Implied Volatilities - PDF Free Download

Implied Volatilities Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 April 1, 2017 Christopher Ting QF 604 Week 2 April 1, 2017 1/59

Table of Contents 1 Introduction 2 VIX 3 Model-Free Formula 4 Proof 5 Market Realty 6 Proposed Method 7 Variance Risk Premiums 8 Takeaways Christopher Ting QF 604 Week 2 April 1, 2017 2/59

Two Uncertainties Market Uncertainty Price fluctuation = volatility Correlation of volatility with return Volatility Uncertainty Volatility is also fluctuating Volatility clusters Ex ante increase or decrease in volatility is a risk Christopher Ting QF 604 Week 2 April 1, 2017 3/59

Important Questions How to forecast ex ante volatility? Volatility index fear gauge VIX, VXD, VXN, RVX, VDAX-NEW, VSMI, VSTOXX,... How to estimate the volatility risk premium? Investors who bear the volatility risk demand a volatility risk premium. Investors who don t want to bear the higher volatility forcasted pay the volatility risk premium. Christopher Ting QF 604 Week 2 April 1, 2017 4/59

VIX: Fear Gauge? Christopher Ting QF 604 Week 2 April 1, 2017 5/59

What is VIX? Created in 1993 VIX is the ticker symbol for the CBOE Volatility Index for S&P 500 Index. VIX quantifies option traders expectation of future volatility for the next 30 calendar days. The old version of VIX relied on the Black-Scholes model to back out an implied volatility for each of the 8 options that are near-the-money. Old VIX is the average of these implied volatilities. Current new version is model free, and uses as many out-of-the-money S&P 500 index options as possible. The formula for model-free VIX is beautiful. Why called the fear gauge? Contributions of OTM put options are larger than OTM call options, S&P 500 Index tends to be a lot lower when VIX is higher. Christopher Ting QF 604 Week 2 April 1, 2017 6/59

Relation with the Underlying S&P 500 Index Christopher Ting QF 604 Week 2 April 1, 2017 7/59

VIX: A Success Story Speculation on the future level of volatility in a pure manner. short the VIX futures when VIX is unusually high long the VIX futures when VIX is unusually low Hedge against long equity exposure. Hedge against a high correlation market condition, which typically makes stock selection more difficult. Tracking of aggregate credit spread Tracking of carry trade benchmark Christopher Ting QF 604 Week 2 April 1, 2017 8/59

VXO and Dow Jones Industrial Average Index Christopher Ting QF 604 Week 2 April 1, 2017 9/59

VXN and Nasdaq 100 Index Christopher Ting QF 604 Week 2 April 1, 2017 10/59

Applications of VIX Volatility becomes a tradable asset class. CBOE offers futures and options on VIX revenue generation for the exchange. Speculation: Express a view on future volatility through trading. Hedging: Reduction of NAV fluctuation. VIX 2 as the fair value for a 30-day variance swap. The payoff function (same as P&L in this case) of this forward contract of amount A for the buyer is, at maturity T = 30, P&L of Buyer T = A ( Realized Variance - VIX 2), where the realized variance is the variance of future daily returns from day 0 up to T. Christopher Ting QF 604 Week 2 April 1, 2017 11/59

Implied Volatility Implied volatility used to be model-dependent. Black-Scholes option pricing formula Binomial tree Model risk All models are wrong... George Box A smile surface that extends well into the wings, which are suspect of model risk Model-free approach to implied volatility VIX discrete computations Academics smooth computations Christopher Ting QF 604 Week 2 April 1, 2017 12/59

Literature Jian and Tian (2005) Carr and Wu (2009) Lim and Ting (2013) Tee and Ting (2016) Christopher Ting QF 604 Week 2 April 1, 2017 13/59

Model-Free Variance σ 2 MF Direct computation given the midquotes of puts and call options σmft 2 := E Q ( ) 0 V (0, T ) = 2e rt ( F0 0 p(x, S 0, T ) X 2 dx + Strike price: X Time to maturity: T Underlying asset price at time 0: S 0 Forward price of the underlying asset: F 0 Risk-free rate of tenor T : r European put s midquote: p(x, S 0, T ) European call s midquote: c(x, S 0, T ) c(x, S 0, T ) F 0 X 2 ) dx (1) Christopher Ting QF 604 Week 2 April 1, 2017 14/59

Features of Model-Free Approach No requirement for an option pricing model = No model risk! No worry about parameters The only exogenous inputs are risk-free interest rate and dividend yields No bias σ MF reflects volatility across all out-of-the-money strike prices and thus reflects the option skew Uses both put and call options = σ MF is less sensitive to individual option prices. Christopher Ting QF 604 Week 2 April 1, 2017 15/59

