Volatility and Skewness Indices Using State-Preference Pricing Zhangxin Frank Liu Finance Theory Module 2 March 16 th , 2013 1 / 71
Outline FIX the VIX 1 BEX and BUX 2 SIX is SICK 3 Future Research 4 2 / 71
Motivation I • WHY CARE ABOUT VOLATILITY AT ALL? “. . . what distinguishes financial economics is the central role that uncertainty plays in both financial theory and its empirical implementation. The start- ing point for every financial model is the uncer- tainty facing investors, . . . Indeed, in the absence of uncertainty, the problems of financial economics reduce to exercises in basic microeconomics.” Campbell, Lo and MacKinlay (1997) 3 / 71
Volatility Forecasting • Volatility forecasting has been discussed in following con- texts (Poon and Granger, 2003; Andersen et al. (2005)): • Historical volatility • Quick and easy but how far back should one refer to? • ARCH/GARCH volatility • ARCH (Engle, 1982): time-varying function of current observables. • GARCH (Bollerslev, 1986; Taylor, 1986): f ( ω 1 ¯ V , ω 2 ˆ V t − 1 , ω 3 ǫ ; t ) 4 / 71
Volatility Forecasting • . . . • Implied volatility • Implied from option prices • Invert the analytical pricing formula from some option pricing models (if exist); or follow some model-free approaches (Du- mas, Fleming and Whaley (1998)). 5 / 71
VIX History • A brief history of VIX (Carr and Wu, 2006) • Old VIX (VXO) • First introduced by Whaley (1993) • Based on OEX options (American style) • An average of the Black-Scholes implied volatilities on eight near-the-money options at the two nearest maturities • Artificially induced upward bias from the CBOE trading day conversion √ NC TV ( t , T ) = ATMV ( t , T ) √ NT √ NC ≡ ATMV ( t , T ) � NC − 2 × int ( NC / 7 ) 6 / 71
VIX History • . . . • New VIX: • CBOE revised methodology in 2003. • Based on SPX options (European style) • Model-free approach in Demeterfi, Derman, Kamal and Zou (DDKZ, 1999) • Correct artificial upward bias from the previous trading day conversion • Trading of VIX futures contracts from May 2004; VIX options from February 2006 7 / 71
VIX 101 • VIX formula: � F � 2 j = 2 ∆ K i e rT Q ( K i ) − 1 � σ 2 − 1 ∀ j = 1 , 2 K 2 T j T j K 0 i i � � � 365 N T 2 − N 30 N 30 − N T 1 T 1 σ 2 + T 2 σ 2 VIX = 100 1 2 30 N T 2 − N T 1 N T 2 − N T 1 • Principal of DDKZ (1999): realized volatility can be captured by the dynamic hedging of a log contract. 8 / 71
VIX 101 Derivation: Theoretical definition of realized variance for a given price history is � T V = 1 σ 2 ( t , . . . ) dt T 0 Think about pricing a variance swap: F = E ( e − rT ( V − K )) For a zero initial value, �� T � K var = E ( V ) = 1 σ 2 ( t , . . . ) dt T E 0 9 / 71
VIX 101 DDKZ (1999) only assumes that future underlyer evolution is diffusive (i.e. no jumps allowed): dS t = µ ( t , . . . ) dt + σ ( t , . . . ) dZ t S t � � µ − 1 Itô’s lemma 2 σ 2 d ( ln S t ) = ⇒ dt + σ dZ t ⇒ dS t − d ( ln S t ) = 1 2 σ 2 dt S t � dS t � σ 2 dt = 2 or − d ( ln S t ) S t 10 / 71
VIX 101 Now we have = 1 �� T � 0 σ 2 ( t , . . . ) dt K var T E � � σ 2 dt dS t = 2 S t − d ( ln S t ) �� T � � T ∴ E ( V ) = K var = 2 dS t T E d ( ln S t ) − S t 0 0 �� T � � � = 2 − 2 dS t ln S T T E T E S t S 0 0 � �� � � �� � B A 11 / 71
VIX 101 �� T � A = E ( r dt + σ ( t , . . . ) dZ t ) Z t ∼ N ( 0 , t ) 0 = rT � � ln S T B = E S 0 � � ln S T + ln S ∗ = E S ∗ S 0 � �� � Log contract where S ∗ is some arbitrary number to define the boundary of OTM calls and puts. 12 / 71
VIX 101 How to value E ( ln ( S T / S ∗ )) ? Suppose we buy a portfolio of op- tions, Π , spanning all strikes K ∈ ( 0 , ∞ ) with expiration T and weighted inversely proportional to K 2 , we have OTM puts OTM calls � �� � � �� � � S ∗ � ∞ 1 1 Π = K 2 max ( K − S T , 0 ) dK + K 2 max ( S T − K , 0 ) dK 0 S ∗ � � S ∗ 1 K 2 ( K − S T ) dK , if S T < S ∗ S T = � S T 1 K 2 ( S T − K ) dK , if S T ≥ S ∗ S ∗ = − 1 − ln S T + S T + ln S ∗ S ∗ = S T − S ∗ − ln S T S ∗ S ∗ � � � S T − S ∗ � ln S T ∴ E = E − Π S ∗ S ∗ 13 / 71
