Inverse Problems and Stochastic Volatility Models Jorge P . Zubelli - - PowerPoint PPT Presentation

Inverse Problems and Stochastic Volatility Models

Jorge P . Zubelli

IMPA, Rio de Janeiro, Brazil Thanks to A. Leit˜ ao and W. Muniz

May 1, 2014

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.Zubelli (IMPA) May 1, 2014 1 / 69

Outline

1

Intro and Background

2

Problem Statement and Results on Local Vol Models

3

Main Technical Results

4

Numerical Examples w/ Synthetic and w/ Real Data

5

Connections with Exponential Families and Risk Measures

6

Conclusions

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Historical Remarks

Options and Derivatives

600BC Greek Philosopher and Geometer Thales from Miletus: Options and Futures of Olives (for Olive Oil) Call options on olive presses nine months ahead of the next harvest XVI century Dutch fisherman made forward contracts before departing XVII Netherlands - Options on prices of tulips XVII century future and option contracts on rice were negociated in Amsterdan and Osaka on rice. XIX century USA Contracts of to arrive type on flour started to be traded (CBOT) in 1849-1850, standard future contracts on grains were introduced in 1865.

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Historical Remarks

1972, CME started to negociate the first future contracts on currencies (6 currencies) Since the mid 80’s the growth of derivative markets overcame the growth

• f the underlyings

Theoretical Developments: 70’s F. Black - M. Scholes - R. Merton

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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 agreed strike price K on the maturity date T. Its payoff is given by h(XT) =

• XT − K

if XT > K, if XT ≤ K.

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Derivative Contracts

European Put Option: a forward contract that gives the holder the right to sell a unit of the asset for a strike price K at the maturity date T. Its payoff is h(XT) =

• K − XT

if XT < K, if XT ≥ K. At other times, the contract has a value known as the derivative price. The

• ption price at time t with stock price Xt = x is denoted by P(t,x).

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Main Contributions

Historical Remarks

Figure: Thales

Thales (Miletus)

• L. Bachelier (Paris)

P . Samuelson (Nobel 1970)

• F. Black
• M. Scholes (Nobel 1997)
• R. Merton (Nobel 1997)

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Main Contributions

Figure: L. Bachelier

Thales (Miletus)

• L. Bachelier (Paris)

P . Samuelson

• F. Black
• M. Scholes
• R. Merton

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Figure: P . Samuelson

Thales (Miletus)

• L. Bachelier (Paris)

P . Samuelson

• F. Black
• M. Scholes
• R. Merton

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Some Background, Terminology, and Notation

Classical Black-Scholes-Merton

Under highly simplifying assumptions, the call option price C on an underlying X is given by CBS(X,t;K,T,r,σ) = XN(d+)− Ke−r(T−t)N(d−) (1) where N is the cumulative normal distribution. d± = log(Xer(T−t)/K)

σ√

T − t

± σ√

T − t 2 . (2) Some of the Assumptions: Non dividend paying (just for simplicity) Complete and Frictionless Markets Exponential Brownian motion dynamics Constant Volatility

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Plot of the Black-Scholes Price of a Call

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However...

Volatility is not deterministic! It is even a multi-scale phenomena! It is not true that the underlying undergoes an Exponential Brownian Motion Even more so in high frequency contexts... Implied Volatility: The value of the volatility that should be used in the Black-Scholes formula to give the quote market price of a derivative.

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The Concept of Implied Volatility

Recall CBS(X,t;K,T,r,σ0) = XN(d+)− Ke−r(T−t)N(d−) (3) where N is the cumulative normal distribution function and d± = log(Xer(T−t)/K)

σ0 √

T − t

± σ0 √

T − t 2 . (4) Notion of Implied Volatility: Fix everything else and consider

σ − → CBS(X,t;K,T,r,σ)

The implied volatilty is the inverse to this map. IMPLIED VOL: ”wrong number that when plugged into the wrong equation gives the right price”

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Figure: Implied Volatility Surface- (From Bruno Dupire - IMPA talk)

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Remarks

1

Implied Vol is only a way of renaming the price (change of variables).

2

Crucial to find the connection with the underlying dynamics.

3

Crucial to introduce the concept of LOCAL VOLATILITY Concept of Stochastic Volatility: dXt Xt

= µtdt +σtdWt

Fundamental Issue: How to model the process σt? One Possibility: Following Dupire... (Local Volatility Model) assume that

σt = σ(t,Xt)

Another Possibility: Assume that the process σt = f(Yt) undergoes its own dynamics according to a SDE.

