CEV Implied Volatility by VIX Implied Volatility Chien-Hung Chang Dept. of Financial and Computation Mathematics, Providence University, Tiachng, Taiwan May, 21, 2015 Chang (Institute) Implied volatility by VIX May, 21, 2015 1 / 37
Agenda Why volatility? How to extract volatility? How do investors feel VIX? VIX from asset models? Chang (Institute) Implied volatility by VIX May, 21, 2015 2 / 37
Diversification to reduce individual risk Chang (Institute) Implied volatility by VIX May, 21, 2015 3 / 37
Question: What can we do about systematic risk? Answer: Conventially, bonds and gold are used to balance equity portfolio. Chang (Institute) Implied volatility by VIX May, 21, 2015 4 / 37
S&P 500 and its dailly returns Chang (Institute) Implied volatility by VIX May, 21, 2015 5 / 37
Volatility as an asset VOlatility may be a good "asset" to includ in portfolio. Question: Is it possible to trade volatility? Answer: 1 Spreads are used from the beginning of options market. 2 Variance swaps is the first volatility product to trade volatility and actively traded on OTC market. Chang (Institute) Implied volatility by VIX May, 21, 2015 6 / 37
Options for "regular" trader (not volatility trader) European call: the right (no obligation) to buy an asset at preset future date by pre-determined price speculation for up-move protection for up-side price European put: the right to sell at pre-set future date by pre-determined price speculation for down-turn protection for down-side price (liquidity risk) Is call or put more sensitive to fearness? Chang (Institute) Implied volatility by VIX May, 21, 2015 7 / 37
How does asset move for risk-neutral investors? Black-Scholes-Merton assume the price is lognormally distributed and its risk-neutral version for futures price is, with σ > 0, standard Brownain motion W t, (discrete form) F t+ t F t = σf t φ (0, t), φ (a, b) = standard normal r.v. with mean a and varian (continuous form) df t = σf t dw t or d ln F t = σ2 2 dt + σdw t Chang (Institute) Implied volatility by VIX May, 21, 2015 8 / 37
BSM option pricing formula European call with maturity date T and strike price K has pay-off max{f T K, 0} and fair value c 0 = e rt (F 0 N (d 1 ) KN (d 2 )) European put with maturity date T and strike price K has pay-off max{k F T, 0} and fair value p 0 = e rt (KN ( d 2 ) F 0 N ( d 1 )) where r the risk-free rate, N ( ) the cumulate normal distribution and d 1 = ln ( ) F0 K + σ2 2 T σ T, d 2 = d 1 σ T Chang (Institute) Implied volatility by VIX May, 21, 2015 9 / 37
Implied volatility of BSM Excepting σ (volatility), parameters, F 0, K, T, r are easier to observe. To fit BSM model for the market price of options, we treat c (σ) as a function of σ and solve σ imp for market price c M as or c (σ imp ) = c M p (σ imp ) = p M Both equations will have "same" solution σ imp. from put-call parity. Chang (Institute) Implied volatility by VIX May, 21, 2015 10 / 37
Implied volatility Question: Can we directly "see" the implied volatility from market? Answer: No, we solve the BSM formula (moel dependent) by numerical method for a particular option price. Solving implied BSM volatility (numerically) is a basic and must-have function for financial toolbox programming. In real market, the implied volatility is not the same for different strike price and shows stochastic property. Chang (Institute) Implied volatility by VIX May, 21, 2015 11 / 37
