Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
|
|
- Ilene Howard
- 6 years ago
- Views:
Transcription
1 Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University University of Missouri-Kansas City Department of Mathematics and Statistics April 5th, 2012 (Joint work with Jeff Nisen)
2 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Power Estimators 3 Main Results 4 Extensions 5 Conclusions
3 Set-up 1 Continuous-time stochastic process t X t with dynamics where dx t = γ t dt + σ t dw t + dj t, t W t is a standard Brownian motion; t J t := N t j=1 ζ j is a piece-wise constant process of finite jump activity; t γ t and t σ t are adapted processes; 2 Finite-jump activity Lévy model: N t X t = γt + σw t + ζ j, where {N t } t 0 is a homogeneous Poisson process with jump intensity λ, {ζ j } j 0 are i.i.d. with density f ζ : R R +, and the triplet ({W t }, {N t }, {ξ j }) are mutually independent. j=1
4 Statistical Problems Given a discrete record of observations, X t0, X t1,..., X tn, from the process during a fixed finite time-horizon [0, T ], the following problems are of interest: 1 Estimating the integrated variance (or quadratic variation): σ 2 T = T 0 σ 2 t dt. 2 Estimating the jump features of the process: Jump times τ 1 < τ 2 < < τ NT Jump sizes ζ 1 < ζ 2 < < ζ NT 3 Jump detection during a given time interval [s, t] [0, T ]
5 Two main classes of estimators Precursor. Realized Quadratic Variation: n 1 ( QV (X) π := Xti+1 X ) 2 t i, (π : 0 = t0 < < t n = T ). i=0 Under very general conditions: RV (X) π mesh(π) 0 σ 2 T + N T j=1 ζ2 j. 1 Multipower Realized Variations (Barndorff-Nielsen and Shephard (2004)): n 1 BPV (X) π := Xti+1 X ti Xti+2 X ti+1, MPV (X) (r 1,...,r k ) π := i=0 n k X ti+1 X ti r 1... X ti+k X ti+k 1 r k. i=0 2 Threshold Realized Variations (Mancini (2003)): n 1 ( TRV (X)[B] π := Xti+1 X ) 2 t i 1{ Xti+1 Xti B}, (B (0, )). i=0
6 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Easy to implement; Exhibit high" bias in the presence of jumps: ] E [MPV (X) (r 1,...,r k ) C r σ T 2 2 Threshold Realized Variations (TRV): π tc r, with Cr = n max i r i Can be adapted for estimating other jump features: k i=1 E Z i r i r r k = 2. n 1 n 1 ( ) N[B] π := 1 { 2 >B} Xti+1, Ĵ[B] X π := Xti+1 X ti 1 { ti Xti+1 Xti i=0 i=0 } >B Its performance strongly depends on a good" choice of the threshold level B; e.g., given a sequence π n of sampling schemes with mesh(π n) 0, L TRV (X)[B] 2 πn σ 2 T B(π n) 0, Ad-hoc thresholds proposed in the literature: B(π n) mesh(πn). B(π) := α mesh(π) ω, B n(π) = α mesh 1 2 Φ 1 (1 β mesh(π))
7 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Easy to implement; Exhibit high" bias in the presence of jumps: ] E [MPV (X) (r 1,...,r k ) C r σ T 2 2 Threshold Realized Variations (TRV): π tc r, with Cr = n max i r i Can be adapted for estimating other jump features: k i=1 E Z i r i r r k = 2. n 1 n 1 ( ) N[B] π := 1 { 2 >B} Xti+1, Ĵ[B] X π := Xti+1 X ti 1 { ti Xti+1 Xti i=0 i=0 } >B Its performance strongly depends on a good" choice of the threshold level B; e.g., given a sequence π n of sampling schemes with mesh(π n) 0, L TRV (X)[B] 2 πn σ 2 T B(π n) 0, Ad-hoc thresholds proposed in the literature: B(π n) mesh(πn). B(π) := α mesh(π) ω, B n(π) = α mesh 1 2 Φ 1 (1 β mesh(π))
8 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Easy to implement; Exhibit high" bias in the presence of jumps: ] E [MPV (X) (r 1,...,r k ) C r σ T 2 2 Threshold Realized Variations (TRV): π tc r, with Cr = n max i r i Can be adapted for estimating other jump features: k i=1 E Z i r i r r k = 2. n 1 n 1 ( ) N[B] π := 1 { 2 >B} Xti+1, Ĵ[B] X π := Xti+1 X ti 1 { ti Xti+1 Xti i=0 i=0 } >B Its performance strongly depends on a good" choice of the threshold level B; e.g., given a sequence π n of sampling schemes with mesh(π n) 0, L TRV (X)[B] 2 πn σ 2 T B(π n) 0, Ad-hoc thresholds proposed in the literature: B(π n) mesh(πn). B(π) := α mesh(π) ω, B n(π) = α mesh 1 2 Φ 1 (1 β mesh(π))
9 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Easy to implement; Exhibit high" bias in the presence of jumps: ] E [MPV (X) (r 1,...,r k ) C r σ T 2 2 Threshold Realized Variations (TRV): π tc r, with Cr = n max i r i Can be adapted for estimating other jump features: k i=1 E Z i r i r r k = 2. n 1 n 1 ( ) N[B] π := 1 { 2 >B} Xti+1, Ĵ[B] X π := Xti+1 X ti 1 { ti Xti+1 Xti i=0 i=0 } >B Its performance strongly depends on a good" choice of the threshold level B; e.g., given a sequence π n of sampling schemes with mesh(π n) 0, L TRV (X)[B] 2 πn σ 2 T B(π n) 0, Ad-hoc thresholds proposed in the literature: B(π n) mesh(πn). B(π) := α mesh(π) ω, B n(π) = α mesh 1 2 Φ 1 (1 β mesh(π))
