An Overview of Calibration Methods for Local Volatility Surfaces

Transcription

1 An Overview of Calibration Methods for Local Volatility Surfaces Jorge P. Zubelli IMPA Joint work with V. Albani (IMPA) A. De Cezaro (FURG,Brazil) O. Scherzer (U.Vienna, Austria) July 11, 2012 Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

2 Outline 1 Intro and Background 2 Problem Statement and Results on Local Vol Models 3 Main Technical Results 4 Numerical Examples 5 Conclusions Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

3 Figure: Example of Data from IBOVESPA Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

4 Central Problems Understand volatility behavior at different time scales. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

5 Central Problems Understand volatility behavior at different time scales. Protect portfolios against volatility oscilations. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

6 Central Problems Understand volatility behavior at different time scales. Protect portfolios against volatility oscilations. Find parsimonious and efficient models Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

7 Central Problems Understand volatility behavior at different time scales. Protect portfolios against volatility oscilations. Find parsimonious and efficient models (simple but not too simple!) Calibrate such models in a robust and effective way. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

8 Central Problems Understand volatility behavior at different time scales. Protect portfolios against volatility oscilations. Find parsimonious and efficient models (simple but not too simple!) Calibrate such models in a robust and effective way. Validate such models across different time scales. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

9 The Concept of Implied Volatility Recall BS Formula: C BS (X,t;K,T,r,σ 0 ) = XN(d + ) Ke r(t t) N(d ) (1) where N is the cumulative normal distribution function and d ± = log(xer(t t) /K ) ± σ 0 T t σ 0 T t 2. (2) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

10 The Concept of Implied Volatility Recall BS Formula: C BS (X,t;K,T,r,σ 0 ) = XN(d + ) Ke r(t t) N(d ) (1) where N is the cumulative normal distribution function and d ± = log(xer(t t) /K ) ± σ 0 T t σ 0 T t 2. (2) Notion of Implied Volatility: Fix everything else and consider σ C BS (X,t;K,T,r,σ) The implied volatility is the inverse to this map. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

11 The Concept of Implied Volatility Recall BS Formula: C BS (X,t;K,T,r,σ 0 ) = XN(d + ) Ke r(t t) N(d ) (1) where N is the cumulative normal distribution function and d ± = log(xer(t t) /K ) ± σ 0 T t σ 0 T t 2. (2) Notion of Implied Volatility: Fix everything else and consider σ C BS (X,t;K,T,r,σ) The implied volatility is the inverse to this map. IMPLIED VOL: wrong number that when plugged into the wrong equation gives the right price Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

12 Figure: Implied Volatility Surface- (From Bruno Dupire - IMPA talk) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

13 Motivation and Goals Good model selection is crucial for modern sound financial practice. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

14 Motivation and Goals Good model selection is crucial for modern sound financial practice. Focus Dupire [Dup94] local volatility models Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

15 Motivation and Goals Good model selection is crucial for modern sound financial practice. Focus Dupire [Dup94] local volatility models Applications risk management hedging evaluation of exotic derivatives. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

16 Local Volatility Models Idea: Assume that the volatility is given by σ = σ(t,x) i.e.: it depends on time and the asset price. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

17 Local Volatility Models Idea: Assume that the volatility is given by σ = σ(t,x) i.e.: it depends on time and the asset price. Easy to check that the Black-Scholes eq. holds. P t + 1 ( 2 σ(t,x)2 x 2 2 P x + 2 r x P ) x P = 0 (3) P(T,x) = h(x) (4) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

18 Local Volatility Models Idea: Assume that the volatility is given by σ = σ(t,x) i.e.: it depends on time and the asset price. Easy to check that the Black-Scholes eq. holds. P t + 1 ( 2 σ(t,x)2 x 2 2 P x + 2 r x P ) x P = 0 (3) P(T,x) = h(x) or in the case you have dividends: P t σ(t,x)2 x 2 2 P P + (r d)x x 2 x rp = 0 P(T,x) = h(x) (4) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

19 Local Volatility Models Idea: Assume that the volatility is given by σ = σ(t,x) i.e.: it depends on time and the asset price. Easy to check that the Black-Scholes eq. holds. P t + 1 ( 2 σ(t,x)2 x 2 2 P x + 2 r x P ) x P = 0 (3) P(T,x) = h(x) or in the case you have dividends: P t σ(t,x)2 x 2 2 P P + (r d)x x 2 x rp = 0 P(T,x) = h(x) (4) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

20 The Direct and the Inverse Problem The Direct Problem Given σ = σ(t,x) and the payoff information, determine P = P(t,x,T,K ;σ) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