Log-contracts Question: Can we directly see the implied volatility from market? Answer: Yes, if we have log-contracts on market (not by mathematics and mofel-free) Reason: E Q [ln (F T /F 0 )] = σ2 2 T or σ 2 = 2 T Question: Do we need a market for log-contract? price (log contract) Chang (Institute) Implied volatility by VIX May, 21, 2015 12 / 37
Static replication by OTM calls and puts For an European claim with payoff g(f T ), the value, V (g (F T )) can be represented as V (g (F T )) = e rt g (F 0 ) + g F0 (K ) C (K ) dk + g (K ) P (K ) dk. F 0 0 For g (F T ) = ln (F T ), we have e rt E Q [ln (F T /F 0 )] = F 0 c (K ) /K 2 dk + F0 0 p (K ) /K 2 dk. thus σ 2 = 2e rt T [ c (K ) /K 2 dk + F0 p (K ) /K 2 dk ] F 0 0 Chang (Institute) Implied volatility by VIX May, 21, 2015 13 / 37
Construction of implied volatility Question: Can we directly see the implied volatility from market? Answer: Yes, we use all OTM calls and puts without knowing the behavior of volatility, apriorially. Reason: σ 2 = 2 T [ F0 c (K ) /K 2 dk + p (K ) /K 2 dk ] F 0 0 2 T K i K i >F 0 Ki 2 e rt Call (K i ) + 2 T K i K i F 0 Ki 2 e rt Put (K i ) Chang (Institute) Implied volatility by VIX May, 21, 2015 14 / 37
VIX of CBOE The VIX 2 of CBOE estimates annualized variance of S&P 500 by OTM calls and puts VIX 2 = 100 2 { 2 T K i K i K 0 Ki 2 e rt Call (K i ) + 2 T K i K i K 0 Ki 2 e rt Put (K i ) 1 T ( F0 K 0 1 ) 2 }, where K 0 is the first strike below forward level F 0. The VIX VIX 2 The options always choose expiring around 30 days from calculating time. That meas VIX 2 measuring the next 30 days annualized variance and VIX tells the volatility of the next 30 days. The calculation of VIX is model-free. Chang (Institute) Implied volatility by VIX May, 21, 2015 15 / 37
S&P 500 price Chang (Institute) Implied volatility by VIX May, 21, 2015 16 / 37
Daily returns of S&P 500 Chang (Institute) Implied volatility by VIX May, 21, 2015 17 / 37
VIX chart VIX seems a good index to measure market fluctuation. Chang (Institute) Implied volatility by VIX May, 21, 2015 18 / 37
The Hathaway Effect: How Anne Gives Warren Buffett a Rise Chang (Institute) Implied volatility by VIX May, 21, 2015 19 / 37
Investor s feeling about VIX (Google Trends) Chang (Institute) Implied volatility by VIX May, 21, 2015 20 / 37
The beta of VIX Beta of VIX is negative CAPM: E (r VIX ) = r f + β(e (r SPX ) r f ) CAPM implies small or even negative returns of VIX Empirically, long position on VIX futures losses money on most of time. Why we focus on an "asset" with negative returns? Chang (Institute) Implied volatility by VIX May, 21, 2015 21 / 37
Optimal portfolio of two risky assets Chang (Institute) Implied volatility by VIX May, 21, 2015 22 / 37
Derivatives on VIX VIX futures, VIX options are treaded in CBOE ETNs on VIX by financial instituion Chang (Institute) Implied volatility by VIX May, 21, 2015 23 / 37
Example of strategies including VIX ETN (illustraion purpose only) Harry Long (Oct. 2014, inventor of Structural Arbitrage and Hedged Convexity Capture and is the Managing Partner of ZOMMA, seeking alpha): 1. Buy the Direxion Daily S&P 500 Bull 3X Shares ETF (NYSEARCA:SPXL) with 50% of the dollar value of the portfolio. 2. Buy the Direxion Daily 30-Year Treasury Bull 3x Shares ETF (NYSEARCA:TMF) with 40% of the dollar value of the portfolio. 3. Buy the VelocityShares Daily 2x VIX Short-Term ETN (NASDAQ:TVIX) with 10% of the dollar value of the portfolio. 4. Rebalance annually to maintain the 50%/40%/10% dollar value split between the positions. That Beats The S&P 500 Every Year from: http://seekingalpha.com/article/2616495-a-weird-all-long-strategythat-beats-the-s-and-p-500-every-year-ii Chang (Institute) Implied volatility by VIX May, 21, 2015 24 / 37
