On Non-stability of some Inverse Problem in Inverse Problem in - PowerPoint PPT Presentation

On Non-stability of some On Non-stability of some Inverse Problem in Inverse Problem in Option Pricing Option Pricing Peter Mathé Introduction, main results Peter Mathé Modulus of continuity for classes of Weierstrass Institute, Berlin Nemytski˘ ı operators joint work with R. Krämer: Modulus of Continuity of Nemytski˘ ı operators with application to the problem of option pricing , J. Inv. Ill-Posed Problems Local analysis (16):435–461, 2008 of forward and backward Black-Scholes kernels E-mail: mathe@wias-berlin.de Homepage: http://www.wias-berlin.de/people/mathe Summary, prospective Linz, October 30, 2008

Outline On Non-stability of some Inverse Problem in Option Pricing Introduction, main results 1 Peter Mathé Introduction, Modulus of continuity for classes of Nemytski˘ ı operators 2 main results Modulus of continuity for classes of Local analysis of forward and backward Black-Scholes 3 Nemytski˘ ı operators kernels Local analysis of forward and backward Black-Scholes Summary, prospective 4 kernels Summary, prospective

Problem formulation On We consider a time-dependent Black-Scholes model Non-stability of some Inverse dP τ = µ P τ d τ + σ ( τ ) P τ dW τ , Problem in Option Pricing for a time-dependent volatility σ ( t ) > 0 on a finite Peter Mathé time-horizon 0 ≤ τ ≤ T . Introduction, main results Modulus of continuity for classes of Nemytski˘ ı operators Local analysis of forward and backward Black-Scholes kernels Summary, prospective

Problem formulation On We consider a time-dependent Black-Scholes model Non-stability of some Inverse dP τ = µ P τ d τ + σ ( τ ) P τ dW τ , Problem in Option Pricing for a time-dependent volatility σ ( t ) > 0 on a finite Peter Mathé time-horizon 0 ≤ τ ≤ T . The price of a European call is Introduction, then obtained as C ( t ) = u call BS ( P , K , r , t , S ( t )) where main results Modulus of P is the actual asset price continuity for classes of K is the strike price Nemytski˘ ı operators r is the short interest rate Local analysis t is the maturity, and of forward and backward � t 0 σ 2 ( τ ) d τ is the integrated volatility. S ( t ) := Black-Scholes kernels Summary, Remark prospective This is the same model problem as considered in Bernd Hofmann’s talk.

The Black-Scholes function Below we keep K , P , r fixed! On Non-stability The function of some Inverse P Φ( d + √ s ) − Ke − rt Φ( d ) Problem in � , if s > 0 , Option Pricing u call BS ( P , K , r , t , s ) = � P − Ke − rt � Peter Mathé , for s = 0 , + Introduction, with main results Modulus of P and d ( t , s ) := c ( t ) − s / 2 continuity for √ s c ( t ) := log Ke − rt , , classes of Nemytski˘ ı operators defines a kernel k ( t , s ) , t ∈ [ 0 , T ] , s > 0, C ( t ) = k ( t , S ( t )) . Local analysis of forward and backward Black-Scholes kernels Summary, prospective

The Black-Scholes function Below we keep K , P , r fixed! On Non-stability The function of some Inverse P Φ( d + √ s ) − Ke − rt Φ( d ) Problem in � , if s > 0 , Option Pricing u call BS ( P , K , r , t , s ) = � P − Ke − rt � Peter Mathé , for s = 0 , + Introduction, with main results Modulus of P and d ( t , s ) := c ( t ) − s / 2 continuity for √ s c ( t ) := log Ke − rt , , classes of Nemytski˘ ı operators defines a kernel k ( t , s ) , t ∈ [ 0 , T ] , s > 0, C ( t ) = k ( t , S ( t )) . Local analysis of forward and Remark backward Black-Scholes kernels The option price depends on the volatility only through Summary, the integrated volatility S ( t ) ! prospective The pricing is a composition of the linear operator S ( t ) and the non-linear Black-Scholes mapping.