21 The Direct and the Inverse Problem The Direct Problem Given σ = σ(t,x) and the payoff information, determine P = P(t,x,T,K ;σ) The Inverse Problem Given a set of observed prices find the volatility σ = σ(t,x). {P = P(t,x,T,K ;σ)} (T,K ) S The set S is taken typically as [T 1,T 2 ] [K 1,K 2 ]. In Practice: Very limited and scarce data Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

22 The Direct and the Inverse Problem The Direct Problem Given σ = σ(t,x) and the payoff information, determine P = P(t,x,T,K ;σ) The Inverse Problem Given a set of observed prices {P = P(t,x,T,K ;σ)} (T,K ) S find the volatility σ = σ(t,x). The set S is taken typically as [T 1,T 2 ] [K 1,K 2 ]. In Practice: Very limited and scarce data Note: To price in a consistent way the so-called exotic derivatives one has to know σ and not only the transition probabilities Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

23 The Smile Curve and Dupire s Equation Assuming that there exists a local volatility function σ = σ(x, t) for which (3) holds Dupire(1994) showed that the call price satisfies { T C 1 2 σ2 (K,T )K 2 2 K C + rs K C = 0, S > 0, t < T C(K,T = 0) = (X K ) +, (5) Theoretical: way of evaluating the local volatility Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

24 The Smile Curve and Dupire s Equation Assuming that there exists a local volatility function σ = σ(x, t) for which (3) holds Dupire(1994) showed that the call price satisfies { T C 1 2 σ2 (K,T )K 2 2 K C + rs K C = 0, S > 0, t < T C(K,T = 0) = (X K ) +, (5) Theoretical: way of evaluating the local volatility ( ) T C + rk K C σ(k,t ) = 2 K 2 2 K C (6) In practice To estimate σ from (5), limited amount of discrete data and thus interpolate. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

25 The Smile Curve and Dupire s Equation Assuming that there exists a local volatility function σ = σ(x, t) for which (3) holds Dupire(1994) showed that the call price satisfies { T C 1 2 σ2 (K,T )K 2 2 K C + rs K C = 0, S > 0, t < T C(K,T = 0) = (X K ) +, (5) Theoretical: way of evaluating the local volatility ( ) T C + rk K C σ(k,t ) = 2 K 2 2 K C (6) In practice To estimate σ from (5), limited amount of discrete data and thus interpolate. Numerical instabilities! Even to keep the argument positive is hard. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

26 Many interpretations of Local Vol Models 1 Stochastic Clock (time of trading) 2 Local variance as a (risk neutral) weighted average of the instantaneous variance over all possible scenarios. (IMPORTANT RESULT!!!) σ 2 (K,T,X 0 ) = E[v T X T = K ], where v T is the instantaneous variance. In words: Local variance can be though of as: The risk-neutral expectation of the instantaneous variance conditional on the final stock price X T = K Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

27 Many interpretations of Local Vol Models 1 Stochastic Clock (time of trading) 2 Local variance as a (risk neutral) weighted average of the instantaneous variance over all possible scenarios. (IMPORTANT RESULT!!!) σ 2 (K,T,X 0 ) = E[v T X T = K ], where v T is the instantaneous variance. In words: Local variance can be though of as: The risk-neutral expectation of the instantaneous variance conditional on the final stock price X T = K Remark: Good estimation of the local volatility is crucial for the consistent pricing of exotics. In fact, prices of exotics based on constant volatility can lead to pretty wrong results. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

28 Literature Very vast!!! Avellaneda et al. [ABF + 00, Ave98c, Ave98b, Ave98a, AFHS97] Bouchev & Isakov [BI97] Crepey [Cré03] Derman et al. [DKZ96] Egger & Engl [EE05] Hofmann et al. [HKPS07, HK05] Jermakyan [BJ99] Achdou & Pironneau (2004) Roger Lee (2001,2005) Abken et al. (1996) Ait Sahalia, Y & Lo, A (1998) Berestycki et al. (2000) Buchen & Kelly (1996) Coleman et al. (1999) Cont, Cont & Da Fonseca (2001) Jackson et al. (1999) Jackwerth & Rubinstein (1998) Jourdain & Nguyen (2001) Lagnado & Osher (1997) Samperi (2001) Stutzer (1997) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