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Figure: Example of Data from IBOVESPA

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Limitations of Classical Black-Scholes

log-normality of asset prices is not verified by statistical tests

• ption prices are subjet to the smile effects

volatility tends to fluctuate with time

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Local Volatility Models

Idea: Assume that the volatility is given by

σ = σ(t,x)

i.e.: it depends on time and the asset price. Easy to check that the Black-Scholes eq. holds.

∂P ∂t + 1

2σ(t,x)2x2 ∂2P

∂x2 + r

• x ∂P

∂x − P

• = 0

(5) P(T,x) = h(x) (6)

• r in the case you have dividends:

∂P ∂t + 1

2σ(t,x)2x2 ∂2P

∂x2 +(r − d)x ∂P ∂x − rP = 0

P(T,x) = h(x)

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Motivation and Goals

Focus

Dupire [Dup94] local volatility models Goal: Present a unified framework for the calibration of local volatility models Use recent tools of convex regularization of ill-posed Inverse Problems. Present convergence results that include convergence rates w.r.t. noise level in fairly general contexts Go beyond the classical quadratic regularization.

Applications

risk management hedging evaluation of exotic derivatives.

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The Direct and the Inverse Problem

The Direct Problem

Given σ = σ(t,x) and the payoff information, determine P = P(t,x,T,K;σ)

The Inverse Problem

Given a set of observed prices

{P = P(t,x,T,K;σ)}(T,K)∈S

find the volatility σ = σ(t,x). The set S is taken typically as [T1,T2]×[K1,K2]. In Practice: Very limited and scarce data Note: To price in a consistent way the so-called exotic derivatives one has to know σ and not only the transition probabilities

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The Smile Curve and Dupire’s Equation

Assuming that there exists a local volatility function σ = σ(x,t) for which (5) holds Dupire(1994) showed that the call price satisfies

∂T C − 1

2σ2(K,T)K 2∂2 K C + rS∂K C = 0 ,

S > 0 , t < T C(K,T = 0) = (X − K)+ , (7) Theoretical: way of evaluating the local volatility

σ(K,T) =

• 2

∂T C + rK∂K C

K 2∂2

K C

• (8)

In practice To estimate σ from (7), limited amount of discrete data and thus

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

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Literature

Very vast!!!

Avellaneda et al. [ABF+00, Ave98c, Ave98b, Ave98a, AFHS97] Bouchev & Isakov [BI97] Crepey [Cr´ e03] Derman et al. [DKZ96] Egger & Engl [EE05] Hofmann et al. [HKPS07, HK05] Jermakyan [BJ99] Achdou & Pironneau (2004) Roger Lee (2001,2005) Abken et al. (1996) Ait Sahalia, Y & Lo, A (1998) Berestycki et al. (2000) Buchen & Kelly (1996) Coleman et al. (1999) Cont, Cont & Da Fonseca (2001) Jackson et al. (1999) Jackwerth & Rubinstein (1998) Jourdain & Nguyen (2001) Lagnado & Osher (1997) Samperi (2001) Stutzer (1997)

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Problem Statement

Starting Point: Dupire forward equation [Dup94] −∂T U + 1

2σ2(T,K)K 2∂2

K U −(r − q)K∂K U − qU = 0,

T > 0, (9) K = X0ey , τ = T − t , b = q − r , u(τ,y) = eqτUt,X(T,K) (10) and a(τ,y) = 1 2σ2(T −τ;X0ey), (11)

Set q = r = 0 for simplicity to get:

uτ = a(τ,y)(∂2

yu −∂yu)

(12) and initial condition u(0,y) = X0(1− ey)+ (13)

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Problem Statement

The Vol Calibration Problem

Given an observed set

{u = u(t,S,T,K;σ)}(T,K)∈S

find σ = σ(t,S) that best fits such market data Noisy data: u = uδ Admissible convex class of calibration parameters:

D(F) := {a ∈ a0 + H1+ε(Ω) : a ≤ a ≤ a}.

(14) where, for 0 ≤ ε fixed, U := H1+ε(Ω) and a > a > 0.

Parameter-to-solution operator

F : D(F) ⊂ H1+ε(Ω) → L2(Ω) F(a) = u(a) (15)

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Setting of the problem

Theorem (H. Egger-H. Engl[EE05] Crepey[Cr´ e03])

The parameter to solution map F : H1+ε(Ω) → L2(Ω) is weak sequentialy continuous compact and weakly closed Consequences: The inverse problem is ill-posed. We can prove that the inverse problem satisfies the conditions to apply the regularization theory.