How does VIX evolve? We need a stochastic volatiilty model on underlying that is consistency on SPX, SPX options and VIX or even VIX derivatives. Chang (Institute) Implied volatility by VIX May, 21, 2015 25 / 37
Stochastic Volatility Models (SVM) Under risk-neutral world, the joint dynamic of underlying price and variance process is df t = F t Vt dw t, dv t = [q (t) Q (V t ) + s (t) S (V t )]dt + GdB t, where Wt, B t are two standard Brownian motions with correlation ρ, The drift term of V t, q (t) Q (V t ) + s (t) S (V t ), is prefer to obtain mean-verting property on V t. Two types are considered in literatures for function G : 1. local Volatility Model( ρ = 1): G = G (t, F t ). 2. Stochastic Volatility Model ( ρ < 1): G = G (t, V t ) = V γ t, γ > 0 Chang (Institute) Implied volatility by VIX May, 21, 2015 26 / 37
Popular SVMs 1. γ = 1 2, dv t = k (θ V t ) dt + ε V t db t (Heston (1993)) 2. γ = 1, dv t = k (θ V t ) dt + εv t db t (Lewis (2000)) 3. γ = 3 2, dv t = ( kv t + svt 2 (2007)) ) dt + εv 3 2 t db t (Lewis (2000),Carr and Sun Chang (Institute) Implied volatility by VIX May, 21, 2015 27 / 37
CEV exponents Carr and Sun (2007) surveys the CEV exponents with empirical support for groups of statical and risk-neutral process. Using affi ne drift Ishida and Engle (2002) have γ = 1.71 for S&P 500 daily returns measured over 30-year period. Javaheri (2004) estimates S&P 500 daily returns, but with constrainted CEV power of 0.5, 1, or 1.5 He favors power of 1.5. Chacko and Viceira (1999) estimate the CEV exponent 1.1 using weekly data over 35- year period and 1.65 using monthly data over 71-year period. Poteshman (1998) examine S&P 500 index options over 7-year period and fins that both statistical and risk-neutral drift of the instantaneous variance process are not affi ne and the volatility of variance is an increasing convex function of instantaneous variance. Jones (2003) examines S&P 100 returns and implied volatility over a 14-year period. The CEV power under affi ne drift is 1.33 for better fitting on 3- and 6-month options. Chang (Institute) Implied volatility by VIX May, 21, 2015 28 / 37
CEV exponents Ait-Sahalia and Kimmel (2007) suggest the CEV exponent lying between the Heston value of 1/2 and the GARCH value of 1 for dailly data from January 2, 1990 to September 30, 2003. The estimation of CEV exponent for (SPX, VIX ) is 0.6545 and 0.94 for VIX. (table 8.) Gatheral (2008) agrees the CEV power γ = 0.94 on VIX options data. Chang (Institute) Implied volatility by VIX May, 21, 2015 29 / 37
3/2-model 1 Numerous works have been done on Heston models. 2 For 3/2-model. Carr and Sun (2007) obtains closed form solution for Fourier-Laplace transform of asset and quadratic variation by Kummer confluent hypergeometric functions. 3 Chan and Platen (2011) consider long dated variance swap on 3/2-model. 4 Baldeaux and Badran (2012) study consistent modeling on VIX and equity derivatives for 3/2 plus jump model. 5 Itkin (2012) gives solvable stochastic volatility models for pricing volatility derivatives. ( highly related to our work) Chang (Institute) Implied volatility by VIX May, 21, 2015 30 / 37
