Simple Variance Swaps Ian Martin ian.martin@stanford.edu - PowerPoint PPT Presentation

Simple Variance Swaps Ian Martin ian.martin@stanford.edu LSE/Stanford and NBER May, 2013 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 1 / 51

What is the Expected Return on the Market? Ian Martin ian.martin@stanford.edu LSE/Stanford and NBER May, 2013 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 1 / 51

Outline Variance swaps and simple variance swaps VIX and SVIX A lower bound on the equity premium Conditional equity premia Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 2 / 51

A lower bound on the equity premium 1 year horizon, in % 20 15 10 5 0 2000 2005 2010 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 3 / 51

A lower bound on the equity premium 1 month horizon, annualized, in % 60 50 40 30 20 10 0 2000 2005 2010 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 3 / 51

Assumptions option prices call 0, T � K � put 0, T � K � K F 0, T No arbitrage Fixed underlying asset, price S t (today, the S&P 500 index) Perfectly liquid market in options Underlying asset does not pay dividends (can be partially relaxed) Next few slides: constant riskless rate Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 4 / 51

Variance swaps A variance swap is an agreement to exchange � � 2 � � 2 � � 2 log S ∆ log S 2 ∆ S T + + · · · + log S 0 S ∆ S T − ∆ for a pre-agreed “strike” � V , at time T � V is chosen so that no money needs to change hands up front: �� � � 2 � � 2 � � 2 log S ∆ log S 2 ∆ S T E ∗ − � + + · · · + log V = 0 S 0 S ∆ S T − ∆ If the underlying asset price follows any diffusion, then as ∆ → 0, V = E ∗ �� T � we end up with � 0 σ 2 t dt Asterisks indicate risk-neutral expectations etc. E ∗ ( X ) = e rT E ( MX ) Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 5 / 51

Variance swaps Exploiting Itˆ o’s lemma, �� T � � E ∗ σ 2 V = t dt 0 �� T � � T 1 2 E ∗ = dS t − d log S t S t 0 0 2 rT − 2 E ∗ log S T = S 0 We could price the variance swap if a “log security” were traded (Neuberger 1994) Using Breeden–Litzenberger 1978 logic, can work out price of log security from European option prices (Carr–Madan 1998) Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 6 / 51

Variance swaps � V is calculated from option prices �� F T � ∞ � 1 1 � V = 2 e rT K 2 put T ( K ) dK + K 2 call T ( K ) dK 0 F T Hedge: (i) a static position in options expiring at T : hold OTM puts and OTM calls in quantities proportional to 1 / K 2 (ii) a dynamic delta-hedge VIX is calculated from this formula The pricing and hedging of variance swaps is often referred to as model-free Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 7 / 51

Variance swap pricing is not model-free The cataclysm that hit almost all financial markets in 2008 had particularly pronounced effects on volatility derivatives . . . . Dealers learned the hard way that the standard theory for pricing and hedging variance swaps is not nearly as model-free as previously supposed . . . . In particular, sharp moves in the underlying highlighted exposures to cubed and higher-order daily returns. . . . This issue was particularly acute for single names, as the options are not as liquid and the most extreme moves are bigger. As a result, the market for single-name variance swaps has evaporated in 2009. —Carr & Lee, Annual Review of Financial Economics , 2009 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 8 / 51

Simple variance swaps A simple variance swap is an agreement to exchange � S ∆ − S 0 � 2 � S 2 ∆ − S ∆ � 2 � S T − S T − ∆ � 2 + + · · · + F 0 , 0 F 0 , ∆ F 0 , T − ∆ for a pre-agreed strike V , at time T F 0 , t is forward price of the underlying to time t , known at time 0 Any constants known at time 0 could be put in the denominator, but choosing forwards results in an important simplification Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 9 / 51

Payoffs on VS and SVS 80 60 40 20 0 2000 2005 2010 Figure: Realized values of the VS (dashed) and SVS (red solid) payoffs, expressed as annualized volatilities in percent. T = 1 month, ∆ = 1 day Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 10 / 51

Simple variance swaps Result (Pricing, version 1) The strike on a simple variance swap is T / ∆ � e ri ∆ � � Π( i ∆) − ( 2 − e − r ∆ )Π(( i − 1 )∆) V = (1) F 2 0 , ( i − 1 )∆ i = 1 where Π( t ) is given by � F 0 , t � ∞ call 0 , t ( K ) dK + S 2 0 e rt Π( t ) = 2 put 0 , t ( K ) dK + 2 0 F 0 , t Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 11 / 51