Nemytski˘ ı operators (superposition operators) Let k ( t , s ) , t ∈ I := [ 0 , T ] , 0 < s < s max ( t ) be any kernel On Non-stability function. We assign the non-linear mapping of some Inverse Problem in Option Pricing [ NS ]( t ) := k ( t , S ( t )) , 0 ≤ t ≤ T , Peter Mathé with domain of definition Introduction, D + ( N ) := { f ∈ C ( I ) , f ( 0 ) = 0 , 0 < f ( t ) < s max ( t ) , t ∈ I } , main results Modulus of and with range R ( N ) . For the general theory we refer continuity for classes of to [Appell-Zabrejko]. Nemytski˘ ı operators Local analysis of forward and backward Black-Scholes kernels Summary, prospective

Nemytski˘ ı operators (superposition operators) Let k ( t , s ) , t ∈ I := [ 0 , T ] , 0 < s < s max ( t ) be any kernel On Non-stability function. We assign the non-linear mapping of some Inverse Problem in Option Pricing [ NS ]( t ) := k ( t , S ( t )) , 0 ≤ t ≤ T , Peter Mathé with domain of definition Introduction, D + ( N ) := { f ∈ C ( I ) , f ( 0 ) = 0 , 0 < f ( t ) < s max ( t ) , t ∈ I } , main results Modulus of and with range R ( N ) . For the general theory we refer continuity for classes of to [Appell-Zabrejko]. Nemytski˘ ı operators Problem (Pricing) Local analysis of forward and Determine C ( t ) := [ NS ]( t ) , t ∈ I ! backward Black-Scholes kernels Summary, prospective

Nemytski˘ ı operators (superposition operators) Let k ( t , s ) , t ∈ I := [ 0 , T ] , 0 < s < s max ( t ) be any kernel On Non-stability function. We assign the non-linear mapping of some Inverse Problem in Option Pricing [ NS ]( t ) := k ( t , S ( t )) , 0 ≤ t ≤ T , Peter Mathé with domain of definition Introduction, D + ( N ) := { f ∈ C ( I ) , f ( 0 ) = 0 , 0 < f ( t ) < s max ( t ) , t ∈ I } , main results Modulus of and with range R ( N ) . For the general theory we refer continuity for classes of to [Appell-Zabrejko]. Nemytski˘ ı operators Problem (Pricing) Local analysis of forward and Determine C ( t ) := [ NS ]( t ) , t ∈ I ! backward Black-Scholes kernels Problem (Calibration, subject of Bernd’s talk) Summary, prospective Determine σ 2 ( τ ) = S − 1 � N − 1 ( C ( t )) � from option prices C ( t ) , t ∈ I .

Nature of ill-posedness On Non-stability of some Both the pricing and calibration problems were studied in Inverse Problem in several papers, Option Pricing [Hein-diss, Hein/Hofmann, Hofmann/Kraemer]. The Peter Mathé following facts appear important: Introduction, main results The operator S : C ( I ) → C ( I ) is a compact linear Modulus of operator , and hence the inversion is ill-posed . continuity for classes of Both, the non-linear opeators Nemytski˘ ı operators N : D ( N ) ⊂ C ( I ) → C ( I ) , and its inverse Local analysis N − 1 : R ( N ) → C ( I ) of forward and backward are continuous ! Black-Scholes kernels The latter is remarkable and was first proved by R. Summary, prospective Krämer, see [Kraemer/Richter].

Modulus of continuity On To any continuous (non-linear) operator Non-stability of some K : D ( K ) ⊂ X → Y Inverse Problem in one can assign its modulus of continuity at x † ∈ D ( K ) as Option Pricing Peter Mathé ω ( K , x † , δ ) Introduction, main results � � � K ( x ) − K ( x † ) � Y , x ∈ D ( K ) , � x − x † � X ≤ δ := sup . Modulus of continuity for classes of Nemytski˘ ı operators Local analysis of forward and backward Black-Scholes kernels Summary, prospective

Modulus of continuity On To any continuous (non-linear) operator Non-stability of some K : D ( K ) ⊂ X → Y Inverse Problem in one can assign its modulus of continuity at x † ∈ D ( K ) as Option Pricing Peter Mathé ω ( K , x † , δ ) Introduction, main results � � � K ( x ) − K ( x † ) � Y , x ∈ D ( K ) , � x − x † � X ≤ δ := sup . Modulus of continuity for classes of Remark Nemytski˘ ı operators This resembles the modulus of continuity of a real valued Local analysis continuous function of forward and backward Black-Scholes � , x , x ′ ∈ [ a , b ] , � f ( x ) − f ( x ′ ) � x − x ′ � � ≤ δ �� � � � kernels ω ( f , δ ) := sup , Summary, prospective and it is its local counterpart, we refer to approximation theory , see [Korne˘ ıchuk]. When further restricting the domain, then better bounds possible.

Pictures from [Romy’s dissertation] On The following behavior was observed from simulation data: Non-stability of some Inverse −3 7 x 10 Problem in measurements Option Pricing 6 Peter Mathé 5 Introduction, main results 4 Modulus of 3 continuity for classes of 2 Nemytski˘ ı operators 1 0 0.01 0.02 0.03 0.04 0.05 Local analysis δ k of forward and backward Black-Scholes Figure: (left) Behavior of modulus of continuity ω ( N − 1 , u † , δ ) as kernels δ → 0. Summary, prospective