29 Early Results Bouchoev-Isakov Uniqueness and Stability Consider the case of time-independent volatility and working with y = log(k /S(0)) τ = T t Suppose U(y, τ) and a(y) = σ(k ) satisfy: τ U = 1 ( ) 1 2 a2 (y) 2 y U 2 a2 (y) + µ y U + (µ r)u (7) U(y,0) = S(0)(1 e y ) +,y R (8) U(y,τ ) = U (y),y I (9) where I is a sub-interval of R. Then, we have Uniqueness of the volatility Stability of the volatility in the Hölder λ-norm w.r.t. to variations of the data on the 2 + λ-norm. (i.e., one needs TWO extra derivatives of the data). Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

30 Early Results More precisely Theorem (Bouchouev & Isakov) Let U 1 and U 2 be solutions of (7-9) with a = a 1 and a = a 2, resp., and the corresponding final data in Eq. (9) given by U 1 and U 2. Let I 0 be an open interval with I I 0 /0. Then, 1 If U 1 = U 2 on I and a 1(y) = a 2 (y) on I 0 then a 1 (y) = a 2 (y) on I. 2 If, in addition, a 1 (y) = a 2 (y) on I 0 (R \ I) and I is bounded, then C = C( a 1 λ (I), a 2 λ (I),I,I 0,τ ) s.t. a 1 a 2 λ (I) U 1 U 2 2+λ (I) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

31 Early Results More precisely Theorem (Bouchouev & Isakov) Let U 1 and U 2 be solutions of (7-9) with a = a 1 and a = a 2, resp., and the corresponding final data in Eq. (9) given by U 1 and U 2. Let I 0 be an open interval with I I 0 /0. Then, 1 If U 1 = U 2 on I and a 1(y) = a 2 (y) on I 0 then a 1 (y) = a 2 (y) on I. 2 If, in addition, a 1 (y) = a 2 (y) on I 0 (R \ I) and I is bounded, then C = C( a 1 λ (I), a 2 λ (I),I,I 0,τ ) s.t. a 1 a 2 λ (I) U 1 U 2 2+λ (I) Remark: The assumption that a is known on I 0 I makes the problem overdetermined. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

32 Early Results Cont. Theorem (Yamamoto, Bouchouev& Isakov) Suppose that U 1 & U 2 are sols. of (7-9) with a = a 1 and a = a 2, resp., and the corresponding final data (9) are given by U 1 and U 2. Let y 0 be an endpoint of the interval I 1 containg y 0, and a 1 (y),a 2 (y) C (I 1 ). If U 1 (y) = U 2 (y) on I and a 1 (y) = a 2 (y) on I 1 \ I, then a 1 (y) = a 2 (y) on I Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

33 Early Results Cont. Theorem (Yamamoto, Bouchouev& Isakov) Suppose that U 1 & U 2 are sols. of (7-9) with a = a 1 and a = a 2, resp., and the corresponding final data (9) are given by U 1 and U 2. Let y 0 be an endpoint of the interval I 1 containg y 0, and a 1 (y),a 2 (y) C (I 1 ). If U 1 (y) = U 2 (y) on I and a 1 (y) = a 2 (y) on I 1 \ I, then a 1 (y) = a 2 (y) on I Note: proof. In all these results the existence of Dupire s formula is crucial in the Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

34 Problem Statement Starting Point: Dupire forward equation [Dup94] T U σ2 (T,K )K 2 2 K U (r q)k K U qu = 0, T > 0, (10) K = S 0 e y, τ = T t, b = q r, u(τ,y) = e qτ U t,s (T,K ) (11) and a(τ,y) = 1 2 σ2 (T τ;s 0 e y ), (12) Set q = r = 0 for simplicity to get: u τ = a(τ,y)( 2 y u yu) (13) and initial condition u(0,y) = S 0 (1 e y ) + (14) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

35 Problem Statement The Vol Calibration Problem Given an observed set {u = u(t,s,t,k ;σ)} (T,K ) S find σ = σ(t,s) that best fits such market data Noisy data: u = u δ Admissible convex class of calibration parameters: D(F) := {a a 0 + H 1+ε (Ω) : a a a}. (15) where, for 0 ε fixed, U := H 1+ε (Ω) and a > a > 0. Parameter-to-solution operator F : D(F) H 1+ε (Ω) L 2 (Ω) F(a) = u(a) (16) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

36 Setting the problem Theorem (H. Egger-H. Engl[EE05] Crepey[Cré03]) The parameter to solution map F : H 1+ε (Ω) L 2 (Ω) is weak sequentialy continuous compact and weakly closed Consequences: The inverse problem is ill-posed. We can prove that the inverse problem satisfies the conditions to apply the regularization theory. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

37 Well-Posed and Ill-Posed Problems Hadamard s definition of well-posedness: Existence Uniqueness Stability Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

