Risk and Decision Analysis 00 (2017) 1–7 1 DOI 10.3233/RDA-170123 IOS Press Why is VIX a fear gauge?
Peter Carr NYU Tandon School of Engineering, 12 MetroTech Brooklyn, NY 11201, USA E-mail: [email protected]
Abstract. VIX is a widely followed volatility index constructed from the market prices of out-of-the-money (OTM) puts and calls written on the S&P500. VIX is often referred to as a fear gauge. While the market prices of OTM puts clearly reflect the fear that the S&P500 will drop, about half of the options used in constructing VIX are actually OTM calls. The market prices of these OTM calls clearly reflect greed rather than fear. In this note, we offer several explanations as to why VIX can properly be regarded as a fear gauge. Keywords: VIX, volatility, fear gauge
1. Introduction the easiest ways to respect positivity is to start the in- dex at a positive level and to assume that the worst pos- VIX is a widely followed index constructed from sible down move in the index over any period is always the market prices of out-of-the-money (OTM) puts and the same fixed fraction of its beginning of period size. calls written on the S&P500. The V in VIX stands For example, suppose an index starts at 100 and can for volatility. It is well known that since its redesign halve every period, but can never drop by more than in 2003, VIX2 approximates the cost of synthesizing half. Then if the worst possible move occurs every pe- a newly issued 30 day variance swap under a set of riod, the index levels are 100, 50, 25, 12.5 etc., so index widely accepted assumptions. levels remain positive. Besides its well known role as a volatility index, If we couple a constant fractional down move with a VIX is often referred to as a fear gauge. While the mar- constant proportional up move, we obtain a multiplica- ket prices of OTM puts clearly reflect the fear that the tive binomial process. For example, if the index level S&P500 will drop, about half of the options used in can either halve or double each period, then it is fol- constructing VIX are actually OTM calls. The market lowing a multiplicative binomial process. For simplic- prices of these OTM calls clearly reflect greed rather ity, consider the further special case of a multiplica- than fear. In this note, we explain why VIX is a volatil- tive binomial process when the two possible percent- ity index and we also offer several explanations as to age moves differ only in sign. For example, suppose an why VIX can properly be regarded as a fear gauge. index is presently at 100 and can only rise or fall by An overview of this paper is as follows. The next 10% each period. Since the percentage down move is section shows why VIX can properly be regarded as a still between 0 and 100%, the index level will alway be volatility index. In the following section, we focus on positive. why VIX can also be regarded as a fear gauge. The For the remainder of this section,we only assume final section summarizes the paper and offers sugges- that the S&P500 index level is positive and arbitrage- tions for future research. free. However, we will illustrate all of our results in this context by assuming that the index can only rise or fall by the round figure of 10% each period. Let 2. VIX as a volatility index Si > 0 be the closing level of S&P500 on day i. Con- sider the difference Si+1 − Si between the closing in- Market prices of options are widely used to calibrate dex levels on two adjacent dates. The daily return is the stochastic process governing the market price of the defined as the ratio of this difference to the initial index underlying asset. When the underlying is a stock index, level. We refer to this random variable as the forward − the stochastic process must respect positivity; a stock return f ≡ Si+1 Si . In contrast, we define the back- i Si index will never hit zero or become negative. One of ward return as the ratio of the difference to the final in-
