MARTINGALES: VARIANCE. EY = 0 Var (Y ) = σ 2

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1 MARTINGALES: VARIANCE SIMPLEST CASE: X t = Y Y t IID SUM V t = X 2 t σ 2 t IS A MG: EY = 0 Var (Y ) = σ 2 V t+1 = (X t + Y t+1 ) 2 σ 2 (t + 1) = V t + 2X t Y t+1 + Y 2 t+1 σ 2 E(V t+1 F t ) = V t + 2X t E(Y t+1 F t ) = V t = 0 + E(Yt+1 2 F t ) σ 2 = 0 Autumn Per A. Mykland

2 QUADRATIC VARIATION (Q.V.) X t = Y Y t : MG (not iid) OBSERVED Q.V. : [X, X] t = Y Y 2 t V t = X 2 t [X, X] t : MG V t+1 = (X t + Y t+1 ) 2 [X, X] t+1 = X 2 t + 2X t Y t+1 + Y 2 t+1 ([X, X] t + Y 2 t+1) = V t + 2X t Y t+1 E(V t+1 F t ) = V t + 2X t E(Y t+1 F t ) = V t PREDICTABLE Q. V. : X, X t =Var (Y 1 F 0 ) Var (Y t F t 1 ) [X, X] t X, X t = t i=1 X 2 t X, X t = MG, TOO Y 2 i E(Y 2 i F i 1 ) = MG Autumn Per A. Mykland

3 CONTINUOUS MARTINGALES * TWO CONTINUITIES: TIME ITSELF: M t, 0 t T (or 0 t < ) PROCESS PATH: t M t = M t (ω) CONTINUOUS FUNCTION OF TIME * QUADRATIC VARIATION FOR CONTINUOUS MGs: M ti = M ti+1 M ti [M, M] t = lim SAME AS t i+1 t M 2 t i max t i 0 {}}{ 0 t 1 t 2 t 3 t M, M t = lim t i+1 t Var ( M ti F ti ) Autumn Per A. Mykland

4 PROOF THAT M, M t = [M, M] t : BEFORE CONVERGENCE: L t = Mt 2 i t i+1 t t i+1 t L IS A MARTINGALE, AND SO Var ( M ti F ti ) L ti = M 2 t i Var ( M ti F ti ) SO E[ L 2 t i ] = E[( M 2 t i Var ( M ti F ti )) 2 ] 2E[( M 4 t i + Var ( M ti F ti ) 2 ] = 2E[ M 4 t i + E( M 2 t i F ti ) 2 ] 2E[ M 4 t i + E( M 4 t i F ti )] = 4E[ M 4 t i ] Var (L ti ) = E[L, L] ti 4E[ Mt 4 i ] t i+1 t 4E max j M 2 t j t i+1 t = 4E [ max M 2 t i [M, M] t ] Mt 2 i 0 as max t i 0 BY CONTINUITY QED (CF. SHREVE II, THM (p. 102)) Autumn Per A. Mykland

5 CONTINUOUS MG HAVE INFINITE PATH. A MG TOTAL VARIATION: TV (M) t = lim M ti t i+1 <t QUADRATIC VARIATION: t i+1 <t M 2 t i }{{ } [M,M] t >0 max M t1 }{{} 0 (CONTINUITY) M ti t i+1 <t }{{ } T V (M) t HENCE TV (M) t IS INFINITE Autumn Per A. Mykland

6 THE THEORY IS ONLY APPROXIMATELY CORRECT dependence of estimated volatility on sampling frequency volatility of AA, Jan 4, 2001, annualized, sq. root scale sampling frequency Plot of realized volatility (quadratic variation of log price) for Alcoa Aluminum for January 4, The data is from the TAQ database. There are 2011 transactions on that day, on average one every seconds. The most frequently sampled volatility uses all the data, and this is denoted as frequency= 1. Frequency=2 corresponds to taking every second sampling point. Because this gives rise to two estimators of volatility, we have averaged the two. And so on for frequency= k up to 20. The plot corresponds to the average realized volatility. Volatilities are given on an annualized and square root scale. Autumn Per A. Mykland