10 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Easy to implement; Exhibit high" bias in the presence of jumps: ] E [MPV (X) (r 1,...,r k ) C r σ T 2 2 Threshold Realized Variations (TRV): π tc r, with Cr = n max i r i Can be adapted for estimating other jump features: k i=1 E Z i r i r r k = 2. n 1 n 1 ( ) N[B] π := 1 { 2 >B} Xti+1, Ĵ[B] X π := Xti+1 X ti 1 { ti Xti+1 Xti i=0 i=0 } >B Its performance strongly depends on a good" choice of the threshold level B; e.g., given a sequence π n of sampling schemes with mesh(π n) 0, L TRV (X)[B] 2 πn σ 2 T B(π n) 0, Ad-hoc thresholds proposed in the literature: B(π n) mesh(πn). B(π) := α mesh(π) ω, B n(π) = α mesh 1 2 Φ 1 (1 β mesh(π))
11 Numerical illustration Diffusion Volatility Parameter (DVP) Estimates RMPV(1, 1) RMPV(2 3, 2 3, 2 3) RMPV(1 2,, 1 2) RMPV(2 5,, 2 5) RMPV(1 3,, 1 3) Min RV(2) Med RV(2) TBPV(Pow(0.05)) TBPV(Pow(0.15)) TBPV(Pow(0.25)) TBPV(Pow(0.35)) TBPV(Pow(0.45)) TBPV(Pow(0.495)) TBPV(B opt) TBPV(BF(0.05)) TBPV(BH(0.05)) Merton Model: Diffusion Volatility Parameter Estimates Actual DVP Multi Power Variation Style Estimators Thresholded Multi Power Style Estimators Multiple Testing Style Estimators Figure: Box Plots of MC numerical experiments (2500 simulations) based on T = 1 year, 5 min sample observations. Parameters: σ = 0.3, λ = 20, f ζ N ( 0.1, ).
12 Optimal Threshold Realized Estimators 1 Aims Develop a well-posed optimal selection criterion for the threshold B, that minimizes a suitable statistical loss function of estimation. Characterize the optimal threshold B asymptotically when mesh(π) 0. Developed a feasible implementation method for the optimal threhold sequence. 2 Assumptions Finite activity Lévy model: X t = γt + σw t + N t j=1 ζ i.i.d. j, ζ j f ζ. Regular sampling scheme with mesh h n := 1 ; i.e., π : t n i = i. n The jump density function f ζ takes the mixture form: 3 Notation: f ζ (x) = pf +(x)1 {x 0} + qf ( x)1 {x<0} with p + q = 1, f ± : [0, ) R + C 1 b(0, ) C(f ζ ) := pf +(0) + qf (0). Φ and φ are the cdf and pdf of a standard Normal variable, respectively. n i X := i X := X ti X ti 1, n i N := i N := N ti N ti 1
13 Optimal Threshold Realized Estimators 1 Aims Develop a well-posed optimal selection criterion for the threshold B, that minimizes a suitable statistical loss function of estimation. Characterize the optimal threshold B asymptotically when mesh(π) 0. Developed a feasible implementation method for the optimal threhold sequence. 2 Assumptions Finite activity Lévy model: X t = γt + σw t + N t j=1 ζ i.i.d. j, ζ j f ζ. Regular sampling scheme with mesh h n := 1 ; i.e., π : t n i = i. n The jump density function f ζ takes the mixture form: 3 Notation: f ζ (x) = pf +(x)1 {x 0} + qf ( x)1 {x<0} with p + q = 1, f ± : [0, ) R + C 1 b(0, ) C(f ζ ) := pf +(0) + qf (0). Φ and φ are the cdf and pdf of a standard Normal variable, respectively. n i X := i X := X ti X ti 1, n i N := i N := N ti N ti 1
14 Optimal Threshold Realized Estimators 1 Aims Develop a well-posed optimal selection criterion for the threshold B, that minimizes a suitable statistical loss function of estimation. Characterize the optimal threshold B asymptotically when mesh(π) 0. Developed a feasible implementation method for the optimal threhold sequence. 2 Assumptions Finite activity Lévy model: X t = γt + σw t + N t j=1 ζ i.i.d. j, ζ j f ζ. Regular sampling scheme with mesh h n := 1 ; i.e., π : t n i = i. n The jump density function f ζ takes the mixture form: 3 Notation: f ζ (x) = pf +(x)1 {x 0} + qf ( x)1 {x<0} with p + q = 1, f ± : [0, ) R + C 1 b(0, ) C(f ζ ) := pf +(0) + qf (0). Φ and φ are the cdf and pdf of a standard Normal variable, respectively. n i X := i X := X ti X ti 1, n i N := i N := N ti N ti 1
15 Loss Functions 1 Natural Loss Function Loss (1) n (B) := E [ TRV (X)[B]n T σ 2 2] + E [ N[B]n N T 2 ]. 2 Alternative Loss Function 3 Interpretation Loss (2) n ( B) := E nt i=1 ( ) 1 [ n i X >B, n N=0] + 1 i [ n i X B, n N 0]. i Loss (1) n (B) favors sequences that minimizes the estimation errors of both the continuous and the jump component. Loss (2) n (B) favors sequences that minimizes the total number of miss-classifications: flag jump when there is no jump and fail to flag jump when there is a jump. Loss (2) n (B) is much more tractable than Loss (1) n (B).
