Volatility Index: VIX vs. GVIX

Transcription

1 I. II. III. IV. Volatility Index: VIX vs. GVIX "Does VIX Truly Measure Return Volatility?" by Victor Chow, Wanjun Jiang, and Jingrui Li (214) An Ex-ante (forward-looking) approach based on Market Price of Options; NOT an Ex-post (backward-looking) Statistical Estimation. NOT just Indexes, BUT Tradable Financial Instruments (See CBOE sites). Negatively Correlated with Underlying Index s Returns and thus Provide a good Market Risk Hedging Vehicle for Portfolio Management. The Difference between GVIX and VIX indexes (GV-Spread) provides a good Forward-Looking Indicator about "Market Sentiment" 1

2 Outlines 1. The Definition of Volatility 2. The Assumption of Symmetric Return Distribution 3. Geometric Brownian Motion: Foundation of the VIX 4. Core Derivation of VIX 5. Holding-Period Return, Log-Return, and Option Prices 6. Formulation of VIX 7. VIX is NOT a Volatility Index in general 8. GVIX is the True Volatility of Log-Returns 9. The GV-Spread (Empirical Evidence) 1. Correlation Matrix of GV-Spread and Distribution Moments 11. GV-Spread is Mean Reverting 2

3 Definition of Volatility Volatility(σ) µ μ = E(x); σ = E(x μ) 2 = E(x 2 ) [μ] 2 3

4 IS NOT BASED ON THE VOLATILITY DEFINITION. CBOE VIX Formulation BUT BASED ON THE ASSUMPTION OF SYMMETRIC RETURN DISTRIBUTION 4

5 The Assumption of Symmetric Return Distribution BELL CURVE STANDARIZED BELL CURVE Two- Moment Distribution x = μ x + σ x z z = x μ x σ x y = μ y + σ y z x ~ N(μ x, σ x ); y ~ N(μ y, σ y ) z = y μ y σ y z ~ N(,1) 5

6 Geometric Brownian Motion: Foundation of the VIX Diffusion Process ds t S t = μdt + σdz t d[ln(s t )] = f (S t )ds t f"(s t)s t 2 σ 2 dt = 1 S t (μs t dt + σs t dz t ) 1 2 σ2 dt 1. Z = 2. Z t is almost surely everywhere continuous 3. Z t has independent increments with (Z t Z s ) ~ N(, t s) (for s < t) Taylor Expansion (Stop@2 nd Order) & Ito Calculus Log returns follow a symmetric distribution = (μ 1 2 σ2 ) dt + σdz t d[ln(s t )] = (μ 1 2 σ2 ) dt + σdz t A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. 6

7 put-call symmetry "Classic put-call symmetry (Bowie and Carr 1994; Bates 1997) relates the prices of puts and calls at strikes that are unequal but equidistant logarithmically to the forward price. For example, it implies that if a forward price M follows geometric Brownian motion under an appropriate pricing measure, and M = 1, then a 2-strike call on M has time- price equal to two times the price of the 5-strike put at the same expiry. " (see Peter Carr and Roger Lee, Put- Call Symmetry: Extension and Applications, Mathematical Finance, Vol. 19, No. 4 (October 29), ) 7

8 Core Derivation of VIX 1. Given that ds t S t = μdt + σdz t, and Solve for the volatility Sum to T-period of time { d[ln(s t )] = (μ 1 2 σ2 ) dt + σdz t σ 2 dt = 2 { ds t S t d[ln(s t )]} σ 2 = 1 T T σ2 dt = 2 T [ ds t T S t ln ( S T )] 4. Volatility Index (VIX) VIX 2 = E(σ 2 ) = σ 2 = 2 T [E ( ds t T S t ) E [ln ( S T )]] VIX in fact captures the difference between the expected holding-period return and the expected log-return over a T-period of time (e.g. 3-day) 8

9 Holding-Period Return, Log-Return, and Option Prices 1. Holding-Period Return R T = S T 2. Log-Return r T = [ln( S T ) ln( )] = ln ( S T ) Taylor Expansion with remainder. The difference between the two returns ln( S T ) = ln( ) + S T + 1 K 2 (S T K) + dk + 1 K 2 (K S T) + dk R T r T = [ 1 K 2 (S T K) + dk + 1 K 2 (K S T) + dk ] 5. The expected difference E(R T ) E(r T ) = e rt { 1 K 2 C T(K)dK + 1 K 2 P T(K)dK} 6. Expected Log-Return E [ln ( S T )] = E(R S T ) + e rt { 1 K 2 C T (K)dK + 1 K 2 P T (K)dK } 9