7 SOLUTION TO PROBLEM log S ti = martingale + constant t + ǫ i t i = transaction time # i S ti = value of stock at time t i ǫ i : (almost) independent noise (very small) STOCK BEHAVES LIKE A MEASUREMENT OF UN- DERLYING SEMI-MARTINGALE ǫ i is too small for effective arbitrage Autumn Per A. Mykland

8 (ADDITIVE) BROWNIAN MOTION W t ORIGINAL DEFINITION: (1) W 0 = 0 (2) t W t IS CONTINUOUS (3) HAS INDEPENDENT INCREMENTS (4) W t+s W s N(0, t) ALTERNATIVE DEFINITION ( LEVY S THEOREM ): (i) W 0 = 0 (ii) W t is MG, CONTINUOUS (iii) [W, W] t = t PROOF THAT ORIG DEF => (iii): BEFORE CONVERGENCE: W, W ti = W, W ti+1 W, W ti =Var (W ti+1 W ti F ti ) = t i+1 t i AND SO W, W t t AS t i 0 FROM p. 3-4: [W, W] t = W, W t QED Autumn Per A. Mykland

9 MARKOV PROPERTY FOR t s, and W s = w E(f(W t ) F s ) = E(f(W t W s + W s ) F s ) = E(f(W t W s + w) F s ) = E(f(W t W s + w)) since W t W s independent of F s = E(f( t sz + w)) where Z is N(0, 1) = g(w) = g(w s ) HERE g(w) = f( t sz + w)φ(z)dz AS ALWAYS: WE CAN IN PARTICULAR TAKE TO GET (FOR SOME g): f(w) = I {W A} P(W t A F s ) = g(w s ) CONDITIONAL DISTRIBUTION OF W t GIVEN F s DEPENDS ONLY ON W s Autumn Per A. Mykland

10 FIRST PASSAGE TIMES m IS A BARRIER, m > 0 τ = inf{t 0 : W t = m} = FIRST TIME THAT W t HITS m τ IS A STOPPING TIME B-S MARTINGALE: S t = exp(σw t 1 2 σ2 t) W t τ m: ON THE SET {τ = }: S t τ exp(σm 1 2 σ2 (t τ)) 0 when t OPTIONAL STOPPING: 1 = E[ S t τ ] E[ S τ I {τ< } ] when t BECAUSE OF DOMINATED CONVERGENCE: W t τ m AND t τ 0 => S t τ exp(σm) Autumn Per A. Mykland

11 FIRST PASSAGE TIMES (cont d) 1 = E[ S τ I {τ< } ] = E[exp(σW τ 1 2 σ2 τ)i {τ< } ] = E[exp(σm 1 2 σ2 τ)i {τ< } ] LET σ 0: 1 = EI {τ< } = P(τ < ) SIMPLIFIED FORMULA: 1 = E[exp(σm 1 2 σ2 τ)] = exp(σm)e[exp( 1 2 σ2 τ)] OR: E[exp( 1 2 σ2 τ)] = exp( σm) SET α = σ 2 /2 E[exp( ατ)] = exp( m 2α) THIS IS THE LAPLACE TRANSFORM OF THE DISTRIBUTION OF τ Autumn Per A. Mykland

12 LAPLACE TRANSFORM: E[exp( ατ)] = exp( m 2α) BECAUSE: COROLLARY: E[τ] = d E[exp( ατ)] = E[τ exp( ατ)] E[τ] as α 0 dα d m E[exp( ατ)] = exp( m 2α) as α 0 dα 2α QED Autumn Per A. Mykland

13 REFLECTION PRINCIPLE AS BEFORE: τ = inf{t 0 : W t = m} = FIRST TIME THAT W t HITS m PRINCIPLE (for 0, w m): P(τ t and W t w) = P(W t 2m w) REASON FOR THIS FORMULA STRONG MARKOV PROPERTY: W t = W t+τ W τ, t 0, IS A BROWNIAN MOTION (PROOF: OPTIONAL STOPPING THEOREM; AL- TERNATIVE DEFINITION OF BROWNIAN MOTION) THEN BY SYMMETRY, ON {τ t}, FOR x 0: P(W t m x F τ ) = P(W t W τ x F τ ) = P(W t W τ x F τ ) = P(W t m x F τ ) Autumn Per A. Mykland