16 Asymptotic Comparison of Loss Functions Theorem (FL & Nisen (2013)) Given a threshold sequence (B n ) n satisfying B n 0 and B n n, there exists a positive sequence (C n ) n, with lim n C n = 0, such that Loss (2) n (B) + R n (B) Loss (1) n (B) (1 + C n (B))Loss (2) n (B) + R n (B) + R n (B), where, as n, R n (B) T ( ( ) 2 λ2 2σ n 2n + T 2 nbn φ λb n C(f )), B n σ [ ] 6T σ 4 R n (B) + 3B 6 n nt 2 λ 2 C(f ) 2. Furthermore, lim n inf B>0 Loss(1) n (B) = 1. (1) inf B>0 Loss(2) n (B)
17 Well-posedness and asymptotic characterization Theorem (FL & Nisen (2013)) There exists an N N such that for all n N, the loss function Loss (2) n (B) is quasi-convex and possesses a unique global minimum Bn: Bn := arg inf B>0 Loss (2) n (B). Furthermore, the optimal threshold sequence (Bn) n is such that Bn 3σ2 ln(n) ( ) = + o ln(n)/n, (n ). n
18 Remarks 1 The leading term of the optimal sequence is proportional to the Lévy modulus of Brownian motion: lim sup h 0 1 2h ln(1/h) 2 The leading order sequence sup W t W s = 1, a.s. t s <h,s,t [0,1] B,1 n := 3σ2 ln(n), n provides a blueprint" to look for a suitable threshold sequence. 3 Merton Model: ζ N (0, δ 2 ). With αn 2 := σ2 n + δ2, ( ) Bn = 3σ2 ln(n) 2σ 2 σλ ln α n + 3σ4 ln(n) n n n 2 δ 2 2σ4 ln(σλ/α n ) n 2 δ 2 4 The performance of the leading term (compared to the optimal threshold) will depend on the quantity: σλ δ. 1/2.
19 A Feasible Iterative Algorithm to Find B n 1 Key Issue: The optimal threshold B would allow us to find an optimal estimate ˆσ for σ 2 of the form but B depends on precisely σ 2. Set σ 2 n,0 := 1 T ˆσ 2 := 1 T TRV (X)[B (σ 2 )] n, 2 The previous issue suggests a fixed-point type of implementation: ( ) 1/2 nt i=1 X t i X ti 1 2 and B n,0 := while σ 2 n,k 1 > σ2 n,k do σ 2 n,k+1 1 T TRV (X)[ B n,k ] n and B n,k+1 ( 3 σ 2 n,0 ln(n) n ) 1/2 3 σ 2 n,k+1 ln(n) n end while { } Let kn := inf k 1 : σ n,k+1 2 = σ2 n,k and take σ n,k 2 as the final n estimate for σ and the corresponding B n,k as an estimate for n B n. 3 The previous algorithm generates a non-increasing sequence of estimators { σ 2 n,k } k and finish in finite time.
20 A numerical illustration Merton Model: 4-year / 1-day σ = 0.3 λ = 5 µ = 0, δ = 0.6 Method TRV S TRV Loss S Loss B n,k n Pow BF Table: Finite-sample performance of the threshold realized variation (TRV) estimators i.i.d. based on K = 5, 000 sample paths for the Merton model ζ i N (µ, δ 2 ). Loss represents the total number of Jump Misclassification Errors, while TRV, Loss, S TRV, and S Loss denote the corresponding sample means and standard deviations, respectively.
21 A numerical illustration (S2) Kou Model: 1-week / 5-minute σ = 0.5 λ = 50 p = 0.45, α + = 0.05, α = 0.1 Method TRV S TRV Loss S Loss B n,k n Pow BF Table: Finite-sample performance of the threshold realized variation (TRV) estimators based on K = 5, 000 sample paths for the Kou model: f Kou (x) = p α + e x/α+ 1 [x 0] + (1 p) α e x /α 1 [x<0]. Loss represents the total number of Jump Misclassification Errors, while TRV, Loss, S TRV, and S Loss denote the corresponding sample means and standard deviations, respectively.
22 A numerical illustration (S3) Kou Model: 1-year / 5-minute σ = 0.4 λ = 1000 p = 0.5, α + = α = 0.1 Method TRV S TRV Loss S Loss B n,k n Pow BF Table: Finite-sample performance of the threshold realized variation (TRV) estimators based on K = 5, 000 sample paths for the Kou model: f ζ (x) = p α + e x/α+ 1 [x 0] + q α e x /α 1 [x<0]. Loss represents the total number of Jump Misclassification Errors, while TRV, Loss, S TRV, and S Loss denote the corresponding sample means and standard deviations, respectively.
