Heinz W. Engl. Industrial Mathematics Institute Johannes Kepler Universität Linz, Austria

Transcription

1 Some Identification Problems in Finance Heinz W. Engl Industrial Mathematics Institute Johannes Kepler Universität Linz, Austria Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Industrial Mathematics Competence Center

2 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

3 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

4 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.

5 Transformation Dupire Equation

6 Least-Squares approach Find σ such that Example: % data noise (rounding) 0.8 t S Reason for the instabilities: ill-posedness

7 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.

8 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 -

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

10 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

11 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)

12 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:

13 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

14 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)

15 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).

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

17 Convergence rates: Theorem (Engl-Kunisch-Neubauer): D(F) convex, let x be an x*-mns. If

18 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)

19 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 :

20 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 )

21 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

22 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*

23 Example % data noise (rounding) t S

24 Example S & P 500 Index: values from 00/08/19 8 maturities ~ 50 strikes S t

25 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

26 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 ) ( ) ( ) ( = 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

27 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)

28 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

29 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

30 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

31 Example 3: Model Calibration Stability: market data versus perturbed market data (1%) 1 1 days days days days