call Option: a forward contract that gives the holder the right, but not the obligation, to buy one unit of an underlying asset for an agreedstrike price K on thematuritydateT.

Its payoff is given by h(X_T) =

X_T−K ifX_T>K, 0 ifXT≤K.

Fundamental QuestionHow to price such an obligation given today’s information?

(4)

Derivative Contracts

European Call Option: a forward contract that gives the holder the right, but not the obligation, to buy one unit of an underlying asset for an agreedstrike price K on thematuritydateT.

Its payoff is given by h(X_T) =

X_T−K ifX_T>K, 0 ifXT≤K.

Fundamental QuestionHow to price such an obligation given today’s information?

(5)

Main Contributions

L. Bachelier (Paris) P. Samuelson F. Black M. Scholes R. Merton

recognized by the Nobel prize in Economics award to R. Merton and M. Scholes

(6)

Black-Scholes Market Model

We consider the Black-Scholes equation (see Black-Scholes-1973):

Ut+¹

2σ²(t,X)X²UXX+ (r−q)XUX =rU, (1) U^t^,^S(t=T,X) =max(X−K,0) European Call (2) Here,X =X(t)denotes the spot price,K is called strike,T the maturity,ris the interesting rate,qthe dividends andσ(t,X)is the local volatility.

Note 1U=U(t,x;σ,r)fort≤T. Note 2Final Value Problem

RemarkConstant volatility model - Nobel Prize: Robert Merton and Myron Scholes

(7)

Ut+¹

Note 1U=U(t,x;σ,r)fort≤T.

Note 2Final Value Problem

(8)

Ut+¹

Note 1U=U(t,x;σ,r)fort≤T. Note 2Final Value Problem

(9)

Smile effect: Empirical remark that calls having different strikes, but otherwise identical, have different implied volatilities.

Before the 1987 crash the graph ofI(K)(fixt,X,T) had aUshape with a minimum close toK =X.

Since 1987 it is a decreasing function in the range

95%<K/X<105%then (forK >>X) it bends upwards. Limitations ...

(10)

Smile effect: Empirical remark that calls having different strikes, but otherwise identical, have different implied volatilities. Before the 1987 crash the graph ofI(K)(fixt,X,T) had aUshape with a minimum close toK =X.

95%<K/X<105%then (forK >>X) it bends upwards. Limitations ...

(11)

95%<K/X<105%then (forK >>X) it bends upwards.

Limitations ...

(12)

95%<K/X<105%then (forK >>X) it bends upwards.

Limitations ...

(13)

Local Volatility Models

IdeaAssume that the volatility is given by σ=σ(t,x) i.e.: it depends on time and the asset price.

The Direct Problem

Givenσ=σ(t,x)and the payoff information, determine U=U(t,x,T,K;σ)

(14)

The Inverse Problem

Given a set of observed prices

{U=U(t,x,T,K;σ)}₍_T_,_K_)∈_S find the volatilityσ=σ(t,x).

The setS is taken typically as[T1,T2]×[K1,K2]. In PracticeVery limited and scarce data

(15)

The Smile Curve and Dupire’s Equation

Assuming that there exists a local volatility functionσ=σ(S,t) Dupire(1994) showed that the call price satisfies

∂TU−¹₂σ²(K,T)K²∂²_KU+rS∂KU=0 , S>0 ,t<T

U(K,T =0) = (S−K)⁺, (3)

Theoreticalway of evaluating the local volatility

σ(K,T) = s

2

∂TU+rK∂KU K²∂²_KU

(4) In practiceTo estimateσfrom (3), limited amount of discrete data and thus interpolate.Numerical instabilities!Even to keep the argument positive is hard.

(16)

U(K,T =0) = (S−K)⁺, (3)

Theoreticalway of evaluating the local volatility σ(K,T) =

s 2

(4) In practiceTo estimateσfrom (3), limited amount of discrete data and thus interpolate.

Numerical instabilities!Even to keep the argument positive is hard.

(17)

U(K,T =0) = (S−K)⁺, (3)

Theoreticalway of evaluating the local volatility σ(K,T) =

s 2

(4) In practiceTo estimateσfrom (3), limited amount of discrete data and thus interpolate.Numerical instabilities!Even to keep the argument positive is hard.

(18)

Early Results

Bouchoev-Isakov Uniqueness and “Stability”

Consider the case oftime-independent volatilityand working with y=log(K/S(0)) τ=T−t. SupposeU(y,τ)anda(y) =σ(K) satisfies (3) with

U(y,0) =S(0)(1−e^y)⁺,y ∈R ⁽⁵⁾ U(y,τ^∗) =U^∗(y),y∈I (6) whereIis a sub-interval ofR. Then, we have

Uniqueness of the volatility

Stability of the volatility in the H ¨olderλ-norm w.r.t. to variations of the data on the 2+λ-norm. (i.e., one needs TWO extra

derivatives of the data).