23 Additive Processes 1 The model X s := s 0 γ(u)du + s 0 N s σ(u)dw u + ζ j =: Xs c + J s, where (N s ) s 0 Poiss ({λ(s)} s 0 ), independent of W, and deterministic smooth functions σ, λ : [0, ) R + and γ : [0, ) R with σ and λ bounded away from 0. 2 Optimal Threshold Problem Given a sampling scheme π : t 0 < < t n = T, determine the vector B π, = ( B π, t 1 inf E B=(Bt1,...,B tn ) R m + n = i=1 j=1,..., B π, t n ) that minimizes the problem n i=1 ( 1 [ Xti X ti 1 >B ti,n ti N ti 1 =0] + 1 [ Xti X ti 1 B ti,n ti N ti 1 0] inf {P ( i X > B ti, i N = 0) + P ( i X B ti, i N 0)}, B ti ( i X := X ti X ti 1, i N := N ti N ti 1 ) )
24 Optimal Threshold Spot Volatility Estimation Notation: h i = t i t i 1 (Mesh), K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm: For each i {1, 2,..., n}, set σ 2 0(t i ) := l j= l 1 h i+j i+j X 2 K θ (t i t i+j ) and B,0 t i := [ 3 σ 2 0(t i )h i ln(1/h i ) ] 1/2 while there exists i {1, 2,..., m} such that σ k 1 2 (t i) > σ k 2(t i) do σ k+1 2 (t i) l 1 j= l h i+j i+j X 2 1 [ ] K i+j X B,k θ (t i t i+j ) and t i+j B,k+1 t i [ 3 σ k+1 2 (t i)h i ln(1/h i ) ] 1/2 end while Let k (π) := inf { k 1 : σ k+1 2 (t i) = σ k 2(t i); for all i = 1, 2,..., n } and take σ k 2 (t i) as the final estimate for σ(t m i ) and the corresponding B,k m t i estimate for Bt i. as an The previous algorithm generates a non-increasing sequence of estimators { σ 2 k (t i)} k,i and finish in finite time.
25 Numerical Illustration (A) Initial Estimates Actual Spot Volatility Est. Spot Vol. (Uniform) Est. Spot Vol. (Quad) Sample Increments Time Horizon Figure: Spot Volatility Estimation using Adaptive Kernel Weighted Realized Volatility.
26 Numerical Illustration (B) Intermediate Estimates Actual Spot Volatility Est. Spot Vol. (Uniform) Est. Spot Vol. (Quad) Sample Increments Time Horizon Figure: Spot Volatility Estimation using Adaptive Kernel Weighted Realized Volatility.
27 Numerical Illustration (C) Terminal Estimates Actual Spot Volatility Est. Spot Vol. (Uniform) Est. Spot Vol. (Quad) Sample Increments Time Horizon Figure: Spot Volatility Estimation using Adaptive Kernel Weighted Realized Volatility.
28 Numerical Illustration (D) Estimation Variability Time Horizon Actual Spot Volatility Figure: Spot Volatility Estimation using Adaptive Kernel Weighted Realized Volatility. 50 simulations
29 Conclusions 1 Introduce an objective threshold selection procedure based on statistical optimality reasoning via a well-posed optimization problem. 2 Characterize precisely the infill asymptotic behavior of the optimal threshold sequence. 3 Proposed an iterative algorithm to find the optimal threshold sequence. 4 Extend the approach to more general stochastic models, which allows time-varying volatility and jump intensity.
30 For Further Reading I Figueroa-López & Nisen. Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models To appear in Stochastic Processes and their Applications, Available at figueroa. Figueroa-López & Nisen. Optimality properties of thresholded multi power variation estimators. In preparation, 2013.
Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University High Dimensional Probability VII Institut d Études Scientifiques
More informationOptimally Thresholded Realized Power Variations for Stochastic Volatility Models with Jumps
Optimally Thresholded Realized Power Variations for Stochastic Volatility Models with Jumps José E. Figueroa-López 1 1 Department of Mathematics Washington University ISI 2015: 60th World Statistics Conference
More informationOptimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University Financial Statistics Stevanovich Center The University
More informationAsymptotic Methods in Financial Mathematics
Asymptotic Methods in Financial Mathematics José E. Figueroa-López 1 1 Department of Mathematics Washington University in St. Louis Statistics Seminar Washington University in St. Louis February 17, 2017
More informationOptimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error
Optimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error José E. Figueroa-López Department of Mathematics Washington University in St. Louis Spring Central Sectional Meeting
More informationShort-Time Asymptotic Methods in Financial Mathematics
Short-Time Asymptotic Methods in Financial Mathematics José E. Figueroa-López Department of Mathematics Washington University in St. Louis Probability and Mathematical Finance Seminar Department of Mathematical
More informationA Simulation Study of Bipower and Thresholded Realized Variations for High-Frequency Data
Washington University in St. Louis Washington University Open Scholarship Arts & Sciences Electronic Theses and Dissertations Arts & Sciences Spring 5-18-2018 A Simulation Study of Bipower and Thresholded
More informationVolatility. Roberto Renò. 2 March 2010 / Scuola Normale Superiore. Dipartimento di Economia Politica Università di Siena
Dipartimento di Economia Politica Università di Siena 2 March 2010 / Scuola Normale Superiore What is? The definition of volatility may vary wildly around the idea of the standard deviation of price movements
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More informationShort-Time Asymptotic Methods In Financial Mathematics
Short-Time Asymptotic Methods In Financial Mathematics José E. Figueroa-López Department of Mathematics Washington University in St. Louis Department of Applied Mathematics, Illinois Institute of Technology
More informationSmall-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias
Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias José E. Figueroa-López 1 1 Department of Statistics Purdue University Computational Finance Seminar Purdue University
More informationShort-Time Asymptotic Methods In Financial Mathematics
Short-Time Asymptotic Methods In Financial Mathematics José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis School Of Mathematics, UMN March 14, 2019 Based
More informationApplications of short-time asymptotics to the statistical estimation and option pricing of Lévy-driven models