14 REWRITE AS INDICATOR FUNCTION: AND SO E(I {Wt m x} F τ ) = E(I {Wt m x} F τ ) E(I {Wt m x}i {τ t} ) = E[E(I {Wt m x}i {τ t} ) F τ )] (tower property) = E[E(I {Wt m x} F τ )]I {τ t} ) = E[E(I {Wt m x} F τ )]I {τ t} ) = E[E(I {Wt m x}i {τ t} ) F τ )] = E[E(I {Wt m x} F τ )] since{w t m x} {τ t} = E(I {Wt m x}) REWRITE AS PROBABILITIES: P(W t m x AND τ t) = P(W t m x) SUBSTITUTE w = m x TO OBTAIN PRINCIPLE (QED) Autumn Per A. Mykland

15 CONSEQUENCE: DISTRIBUTION OF τ P(τ t) = 2 exp{ y2 }dy for t 0 ( ) 2π m 2 t BECAUSE: SET w = m IN REFLECTION PRINCI- PLE, OBTAIN: P(τ t AND W t m) = P(W t m) ALSO, SINCE {W t m} {τ t}: P(τ t AND W t m) = P(W t m) ADD THESE TWO: P(τ t) = P(τ t AND W t m) + P(τ t AND W t m) = P(W t m) + P(W t m) = 2P(W t m) THIS SHOWS (*) DENSITY: f τ (t) = d m dtp(τ t) = t 2πt exp{ m2 2t } Autumn Per A. Mykland

16 DISTRIBUTION OF MAXIMUM OF BROWNIAN MOTION M t = max 0 s t W s USE M t m IF AND ONLY IF τ t FROM PREVIOUS PAGE: P(M t m) = P(τ t) = 2P(W t m) = P( W t m) THEREFORE: M t HAS SAME DISTRIBUTION AS W t Autumn Per A. Mykland

17 USE JOINT DISTRIBUTION BROWNIAN MOTION AND ITS MAXIMUM M t = max 0 s t W s M t m IF AND ONLY IF τ t REFLECTION PRINCIPLE BECOMES (0, w m): P(M t m and W t w) = P(τ t and W t w) = P(W t 2m w) REWRITE WITH DENSITIES m w f M(t),W(t) (x, y)dxdy = 1 2πt 2m w exp{ z2 2t }dz / m: w f M(t),W(t) (m, y) = 2 (2m w)2 exp{ } 2πt 2t / w: 2(2m w) f M(t),W(t) (m, w) = t 2πt exp{ (2m w)2 2t } Autumn Per A. Mykland

18 FINAL FORM OF DENSITY f M(t),W(t) (m, w) = 2(2m w) t 2πt exp{ (2m w)2 2t } for 0, w m (otherwise f M(t),W(t) (m, w) = 0) CONDITIONAL DISTRIBUTION f M(t) W(t) (m w) = f M(t),W(t)(m, w) f W(t) (w) = = 2(2m w) t 2πt exp{ (2m w)2 2t } 1 2πt exp{ w2 2t } 2(2m w) t exp{ 2m(m w) } 2t Autumn Per A. Mykland

19 STOPPED σ-fields WHAT MEANING CAN WE GIVE TO F τ? (F t ) IS FILTRATION τ IS (F t )-STOPPING TIME DEFINE: F τ = {A F : A {τ t} F t for all t} F τ : THE SETS A WHICH CAN BE DETERMINED ONCE τ HAS OCCURRED, OR THE INFORMATION AVAILABLE AT TIME τ F τ IS A σ-field EXTENDED OPTIONAL STOPPING THEOREM: SUPPOSE τ 1 and τ 2 ARE TWO STOPPING TIMES, τ 1, τ 2 0, τ 1 T. SUPPOSE (M t ) IS A MARTIN- GALE. THEN E(M τ1 F τ2 ) = M τ1 τ 2 Autumn Per A. Mykland