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 %

2000 2005 2010 5 10 15 20

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 3 / 51

A lower bound on the equity premium

1 month horizon, annualized, in %

2000 2005 2010 10 20 30 40 50 60

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 3 / 51

Assumptions

F0,T K

• ption prices

call0,TK put0,TK

No arbitrage Fixed underlying asset, price St (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

• log S∆

S0 2 +

• log S2∆

S∆ 2 + · · · +

• log

ST ST−∆ 2 for a pre-agreed “strike” V, at time T

• V is chosen so that no money needs to change hands up front:

E∗

• log S∆

S0 2 +

• log S2∆

S∆ 2 + · · · +

• log

ST ST−∆ 2 − V

• = 0

If the underlying asset price follows any diffusion, then as ∆ → 0, we end up with V = E∗ T

0 σ2 t dt

• Asterisks indicate risk-neutral expectations etc. E∗(X) = erT E(MX)

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 5 / 51

Variance swaps

Exploiting Itˆ

• ’s lemma,
• V

= E∗ T σ2

t dt

• =

2 E∗ T 1 St dSt − T d log St

• =

2rT − 2 E∗ log ST S0 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
• V = 2erT

FT 1 K2 putT(K) dK + ∞

FT

1 K2 callT(K) dK

• Hedge:

(i) a static position in options expiring at T: hold OTM puts and OTM calls in quantities proportional to 1/K2 (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∆ − S0 F0,0 2 + S2∆ − S∆ F0,∆ 2 + · · · + ST − ST−∆ F0,T−∆ 2 for a pre-agreed strike V, at time T F0,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

2000 2005 2010 20 40 60 80

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 V =

T/∆

• i=1

eri∆ F2

0,(i−1)∆

• Π(i∆) − (2 − e−r∆)Π((i − 1)∆)
• (1)

where Π(t) is given by Π(t) = 2 F0,t put0,t(K) dK + 2 ∞

F0,t

call0,t(K) dK + S2

0ert

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 11 / 51

Simple variance swaps

Result (Pricing, version 2)

In the limit as ∆ → 0, V = 2erT F2

0,T

F0,T put0,T(K) dK + ∞

F0,T

call0,T(K) dK

• 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 V = 2erT F2

0,T

F0,T put0,T(K) dK + ∞

F0,T

call0,T(K) dK

• 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∆ − S0 F0,0 2 + S2∆ − S∆ F0,∆ 2 + · · · + ST − ST−∆ F0,T−∆ 2 − φ(ST) where φ(ST) =         

• A−ST

F0,T−∆

2 if ST < A if A ≤ ST ≤ B

• ST−B

F0,T−∆

2 if ST > B 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?

2000 2005 2010 500 1000 1500

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 ST = 0, the total payoff will be

2 F2

0,T

F0,T max {0, K − ST} dK = 2 F2

0,T

F0,T K dK = 1.

◮ 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: VIX2 = 2erT T F0,T 1 K2 put0,T(K) dK + ∞

F0,T

1 K2 call0,T(K) dK

• This is a definition, not a statement about pricing

Generally interpreted as a measure of risk-neutral variance E∗ T

0 σ2 t dt but, in the presence of jumps. . .

◮ VIX2 = E∗ T

0 σ2 t dt

◮ variance swap strike,

V = E∗ T

0 σ2 t dt

◮

V = VIX2

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 RT = ST/S0, VIX2 = 2 T (log E∗ RT − E∗ log RT) If RT is lognormal, then VIX2 = 1 T var∗ log RT ≈ 1 T var∗ RT But, in general, with jumps and/or time-varying volatility, VIX depends on all of the (annualized, risk-neutral) cumulants of log returns, VIX2 = 2

∞

• n=2

κ∗

n

n! = κ∗

2 + κ∗ 3

3 + κ∗

4

12 + κ∗

5

60 + · · ·

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 19 / 51

What does VIX measure?

Initially surprising observation: negative skewness drives VIX down VIX2 = 2

∞

• n=2

κ∗

n

n! = κ∗

2 + κ∗ 3

3 + κ∗

4

12 + κ∗

5

60 + · · ·

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 20 / 51

What does VIX measure?

Initially surprising observation: negative skewness drives VIX down VIX2 = 2

∞

• n=2

κ∗

n

n! = κ∗

2 + κ∗ 3

3 + κ∗

4

12 + κ∗

5

60 + · · · But this is skewness calculated with risk-neutral probabilities To see how VIX is linked to real-world probabilities, suppose that there is a representative investor with log utility

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 20 / 51

What does VIX measure?