38 Well-Posed and Ill-Posed Problems Hadamard s definition of well-posedness: Existence Uniqueness Stability The problem under consideration: Ill-posed. Equation: F(a) = u Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

39 Well-Posed and Ill-Posed Problems Hadamard s definition of well-posedness: Existence Uniqueness Stability The problem under consideration: Ill-posed. Equation: F(a) = u Need Regularization: Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

40 Approach Convex Tikhonov Regularization For given convex f minimize the Tikhonov functional F β,u δ(a) := F(a) u δ 2 L 2 (Ω) + βf(a) (17) over D(F), where, β > 0 is the regularization parameter. Remark that f incorporates the a priori info on a. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

41 Approach Convex Tikhonov Regularization For given convex f minimize the Tikhonov functional F β,u δ(a) := F(a) u δ 2 L 2 (Ω) + βf(a) (17) over D(F), where, β > 0 is the regularization parameter. Remark that f incorporates the a priori info on a. ū u δ L2 (Ω) δ, (18) where ū is the data associated to the actual value â D(F). Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

42 Approach Convex Tikhonov Regularization For given convex f minimize the Tikhonov functional F β,u δ(a) := F(a) u δ 2 L 2 (Ω) + βf(a) (17) over D(F), where, β > 0 is the regularization parameter. Remark that f incorporates the a priori info on a. ū u δ L2 (Ω) δ, (18) where ū is the data associated to the actual value â D(F). Assumption (very general!) Let ε 0 be fixed. f : D(f) H 1+ε (Ω) [0, ] is a convex, proper and sequentially weakly lower semi-continuous functional with domain D(f) containing D(F). Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

43 Questions Questions: Does there exist a minimizer of the regularized problem? Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

44 Questions Questions: Does there exist a minimizer of the regularized problem? Suppose that the noise level goes to zero... How fast does the regularized go to the true solution? Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

45 Questions Questions: Does there exist a minimizer of the regularized problem? Suppose that the noise level goes to zero... How fast does the regularized go to the true solution? Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

46 Main Theoretical Result F(a) = u(a) (16) F β,u δ(a) := F(a) u δ 2 L 2 (Ω) + βf(a) (17) Theorem (Existence, Stability, Convergence) For the regularized inverse problem we have: minimizer of F β,u δ. If (u k ) u in L 2 (Ω), then a seq. (a k ) s.t. F(a) = u (19) a k argmin { F β,uk (a) : a D } has a subsequence which converges weakly to ã ã argmin { F β,u (a) : a D } Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

47 Main Theoretical Result (cont) F(a) = u(a) (16) F β,u δ(a) := F(a) u δ 2 L 2 (Ω) + βf(a) (17) Theorem (cont.) NOISY CASE Take β = β(δ) > 0 and assume Then, β(δ) satisfies β(δ) 0 and δ2 0, as δ 0. (20) β(δ) The seq. (δ k ) converges to 0, and that u k := u δ k satisfies ū u k δ k. 1 Every seq. (a k ) argminf βk,u k, has weak-convergent subseq. (a k ). 2 The limit a := w lima k is an f -minimizing solution of (16), and f(a k ) f(a ). 3 If the f -minimizing solution a is unique, then a k a weakly. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

48 Bregman distance Let f be a convex function. For a D(f) and f(a) the subdifferential of the functional f at a. We denote by D( f) = {ã : f(ã) /0} the domain of the subdifferential. The Bregman distance w.r.t ζ f(a 1 ) is defined on D(f) D( f) by D ζ (a 2,a 1 ) = f(a 2 ) f(a 1 ) ζ,a 2 a 1. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

49 Bregman distance Let f be a convex function. For a D(f) and f(a) the subdifferential of the functional f at a. We denote by D( f) = {ã : f(ã) /0} the domain of the subdifferential. The Bregman distance w.r.t ζ f(a 1 ) is defined on D(f) D( f) by D ζ (a 2,a 1 ) = f(a 2 ) f(a 1 ) ζ,a 2 a 1. Assumption (1) We assume that 1 an f -minimizing sol. a of (16), a D B (f). 2 β 1 [0,1), β 2 0, and ζ f(a ) s.t. ζ,a a β 1 D ζ (a,a ) + β 2 F(a) F(a ) 2 L (Ω) for a M β max (ρ), (21) where ρ > β max f(a ) > 0. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