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Well-Posed and Ill-Posed Problems

Hadamard’s definition of well-posedness: Existence Uniqueness Stability The problem under consideration: Ill-posed. Equation: F(a) = u Need Regularization:

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Approach

Convex Tikhonov Regularization

For given convex f minimize the Tikhonov functional

Fβ,uδ(a) := ||F(a)− uδ||2

L2(Ω) +βf(a)

(16)

• ver D(F), where, β > 0 is the regularization parameter.

Remark that f incorporates the a priori info on a.

||¯

u − uδ||L2(Ω) ≤ δ, (17) where ¯ u is the data associated to the actual value ˆ a ∈ D(F).

Assumption (very general!)

Let ε ≥ 0 be fixed. f : D(f) ⊂ H1+ε(Ω) −

→ [0,∞] is a convex, proper, coercive

and sequentially weakly lower semi-continuous functional with domain D(f) containing D(F).

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Questions

Theoretical Questions:

Does there exist a minimizer of the regularized problem? Suppose that the noise level goes to zero... How fast does the regularized go to the true solution? Results obtained in joint work with D. Cezaro and O. Scherzer. Published in J. Nonlinear Analysis, 2012 [DCSZ12]

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Questions

Practical Questions:

Can we devise an iterative algorithm to compute the solution? Does this algorithm converge? Can we regularize by stopping the iteration judiciously? We proved:

1

A tangential cone condition that ensures convergence of the Landwebber iteration. Joint work with D. Cezaro. (IMA J. of Applied Math.)

2

Obtained a Morozov-type criterium to stop the iteration.

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Main Theoretical Result

F(a) = u(a) (15)

Fβ,uδ(a) := ||F(a)− uδ||2

L2(Ω) +βf(a)

(16)

Theorem (Existence, Stability, Convergence)

For the regularized inverse problem F(a) = u (18) we have:

∃ minimizer of Fβ,uδ.

If (uk) → u in L2(Ω), then ∃ a seq. (ak) s.t. ak ∈ argmin

Fβ,uk(a) : a ∈ D

• has a subsequence which converges weakly to

a

• a ∈ argmin

Fβ,u(a) : a ∈ D

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Main Theoretical Result (cont)

F(a) = u(a) (15)

Fβ,uδ(a) := ||F(a)− uδ||2

L2(Ω) +βf(a)

(16)

Theorem (cont.) NOISY CASE

Take β = β(δ) > 0 and assume

β(δ) satisfies β(δ) → 0 and δ2 β(δ) → 0, as δ → 0 .

(19) The seq. (δk) converges to 0, and that uk := uδk satisfies ¯ u − uk ≤ δk. Then,

1

Every seq. (ak) ∈ argminFβk,uk , has weak-convergent subseq. (ak′).

2

The limit a† := w − limak′ is an f-minimizing solution of (15), and f(ak) → f(a†).

3

If the f-minimizing solution a† is unique, then ak → a† weakly.

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Bregman distance

Let f be a convex function. For a ∈ D(f) and ∂f(a) the subdifferential of the functional f at a. We denote by D(∂f) = {˜ a : ∂f(˜ a) = /

0} the domain of the subdifferential.

The Bregman distance w.r.t ζ ∈ ∂f(a1) is defined on D(f)×D(∂f) by Dζ(a2,a1) = f(a2)− f(a1)−ζ,a2 − a1 .

Assumption (1)

We assume that

1

∃ an f-minimizing sol. a† of (15), a† ∈ DB(f).

2

∃β1 ∈ [0,1), β2 ≥ 0, and ζ† ∈ ∂f(a†) s.t. ζ†,a† − a ≤ β1Dζ†(a,a†)+β2

• F(a)− F(a†)
• L2(Ω) for a ∈ Mβmax(ρ) ,

(20) where ρ > βmaxf(a†) > 0.

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Convergence rates [SGG+08]

Theorem (Convergence rates [SGG+08])

Let F, f, D, H1+ε(Ω), and L2(Ω) satisfy Assumption 1. Moreover, let

β : (0,∞) → (0,∞) satisfy β(δ) ∼ δ. Then

Dζ†(aδ

β,a†) = O(δ),

• F(aδ

β)− uδ

• L2(Ω) = O(δ) ,

and there exists c > 0, such that f(aδ

β) ≤ f(a†)+δ/c for every δ with

β(δ) ≤ βmax.

Example: The regularization functional f as the Boltzmann-Shannon entropy f(a) =

• Ω

alog(a)dx , a ∈ D(F),

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Example of the Convergence Rate

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How about the assumptions?