Result (VIX in equilibrium)

If there is a representative agent with log utility, then VIX can be expressed in terms of the cumulants of log RT under the real-world probabilities, VIX2 = 2

∞

• n=2

(−1)n(n − 1)κn n! = κ2 − 2 3κ3 + κ4 4 − κ5 15 + · · · Paper does this with power utility, but the point here is qualitative

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 21 / 51

What does SVIX measure?

F0,T K

• ption prices

call0,TK put0,TK

Analogously, we can define an index, SVIX, based on the strike of a simple variance swap: SVIX2 = 2erT T · F2

0,T

F0,T put0,T(K) dK + ∞

F0,T

call0,T(K) dK

• Ian Martin (LSE/Stanford)

Simple Variance Swaps May, 2013 22 / 51

What does SVIX measure?

Result (Interpretation of SVIX)

SVIX measures the risk-neutral variance of the simple return: SVIX2 = 1 T var∗ RT Rf,T

• Ian Martin (LSE/Stanford)

Simple Variance Swaps May, 2013 23 / 51

VIX and SVIX

2000 2005 2010 10 20 30 40 50 60 70

Figure: VIX (dotted) and SVIX (solid). Jan 4, 1996–Jan 31, 2012 Figure shows 10-day moving average. T = 1 month

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 24 / 51

VIX minus SVIX

2000 2005 2010 1 2 3 4 5 6 7

Figure: VIX minus SVIX. Jan 4, 1996–Jan 31, 2012 Figure shows 10-day moving average. T = 1 month

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 25 / 51

VIX minus SVIX

If returns and the SDF are conditionally lognormal with return volatility σR then we can calculate VIX and SVIX in closed form: SVIX2 = 1 T

• eσ2

RT − 1

• VIX2

= σ2

R

VIX would be lower than SVIX No conditionally lognormal model is consistent with observed

• ption prices

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 26 / 51

A lower bound on the equity premium

SVIX gives a bound on the equity premium perceived by an investor

◮ who is unconstrained, ◮ who holds the market over the horizon of interest, and ◮ whose relative risk aversion (which need not be constant) is at least

• ne

These assumptions can be relaxed to some extent But I have nothing to say about

◮ Constrained investors ◮ Irrational investors ◮ The connection between prices and aggregate cashflows or

consumption

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 27 / 51

A lower bound on the equity premium

Notation:

1 Rf,T E∗ XT = E MTXT. Remember, E MTRT = 1 for any

return RT Start from an identity: var∗ RT Rf,T = 1 Rf,T E∗ R2

T −

1 Rf,T (E∗ RT)2 = E(MTR2

T) − Rf,T

= E RT − Rf,T + E(MTR2

T) − E RT

= E RT − Rf,T + cov(MTRT, RT)

• ≤0

This connects something we can measure to something interesting plus something we can control

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 28 / 51

A lower bound on the equity premium

Definition (Negative correlation condition)

Given a gross return RT and stochastic discount factor MT, the negative correlation condition (NCC) holds if cov (MTRT, RT) ≤ 0 This would fail badly in a risk-neutral world (MT deterministic) Under lognormality, NCC is equivalent to − corr(log MT, log RT) σ(log MT) ≥ σ(log RT)

Result (SVIX and the equity premium)

If the NCC holds, then E RT − Rf,T ≥ Rf,T · SVIX2

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 29 / 51

A lower bound on the equity premium

In particular, the NCC holds if any of the following conditions holds. B1 There is a one-period marginal investor who maximizes expected utility, who holds the market, and whose relative risk aversion γ(C) ≡ − Cu′′(C)

u′(C) ≥ 1 (not necessarily constant)

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 30 / 51

A lower bound on the equity premium

In particular, the NCC holds if any of the following conditions holds. B2 There is an intertemporal marginal investor with separable utility who holds the market, whose value function V can be defined recursively as a function of wealth W0, V

• W0
• = max

C0,{wi} u(C0)+β E V

• (W0−C0)
• i

wiR(i)

T

• s.t.
• i

wi = 1, and whose relative risk aversion Γ(W) ≡ − WV′′[W]

V′[W]

≥ 1

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 30 / 51

A lower bound on the equity premium

In particular, the NCC holds if any of the following conditions holds. B3 There is an Epstein–Zin (1989) marginal investor who holds the market, and has a constant consumption-wealth ratio and risk aversion γ ≥ 1

