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Lecture III: Review of Classic Quadratic Variation Results and Relevance to Statistical Inference in Finance

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Lecture III: Review of Classic Quadratic Variation Results and Relevance to Statistical Inference in Finance Lecture III: Review of Classic Quadratic Variation Results and Relevance to Statistical Inference in Finance Christopher P. Calderon Rice University / Numerica Corporation Research Scientist Chris Calderon, PASI, Lecture 3 Outline I Refresher on Some Unique Properties of Brownian Motion II Stochastic Integration and Quadratic Variation of SDEs III Demonstration of How Results Help Understand ”Realized Variation” Discretization Error and Idea Behind “Two Scale Realized Volatility Estimator” Chris Calderon, PASI, Lecture 3 Part I Refresher on Some Unique Properties of Brownian Motion Chris Calderon, PASI, Lecture 3 Outline For Items I & II Draw Heavily from: Protter, P. (2004) Stochastic Integration and Differential Equations, Springer-Verlag, Berlin. Steele, J.M. (2001) Stochastic Calculus and Financial Applications, Springer, New York. For Item III Highlight Results from: Barndorff-Nielsen, O. & Shepard, N. (2003) Bernoulli 9 243-265. Zhang, L.. Mykland, P. & Ait-Sahalia,Y. (2005) JASA 100 1394-1411. Chris Calderon, PASI, Lecture 3 Assuming Some Basic Familiarity with Brownian Motion (B.M.) • Stationary Independent Increments • Increments are Gaussian Bt Bs (0,t s) • Paths are Continuous but “Rough”− / “Jittery”∼ N − (Not Classically Differentiable) • Paths are of Infinite Variation so “Funny” Integrals Used in Stochastic Integration, e.g. t 1 2 BsdBs = 2 (Bt t) 0 − R Chris Calderon, PASI, Lecture 3 Assuming Some Basic Familiarity with Brownian Motion (B.M.) The Material in Parts I and II are “Classic” Fundamental & Established Results but Set the Stage to Understand Basics in Part III. The References Listed at the Beginning Provide a Detailed Mathematical Account of the Material I Summarize Briefly Here. Chris Calderon, PASI, Lecture 3 A Classic Result Worth Reflecting On N 2 lim (Bti Bti 1 ) = T N − →∞ i=1 − T Implies Something Not t = i Intuitive to Many Unfamiliar i N with Brownian Motion … Chris Calderon, PASI, Lecture 3 Discretely Sampled Brownian Motion Paths 2 1.8 Fix Terminal Time, “T” and Refine Mesh by 1.6 Increasing “N”. 1.4 t Z Shift IC to 1.2 Zero if “Standard 1 Brownian Motion” Desired 0.8 0 1 t Chris Calderon, PASI, Lecture 3 Discretely Sampled Brownian Motion Paths 2 Fix Terminal Time, “T” δ T 1.8 N and Refine Mesh by ≡ T Increasing “N”. ti i N 1.6 ≡ Suppose You are Interested in Limit of: 1.4 t k(t) Z 2 1.2 (Bti Bti 1 ) − 1 i=1 − k(t)=max i : ti+1 t 0.8 { ≤ } 0 1 t Chris Calderon, PASI, Lecture 3 Different Paths, Same Limit k(t) 2 lim (Bti Bti 1 ) = t 2 N − − →∞ i=1 1.8 [ProofP Sketched on Board] 1.6 B (ω ) 1.4 t 1 t Z 1.2 θˆ1 1 Bt(ω2) 0.8 0 1 t Chris Calderon, PASI, Lecture 3 Different Paths, Same Limit k(t) 2 lim (Bti Bti 1 ) = t 2 N − − →∞ i=1 1.8 [ProofP Sketched on Board] 1.6 B (ω ) 1.4 t 1 Compare To t Z Central Limit 1.2 Theorem for “Nice” θˆ1 1 iid Random Bt(ω2) Variables. 