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Si+1−Si dex level bi ≡ . If time ran backward, then the ward return, plus an error whose leading order is a Si+1 backward return would become the forward return in cubed percentage return: the reversed clock. With time running forward, we now − = 2 + 3 show that the backward return is always arithmetically fi bi fi O(fi) . (1) 1 lower than the forward return, i.e. bi fi. If the index rises, then the backward return divides In continuous time, and for continuous sample path the positive gain by a larger denominator than the for- stochastic processes, the cubes and higher powers of ward return, so the fraction is less positive. For exam- returns vanish. As a result, the difference between the ple, if the index rises from 100 to 110, then the for- forward return and the backward return is the square of = the forward return. ward return is 10/100 0.1 or 10%, while the back- − = The forward return f ≡ Si+1 Si can be manufac- ward return is less positive at 10/110 0.0909 or i Si 9.09%. If the index falls, then the backward return di- tured by holding 1 units of the index at time i and Si vides the negative gain by a smaller denominator than borrowing the cost. Assuming no dividends and no in- the forward return, so the fraction is more negative. terest, the realized gain on this position at time i + 1 1 For example, if the index instead falls from 100 to is (S + − S ) = f , the forward return. How can Si i 1 i i 90, then the forward return is −10/100 =−0.1, or we manufacture the backward return? The backward −10%, while the backward return is more negative at return cannot be manufactured by holding 1 units Si+1 −10/90 =−0.1111 or −11.11%. of the index at time i because the required holding in The square of the daily return is an estimate of the the index is anticipating. Fortunately, the gains from a variance of the daily return. It turns out that the square static position in an option can be interpreted as equiv- of the forward return is quite close to the difference alent to the gains from a particular anticipating dy- between the forward return and the backward return. namic trading strategy in its underlying index, even For example, if the index rises from 100 to 110, then when the volatility process is unknown. For example, 2 = the squared return is (0.1) 0.0100, while the dif- consider the gain in intrinsic value |Si+1 − Si| that ference between the forward return and the backward arises from being long one ATM straddle at time i.Re- return is 0.1 − 0.0909 = 0.0091. If the index in- gardless of the S dynamics, this gain can always be a − stead falls from 100 to 90, then the squared return represented as Ni (Si+1 Si), where the number of − 2 = + a = is still ( 0.1) 0.0100, while the difference be- shares held from time i to time i 1isNi 1(Si+1 tween the forward return and the backward return is Si) − 1(Si+1
1 3 As a mnemonic, notice that the letter b comes before the letter f (S ) e =− i A (m ), (3) in the alphabet. 12 i P. Carr / Why is VIX a fear gauge? 3 with mi between Si+1 and Si. In words, (2) says that ward return which is half the squared return: the change in the area under a function such as A (S) is approximated by the product of the average height 1 1 3 fi − ci = fi − fi + bi + O f of the function and its base. From (3), the leading term 2 2 i in the error is the cube of the index change. 1 = = (f − b ) + O f 3 To illustrate, suppose A(S) ln S. Then since 2 i i i = 1 A (S) S : = 1 2 + 3 fi O fi , (6) 2 = 1 1 + 1 1 + 3 (ln S)i Si O fi . (4) 2 Si 2 Si+1 from (1). It follows that if we double the difference be- tween the forward return and the continuously com- The LHS is just the log price relative (ln S)i = pounded return, we approximate the squared return: Si+1 ln( ) ≡ ci, which is the continuously compounded Si − = 2 + 3 rate of return. Hence, (4) shows that the continuously 2(fi ci) fi O fi . (7) compounded rate of return is approximated by the blended return that arises by averaging the forward re- This observation lies at the heart of the following VIX turn with the backward return: construction. Summing over 30 days implies: = 1 + 1 + 3 ci fi bi O fi . (5) 29 29 29 2 2 − = 2 + 3 2(fi ci) fi O fi . (8) i=0 i=0 i=0 Since bi fi, we obviously have bi ci fi when O(f 3) the i term can be ignored. In words, the contin- The LHS contains the continuously compounded re- 2 uously compounded return