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 30 / 51

A lower bound on the equity premium

In particular, the NCC holds if any of the following conditions holds. B4 The SDF and market return are conditionally jointly lognormal, and the market’s conditional Sharpe ratio exceeds its conditional volatility

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 30 / 51

A lower bound on the equity premium

These cover some of the leading macro-finance models

◮ Campbell–Cochrane 1999 ◮ Bansal–Yaron 2004 ◮ Barro 2006 ◮ Wachter 2012

More generally, one has to check that the NCC holds

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 31 / 51

A lower bound on the equity premium

Result (Version B1)

If there is a one-period investor who holds the market from time 0 to time T, and whose risk aversion γ(C) ≡ −Cu′′(C)/u′(C) ≥ 1, then 1 T E

• RT − Rf,T
• ≥ Rf,T · SVIX2

With log utility, γ(C) ≡ 1, this holds with equality Does not require power utility: γ doesn’t have to be constant Does not require lognormality

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 32 / 51

A lower bound on the equity premium

Proof, version B1.

The given assumption implies that the SDF is proportional to u′(W0RT), so we must show that cov (RTu′(W0RT), RT) ≤ 0 This holds because RTu′(W0RT) is decreasing in RT: its derivative is u′(W0RT) + W0RTu′′(RT) = −u′(W0RT) [γ(W0RT) − 1] ≤ 0

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 33 / 51

A lower bound on the equity premium

var∗ RT Rf,T ≤ E RT − Rf,T ≤ Rf,T · σ(M) · σ(RT) Left-hand inequality is the new result

◮ Good: relates unobservable equity premium to an observable

quantity

◮ Bad: requires the negative correlation condition

Right-hand inequality is the Hansen–Jagannathan bound

◮ Good: no assumptions ◮ Bad: neither side is observable Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 34 / 51

A lower bound on the equity premium

Merton (1980) estimated equity premium from instantaneous risk premium = γσ2 Assumes power utility and the market’s price follows a diffusion No distinction between risk-neutral and real-world variance in a diffusion-based model (Girsanov’s theorem) Beyond diffusions, the appropriate generalization relates the risk premium to the risk-neutral variance

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 35 / 51

A lower bound on the equity premium

1mo horizon, annualized, 10-day moving avg. Mid prices in black, bid prices in red

2000 2005 2010 10 20 30 40

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 36 / 51

A lower bound on the equity premium

3mo horizon, annualized, 10-day moving avg. Mid prices in black, bid prices in red

2000 2005 2010 5 10 15 20 25 30

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 36 / 51

A lower bound on the equity premium

1yr horizon, annualized, 10-day moving avg. Mid prices in black, bid prices in red

2000 2005 2010 5 10 15 20

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 36 / 51

A lower bound on the equity premium

horizon mean s.d. min 1% 10% 25% 50% 75% 90% 99% max 1 mo 4.92 4.53 0.81 1.01 1.52 2.41 3.85 5.65 8.85 25.4 54.2 2 mo 4.92 3.94 0.97 1.17 1.62 2.57 4.05 5.82 8.41 23.2 45.4 3 mo 4.87 3.55 1.03 1.25 1.70 2.65 4.15 5.83 8.05 21.1 38.6 6 mo 4.77 2.93 1.24 1.45 1.88 2.83 4.29 5.88 7.57 16.7 28.6 1 yr 4.48 2.41 1.03 1.53 1.95 2.71 4.19 5.48 7.02 13.7 21.2

Table: Mean, standard deviation, and quantiles of EP bound (in %)

These numbers allow for dividends

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 37 / 51

The equity premium

Figure from John Campbell’s Princeton Lecture in Finance

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 38 / 51

A lower bound on the equity premium

Equity premium was high from 1998–2000

◮ Forecasts based on market valuation ratios incorrectly predicted a

low or even negative equity premium during this period

◮ By construction, the lower bound can never be less than zero

Most important: no out-of-sample issues (Ang–Bekaert, Goyal–Welch) because no parameter estimation is required

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 39 / 51

A lower bound on the equity premium

Suppose you think this just reflects “market segmentation”. What trade should you have done in November 2008?

