August 2005

SomeSome IdentificationIdentification Problems in Problems in FinanceFinance

Heinz W. Engl

Industrial Mathematics InstituteJohannes Kepler Universität Linz, Austriawww.indmath.uni-linz.ac.at

Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of Scienceswww.ricam.oeaw.ac.at

Industrial Mathematics Competence Centerwww.mathconsult.co.at/imcc

August 2005

European Call Option C provides the right to buy the underlying (stock) at maturity T for the strike price K,no-arbitrage arguments and Ito's formula yield the Black-Scholes Equation for CK,T(S,t)

→ convection – diffusion - reaction equation

Black-Scholes world: stock S satisfies SDE

Inverse Problems in FinanceInverse Problems in Finance

r … interest rateq … dividend yieldσ … volatility

August 2005

if volatility σ and drift rate µ are assumed to be constant:

→ closed form solution (Black-Scholes formula)

solve for σ: “implied volatility” should be constant, but depends on K,T → “volatility smile”

→ alternative: compute volatility surface σ(S,t) via parameter identification in the PDE from observed prices

August 2005

Parameter Parameter IdentificationIdentificationIdentify diffusion parameter σ = σ(S,t) in BS-Equation

from given (observed) values Cki,Tj(S,t)References:• Jackson, Süli, and Howison. Computation of deterministic volatility surfaces.

J. Mathematical Finance,1998.• Lishang and Youshan. Identifying the volatility of unterlying assets from

option prices. Inverse Problems, 2001• Lagnado and Osher. A technique for calibrating derivative security, J. Comp.

Finance, 1997• Crépey. Calibration of the local volatility in a generalized Black-Scholes

model using Tikhonov regularization. SIAM J. Math. Anal., 2003.• Egger and Engl.Tikhonov Regularization Applied to the Inverse Problem of

Option Pricing: Convergence Analysis and Rates, Inverse Problems, 2005.

August 2005

Transformation Transformation –– DupireDupire EquationEquation

August 2005

LeastLeast--SquaresSquares approachapproach

Find σ such that

Example:

1 % data noise (rounding)

Reason for the instabilities: ill-posedness0

50100

150200

250300

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

St

August 2005

„Inverse Problems“: Looking for causes of an observedor desired effect?

Inverse Probelms are usually „ill-posed“:

Due to J. Hadamard (1923), a problem is called„well posed“ if(1) for all data, a solution exists.(2) for all data, the solution is unique.(3) the solution depends continuously on the data.

„Correct modelling of a physically relevant problemleads to a well-posed problem.“

August 2005

A.Tikhonov (~ 1936): geophysical (ill-posed) problems.F.John: „The majority of all problems is ill-posed,especially if one wants numerical answers“.Examples:- Computerized tomography (J. Radon)- (medical) imaging- inverse scattering- inverse heat conduction problems- geophysics / geodesy- deconvolution- parameter identification- …

August 2005

Linear inverse problems frequently lead to „integral equations of the first kind“:Linear (Fredholm) integral equation:

August 2005

Tx = yT: bounded linear operator between Hilbert spaces X,Y„solution“:

R(T) non closed, e.g.:dim X = ∞, T compact and injective ⇒ T† unbounded and densely defined, i.e., problem ill-posed

August 2005

„Regularization“: replacing an ill-posed problem by a(parameter dependent) family of well-posedneighbouring problems.

Regularization by:

(1) Additional information (restrict to a compact set)(2) Projection(3) Shifting the spectrum(4) Combination of (2) and (3)

August 2005

T compact with singular system {σ; un, vn}

→ amplification of high-frequency errors, since (σn) → 0.The worse, the faster the (σn) decay (i.e., thesmoother the kernel). Necessary and sufficient forexistence:

August 2005

General (spectral theoretic) construction for linear regularization methods, contains e.g.,„Tikhonov regularization“

equivalent characterization:

(yδ: noisy data, || y – yδ|| δ; alternative: stochastic noiceconcepts)Contains many methods, also iterative ones! Not: - conjugate gradients (nonlinear method), → Hanke- maximum entropy, BV-regularization

August 2005

FunctionalFunctional analyticanalytic theorytheory of of nonlinearnonlinear illill--posedposed problemsproblems

where F: D(F) ⊂ X → Y is a nonlinear operator betweenHilbert spaces X and Y; assume that

- F is continuous and- F is weakly (sequentially) closed, i.e., for any sequence

{xn}⊂ D (F), weak convergence of xn to x in X and weak convergence of F (xn) to y in Y imply that x ∈ D (F)and F (x) = y.

F: forward operator for an inverse problem, e.g.- parameter-to-solution map for a PDE

(→ parameter identification)- maps domain to the far field in a scattering problem

(→ inverse scattering)

August 2005

Notion of a „soluton“: „x*-minimum-norm-least-squaressolution x†“:

and

need not exist, if it does: need not be unique!Choice of x* crucial: Available a-priori information has to enter into the selection criterion.

Thus: Compactness and local injectivity → ill-posedness(like in the linear case).

