MARTINGALES: VARIANCE SIMPLEST CASE: X t = Y 1 +... + 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 2008 1 Per A. Mykland
QUADRATIC VARIATION (Q.V.) X t = Y 1 +... + Y t : MG (not iid) OBSERVED Q.V. : [X, X] t = Y 2 1 +... + 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 2008 2 Per A. Mykland
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 2008 3 Per A. Mykland
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 3.4.3 (p. 102)) Autumn 2008 4 Per A. Mykland
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 2008 5 Per A. Mykland
THE THEORY IS ONLY APPROXIMATELY CORRECT dependence of estimated volatility on sampling frequency volatility of AA, Jan 4, 2001, annualized, sq. root scale 0.6 0.8 1.0 1.2 5 10 15 20 sampling frequency Plot of realized volatility (quadratic variation of log price) for Alcoa Aluminum for January 4, 2001. The data is from the TAQ database. There are 2011 transactions on that day, on average one every 13.365 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 2008 6 Per A. Mykland
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 2008 7 Per A. Mykland
(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 2008 8 Per A. Mykland
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 2008 9 Per A. Mykland
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 2008 10 Per A. Mykland
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 2008 11 Per A. Mykland
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 2008 12 Per A. Mykland
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 2008 13 Per A. Mykland
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 2008 14 Per A. Mykland
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 2008 15 Per A. Mykland
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 2008 16 Per A. Mykland
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 2008 17 Per A. Mykland
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 2008 18 Per A. Mykland
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 2008 19 Per A. Mykland