ci is always between the turn ci = ln Si+1 − ln Si. The summing causes tele- backward return bi and the forward return fi.From(5), scoping so that: the leading term in the error in treating ci as a simple 3 average of bi and fi is the cube of the return, fi , which 29 is typically quite small. −2lnS30 + 2lnS0 + 2fi To see how small this approximation error is in prac- i=0 tice, let’s examine the error when the daily percentage 29 29 moves are ±10%, which corresponds to a huge annu- = 2 + 3 √ fi O fi . (9) alized volatility of 252 × 0.1 = 158%. If the in- i=0 i=0 dex rises from 100 to 110, then the continuously com- pounded non-annualized return is ln(1.1) = 0.0953 Solving for the sum of squared returns: or 9.53%. The simple average of the forward return 29 of 10% and the backward return of 9.09% is 9.545%, f 2 ≈−2ln(S /S ) which is quite close. If the index instead falls from i 30 0 i=0 100 to 90, then the continuously compounded return is (10) ln(0.9) =−0.1053, or 10.53%. The simple average of 29 + 2 − the forward return of −10% and the backward return (Si+1 Si), = Si of −11.11% is −10.55%, which is also quite close. i 0 Now consider the difference between the forward re- Si+1−Si since fi = . The first term on the RHS is path- turn and the continuously compounded return. Since Si independent, so can be spanned by a static position in the continuously compounded return is quite close to 30 day OTM options. Carr and Madan[2]show: the blended return, this difference is quite close to half the difference between the forward return and the back- −2ln(S30/S0) S 2As a mnemonic, notice that the letter c is between the letter b and 2 0 2 + =− (S − S ) + (K − S ) dK the letter f in the alphabet. 30 0 2 30 S0 0 K 4 P. Carr / Why is VIX a fear gauge?
∞ 2 + = + (S − K) dK and call if Ki Ko and Q(Ki) is the midpoint of the 2 30 S0 K bid-ask spread for each option with strike Ki. It differs from VIX2 for several reasons: 2 29 =− (Si+1 − Si) 1. We have assumed that R = 0. S0 i=0 2. VIX2 uses options of two maturities and then lin- S early interpolates to achieve a 30 day expiration. 0 2 + + (K − S30) dK As long as the convexity of the term structure of K2 0 variance rates is small, the impact of this differ- ∞ 2 + ence is small. + (S − K) dK. 2 30 (11) 2 S0 K 3. VIX separates puts from calls using the initial forward at their common maturity rather than the 252 initial spot. As long as short term interest rates Substituting (11)in(10) and multiplying by 30 im- plies that an estimate of the annualized daily variance and short term dividends are small, the impact of rate is: this difference is small. 4. VIX2 uses discrete strikes rather than a contin- 252 29 uum of strikes. As long as there is a fine grid and f 2 a wide range of S&P500 strikes trading, the im- 30 i i=0 pact of this difference is small. S 0 252 + This section has shown why VIX can properly be ≈ (K − S ) dK 2 30 0 15K regarded as a volatility index. In the next section, we ∞ focus on why VIX can also be regarded as a fear 252 + + (S − K) dK gauge. 2 30 S0 15K 29 252 1 1 + − (Si+1 − Si). (12) 3. VIX as a fear gauge 15 Si S0 i=0 There are a few obvious reasons why VIX can also In words, an estimate of the annualized daily variance be regarded as a fear gauge. First, the market as a is approximated by the P&L arising from combining whole is usually considered to be long the S&P500 and a static position in OTM puts and calls with daily dy- risk-averse. Suppose that the S&P500 moves up and namic trading in the index. down relative to its expected level by the same percent- Under zero interest rates and dividends, the last term age amount of this level. When the magnitude of these on the RHS of (12) can be synthesized at zero cost by daily moves increases from e.g. ±1% to ±2%, then holding 252 ( 1 − 1 ) units of the index and borrowing VIX obviously rises and the utility of risk-averse mar- 15 Si S0 the cost. In contrast, the first two terms on the RHS ket participants falls. Alternatively, if the daily moves of (12) each cost money to create. For K
