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
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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