VIX 101 � � S T − S ∗ � � K var = 2 T ( rT ) − 2 + ln S ∗ E − Π T S ∗ S 0 � �� S ∗ � S T � = 2 1 − 1 K 2 max ( K − S T , 0 ) dK + rT − E + E T S ∗ 0 � ∞ � � 1 − ln S ∗ K 2 max ( S T − K , 0 ) dK S 0 S ∗ a � �� � � S 0 e rT � = 2 − ln S ∗ rT − − 1 + T S ∗ S 0 b � �� � � S ∗ � ∞ P ( K ) C ( K ) e rT dK + e rT dK K 2 K 2 0 S ∗ 14 / 71
VIX 101 � � S 0 e rT � � a = 2 � e rT � ln − 1 − ln ( S ∗ ) + ln ( S 0 ) − T S ∗ � � S 0 e rT � � S 0 e rT �� = 2 ln − 1 − T S ∗ S ∗ � F � F � � �� = 2 where F = S 0 e rT ln − 1 − T S ∗ S ∗ � F � F � F � � 2 � ≈ 2 − 1 − 1 − 1 − − 1 T S ∗ 2 S ∗ S ∗ � �� � Taylor expansion of ln ( F / S ∗ ) � F � 2 = − 1 − 1 where S ∗ ≡ K 0 T S ∗ 15 / 71
VIX 101 �� S ∗ � � ∞ b = 2 e rT P ( K ) C ( K ) dK + dK K 2 K 2 T 0 S ∗ � S ∗ � K H ≈ 2 e rT P ( K ) C ( K ) dK + dK K 2 K 2 T K L S ∗ � �� � � �� � truncation error ∞→ K H truncation error 0 → K L ≈ 2 ∆ K i � e rT Q ( K i ) K 2 T i i � �� � discretization error Hence we obtain the VIX formula � F �� � � 2 K var = E ( V ) ≈ 2 − 1 ∆ K i e rT Q ( K i ) = σ 2 − 1 VIX K 2 T T S ∗ i i 16 / 71
Why the switch? • SPX options are more popular • “Model-free approach” • One can replicate the payoff of VIX futures and options • VIX futures and options can be traded for volatility hedging purposes 17 / 71
Any Drawbacks? • Truncation and Discretization errors (Jiang and Tian, 2007) • Linear interpolation may induce an error, if model-free im- plied variance does not follow a linear function of maturity. • Mechanically higher weights are allocated to OTM puts i.e. VIX may be manipulable by trading relatively cheaper Deep- OTM put options. • Why not consider trade volume? 18 / 71
FIX the VIX Let’s FIX the VIX. 19 / 71
FIX the VIX • A forward-looking volatility index (FIX) as a proxy for market realized volatility over the next 30 days. • State-Preference Pricing Approach • Arrow (1964) and Debreu (1959) S � P t = (Φ s , t + 1 d s , t + 1 ) s = 1 • View FIX 2 as a financial asset pays you this dollar amount: � � S T + 0 . 05 �� 2 ln S 0 • How to define state prices? 20 / 71
FIX the VIX • State prices (Breeden and Litzenberger, 1978): Φ( T , . . . ) = ∂ 2 C ( K , T ) = ∂ 2 P ( K , T ) ∂ K 2 ∂ K 2 • To see this, construct a butterfly spread to long one call with strike M − ∆ M , long one call with strike M + ∆ M and short two calls with strike M (Barraclough, 2008). S T < M − ∆ M M − ∆ M < S T < M M < S T < M + ∆ M M + ∆ M < S T Long 1 call with M − ∆ M 0 S T − ( M − ∆ M ) S T − ( M − ∆ M ) S T − ( M − ∆ M ) Short 2 calls with M 0 0 − 2 ( S T − M ) − 2 ( S T − M ) Long 1 call with M + ∆ M 0 0 0 S T − ( M + ∆ M ) Total at t = T 0 ∆ M + ( S T − M ) ∆ M − ( S T − M ) 0 • Payoff is $∆ M if S T = M at maturity. 21 / 71
FIX the VIX • Thus the cost of butterfly spread that produces a payment of $1 if the future state is S T = M is: P ( M ; ∆ M ) = C ( M − ∆ M , T ) − 2 C ( M , T ) + C ( M + ∆ M , T ) ∆ M • Divide the above by the step size ∆ M and in the limit as ∆ M → 0 yields: C ( M − ∆ M , T ) − 2 C ( M , T ) + C ( M + ∆ M , T ) P ( M ; ∆ M ) lim = lim ∆ M 2 ∆ M ∆ M → 0 ∆ M → 0 � = ∂ 2 C ( K , T ) � � ∂ K 2 � K = M 22 / 71
FIX the VIX • Thus the price of a security f with payoff d f M at some future state M is � ∂ 2 C ( K , T ) � � � � P f = � d f d f P ( M ; dM ) = dM � M M ∂ K 2 � ���� � �� � ���� M M K = M � �� � payoff state price payoff state price • As an example, let’s have a look at pricing a European put option. We know the price of the put option can be found as: � ∞ P = E ( e − rT ( K − S T ) + ) = ( K − S T ) + e − rT f ( S T ) dS T � �� � � �� � 0 payoff state price � K ( K − S T ) e − rT f ( S T ) dS T = 0 23 / 71
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