OTM puts than OTM calls in synthesizing a variance It is also straightforward to value the short position in swap. the 2 zero strike calls on XT : Let XT ≡ ln(ST /S0) be the non-annualized con- tinuously compounded return over the period [0,T]. Q − − + Recall that the cost of creating the floating leg of a E0 2(XT 0) √ √ variance swap matches the cost of creating the path- √ σ T σ T independent payoff −2X : =−2σ TN + σ 2TN − . T 2 2 T (22) Q = Q − E0 Vt dt E0 ( 2XT ). (16) 0 Each position is the sum of two terms. The first term Now the path-independent payoff −2XT can be de- on the RHS of (21) is positive, while the first term on composed into the contribution from negative XT and the RHS of (22) is negative and has the same abso- the contribution from non-negative XT : lute value. Clearly, the first term on the RHS of (21) is larger than its counterpart in (22). The last term on −2XT =−2XT 1(XT < 0) − 2XT 1(XT 0) the RHS of (21) is larger than the last term on the + + RHS of (22), since N(·) is increasing. As a result, = 2[0 − XT ) − 2(XT − 0) . (17) the long position in the two zero strike puts is clearly 3 Substituting (17)in(16) implies that more valuable than the short position in the two zero strike calls. The sum of the two values is σ 2T , since √ √ T σ T + − σ T = Q N( 2 ) N( 2 ) 1. E0 Vt dt 0 = Q − + − Q − + 2E0 (0 XT ) 2E0 (XT 0) . (18) 4. Summary and future research
The results of Carr and Madan imply that: VIX is both a volatility index and a fear gauge. It Q + E 2(0 − X ) is well known that it arises as a volatility index be- 0 T 2 cause VIX is essentially the cost of replicating the 2 S0 2 floating leg of a variance swap. In this paper, we ar- = P (S ) + P (K) dK 0 0 2 0 (19) S0 −∞ K gued that the reason that the path-independent pay- XT − − off 2(e 1 XT ) has the same value as the path- and: = T 2 dependent payoff X T 0 (dXt ) is that incre- (eXt − − X ) Q − − + ments of 2 1 t capture the difference be- E0 2(XT 0) tween the forward and backward returns. When the un- 2 ∞ 2 derlying price process has positive continuous sample =− C (S ) + C (K) dK. (20) 0 0 2 0 paths in continuous time, the difference between the S0 S0 K forward and backward returns is just the variance rate Thus, the 2 zero strike puts on XT are created from at of X. and OTM puts on ST . Similarly, the short position in We also gave four reasons why VIX can properly the 2 zero strike calls on XT are created from at and be regarded as a fear gauge. First, assuming that risk- OTM calls on ST . averse investors are long the S&P500, increases in ex- In the Black Scholes model, it is straightforward to pected variance or risk aversion raise VIX and lower 2 value the 2 zero strike puts on XT : expected utility. Second, we argued that VIX T has the same value as the path-independent claim paying Q[ − ]+ ln(ST /S0) at T and that the latter claim has negative 2E0 0 XT √ √ delta. Third, the leverage effect implies that abnor- √ σ T 2 σ T = 2σ TN + σ TN . 3 2 2 Notice we are not claiming that the last term in (21) has greater value than the last term in (22). The inclusion of the ATM options on (21) ST is essential for our results. P. Carr / Why is VIX a fear gauge? 7 mally high VIX levels tend to be accompanied by ab- Acknowledgement normally low S&P500 levels. Fourth, when the Black Scholes model is holding, the VIX construction has I am grateful to Maggie Copeland and Charles more dollars invested in at and OTM puts than in at and Tapiero for comments. They are not responsible for any errors. OTM calls. If the at and OTM puts reflect fear while the at and OTM calls reflect greed, then VIX is more about fear than greed. When the negative skew is taken References into account, this fear to greed ratio increases. We con- clude that there are valid reasons for regarding VIX as [1] P. Carr and R. Lee, Volatility derivatives, Annual Review of both a volatility index and a fear gauge. Financial Economics 1 (2009), 319–339. doi:10.1146/annurev. financial.050808.114304. Future research should focus on weaker or alterna- [2] P. Carr and D. Madan, Optimal positioning in derivative se- tive sufficient conditions on the risk-neutral index dy- curities, Quantitative Finance 1 (2001), 19–37. doi:10.1080/ namics which lead to greater aggregate investment in 713665549. at and OTM puts than calls.