CEV state variable We rewrite the variance process, V t,to obtain an equivalent model as above into a form of CEV process. Under risk-neutral probability, we assume the joint dynamic of futures price and volatility driving process is df t = v β 2 F t dw t, t dv t = k (θ v t ) dt + ε v t db t, where γ = 0 and σ > 0 are constants, k, θ, and ε are constants with 2k θ > 1 to ensure the positivity of v ε 2 t if v 0 > 0, W t, B t are two standard Brownian motions with correlation ρ. There two categories divideded by ρ = 1 and ρ = 1. If ρ = 1, the random source of asset and variance are the same Brownian motion and the sochastic volatility model becomes constant elasticity of variance (CEV) model after proper choosing the parameters k, θ, and, ε. Chang (Institute) Implied volatility by VIX May, 21, 2015 31 / 37
non-linear drifted CEV variance process We will focus on ρ = 1 since some famous stochastic volatility models are included. For β = 1, this is the Heston models. For the general setting, letting V t = v β, the asset price is df t F t = V t dw t, and the variance process, by Ito s formula, is (( ) ) dv t = kβ θ ε2 (β 1) V 1 1 β t V t dt + εβv 1 2β 1 t db t. (CEV IV) 2k (( For case of β = 1, dv t = kv t θ + 2ε2 2k ) ) 2 V t 1 εv 3 t db t, our setting is 3/2 model. The case of β = 1 2, dv t = k ) 2 Vt ((θ + ε2 2k ) V t dt + εdb t, shares the same CEV exponent of OU varaiance process. Chang (Institute) Implied volatility by VIX May, 21, 2015 32 / 37
Special functions We summarize the definitions of special functions used in this paper. The Gamma function Γ (z) = x z 1 e x dx. 0 The modified Bessel function of first kind in the form of infinite series 1 ( x ) 2m+α I α (y) =. m!γ (m + α + 1) 2 m=0 The infinite series representation of Kummer conflument hypergeometric functions, for γ = 0, 1, 2,..., Φ (α, γ; z) = 1 + α γ z α (α + 1) z + 2 a (α + 1) (α + 2) z + 3 1! γ (γ + 1) 2! γ (γ + 1) (γ + 2) 3! +... that converges for z < and an inergal form, for b > a > 0, Φ (a, b, z) = Γ (b) Γ (a) Γ (b a) 1 0 e zx x a 1 (1 x) b a 1 dx. Chang (Institute) Implied volatility by VIX May, 21, 2015 33 / 37
Non-central chi-square distribution Note that the CIR process is non-cnetral χ 2 -distrbuted and the transition probability density from v t = x to v s = y, τ = s t, is ( ye k τ P τ (x, y) = c x ) 2 1 ( 2k θ 1) ε 2 ( ( exp c y + xe k τ)) ( I 2k θ ε 2 1 2c ) xye k τ where c = c (τ) = 2k ε 2 ( 1 e k τ ) 1. The conditional characteristic function of v s at time t, τ = s t, has the form of φ (u, v t, τ) = E [ e iuv s v t ] = (1 ε2 ( 1 e k τ) ) 2k θ ε 2 iu 2k e e k τ iu v 1 2k ε2 t (1 e k τ )iu. we can add jumps in the closed-form of CF fucntion by multiplied by a jump part factor. Chang (Institute) Implied volatility by VIX May, 21, 2015 34 / 37
VIX on CEV variance process For S&P 500 futures price modeled by the risk-neutral stochastic model (CEV IV), if β + 2k θ > 0, and the current instantaneous vairance level ε 2 v t = x, then VIXt 2 = 1 Γ ( β + 2k θ ) ( ) ε ε 2 β 2 τ Γ ( ) 2k θ 2k ε 2 τ ( ) ( e xe ks 1 e ks β Φ β + 2kθ ε 2, 2kθ ε 2 ; x 2k e ks ) ε 2 1 e ks ds. 0 where τ = 30 365, Γ (z) the Gamma functions, and Φ (a, b, z) the confluent hypergeometric functions. Chang (Institute) Implied volatility by VIX May, 21, 2015 35 / 37
Conclusions VIX is designed to measure forward volatility. VIX is highly negatively correlative with S&P 500. VIX is a fear gauge of investors and is a candidate to "diversify" systematic risk. We use CEV-type volatility to capture the complicated behavior of VIX (volatility). Chang (Institute) Implied volatility by VIX May, 21, 2015 36 / 37
Thanks for your attention. Chang (Institute) Implied volatility by VIX May, 21, 2015 37 / 37