Applications of short-time asymptotics to the statistical estimation and option pricing of Lévy-driven models José Enrique Figueroa-López 1 1 Department of Statistics Purdue University CIMAT and Universidad
More informationShort-time asymptotics for ATM option prices under tempered stable processes
Short-time asymptotics for ATM option prices under tempered stable processes José E. Figueroa-López 1 1 Department of Statistics Purdue University Probability Seminar Purdue University Oct. 30, 2012 Joint
More informationOn modelling of electricity spot price
, Rüdiger Kiesel and Fred Espen Benth Institute of Energy Trading and Financial Services University of Duisburg-Essen Centre of Mathematics for Applications, University of Oslo 25. August 2010 Introduction
More informationEquity correlations implied by index options: estimation and model uncertainty analysis
1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to
More informationOptimal Kernel Estimation of Spot Volatility of SDE
Optimal Kernel Estimation of Spot Volatility of SDE José E. Figueroa-López Department of Mathematics Washington University in St. Louis figueroa@math.wustl.edu (Joint work with Cheng Li from Purdue U.)
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November
More informationMgr. Jakub Petrásek 1. May 4, 2009
Dissertation Report - First Steps Petrásek 1 2 1 Department of Probability and Mathematical Statistics, Charles University email:petrasek@karlin.mff.cuni.cz 2 RSJ Invest a.s., Department of Probability
More informationBROWNIAN MOTION Antonella Basso, Martina Nardon
BROWNIAN MOTION Antonella Basso, Martina Nardon basso@unive.it, mnardon@unive.it Department of Applied Mathematics University Ca Foscari Venice Brownian motion p. 1 Brownian motion Brownian motion plays
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationI Preliminary Material 1
Contents Preface Notation xvii xxiii I Preliminary Material 1 1 From Diffusions to Semimartingales 3 1.1 Diffusions.......................... 5 1.1.1 The Brownian Motion............... 5 1.1.2 Stochastic
More informationTesting for non-correlation between price and volatility jumps and ramifications
Testing for non-correlation between price and volatility jumps and ramifications Claudia Klüppelberg Technische Universität München cklu@ma.tum.de www-m4.ma.tum.de Joint work with Jean Jacod, Gernot Müller,
More informationModeling the dependence between a Poisson process and a continuous semimartingale
1 / 28 Modeling the dependence between a Poisson process and a continuous semimartingale Application to electricity spot prices and wind production modeling Thomas Deschatre 1,2 1 CEREMADE, University
More informationExact Sampling of Jump-Diffusion Processes
1 Exact Sampling of Jump-Diffusion Processes and Dmitry Smelov Management Science & Engineering Stanford University Exact Sampling of Jump-Diffusion Processes 2 Jump-Diffusion Processes Ubiquitous in finance
More informationWeierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions
Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions Hilmar Mai Mohrenstrasse 39 1117 Berlin Germany Tel. +49 3 2372 www.wias-berlin.de Haindorf
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Other Miscellaneous Topics and Applications of Monte-Carlo Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationLecture 7: Bayesian approach to MAB - Gittins index
Advanced Topics in Machine Learning and Algorithmic Game Theory Lecture 7: Bayesian approach to MAB - Gittins index Lecturer: Yishay Mansour Scribe: Mariano Schain 7.1 Introduction In the Bayesian approach
More informationinduced by the Solvency II project
Asset Les normes allocation IFRS : new en constraints assurance induced by the Solvency II project 36 th International ASTIN Colloquium Zürich September 005 Frédéric PLANCHET Pierre THÉROND ISFA Université
More informationAsymptotic results discrete time martingales and stochastic algorithms
Asymptotic results discrete time martingales and stochastic algorithms Bernard Bercu Bordeaux University, France IFCAM Summer School Bangalore, India, July 2015 Bernard Bercu Asymptotic results for discrete
More informationStochastic Dynamical Systems and SDE s. An Informal Introduction
Stochastic Dynamical Systems and SDE s An Informal Introduction Olav Kallenberg Graduate Student Seminar, April 18, 2012 1 / 33 2 / 33 Simple recursion: Deterministic system, discrete time x n+1 = f (x
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Haindorf, 7 Februar 2008 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar
More informationEconomics 883: The Basic Diffusive Model, Jumps, Variance Measures. George Tauchen. Economics 883FS Spring 2015
Economics 883: The Basic Diffusive Model, Jumps, Variance Measures George Tauchen Economics 883FS Spring 2015 Main Points 1. The Continuous Time Model, Theory and Simulation 2. Observed Data, Plotting
More informationControl. Econometric Day Mgr. Jakub Petrásek 1. Supervisor: RSJ Invest a.s.,
and and Econometric Day 2009 Petrásek 1 2 1 Department of Probability and Mathematical Statistics, Charles University, RSJ Invest a.s., email:petrasek@karlin.mff.cuni.cz 2 Department of Probability and
More informationArbitrage of the first kind and filtration enlargements in semimartingale financial models. Beatrice Acciaio
Arbitrage of the first kind and filtration enlargements in semimartingale financial models Beatrice Acciaio the London School of Economics and Political Science (based on a joint work with C. Fontana and
More informationAn Introduction to Stochastic Calculus
An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2-3 Haijun Li An Introduction to Stochastic Calculus Week 2-3 1 / 24 Outline
More informationParametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen
Parametric Inference and Dynamic State Recovery from Option Panels Torben G. Andersen Joint work with Nicola Fusari and Viktor Todorov The Third International Conference High-Frequency Data Analysis in
More informationOptimal Placement of a Small Order Under a Diffusive Limit Order Book (LOB) Model
