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
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 C K,T (S,t) convection diffusion - reaction equation r interest rate q dividend yield σ volatility
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
Parameter Identification Identify diffusion parameter σ = σ(s,t) in BS-Equation from given (observed) values C k i,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, 001 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., 003. Egger and Engl.Tikhonov Regularization Applied to the Inverse Problem of Option Pricing: Convergence Analysis and Rates, Inverse Problems, 005.
Transformation Dupire Equation
Least-Squares approach Find σ such that Example: 0.8 0.6 0.4 0. 1 1 % data noise (rounding) 0.8 t 0.6 0.4 0. 0 0 50 100 150 S 00 50 300 Reason for the instabilities: ill-posedness
Inverse Problems : Looking for causes of an observed or desired effect? Inverse Probelms are usually ill-posed : Due to J. Hadamard (193), a problem is called well posed if (1) for all data, a solution exists. () 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.
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 -
Linear inverse problems frequently lead to integral equations of the first kind : Linear (Fredholm) integral equation:
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
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) () Projection (3) Shifting the spectrum (4) Combination of () and (3)
T compact with singular system {σ; u n, v n } amplification of high-frequency errors, since (σ n ) 0. The worse, the faster the (σ n ) decay (i.e., the smoother the kernel). Necessary and sufficient for existence:
General (spectral theoretic) construction for linear regularization methods, contains e.g., Tikhonov regularization equivalent characterization: (y δ : noisy data, y y δ δ; alternative: stochastic noice concepts) Contains many methods, also iterative ones! Not: - conjugate gradients (nonlinear method), Hanke - maximum entropy, BV-regularization
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 {x n } D (F), weak convergence of x n to x in X and weak convergence of F (x n ) 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)
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).
Tikhonov Regularization - stable for α>0 (in a multi-valued sense) - convergence to an x*-minimum-norm solution if (Seidman- Vogel)
Convergence rates: Theorem (Engl-Kunisch-Neubauer): D(F) convex, let x be an x*-mns. If
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* H (mildly ill-posed) - inverse scattering (x : parameterization of unknown boundary of scatter): not even x x* analytic suffices (severely ill-posed)
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 x k :
Tikhonov regularization leads to the Levenberg-Marquardt method: with α k 0 as k, y y δ 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, y δ )
Tikhonov Regularization,applied to volatility identification: a-priori guess a*, noisy data C δ (δ: bound for noise level) (alternative: replace a a * by entropy theory: Engl-Landl, SIAM J. Num. An. 1991, in finance: R. Cont 005) Convergence and Stability: analysis as in general theory
Convergence Rates: (based on Engl and Zou, Stability and convergence analysis of Tikhonov regularization for parameter identification in a parabolic equation, Inverse Problems 000) 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*
Example 1 0.8 1 % data noise (rounding) 0.6 0.4 0. 1 0.8 t 0.6 0.4 0. 0 0 50 100 150 S 00 50 300
Example S & P 500 Index: values from 00/08/19 8 maturities ~ 50 strikes 0.5 0.45 0.4 0.35 0.3 0.5 0. 0.15 0.1 400 600 800 1000 100 1400 1600 S 0 0.5 1 1.5 t
Interest Rate Derivatives - Pricing Hull & White Interest Rate Model (two-factor) dr du with = ( θ ( t) + u ar) dt = budt + σ ( t) dw + σ ( t) dw 1 1 E [dw1,dw ] = ρ dt, 1 < ρ < 1 a and b are mean reversion speeds, σ 1 and σ volatilities, θ is the deterministic drift, dw 1 and dw are increments of Wiener processes with instantaneous correlation ρ
Arbitrage arguments lead to for the price V of different types (determined by different initial and transition conditions) of structured interest rate derivatives 0 ) ) ( ( ) ( 1 ) ( ) ( ) ( 1 1 1 = + + + + + rv u V bu r V a r u t u V t u r V t t r V t t V θ σ σ ρ σ σ Interest Interest Rate Rate Derivatives Derivatives - Pricing Pricing
Interest Rate Derivatives - Model Calibration - identify the drift θ (t) from swap rates - identify a, b, ρ, σ 1 (t) and σ (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 (CalculatedCapSwaptionPrices MarketCapSwaptionPrices)
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
Example 3: Model Calibration Goodness of Fit Cap Prices: price Maturity: years price Maturity: 6 years price Maturity: 1 years strike price Maturity: 0 years strike strike strike
Example 3: Model Calibration Goodness of Fit Swaption Prices: price Expiry: years price Expiry: 3 years swapmaturity swapmaturity price Expiry: 5 years price Expiry: 10 years swapmaturity swapmaturity
Example 3: Model Calibration Stability: market data versus perturbed market data (1%) 1 1 days days days days