August 2005. Some Identification Problems in Finance. Heinz W. Engl. Industrial
Mathematics Institute. Johannes Kepler Universität Linz, Austria.
Some Identification Problems in Finance Heinz W. Engl Industrial Mathematics Institute Johannes Kepler Universität Linz, Austria www.indmath.uni-linz.ac.at Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences www.ricam.oeaw.ac.at Industrial Mathematics Competence Center www.mathconsult.co.at/imcc
August 2005
Inverse Problems in Finance Black-Scholes world: stock S satisfies SDE 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) r … interest rate q … dividend yield σ … volatility
→ convection – diffusion - reaction equation 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 Identification Identify 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 – Dupire Equation
August 2005
Least-Squares approach Find σ such that Example: 0.8
0.6
0.4
0.2
1 0.8
300 250
0.6
1 % data noise (rounding)
200
0.4
t
150 100
0.2
50 0
0
S
Reason for the instabilities: ill-posedness August 2005
„Inverse Problems“: Looking for causes of an observed or 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 problem leads 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 = y T: 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-posed neighbouring 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., the smoother the kernel). Necessary and sufficient for existence:
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 noice concepts) Contains many methods, also iterative ones! Not: - conjugate gradients (nonlinear method), → Hanke - maximum entropy, BV-regularization August 2005
Functional analytic theory of nonlinear ill-posed problems where F: D(F) ⊂ X → Y is a nonlinear operator between Hilbert 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-squares solution 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 an x*-MNS. If
August 2005
„source conditions“ like - a-priori smoothness assumption (related to smoothing properties of the forward map F): only smooth parts of x† – x* can be resolved fast - boundary conditions, i.e., some boundary information about x† is necessary 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 even x† – x* analytic suffices (severely ill-posed)
August 2005
disadvantage of Tikhonov regularization: functional in general not convex, local minima → alternative: iterative regularization methods Iterative 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 the Levenberg-Marquardt method:
with αk→ 0 as k→ ∞, || y – yδ || ≤ δ. Convergence for ill-posed problems: Hanke Iteratively regularized Gauß-Newton method:
Convergence (rates): Bakushinskii, Hanke-Neubauer-Scherzer, Kaltenbacher Landweber method: Convergence (rates): Hanke, Neubauer, Scherzer Crucial: Choice of „stopping index“ n=n(δ, yδ)
August 2005
Tikhonov Regularization,applied 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: analysis as in general theory August 2005
Convergence Rates: (based on Engl and Zou, Stability and convergence analysis of Tikhonov regularization for parameter identification in a parabolic equation, 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
Example 1
0.8
1 % data noise (rounding)
0.6
0.4
0.2
1 0.8
300 250
0.6
t
200
0.4
150 100
0.2
50 0
0
S
August 2005
Example 2 S & P 500 Index: values from 2002/08/19 8 maturities ~ 50 strikes
0.5 0.45 0.4 0.35 0.3 0.25 2
0.2 1.5 0.15 1 0.1 400
0.5 600
800
1000
1200
1400
1600
t
0
S
August 2005
Interest Rate Derivatives - Pricing Hull & White Interest Rate Model (two-factor)
dr = (θ (t ) + u − a r )dt + σ 1 (t )dW1 du = −budt + σ 2 (t )dW2 with E[dW1 , dW2 ] = ρ dt ,
−1 < ρ < 1
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
August 2005
Interest Rate Derivatives - Pricing Arbitrage arguments lead to 2 2 2 ∂V 1 ∂ ∂ 1 ∂ V V V 2 2 + σ 1 (t ) + ρ σ 1 (t )σ 2 (t ) + σ 2 (t ) + 2 2 ∂t 2 ∂r∂u 2 ∂r ∂u ∂V ∂V (θ (t ) + u − a r ) − bu − rV = 0 ∂r ∂u
for the price V of different types (determined by different initial and transition conditions) of structured interest rate derivatives
August 2005
Interest Rate Derivatives - Model 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: minimize 2 (Calculate dCapSwapti onPrices − MarketCapS waptionPri ces ) ∑
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 Calibration Goodness of Fit – Cap Prices: price
price
Maturity: 2 years
price
Maturity: 6 years
strike
Maturity: 12 years
strike
price
strike
Maturity: 20 years
strike
August 2005
Example 3: Model Calibration Goodness of Fit – Swaption Prices: price
price
Expiry: 2 years
Expiry: 3 years
swapmaturity
swapmaturity price
price
Expiry: 5 years
Expiry: 10 years
swapmaturity
swapmaturity
August 2005
Example 3: Model Calibration Stability: market data versus perturbed market data (1%) 1
1
days
2
days
2
days
days
August 2005