0.8 0 1 t Chris Calderon, PASI, Lecture 3 Different Paths, Same Limit N N 2 lim (Bti Bti 1 ) lim Yi = T N 2 − − ≡ N →∞ i=1 →∞ i=1 1.8 P Compare/Contrast to ProperlyP Centered and 1.6 Scaled Sum of “Nice” iid Random Variables: N B (ω ) 1.4 t 1 (θ(ωi) μ) t Z i=1 − SN 1.2 ≡ P √N θˆ1 1 Bt(ω2) 2 SN L (0, σ ) 0.8 → N 0 1 t Chris Calderon, PASI, Lecture 3 Continuity in Time N N 2 lim (Bti Bti 1 ) lim Yi = T N 2 − − ≡ N →∞ i=1 →∞ i=1 1.8 P Above is “Degenerate” but NoteP Time Scaling Used in Our Previous Example Makes Infinite 1.6 Sum Along a Path Seem Similar to 1.4 Computing Variance of Independent RV’s t Z N 1.2 (θ(ωi) μ) ˆ i=1 − θ1 SN 1 ≡ P √N 0.8 2 SN L (0, σ ) 0 1 → N t Chris Calderon, PASI, Lecture 3 B.M. Paths Do Not Have Finite Variation Let 0 <t T ≤ πn = 0 t1 ...ti ... tn = t { ≤ ≤ ≤ ≤ } lim max(ti+1 ti)=0 n i<n →∞ − All Finite Partitionsof [0,t] P ≡ Chris Calderon, PASI, Lecture 3 B.M. Paths Do Not Have Finite Variation [0,t](ω)=sup Bti+1 Bti V π ti π | − | ∈P ∈ Suppose That: ( [0,t] < ) > 0 Then: N P V ∞ 2 t =lim (Bti Bti 1 ) N − − N →∞ i=1 lim Psup Bti Bti 1 Bti Bti 1 − − ≤ N ti πN | − | i=1 | − | →∞ ∈ Chris Calderon, PASI, Lecture 3 P B.M. Paths Do Not Have Finite Variation N 2 t =lim (Bti Bti 1 ) N i=1 − − →∞ N P lim sup Bti Bti 1 Bti Bti 1 − − ≤ N ti πN | − | i=1 | − | →∞ ∈ P t > 0, so lim sup Bti Bti 1 [0,t] − ≤ N ti πN | − |V Probability of →∞ ∈ Tends to Zero Due to Finite Variation =0 a.s. uniform continuity of Must be Zero B.M. Paths Chris Calderon, PASI, Lecture 3 Part II Stochastic Integration and Quadratic Variation of SDEs Chris Calderon, PASI, Lecture 3 Stochastic Integration ( [0,t] < )=0 P tV ∞ t X(ω,t)= b(ω,s)ds + σ(ω,s)dBs 0 0 “Finite Variation Term” “Infinite Variation Term” Chris Calderon, PASI, Lecture 3 Stochastic Integration ( [0,t] < )=0 P tV ∞ t X(ω,t)= b(ω,s)ds + σ(ω,s)dBs 0 0 “Infinite Variation Term” Complicates Interpretting Limit as Lebesgue Integral σ Xt (Bt ) Bt Bt i i | i+1 − i | Chris Calderon, PASI, Lecture 3 Stochastic Differential Equations (SDEs) t t X(ω,t)= b(ω,s)ds + σ(ω,s)dBs 0 0 Written in Abbreviated Form: dX(ω,t)=b(ω,t)dt + σ(ω,t)dBt Chris Calderon, PASI, Lecture 3 Stochastic Differential Equations Driven by B.M. dX(ω,t)=b(ω,t)dt + σ(ω,t)dBt Adapted and Measurable Processes dXt = btdt + σtdBt I will use this shorthand for the above (which is itself shorthand for stochastic integrals…) Chris Calderon, PASI, Lecture 3 Stochastic Differential Equations Driven by B.M. dXt = btdt + σtdBt General Ito Process dXt = b(Xt,t)dt + σ(Xt,t)dBt Diffusion Process dXt =0dt +1dBt Repackaging Brownian Motion Chris Calderon, PASI, Lecture 3 Quadratic Variation for General SDEs (Jump Processes Readily Handled) N 2 RVπN (t) (Xti Xti 1 ) ≡ i=1 − − “RealizedP Variation” Quadratic Variation: Defined if the limit