10 Recall Under No-arbitrage Volatility Index Taylor Expansion nd Order) CBOE VIX Formulation Formulation of VIX T VIX 2 = E(σ 2 ) = σ 2 = 2 T [ E (ds t S t ) E [ln ( S T )]] [ln ( S T )] = E(R S T ) + e rt { 1 K 2 C T (K)dK + 1 K 2 P T (K)dK } T E ( ds t ) = rt, and E(R T ) = e rt 1 S t VIX 2 = 2 T {rt (ert 1) + e rt [ 1 K 2 C T(K)dK + 1 K 2 P T(K)dK]} = 2 T {rt (F 1) ln ( K ) + e rt [ 1 K K K 2 C T (K)dK + 1 K K 2 P T (K)dK ]} [rt ( F 1) ln ( K )] = [ln ( F ) ( F 1)] 1 2 K K K 2 (F 1) K VIX 2 = 2erT T [ 1 K K 2 C T(K)dK + 1 K K 2 P T(K)dK] 1 2 T (F 1) K VIX 2 = 2erT T 1 Q(K 2 i ) K i 1 2 K i i T (F 1) K 1

11 VIX is NOT a Volatility Index in general Key Component of VIX is e rt [ 1 K 2 C T(K)dK + 1 K 2 P T(K)dK] It is the expected return difference. Taylor Expansion (Stop@ N th Order) The expected return difference is Key Component of VIX is E(R T ) E(r T ) = e rt { 1 K 2 C T(K)dK + 1 K 2 P T(K)dK} (1 + R T ) = S T = exp [ln ( S T )] = 1 + N κ=1 1 κ κ! [ln (S T )] + o [ln ( S N T )] E(R T ) E(r T ) = 1 2 E(r T 2 ) E(r T 3 ) E(r T 4 ) + o[e(r 4 T )] 1 2 E(r T 2 ) E(r T 3 ) E(r T 4 ) + o[e(r 4 T )] Let V T = E(r T 2 ), W T = E(r T 3 ), and X T = E(r T 4 ) VIX is a Moment-Combination VIX = 1 T [V T + W T 3 + X T 12 + o(x T)] 2[(e rt 1) rt] 11

12 GVIX is the True Volatility of Log-Returns Definition of Variance μ = E(x), σ 2 = E(x μ) 2 V = E(x 2 ) = E(x 2 ) [E(x)] 2 = V μ 2 GVIX = 1 T V T (μ T ) 2 The Generalized Volatility Index (GVIX) μ T = ln ( K ) + ( F 1) e rt [ 1 K K 2 C T(K)dK + 1 K K 2 P T(K)dK] V T = ln 2 ( K ) + 2ln ( K ) ( F K 1) K [1 ln ( K + 2e rt S )] [ K 2 C T (K)dK + K K S [1 + ln ( K )] K 2 P T (K)dK] 12

13 GV-Spread (Empirical Evidence) Volatility Indexes: VIX vs. GVIX K. Victor Chow, PhD, CFA The Spread VIX 2 GVIX 2 1 T (μ T 2 + W T 3 + X T 12 ) GVIX > VIX indicates that W T < W T = ln 3 ( K ) + 3ln 2 ( K ) ( F [2ln ( K 1) + 3e rt S ) ln 2 ( K [ S )] K K 2 C T (K)dK [2ln ( ) + ln 2 ( S K )] K K K K 2 P T (K)dK] 13

14 VIX 2 GVIX 2 1 T (μ T 2 + W T 3 + X T 12 ) A. GVIX a VIX a (Spread) B. 1 3 Ŵ a 14

15 Correlation Matrix of GV-Spread and Distribution Moments V a Ŵ a X a daily (annualized) ex-ante second moment (Volatility) daily (annualized) ex-ante third moment (Skewness) daily (annualized) ex-ante fourth moment (Kurtosis) 15

16 GV-Spread is Mean-Reverting Volatility Indexes: VIX vs. GVIX K. Victor Chow, PhD, CFA The $Spread Chart (214) Zivot-Andrews Unit-Root Test 16