Variance as a Difference of Two Returns With R t being the simple return, ln ( 1 + R t ) = Rt 1 2 R2 t + O(R 3 t ). In other words, the following approximation holds because at daily frequency or higher, R t is generally very small. R 2 t 2 ( R t ln ( 1 + R t )). (2) Since the mean E ( R t ) 0, E ( R 2 t ) V ( Rt ), i.e. R 2 t may be regarded as the variance σ 2 t of time t. Insight: Twice the difference between the simple return R t and the log return ln ( 1 + R t ) is the variance. Is it guaranteed that R t ln ( 1 + R t ) 0? YES! Christopher Ting QF 604 Week 2 April 1, 2017 16/59

Integrated Variance Next, we consider the integrated variance V (0, T ) defined as V (0, T ) := T 0 σ 2 t dt The variance V (0, T ) is the sum of instantaneous variances σ 2 t realized over time 0 to time T. From (2) T 0 σ 2 t dt = 2 T 0 R t dt 2 T 0 ln(1 + R t ) dt. (3) Christopher Ting QF 604 Week 2 April 1, 2017 17/59

Integrated Variance as Model-Free Variance Now, under the risk neutral measure Q, and assuming a risk-free rate r 0 that remains constant from time 0 to time T, E Q 0 ( T 0 ) T R t dt = E Q ( ) T 0 Rt dt = r 0 dt = r 0 T. 0 0 On the other hand, telescoping sum ln ( ) ( ) ST 1 ST + ln + ln = ln S T 2 0 S T 1 ( S1 ( ) S 0 ST S 0 ) + ln ( S2 S 1 ) + for tiny time interval t = 1 unit results in ( T ) ( ( )) E Q 0 ln(1 + R t ) dt = E Q ST 0 ln. S 0 Christopher Ting QF 604 Week 2 April 1, 2017 18/59

Under Risk-Neutral Measure Putting all terms together, we have σ 2 MF T := EQ 0 ( T 0 ) ( ( )) σt 2 dt = 2r 0 T 2E Q ST 0 ln. (4) S 0 Accordingly, σ 2 MFT := E Q 0 ( V (0, T ) ) = 2r0 T 2E Q 0 ( ( )) ST ln. (5) S 0 Christopher Ting QF 604 Week 2 April 1, 2017 19/59

Forward Price For the second term on the right-hand side in (5), we consider F 0 known at time t = 0, and we express ln(s T /F 0 ) as ( ) ( ST 1 ln = ln S T ln F 0 S T 1 ) + S T 1 F 0 F 0 F 0 = = ST 1 F 0 ST F 0 X dx S T S T X X 2 ST S T F 0 1 X 2 dx + S T F 0 1 dx + S T F 0 1. (6) For any z > 1, ln(1 + z) is a strictly concave function and ln(1 + z) < z. The left side of equation (6) is ln(1 + z) with z := S T F 0 1. Christopher Ting QF 604 Week 2 April 1, 2017 20/59

It follows that the integral hence is strictly positive. Strictly Positive ST F 0 S T X X 2 We can then rewrite the integral as dx equals z ln(1 + z) and ST F 0 S T X ST X 2 dx = 1 ST >F 0 F 0 = 1 ST >F 0 ST = F 0 F 0 (S T X) + X 2 dx + S T X F0 X 2 dx 1 ST <F 0 S T X X 2 dx + 1 ST <F 0 F0 0 S T F0 S T (X S T ) + S T X X 2 X S T X 2 dx dx X 2 dx. (7) In the last step, we have used the fact that the asset price S T, which is unknown at time t = 0, can potentially attain a low value 0, or appreciate substantially to a high value. Christopher Ting QF 604 Week 2 April 1, 2017 21/59

Risk-Neutral Expectation In view of (7), (6) becomes, under the risk-neutral measure Q, E Q 0 ( ( )) ST ln F 0 H = e rt C(S 0, X, T ) F 0 X 2 dx e rt ( ) +E Q ST 0 1 F 0 F0 L P (S 0, X, T ) X 2 dx = e rt H c(x, S 0, T ) F 0 X 2 dx e rt F0 L p(x, S 0, T ) X 2 dx. To arrive at this result, E Q 0 ( ST ) = F0 has been applied. Christopher Ting QF 604 Week 2 April 1, 2017 22/59

Last Step Finally, we write and substituting equation (8) into (5), we obtain σ 2 MFT = 2r 0 T + 2e rt ( H 2 ln F 0 S 0, ln S T S 0 = ln S T F 0 + ln F 0 S 0 (8) F 0 c(x, S 0, T ) X 2 dx + F0 L p(x, S 0, T ) X 2 where H is the highest strike price and L is the lowest strike price. Since F 0 = e r 0T S 0, the first and last terms cancel out and the model-free formula (1) is obtained. ) dx Christopher Ting QF 604 Week 2 April 1, 2017 23/59

Advantages and Limitation The model-free approach incorporates information from out-of-the-money puts and calls (with respect to the forward price F 0 ) to produce a single implied volatility σ MF for a given maturity. 1 Given the weight, out-of-the-money puts contribute more to K2 σ MF, hence fear gauge. The model-free approach to implied volatility is applicable only for European options. Equity index options are typically European but stock options are American. Christopher Ting QF 604 Week 2 April 1, 2017 25/59