◮ Short the portfolio of options, i.e. short an at-the-money-forward

straddle and (equally weighted) out-of-the-money calls and puts

◮ You end up short if the market rallies and long if the market sells off ◮ You’re taking a contrarian position, providing liquidity to the market

At the height of the credit crisis, very high risk premia were available for investors who were prepared to take on this position

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 40 / 51

How not to think about what’s going on

In a lognormal world, you could think of what we’re doing as

◮ Exploiting a relationship between risk premia and P-variance ◮ Q-variance equals P-variance

In the real, non-lognormal, world, this is the wrong intuition

◮ Risk premia sensitive to higher moments as well as P-variance ◮ Q-variance not equal to P-variance Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 41 / 51

The bound is conservative in two respects

Can’t observe deep-OTM option prices

F0,T K

• ption prices

call0,TK put0,TK

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 42 / 51

The bound is conservative in two respects

Even near-the-money, can’t observe a continuum of strikes put0,T K K1 K2 K3 K

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 42 / 51

The bound is conservative in two respects

Both these effects mean that the true lower bound is even higher By ignoring deep-OTM options, we underestimate the true area under the curve Discretization in strike also leads to underestimating the true area, because call0,T(K) and put0,T(K) are both convex in K

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 43 / 51

Labor income

Suppose the investor has some other source of income L(RT) at time T; think of bonds or labor income Then we get the same result under a stronger assumption on risk aversion

◮ If L′(RT) ≥ 0 and L(RT) ≤ κWT then we need risk aversion at least

1 + κ

◮ If the agent has at least as much wealth in the market as labor (or

bond) income between now and time T, L(RT) ≤ WT, then we need risk aversion at least 2

◮ All the results go through—and the numbers are the same too Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 44 / 51

Equity premium bound (rescaled) and S&P 500

2000 2005 2010 500 1000 1500

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 45 / 51

What if the bound were tight?

REPt→t+T

?

≈ α + β · EPBt + εt+T OLS with Hansen–Hodrick standard errors to account for heteroskedasticity and overlapping observations Null hypothesis: bound holds with equality, α = 0, β = 1

horizon

• α

s.e.

• β

s.e. R2 1 mo −0.01 [0.06] 0.77 [1.41] 0.3% 2 mo −0.02 [0.07] 0.95 [1.47] 0.8% 3 mo −0.02 [0.07] 0.90 [1.63] 0.9% 6 mo −0.07 [0.06] 1.90 [0.90] 4.6% 1 yr −0.04 [0.09] 1.54 [1.29] 3.6%

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 46 / 51

The conditional equity premium

SVIX2 = 2Rf,T T · F2

0,T

F0,T putT(K) dK

• “down-SVIX2”

+ 2Rf,T T · F2

0,T

∞

F0,T

callT(K) dK

• “up-SVIX2”

Natural to think about what the calls and puts tell us separately Can we give a nice interpretation to down- and up-SVIX2?

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 47 / 51

The conditional equity premium

Yes, if we think from the perspective of a log investor P

• RT > Rf,T
• =

−Rf,T call′(F0,T) + call(F0,T) S0 1 T E

• (RT − Rf,T)1
• RT > Rf,T
• =

Rf,T T · S0 call(F0,T) + Rf,T up-SVIX2 1 T E

• (RT − Rf,T)1
• RT < Rf,T
• =

− Rf,T T · S0 call(F0,T) + Rf,T down-SVIX2 These are real-world probabilities, P not P∗ Can also do this with power utility

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 48 / 51

Probability of an up-move, P

• RT > Rf,T
• T = 1 mo

2000 2005 2010 0.50 0.55 0.60 0.65 0.70

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 49 / 51

Probability of an up-move, P

• RT > Rf,T
• T = 1 yr

2000 2005 2010 0.50 0.55 0.60 0.65 0.70

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 49 / 51

Conditional equity premium E

• RT − Rf,T
• RT ≷ Rf,T
• Black: up-move. Red: down-move (sign flipped). T = 1 yr

2000 2005 2010 10 20 30 40

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 49 / 51

A comparison: November 1998 vs November 2008

11/98 11/08 E

• RT − Rf,T
• 11%

18% P (up-move) 65% 67% E

• RT − Rf,T
• up-move
• 33%

38% E

• RT − Rf,T
• down-move
• −35%

−24% Internet boom was the only period in the sample when the equity premium was larger (in absolute value) conditional on a down-move than on an up-move

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 50 / 51

Conclusion

Proposed a new derivative contract, the simple variance swap, that can be hedged and therefore priced even if there are jumps SVIX is less than VIX in the data; conditionally lognormal models make the opposite prediction The equity premium was extraordinarily high during the crisis The equity premium is extremely volatile, and variation is faster than business-cycle frequency This has many implications for finance and macroeconomics

Ian Martin (LSE/Stanford) Simple Variance Swaps May, 2013 51 / 51