Optimal Placement of a Small Order Under a Diffusive Limit Order Book (LOB) Model José E. Figueroa-López Department of Mathematics Washington University in St. Louis INFORMS National Meeting Houston, TX
More informationOption Pricing Modeling Overview
Option Pricing Modeling Overview Liuren Wu Zicklin School of Business, Baruch College Options Markets Liuren Wu (Baruch) Stochastic time changes Options Markets 1 / 11 What is the purpose of building a
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Modeling financial markets with extreme risk Tobias Kusche Preprint Nr. 04/2008 Modeling financial markets with extreme risk Dr. Tobias Kusche 11. January 2008 1 Introduction
More informationOptimal Kernel Estimation of Spot Volatility
Optimal Kernel Estimation of Spot Volatility José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis figueroa@math.wustl.edu Joint work with Cheng Li from Purdue
More informationEstimation methods for Levy based models of asset prices
Estimation methods for Levy based models of asset prices José Enrique Figueroa-López Financial mathematics seminar Department of Statistics and Applied Probability UCSB October, 26 Abstract Stock prices
More informationNear-expiration behavior of implied volatility for exponential Lévy models
Near-expiration behavior of implied volatility for exponential Lévy models José E. Figueroa-López 1 1 Department of Statistics Purdue University Financial Mathematics Seminar The Stevanovich Center for
More informationRegression estimation in continuous time with a view towards pricing Bermudan options
with a view towards pricing Bermudan options Tagung des SFB 649 Ökonomisches Risiko in Motzen 04.-06.06.2009 Financial engineering in times of financial crisis Derivate... süßes Gift für die Spekulanten
More informationConditional Density Method in the Computation of the Delta with Application to Power Market
Conditional Density Method in the Computation of the Delta with Application to Power Market Asma Khedher Centre of Mathematics for Applications Department of Mathematics University of Oslo A joint work
More information3.4 Copula approach for modeling default dependency. Two aspects of modeling the default times of several obligors
3.4 Copula approach for modeling default dependency Two aspects of modeling the default times of several obligors 1. Default dynamics of a single obligor. 2. Model the dependence structure of defaults
More informationNumerical Methods for Pricing Energy Derivatives, including Swing Options, in the Presence of Jumps
Numerical Methods for Pricing Energy Derivatives, including Swing Options, in the Presence of Jumps, Senior Quantitative Analyst Motivation: Swing Options An electricity or gas SUPPLIER needs to be capable,
More informationEstimation of High-Frequency Volatility: An Autoregressive Conditional Duration Approach
Estimation of High-Frequency Volatility: An Autoregressive Conditional Duration Approach Yiu-Kuen Tse School of Economics, Singapore Management University Thomas Tao Yang Department of Economics, Boston
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationNormal Inverse Gaussian (NIG) Process
With Applications in Mathematical Finance The Mathematical and Computational Finance Laboratory - Lunch at the Lab March 26, 2009 1 Limitations of Gaussian Driven Processes Background and Definition IG
More informationEffectiveness of CPPI Strategies under Discrete Time Trading
Effectiveness of CPPI Strategies under Discrete Time Trading S. Balder, M. Brandl 1, Antje Mahayni 2 1 Department of Banking and Finance, University of Bonn 2 Department of Accounting and Finance, Mercator
More informationAsymptotic Theory for Renewal Based High-Frequency Volatility Estimation
Asymptotic Theory for Renewal Based High-Frequency Volatility Estimation Yifan Li 1,2 Ingmar Nolte 1 Sandra Nolte 1 1 Lancaster University 2 University of Manchester 4th Konstanz - Lancaster Workshop on
More informationAsymptotic methods in risk management. Advances in Financial Mathematics
Asymptotic methods in risk management Peter Tankov Based on joint work with A. Gulisashvili Advances in Financial Mathematics Paris, January 7 10, 2014 Peter Tankov (Université Paris Diderot) Asymptotic
More informationLarge Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models
Large Deviations and Stochastic Volatility with Jumps: TU Berlin with A. Jaquier and A. Mijatović (Imperial College London) SIAM conference on Financial Mathematics, Minneapolis, MN July 10, 2012 Implied
More informationCredit Risk using Time Changed Brownian Motions
Credit Risk using Time Changed Brownian Motions Tom Hurd Mathematics and Statistics McMaster University Joint work with Alexey Kuznetsov (New Brunswick) and Zhuowei Zhou (Mac) 2nd Princeton Credit Conference
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Generating Random Variables and Stochastic Processes Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationSimulating Stochastic Differential Equations
IEOR E4603: Monte-Carlo Simulation c 2017 by Martin Haugh Columbia University Simulating Stochastic Differential Equations In these lecture notes we discuss the simulation of stochastic differential equations
More informationOptimal Securitization via Impulse Control
Optimal Securitization via Impulse Control Rüdiger Frey (joint work with Roland C. Seydel) Mathematisches Institut Universität Leipzig and MPI MIS Leipzig Bachelier Finance Society, June 21 (1) Optimal
More informationApproximations of Stochastic Programs. Scenario Tree Reduction and Construction
Approximations of Stochastic Programs. Scenario Tree Reduction and Construction W. Römisch Humboldt-University Berlin Institute of Mathematics 10099 Berlin, Germany www.mathematik.hu-berlin.de/~romisch
More informationStock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models
Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models David Prager 1 1 Associate Professor of Mathematics Anderson University (SC) Based on joint work with Professor Qing Zhang,
More informationSelf-Exciting Corporate Defaults: Contagion or Frailty?