(19)

Early Results

More precisely

Theorem (Bouchouev & Isakov)

Let U1and U2be solutions of (3-6) with a=a1and a=a2, resp., and the corresponding final data in Eq. (6) given by U₁^∗and U₂^∗. Let I0be an open interval with I⊃I₀6=0/. Then,

1 If U₁^∗=U₂^∗on I and a1(y) =a2(y)on I0then a1(y) =a2(y)on I.

2 If, in addition, a1(y) =a2(y)on I0∪(R\I)and I is bounded, then

∃C=C(|a₁|_λ(I),|a₂|_λ(I),I,I₀,τ^∗)s.t.

|a1−a2|_λ(I)≤ |U₁^∗−U₂^∗|₂_+λ(I)

RemarkThe assumption thatais known onI0⊂Imakes the problem overdetermined.

(20)

Early Results

More precisely

Theorem (Bouchouev & Isakov)

Let U1and U2be solutions of (3-6) with a=a1and a=a2, resp., and the corresponding final data in Eq. (6) given by U₁^∗and U₂^∗. Let I0be an open interval with I⊃I₀6=0/. Then,

1 If U₁^∗=U₂^∗on I and a1(y) =a2(y)on I0then a1(y) =a2(y)on I.

2 If, in addition, a1(y) =a2(y)on I0∪(R\I)and I is bounded, then

∃C=C(|a₁|_λ(I),|a₂|_λ(I),I,I₀,τ^∗)s.t.

|a1−a2|_λ(I)≤ |U₁^∗−U₂^∗|₂_+λ(I)

RemarkThe assumption thatais known onI0⊂Imakes the problem overdetermined.

Adriano De Cezaro e-mail: decezaro@impa.br Convex Regularization of Local Volatility Models from Option Prices: Convergence Analysis and Rates

(21)

Joint work with J.P. Zubelli & O. Scherzer

Start from

−U_t+¹

2σ²(T,K)K²U_KK−(r−q)KU_K−qU=0 (7) U^t^,^X(T=t,K) = (X−K)⁺, for K>0, (8) SettingK=Xe^y,τ=T−t,u(τ,y) =exp(

RT t

q(s)ds)U(.,T,K), b(τ) =q(τ)−r(τ)and

a(τ,y) =¹

2σ²(τ;K) (9)

yields

(22)

−u_τ+a(τ,y)(u_yy−u_y) +b(τ)u_y =0 (10) u(0,y) =X(1−e^y)⁺. (11) The parameter-to-solution mapF is defined by

F(a) =u(a), (12)

whereu(a)is the solution of (10) fora∈D⁽^F⁾^.

We assume noise datau^δsatisfies the inequality

||u^∗−u^δ||_L2(Ω)≤δ. (13)

(23)

Properties ofF and ill-posedness of the inverse problem

i) There exists ap^∗>2 such thatF :D⁽^F⁾^→^Wp²^,¹(Ω)is continuous and compact for 2≤p<p^∗. Moreover,F is weakly (sequentially) continuous and thus weakly closed.

ii) F is Gateaux differentiable w.r.t. a∈D⁽^F⁾in directionshsuch thata+h∈D⁽^F⁾, and the derivativeF⁰(a)extends as a linear operator toH¹(Ω).

iii) F⁰(a)is injective and compact. MoreoverR⁽^F⁰⁽^a⁾^∗^{) =}^H¹^(Ω)^.

Proof:see DC & Scherzer & Zubelli (2009) and Egger & Engl (2005) or Crepey (2003).

(24)

Ill-posedness in more details

D⁽^F⁾^:=^{^a^∈^a0+H¹ :a≤a,a₀≤a}. a_k *a˜ withu_k =F(a_k)→u˜=F(a˜)

similar pricesuk andu˜ linked with completely different volatilities

ask for regularization!!!! Tikhonov regularization

(25)

similar pricesuk andu˜ linked with completely different volatilities ask for regularization!!!!

Tikhonov regularization

(26)

similar pricesuk andu˜ linked with completely different volatilities ask for regularization!!!!

Tikhonov regularization

(27)

Standard Tikhonov regularizationresidual norm +βtimes penalization penalization=|| · ||².

source wise conditionequivalentwhat we assume know about the inverse solution

convergence rates to Tikhonov for quadratic penalization

||a^δ−a^†||=O⁽^√^δ)

Our approach: convex regularizationβf(·)withf convex, positive, etc...