Simple variance swaps Result (Pricing, version 2) In the limit as ∆ → 0 , �� F 0 , T � � ∞ V = 2 e rT put 0 , T ( K ) dK + call 0 , T ( K ) dK F 2 0 F 0 , T 0 , T Hedge: (i) a static position in options expiring at T : hold OTM puts and OTM calls, equally weighted (ii) a dynamic delta-hedge Hedging and pricing works even if the underlying asset’s price can jump Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 12 / 51

Robustness of simple variance swaps What if ∆ > 0? In the limit as ∆ → 0, we saw that the fair strike on a simple variance swap is �� F 0 , T � � ∞ V = 2 e rT put 0 , T ( K ) dK + call 0 , T ( K ) dK F 2 0 F 0 , T 0 , T In practice, contract has to stipulate ∆ > 0 Paper derives analytic bound showing that the error in approximating V (∆) by V is tiny ◮ Daily sampling: Percentage error < 0 . 001 % ◮ Weekly sampling: Percentage error < 0 . 005 % Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 13 / 51

Robustness of simple variance swaps What if you can only trade strikes in the range ( A , B ) ? Modify payoff with a correction term � S ∆ − S 0 � 2 � S 2 ∆ − S ∆ � 2 � S T − S T − ∆ � 2 + + · · · + − φ ( S T ) F 0 , 0 F 0 , ∆ F 0 , T − ∆ where  � � 2  A − S T  if S T < A   F 0 , T − ∆ φ ( S T ) = 0 if A ≤ S T ≤ B  � � 2    S T − B if S T > B F 0 , T − ∆ Can be replicated with strikes in the range ( A , B ) For 1-month simple variance swaps on S&P 500, correction term was zero on every date in my sample Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 14 / 51

Robustness of simple variance swaps What if you can’t trade deep out-of-the-money strikes? 1500 1000 500 0 2000 2005 2010 Figure: Dashed lines indicate strike range tradable on a given day. Solid line indicates where the market ended up 30 days later Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 15 / 51

Simple variance swaps What about dividends? Analysis goes through if any dividend payments that occur are completely unanticipated. Consider an extreme case . . . ◮ No dividends expected; but at time t = ∆ , the underlying asset is suddenly liquidated via an extraordinary dividend, so S ∆ = 0 ◮ SVS payoff equals 1 ◮ Hedge portfolio: put options go in-the-money ◮ Dynamic position has zero payoff: it was neither long nor short at time 0, and subsequently the asset’s price never moved from zero ◮ Since S T = 0, the total payoff will be � F 0 , T � F 0 , T 2 2 max { 0 , K − S T } dK = K dK = 1 . F 2 F 2 0 0 0 , T 0 , T ◮ Perfect replication! Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 16 / 51

Simple variance swaps What about dividends? Easy to deal with the case of perfectly anticipated dividends Can even handle the fully general case (partially anticipated dividends, unknown size and timing), but need total-return options for the hedge ◮ Total-return options have started to trade OTC but are not particularly liquid at present Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 17 / 51

What does VIX measure? The VIX index is based on the strike on a variance swap: �� F 0 , T � � ∞ VIX 2 = 2 e rT 1 1 K 2 put 0 , T ( K ) dK + K 2 call 0 , T ( K ) dK T 0 F 0 , T This is a definition , not a statement about pricing Generally interpreted as a measure of risk-neutral variance E ∗ � T 0 σ 2 t dt but, in the presence of jumps. . . ◮ VIX 2 � = E ∗ � T 0 σ 2 t dt V � = E ∗ � T ◮ variance swap strike, � 0 σ 2 t dt ◮ � V � = VIX 2 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 18 / 51

What does VIX measure? Result (Interpretation of VIX) VIX measures the entropy of the simple return R T = S T / S 0 , VIX 2 = 2 T ( log E ∗ R T − E ∗ log R T ) If R T is lognormal, then VIX 2 = 1 T var ∗ log R T ≈ 1 T var ∗ R T But, in general, with jumps and/or time-varying volatility, VIX depends on all of the (annualized, risk-neutral) cumulants of log returns, ∞ � κ ∗ 2 + κ ∗ 3 + κ ∗ 12 + κ ∗ VIX 2 = 2 n ! = κ ∗ n 3 4 5 60 + · · · n = 2 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 19 / 51

What does VIX measure? Initially surprising observation: negative skewness drives VIX down ∞ � κ ∗ 2 + κ ∗ 3 + κ ∗ 12 + κ ∗ VIX 2 = 2 n ! = κ ∗ n 3 4 5 60 + · · · n = 2 Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 20 / 51