taken over any partition exists with the mesh going to zero. := Vt { } Chris Calderon, PASI, Lecture 3 Stochastic Integration t t Y (ω,t)= b0(ω,s)ds + σ0(ω,s)dXs 0 0 “Finite Variation Term” “Infinite Variation Term” Quadratic Variation of “X” Can Be Used to Define Other SDEs (A “Good Stochastic Integrator”) Chris Calderon, PASI, Lecture 3 Quadratic Variation Comes Entirely From Stochastic Integral t t X(ω,t)= b(ω,s)ds + σ(ω,s)dBs 0 0 i.e. the drift (or term in front of “ds” makes) no contribution to the quadratic variation [quick sketch of proof on board] Chris Calderon, PASI, Lecture 3 Stochastic Integration ( [0,t] < )=0 P tV ∞ t X(ω,t)= b(ω,s)ds + σ(ω,s)dBs 0 0 “Infinite Variation Term” Can Now Be Dealt With If One Uses Q.V. σ Xt (Bt ) Bt Bt i i | i+1 − i | Chris Calderon, PASI, Lecture 3 Quadratic Variation and “Volatility” t t Xt = bsds + σsdBs 0 0 Typical Notation for QV t 2 [X, X]t Vt = σ dt ≡ s 0 Chris Calderon, PASI, Lecture 3 Quadratic Variation and “Volatility” For Diffusions t 2 [X, X]t Vt = σ dt ≡ s 0 More Generally for Semimartingales t (Includes Jump Processes) 2 [X, X]t := X 2 X dXs t − s− 0 Chris Calderon, PASI, Lecture 3 Part III Demonstration of How Results Help Understand ”Realized Variation” Discretization Error and Idea Behind “Two Scale Realized Volatility Estimator” Chris Calderon, PASI, Lecture 3 Most People and Institutions Don’t Sample Functions (Discrete Observations) N 2 RVπN (t) (Xti Xti 1 ) ≡ i=1 − − Assuming Process is Pa Genuine Diffusion, How Far is Finite N Approximation From Limit? [X, X]t Chris Calderon, PASI, Lecture 3 Discretization Errors and Their Limit Distributions t 1/2 L 4 N RVπN (t) [X, X]t σt (0,C σs ds) − | → N 0 ¡ ¢ R Paper Below Derived Explicit Expressions for Variance for Fairly General SDEs Driven by B.M. Barndorff-Nielsen, O. & Shepard, N. (2003) Bernoulli 9 243-265. Chris Calderon, PASI, Lecture 3 Discretization Errors and Their Limit Distributions t 1/2 L 4 N RVπN (t) [X, X]t σt (0,C σs ds) − | → N 0 ¡ ¢ R Their paper goes over some nice classic moment generating function results for wide class of spot volatility processes (and proves things beyond QV) Barndorff-Nielsen, O. & Shepard, N. (2003) Bernoulli 9 243-265. Chris Calderon, PASI, Lecture 3 Discretization Errors and Their Limit Distributions δ t One Condition That They Assume: ≡ N N 1/2 2 2 lim δ σt +δ σt =0 N 0 i i → i=1 | − | Barndorff-Nielsen, O. & Shepard, N. (2003) Bernoulli 9 243-265. Chris Calderon, PASI, Lecture 3 Discretization Errors and Their Limit Distributions t δ N Good for Many Stochastic Volatility Models … BUT ≡ Some Exceptions are Easy to Find (See Board) N 1/2 2 2 lim δ σt +δ σt =0 N 0 i i → i=1 | − | Barndorff-Nielsen, O. & Shepard, N. (2003) Bernoulli 9 243-265. Chris Calderon, PASI, Lecture 3 What Happens When One Has to Deal with “Imperfect” Data? Suppose “Price” Process Is Not Directly Observed But Instead One Has: yti = Xt1 + ²i Zhang, L.
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