1 Self-Exciting Corporate Defaults: Contagion or Frailty? Kay Giesecke CreditLab Stanford University giesecke@stanford.edu www.stanford.edu/ giesecke Joint work with Shahriar Azizpour, Credit Suisse Self-Exciting
More informationRisk Measurement in Credit Portfolio Models
9 th DGVFM Scientific Day 30 April 2010 1 Risk Measurement in Credit Portfolio Models 9 th DGVFM Scientific Day 30 April 2010 9 th DGVFM Scientific Day 30 April 2010 2 Quantitative Risk Management Profit
More informationParameters Estimation in Stochastic Process Model
Parameters Estimation in Stochastic Process Model A Quasi-Likelihood Approach Ziliang Li University of Maryland, College Park GEE RIT, Spring 28 Outline 1 Model Review The Big Model in Mind: Signal + Noise
More informationEstimating Bivariate GARCH-Jump Model Based on High Frequency Data : the case of revaluation of Chinese Yuan in July 2005
Estimating Bivariate GARCH-Jump Model Based on High Frequency Data : the case of revaluation of Chinese Yuan in July 2005 Xinhong Lu, Koichi Maekawa, Ken-ichi Kawai July 2006 Abstract This paper attempts
More informationUQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions.
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions. Random Variables 2 A random variable X is a numerical (integer, real, complex, vector etc.) summary of the outcome of the random experiment.
More informationGeneralized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models
Generalized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models Bilkan Erkmen (joint work with Michael Coulon) Workshop on Stochastic Games, Equilibrium, and Applications
More informationVolume and volatility in European electricity markets
Volume and volatility in European electricity markets Roberto Renò reno@unisi.it Dipartimento di Economia Politica, Università di Siena Commodities 2007 - Birkbeck, 17-19 January 2007 p. 1/29 Joint work
More informationA Continuity Correction under Jump-Diffusion Models with Applications in Finance
A Continuity Correction under Jump-Diffusion Models with Applications in Finance Cheng-Der Fuh 1, Sheng-Feng Luo 2 and Ju-Fang Yen 3 1 Institute of Statistical Science, Academia Sinica, and Graduate Institute
More informationOperational Risk. Robert Jarrow. September 2006
1 Operational Risk Robert Jarrow September 2006 2 Introduction Risk management considers four risks: market (equities, interest rates, fx, commodities) credit (default) liquidity (selling pressure) operational
More informationLogarithmic derivatives of densities for jump processes
Logarithmic derivatives of densities for jump processes Atsushi AKEUCHI Osaka City University (JAPAN) June 3, 29 City University of Hong Kong Workshop on Stochastic Analysis and Finance (June 29 - July
More informationChapter 2 Uncertainty Analysis and Sampling Techniques
Chapter 2 Uncertainty Analysis and Sampling Techniques The probabilistic or stochastic modeling (Fig. 2.) iterative loop in the stochastic optimization procedure (Fig..4 in Chap. ) involves:. Specifying
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationRMSC 4005 Stochastic Calculus for Finance and Risk. 1 Exercises. (c) Let X = {X n } n=0 be a {F n }-supermartingale. Show that.
1. EXERCISES RMSC 45 Stochastic Calculus for Finance and Risk Exercises 1 Exercises 1. (a) Let X = {X n } n= be a {F n }-martingale. Show that E(X n ) = E(X ) n N (b) Let X = {X n } n= be a {F n }-submartingale.