50 Convergence rates [SGG + 08] Theorem (Convergence rates [SGG + 08]) Let F, f, D, H 1+ε (Ω), and L 2 (Ω) satisfy Assumption 1. Moreover, let β : (0, ) (0, ) satisfy β(δ) δ. Then D ζ (aβ δ,a ) = O(δ), F(aβ δ ) uδ = O(δ), L 2 (Ω) and there exists c > 0, such that f(a δ β ) f(a ) + δ/c for every δ with β(δ) β max. Example: The regularization functional f as the Boltzmann-Shannon entropy f(a) = a log(a)dx, a D(F), Ω Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

51 Putting it all together NOTE: We have proved We have also proved a tangential cone condition for this problem, which implies that the Landwever iteration converges in a suitable neighborhood. Landweber Iteration [EHN96]: a δ k+1 = a δ k + cf (a δ k) (u δ F(a δ k)). (22) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

52 Putting it all together NOTE: We have proved We have also proved a tangential cone condition for this problem, which implies that the Landwever iteration converges in a suitable neighborhood. Landweber Iteration [EHN96]: a δ k+1 = a δ k + cf (a δ k) (u δ F(a δ k)). (22) Discrepancy Principle: u δ F(a δ k (δ,y δ ) ) u rδ < δ F(ak) δ, (23) where r > η 1 2η, (24) is a relaxation term. If the iteration is stopped at index k (δ,y δ ) such that for the first time, the residual becomes small compared to the quantity r δ. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

53 Numerical Examples with Simulated Data Description of the Examples Using a Landweber iteration technique we implemented the calibration. Produced for different test variances a the option prices and added different levels of multiplicative noise. The examples consisted of perturbing a = 1 during a period of T = 0,,0.2 and log-moneyness y varying between 5 and 5. Initial guess: Constant volatility. Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

54 Numerical Examples - Exact Solution Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

55 Numerical Examples - Exact Solution Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

56 Numerical Examples 1 - noiseless steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

57 Numerical Examples 1 - error steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

58 Numerical Examples 1 - error steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

59 Numerical Examples 1 - error steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

60 Numerical Examples 1 - error steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

61 Numerical Examples 1 - error steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

62 Numerical Examples 1 - error steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

63 Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

64 Numerical Examples 2-5% noise level steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

65 Numerical Examples 2-5% noise level steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

66 Numerical Examples 2-5% noise level steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

67 Numerical Examples 2-5% noise level steps Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

68 Numerical Examples 2-5% noise level - Stopping criteria Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

69 Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

70 Numerical Examples 2-5% noise level iterations Too many!!! Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

71 Numerical Examples: with Real Data Reconstruction of a = σ 2 /2 with PBR Stock Data (implemented by Vinicius L. Albani/IMPA) Figure: Minimal Entropy functional / Landweber Method / a priori Implied Vol / maturities: Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

72 Numerical Examples: with Real Data Reconstruction of a with PBR Stock Data (implemented by Vinicius L. Albani/IMPA) Figure: Minimal Entropy functional / Minimization (Levenberg-Marquadt) Method / Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

73 Conclusions The problem of volatility surface calibration is a classical and fundamental one in Quantitative Finance Unifying framework for the regularization that makes use of tools from Inverse Problem theory and Convex Analysis. Establishing convergence and convergence-rate results. Obtain convergence of the regularized solution with respect to the noise level in L 1 (Ω) The connection with exponential families opens the door to recent works on entropy-based estimation methods. The connection with convex risk measures required the use of techniques from Malliavin calculus. Implemented a Landweber type calibration algorithm. Future Possibilities: 1 Use a priori asymptotic behavior (e.g.: following Roger Lee s results for the implied volatility) 2 Incorporate more and more data as time flows... (online estimation... see V. Albani s talk) Local-Volatility. Calibration c J.P.Zubelli (IMPA) July 11, / 48

74 THANK YOU FOR YOUR ATTENTION!!! Collaborators: V. Albani (IMPA), A. de Cezaro (FURG), O. Scherzer (Vienna) Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

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76 In Trois applications des mathématiques, volume 1998 of SMF Journ. Annu., pages Soc. Math. France, Paris, Marco Avellaneda. The minimum-entropy algorithm and related methods for calibrating asset-pricing models. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), number Extra Vol. III, pages (electronic), I. Bouchouev and V. Isakov. The inverse problem of option pricing. Inverse Problems, 13(5):L11 L17, James N. Bodurtha, Jr. and Martin Jermakyan. Nonparametric estimation of an implied volatility surface. Journal of Computational Finance, 2(4), Summer S. Crépey. Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization. SIAM J. Math. Anal., 34(5): (electronic), Local-Volatility Calibration c J.P.Zubelli (IMPA) July 11, / 48

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