(28)

||a^δ−a^†||=O⁽^√^δ)

(29)

||a^δ−a^†||=O⁽^√^δ)

(30)

||a^δ−a^†||=O⁽^√^δ)

(31)

State of the art from the literature

Let the standard choice of regularization parameterβ=β(δ)∼δ. (i) Iff(·) =|| · ||²_H1(Ω), then we have the convergence rate results

(Egger & Engl(2005))

||a^δ_β−a^†||_H1(Ω)=O⁽^√^δ)ând^||^F⁽â^δ_β⁾⁻û^δ^||L²(Ω)=O^(δ), ⁽¹⁴⁾

assuming the source-wise representationa^∗−a^†=F⁰(a^†)^∗w.

(ii) In (Hoffman & Kramer (2005)), with

f(a(τ)) =^R_I{a(τ)ln^a_a_¯^(τ)_(τ)+a¯(τ)−a(τ)}dτthe convergence rate results

||a^δ_β−a^†||_L¹_([₀_,_T_])=O⁽^√^δ) ⁽¹⁵⁾

using a source-wise representationln^a_a^†∗ =F⁰(a^†)^∗w.

(32)

iii) Withf(·) =|| · ||²_L2([0,T]), (Hoffman et al. (2006)) obtained the same rates of (Hoffman & Kramer (2005) (volatility

time-dependent only) assuming the source wise representation ξ^†=F⁰(a^†)^∗w∈∂f(a^†) (16) in terms of Bregman distances.

(33)

Convergence Convergence rates

Our approach

Minimize the functional

F_β,u^δ(a):=¹

2||F(a)−u^δ||²_L2(Ω)+βf(a), (17) wheref :dom(f)⊂B1→[0,∞]is a convex, proper and lower

semi-continuous stabilization functional.

(34)

Convergence

Theorem (Existence, Stability, Convergence)

Suppose that F , f ,D⁽^F⁾^{as above,}^β^>⁰^and⁽¹³⁾holds. Then There exists a minimizer ofFβ,u^δ.

If(u_k)→u in L²(Ω), then every sequence(a_k)with ak ∈argmin

Fβ,uk(a):a∈D⁽^F⁾

has a subsequence which weak converges. The limit of every w-convergent subsequence(a_k⁰)of(a_k)is a minimizera of˜ Fβ,u, and f(ak⁰)

converges to f(a˜).

(35)

Theorem (Semi-convergence)

If there exists a solution of (12)inD⁽^F⁾, then there exists an f -minimizing solution of (12).

Assume that (12)has a solution inD⁽^F⁾(which implies the existence of an f -minimizing solution) and thatβ:(0,∞)→(0,∞) satisfies

β(δ)→0and δ²

β(δ) →0, asδ→0. (18) Moreover,(δ_k)→0, and that u_k :=u^δ^k satisfiesku¯−u_kk ≤δ_k. Setβk :=β(δk). Then, every sequence(ak)of elements

minimizingFβk,uk, has a subsequence(ak⁰)that is w-convergent.

The limit a^†of any w-convergent subsequence(a_k⁰)is an f -minimizing solution of (12), and f(ak)→f(a^†).

Proof:see DC & Scherzer & Zubelli (2009).

Adriano De Cezaro e-mail: decezaro@impa.br Convex Regularization of Local Volatility Models from Option Prices: Convergence Analysis and Rates

(36)

Lemma

Letζ^†∈∂f(a^†). Then There exists a function w^†∈L²(Ω)and a function r∈H¹(Ω)such that

ζ^†=F⁰(a^†)^∗w^†+r (19) holds. Furthermore,krk_H¹_(Ω)can to be taken arbitrarily small.

(37)

Definition

Let1≤q<∞. Moreover, letU be a subset of H˜ ¹. The Bregman distance D_ζ(·,a˜)of f :H¹→R∪ {+∞}ata˜∈DB(f)andζ∈∂f is said to be q-coercive with constant c>0if

D_ζ(a,a˜)≥cka−a˜k^q_˜

U ∀a∈D⁽^f^). ⁽²⁰⁾

(38)

Convergence rates

Lemma

Letζ^†∈∂f(a^†)satisfy(19)with w^†and r such that c Ckw^†k_L1(Ω)+krk_L2(Ω)

:=β1∈[0,1),

and the Bregman distance with respect to f is1−coercive (as in the Definition 6.1) with_U˜ :=H¹(Ω). Then,

hζ^†,a^†−ai ≤β1D_ζ†(a,a^†) +β2kF(a)−F(a^†)k_L2(Ω), (21) for a∈Mβmax(ρ), whereβmax,ρ>0satisfy the relationρ>βmaxf(a^†).