Although Assumption 1 may seem too restrictive, the next result reveals that it can be obtained from rather classical ones:

Proposition

Assume that ∃ an f-minimizing solution a† of F(a) = u and that F is Gˆ ateaux differentiable at a†. Moreover, assume that ∃γ ≥ 0 and ω† ∈ L2(Ω)∗ with γ

• ω†

< 1, s.t. ζ† := F ′(a†)∗ω† ∈ ∂f(a†)

(21) and ∃βmax > 0 satisfying ρ > βmaxf(a†) such that

• F(a)− F(a†)− F ′(a†)(a− a†)
• ≤ γDζ†(a,a†), for a ∈ Mβmax(ρ) .

(22) Then, Assumption 1 holds.

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How about algorithms?

NOTE: We have proved

We have also proved a tangential cone condition for this problem, which implies that the Landwever iteration converges in a suitable neighborhood. Landweber Iteration [EHN96]: aδ

k+1 = aδ k + cF ′(aδ k)∗(uδ − F(aδ k)).

(23) Discrepancy Principle:

• uδ − F(aδ

k∗(δ,yδ))

• ≤ rδ <
• uδ − F(aδ

k)

• ,

(24) where r > 2 1+η 1− 2η , (25) is a relaxation term. If the iteration is stopped at index k∗(δ,yδ) such that for the first time, the residual becomes small compared to the quantity rδ.

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Numerical Examples with Simulated Data

Description of the Examples

Using a Landweber iteration technique we implemented the calibration. Produced for different test variances a the option prices and added different levels of multiplicative noise. The examples consisted of perturbing a = 1 during a period of T = 0,··· ,0.2 and log-moneyness y varying between −5 and 5. Initial guess: Constant volatility.

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Numerical Examples - Exact Solution

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Numerical Examples - Exact Solution

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Numerical Examples 1 - noiseless - 4000 steps

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Numerical Examples 1 - error - 100 steps

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Numerical Examples 1 - error - 300 steps

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Numerical Examples 1 - error - 500 steps

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Numerical Examples 1 - error - 1000 steps

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Numerical Examples 1 - error - 2000 steps

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Numerical Examples 1 - error - 4000 steps

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Numerical Examples 2 - 5% noise level - 100 steps

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Numerical Examples 2 - 5% noise level - 200 steps

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Numerical Examples 2 - 5% noise level - 300 steps

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Numerical Examples 2 - 5% noise level - 400 steps

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Numerical Examples 2 - 5% noise level - Stopping criteria

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Numerical Examples 2 - 5% noise level - 2000 iterations

Too many!!!

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Local Vol Surface from Heston Model

Figure: Local Vol Surface associated to Heston Model Calibrated on PBR data

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Local Vol Surface from Heston Model

Figure: Local Vol Surface associated to Heston Model Calibrated on SPX data

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Local Vol Surface for WTI Crude Oil

totally nonparametric

Figure: Local Vol Surface associated to Heston Model Calibrated on SPX data

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Numerical Examples: with Real Data

Reconstruction of a = σ2/2 with PBR Stock Data (implemented by Vinicius L. Albani/IMPA)

Figure: Minimal Entropy functional / Landweber Method / a priori Implied Vol / maturities: 2010-11

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Numerical Examples: with Real Data

Reconstruction of a with PBR Stock Data (implemented by Vinicius L. Albani/IMPA)

Figure: Minimal Entropy functional / Minimization (Levenberg-Marquadt) Method /

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Comparison Kullback-Leibler × Quadratic Regularization

WTI Data

Figure: Left Kullback-Leibler Regularization and Right Quadratic Regularization

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Comparison Kullback-Leibler × Quadratic Regularization

WTI Data

Figure: Left Kullback-Leibler Regularization and Right Quadratic Regularization

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Local Vol Surface for Henry Hub Natural Gas

In the next plots we show an online approach (joint work w/ V. Albani). We performed the following: We consider the evolution of prices of futures and options for several days but kept the maturity dates and all the other features of the options. Calibrated using the extra information. This is part of an extension of the above results that leads to incorporating the flow of information.