August 2005

Tikhonov Regularization

- stable for α>0 (in a multi-valued sense)- convergence to an x*-minimum-norm solution if

(Seidman- Vogel)

August 2005

Convergence rates:Theorem (Engl-Kunisch-Neubauer): D(F) convex, let x† be anx*-MNS. If

August 2005

„source conditions“ like

- a-priori smoothness assumption (related to smoothingproperties of the forward map F): only smooth parts of x† – x* canbe resolved fast

- boundary conditions, i.e., some boundary information about x† isnecessary

Severeness depends on smoothing properties of forward map:- identification of a diffusion coefficient: essentially x† – x* ∈ H2

(mildly ill-posed)- inverse scattering (x†: parameterization of unknown boundary of

scatter): not evenx† – x* analytic

suffices (severely ill-posed)

August 2005

disadvantage of Tikhonov regularization:functional in general not convex, local minima

→ alternative: iterative regularization methodsIterative methods:Newton´s method for nonlinear well-posed problems:fast local convergence. For ill-posed problems?Linearization of F(x) = y at a current iterate xk:

August 2005

Tikhonov regularization leads to theLevenberg-Marquardt method:

with αk→ 0 as k→∞, || y – yδ || δ.Convergence for ill-posed problems: HankeIteratively regularized Gauß-Newton method:

Convergence (rates): Bakushinskii, Hanke-Neubauer-Scherzer, KaltenbacherLandweber method:

Convergence (rates): Hanke, Neubauer, ScherzerCrucial: Choice of „stopping index“ n=n(δ, yδ)

≤

August 2005

TikhonovTikhonov Regularization,appliedRegularization,applied to volatility identification:to volatility identification:

a-priori guess a*, noisy data Cδ (δ: bound for noise level)(alternative: replace || a – a*|| by entropy theory: Engl-Landl, SIAM J. Num. An. 1991, in finance: R. Cont 2005)

Convergence and Stability:Convergence and Stability:

analysis as in general theory

August 2005

ConvergenceConvergence Rates:Rates:(based on Engl and Zou, Stability and convergence analysis of Tikhonov regularization for parameter identification in a parabolicequation, Inverse Problems 2000)

In general, convergence may be arbitrarily slow.Assumptions:- continuous data (for all strikes)- observation for arbitrarily small time interval

then- under a smoothness and decay condition (→ source

condition) on a† – a*

August 2005

ExampleExample 11

1 % data noise (rounding)

050

100150

200250

300

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

St

August 2005

ExampleExample 22

S & P 500 Index:values from 2002/08/198 maturities~ 50 strikes

400 600 800 1000 1200 1400 16000

0.5

1

1.5

2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

S

t

August 2005

InterestInterest Rate Rate DerivativesDerivatives -- PricingPricing

Hull & White Interest Rate Model (two-factor)

with

a and b are mean reversion speeds, σ1 and σ2 volatilities,θ is the deterministic drift, dW1 and dW2 are increments of Wiener processes with instantaneous correlation ρ

22

11

)()())((

dWtudtbdudWtdtrautdr

σσθ

+−=+−+=

11,]d,d[E 21 <<−= ρρ dtWW

August 2005

Arbitrage arguments lead to

for the price V of different types (determined by different initial and transition conditions) of structured interest rate derivatives

0))((

)(21)()()(

21

2

22

2

2

212

22

1

=−∂∂−

∂∂−+

+∂∂+

∂∂∂+

∂∂+

∂∂

rVuVub

rVraut

uVt

urVtt

rVt

tV

θ

σσσρσ

InterestInterest Rate Rate DerivativesDerivatives -- PricingPricing

August 2005

InterestInterest Rate Rate DerivativesDerivatives -- Model CalibrationModel Calibration

- identify the drift θ (t) from swap rates- identify a, b, ρ , σ1 (t) and σ2 (t) from cap / swaption

matrices

two level calibration:inner loop: given reversion speeds, volatilities, and

correlation, identify drift. This can be done uniquely from money market/swap rates (in the space of piecewise constant functions) → first kind integral equation

outer loop: minimize2)ceswaptionPriMarketCapSonPricesdCapSwapti(Calculate∑ −

August 2005

regularization by iteration with “early stopping“: Newton -CG algorithm

closed form solutions for cap and swaption prices enables fast calibration

minimization in two steps: determination of starting values based on cap prices only, final minimization based on cap and swaption prices

input data: Black76 cap and at-the-money swaption volatilities

August 2005

Example 3: Model CalibrationExample 3: Model Calibration

Goodness of Fit – Cap Prices:Maturity: 2 years

Maturity: 20 yearsMaturity: 12 years

Maturity: 6 years

strike

strikestrike

strike

price

priceprice

price

August 2005

Example 3: Model CalibrationExample 3: Model Calibration

Goodness of Fit – Swaption Prices:Expiry: 2 years

Expiry: 10 yearsExpiry: 5 years

Expiry: 3 years

swapmaturity

swapmaturityswapmaturity

swapmaturity

price

priceprice

price

August 2005

Example 3: Model CalibrationExample 3: Model Calibration

Stability: market data versus perturbed market data (1%)

days days

days days

1 1

22