More informationOn Using Shadow Prices in Portfolio optimization with Transaction Costs
On Using Shadow Prices in Portfolio optimization with Transaction Costs Johannes Muhle-Karbe Universität Wien Joint work with Jan Kallsen Universidad de Murcia 12.03.2010 Outline The Merton problem The
More informationLikelihood Estimation of Jump-Diffusions
Likelihood Estimation of Jump-Diffusions Extensions from Diffusions to Jump-Diffusions, Implementation with Automatic Differentiation, and Applications Berent Ånund Strømnes Lunde DEPARTMENT OF MATHEMATICS
More informationNumerical valuation for option pricing under jump-diffusion models by finite differences
Numerical valuation for option pricing under jump-diffusion models by finite differences YongHoon Kwon Younhee Lee Department of Mathematics Pohang University of Science and Technology June 23, 2010 Table
More informationExam Quantitative Finance (35V5A1)
Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D
More informationHomework Assignments
Homework Assignments Week 1 (p. 57) #4.1, 4., 4.3 Week (pp 58 6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15 19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9 31) #.,.6,.9 Week 4 (pp 36 37)
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationPolynomial processes in stochastic portofolio theory
Polynomial processes in stochastic portofolio theory Christa Cuchiero University of Vienna 9 th Bachelier World Congress July 15, 2016 Christa Cuchiero (University of Vienna) Polynomial processes in SPT
More informationEFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS
Commun. Korean Math. Soc. 23 (2008), No. 2, pp. 285 294 EFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS Kyoung-Sook Moon Reprinted from the Communications of the Korean Mathematical Society
More informationDynamic Replication of Non-Maturing Assets and Liabilities
Dynamic Replication of Non-Maturing Assets and Liabilities Michael Schürle Institute for Operations Research and Computational Finance, University of St. Gallen, Bodanstr. 6, CH-9000 St. Gallen, Switzerland
More informationOption pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard
Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications
More informationModeling the extremes of temperature time series. Debbie J. Dupuis Department of Decision Sciences HEC Montréal
Modeling the extremes of temperature time series Debbie J. Dupuis Department of Decision Sciences HEC Montréal Outline Fig. 1: S&P 500. Daily negative returns (losses), Realized Variance (RV) and Jump
More informationSaddlepoint Approximation Methods for Pricing. Financial Options on Discrete Realized Variance
Saddlepoint Approximation Methods for Pricing Financial Options on Discrete Realized Variance Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology Hong Kong * This is
More informationOptimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing
Optimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing Prof. Chuan-Ju Wang Department of Computer Science University of Taipei Joint work with Prof. Ming-Yang Kao March 28, 2014
More informationSemi-Markov model for market microstructure and HFT
Semi-Markov model for market microstructure and HFT LPMA, University Paris Diderot EXQIM 6th General AMaMeF and Banach Center Conference 10-15 June 2013 Joint work with Huyên PHAM LPMA, University Paris
More informationAnumericalalgorithm for general HJB equations : a jump-constrained BSDE approach
Anumericalalgorithm for general HJB equations : a jump-constrained BSDE approach Nicolas Langrené Univ. Paris Diderot - Sorbonne Paris Cité, LPMA, FiME Joint work with Idris Kharroubi (Paris Dauphine),
More informationDependence Structure and Extreme Comovements in International Equity and Bond Markets
Dependence Structure and Extreme Comovements in International Equity and Bond Markets René Garcia Edhec Business School, Université de Montréal, CIRANO and CIREQ Georges Tsafack Suffolk University Measuring
More informationVariable Annuities with Lifelong Guaranteed Withdrawal Benefits
Variable Annuities with Lifelong Guaranteed Withdrawal Benefits presented by Yue Kuen Kwok Department of Mathematics Hong Kong University of Science and Technology Hong Kong, China * This is a joint work
More informationDiscrete time interest rate models
slides for the course Interest rate theory, University of Ljubljana, 2012-13/I, part II József Gáll University of Debrecen, Faculty of Economics Nov. 2012 Jan. 2013, Ljubljana Introduction to discrete
More informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationJumps in Equilibrium Prices. and Market Microstructure Noise
Jumps in Equilibrium Prices and Market Microstructure Noise Suzanne S. Lee and Per A. Mykland Abstract Asset prices we observe in the financial markets combine two unobservable components: equilibrium
More informationHedging Under Jump Diffusions with Transaction Costs. Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo
Hedging Under Jump Diffusions with Transaction Costs Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo Computational Finance Workshop, Shanghai, July 4, 2008 Overview Overview Single factor
More informationSHORT-TIME IMPLIED VOLATILITY IN EXPONENTIAL LÉVY MODELS
SHORT-TIME IMPLIED VOLATILITY IN EXPONENTIAL LÉVY MODELS ERIK EKSTRÖM1 AND BING LU Abstract. We show that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential
More informationLecture 1: Lévy processes
Lecture 1: Lévy processes A. E. Kyprianou Department of Mathematical Sciences, University of Bath 1/ 22 Lévy processes 2/ 22 Lévy processes A process X = {X t : t 0} defined on a probability space (Ω,
More informationOptimal Hedging of Variance Derivatives. John Crosby. Centre for Economic and Financial Studies, Department of Economics, Glasgow University
Optimal Hedging of Variance Derivatives John Crosby Centre for Economic and Financial Studies, Department of Economics, Glasgow University Presentation at Baruch College, in New York, 16th November 2010
More informationEconomics 883: The Basic Diffusive Model, Jumps, Variance Measures, and Noise Corrections. George Tauchen. Economics 883FS Spring 2014
Economics 883: The Basic Diffusive Model, Jumps, Variance Measures, and Noise Corrections George Tauchen Economics 883FS Spring 2014 Main Points 1. The Continuous Time Model, Theory and Simulation 2. Observed
More information