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Figure: Local Vol Surface associated to Henry Hub Gas Prices

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Figure: Local Vol Surface associated to Henry Hub Gas Prices

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Figure: Local Vol Surface associated to Henry Hub Gas Prices

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Figure: Local Vol Surface associated to Henry Hub Gas Prices

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Figure: Local Vol Surface associated to Henry Hub Gas Prices

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Figure: Local Vol Surface associated to Henry Hub Gas Prices

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Connection with Statistics and Exponential Families

Regular Exponential Families:

family of probability distribution functions pψ,θ : R → R+ defined by pψ,θ(s) := exp(s ·θ−ψ(θ))p0(s) where ψ : R → R∪{+∞} is convex and p0 : R → R+ is continuous.

Example:

Gaussians parametrized by the mean. The Darmois-Koopman-Pitman Thm: Under certain regularity conditions on the probability density, a necessary and sufficient condition for the existence of a sufficient statistic1 of fixed dimension is that the probability density belongs to the exponential family [And70].

1no other statistic which b can be calculated from the same sample provides any

additional information as to the value of the parameter

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Connection with Statistics and Exponential Families(cont.)

Recall the Fenchel Conjugate

Given a function f : X → R∪{+∞}, the Fenchel dual f ∗X ∗ → R∪{+∞} is defined by f ∗(x∗) := sup{x∗,x− f(x) | x ∈ X}

Theorem (Banerjee et al. [BMDG05])

Let ψ∗ denote the Fenchel transform of ψ, which we assume to be

• differentiable. Using the Bregman distance w.r.t. ψ∗

Dψ∗(ˆ a,˜ a) = ψ∗(ˆ a)−ψ∗(˜ a)−ψ∗′(˜ a)(ˆ a−˜ a) , if we assume that a(θ) ∈ int(dom(ψ∗)), then pψ,θ(a) = exp

• − Dψ∗ (a,a(θ))
• exp
• ψ∗(a)
• p0(a) .

(26)

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Connection with Statistics and Exponential Families(cont.)

Example (Exponential Families and their Fenchel conjugates)

For a Gaussian distribution ψ(θ) = ϖ2

2 θ2, then ψ∗(a) = a2 2ϖ2 . For Poisson

distribution ψ(θ) = exp(θ) we have ψ∗(a) = alog(a)− a.

Example

According to Example 1, if we use the exponential family associated to Poisson distributions, we obtain Kullback-Leibler regularization, consisting in minimization of a −

→ Fβ,uδ(a) :=

• F(a)− uδ
• 2

L2(Ω) +βKL(ˆ

a,a) , (27) where KL(ˆ a,a) =

• Ω

alog(ˆ a/a)−(ˆ a− a)dx . We note that the Kullback-Leibler distance is the Bregman distance associated to the Boltzmann-Shannon entropy

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Connection with Statistics and Exponential Families(cont.)

Lemma (3)

Let Ω be a bounded subset of R2 with Lipschitz boundary. Moreover, assume that F is continuous w.r.t. the weak topologies on L1(Ω) and L2(Ω), respec.

1

Let a,b ∈ D(G). Then

a− b2

L1(Ω) ≤

• 2

3aL1(Ω) + 4 3bL1(Ω)

• KL(a,b) .

(29) (Convention: 0·(+∞) = 0)

2

Let 0 = ˆ a ∈ DB(G), then the sets

Mβ,uδ(M) := {a ∈ DB(G) : Fβ,uδ(a) ≤ M}

are τ˜

U sequentially compact.

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Connection with Statistics and Exponential Families(cont.)

An important consequence of (29) and Theorem 2 is that

• aδ

β − a†

• L1(Ω) = O(

√ δ).

(30) Now, let δk be a sequence converging to zero and ak = aδk

βk the respective

minimizers of the Tikhonov functional (16). Take bk = a† for all k ∈ N. Then, from Lemma 2

• ak − a†
• L1(Ω) → 0,

as

δk → 0.

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Conclusions

Volatility surface calibration is a classical and fundamental problem in Quantitative Finance We developed a unifying framework for the regularization that makes use

• f tools from Inverse Problem theory and Convex Analysis and

established:

1

Convergence and convergence-rate results.

2

Convergence of the regularized sol. w.r.t the noise level in L1(Ω) (when Ω is compact)

3

The connection with exponential families.

4

The source condition connection with convex risk measures

5

Implemented a Landweber type calibration algorithm.

Developed an Online Calibration Methodology Future Possibilities:

1

Use a priori asymptotic behavior (e.g.: following Roger Lee’s results for the implied volatility)

2

Enhance the model to incorporate jumps and perhaps more sofisticated models such as cogarch (for example).

.

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THANK YOU FOR YOUR ATTENTION!!! Collaborators:

• V. Albani (IMPA), A. de Cezaro (FURG), O. Scherzer (Vienna)

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