Issues in Implementation Strike price is not continuous but discrete. Strike prices in the option chain are not from 0 to. Most options are illiquid and most have only ask prices but not bid prices. Christopher Ting QF 604 Week 2 April 1, 2017 26/59

Calls Puts Bid Ask Strike Bid Ask 28 32.8 40 0 4.8 23 27.8 45 0 4.8 20.5 25.3 47.5 0 4.8 18 22.8 50 0 4.8 15.5 20.3 52.5 0 4.8 13.1 17.9 55 0 4.8 10.6 15.4 57.5 0 4.8 8.1 12.9 60 0 4.8 5.6 10.4 62.5 0 4.8 3.2 8 65 0 4.8 1.15 5.8 67.5 0 4.8 0.05 4.8 70 0 4.8 0 4.8 72.5 0.25 4.8 0 4.8 75 2.15 6.9 0 4.8 77.5 4.6 9.4 0 4.8 80 7.1 11.9 0 4.8 82.5 9.6 14.4 0 4.8 85 12.1 16.9 0 4.8 90 17.1 21.9 0 5 95 21.5 26.5 0 5 100 26.5 31.5 Market Reality Option chain of BKX, KBW Nasdaq Bank Index (@ 70.61) Expiration: Sep 15, 2016 Source: Optionetics, as of Aug 22, 2016 Discrete strike price Limited strike range lowest strike L=$40 highest strike H=$100 Not liquid But quotes are firm, ready for trades Christopher Ting QF 604 Week 2 April 1, 2017 27/59

CBOE s Implementation According to a CBOE s white paper, the generalized formula used in the VIX calculation is σ 2 CBOE = 2er 0T K i T Ki 2 Q(K i ) 1 ( ) F 2 1, T K 0 i=1 where K 0 is the first strike below the forward index level, F ; K i is the strike price of the i-th out-of-the-money option; a call if K i > K 0 ; and a put if K i < K 0 ; both put and call if K i = K 0 ; K i is the interval between strike prices half the difference between the strike on either side of K i ; Q(K i ) is the midpoint of the bid-ask spread for each option with strike K i. Christopher Ting QF 604 Week 2 April 1, 2017 28/59

CBOE s Method in Detail SPX option selection criteria Out-of-the-money with respect to K 0 Non-zero bid price Once two puts (calls) with consecutive strike prices are found to have zero bid prices, no puts (calls) with lower (higher) strikes are considered for inclusion. Determine the forward SPX level, F Identify the strike price K s at which the absolute difference between the call and put prices is smallest. Then F = K s + e r0t ( c(k s ) p(k s ) ) determine K 0 as the strike price immediately below F Christopher Ting QF 604 Week 2 April 1, 2017 29/59

Current Academic Practice Compute the midquotes. Convert the midquotes into implied volatilities with either Black-Scholes or binomial tree pricing model. Interpolate the implied volatilities with a spline into a smirk/smile. Sample as many synthetic implied volatilities as possible from the volatility smirk/smile. Apply the pricing model to obtain the synthetic option prices. Christopher Ting QF 604 Week 2 April 1, 2017 30/59

Fully Model-Free and Exact Don t involve any option pricing models at all stages of computation. Rely exclusiely on put-call parity c 0 p 0 = S 0 e qt Ke rt. The synthetic option o k (X, T ) over any small sub-interval (X k, X k+1 ] of strikes is represented locally as a cubic polynomial function: o k (X, T ) = s k 1X 3 + s k 2X 2 + s k 3X + s k 4. (9) Christopher Ting QF 604 Week 2 April 1, 2017 31/59

Fully Model-Free and Exact (Cont d) Every cubic spline is defined by its coefficients s k 1 to sk 4. Integration over each sub-interval (X k, X k+1 ] admits a closed form expression: Xk+1 o k (X, T ) X k X 2 dx = s k Xk+1 2 X2 k 1 + s k 2 2(X k+1 X k ) ( ) ( + s k Xk+1 1 3 ln s k 4 1 ). X k X k+1 X k (10) Christopher Ting QF 604 Week 2 April 1, 2017 32/59

Exact Representation Let there be M sub-intervals for the integration from L to F 0 for puts, and N sub-intervals for the integration from F 0 to H for calls in the model-free formula, equation (1). We obtain an exact representation of equation (1) as follows: ( 1 σmft 2 = 2e rt i= M ( N 1 + 2e rt i=0 p i Xi+1 2 X2 i 1 2 + p i 3 ln c i Xi+1 2 X2 i 1 2 +c i 3 ln + p i 2(X i+1 X i ) ( Xi+1 X i ( Xi+1 ) ( 1 p i 4 1 )) X i+1 X i + c i 2(X i+1 X i ) X i ) ( 1 c i 4 1 )) (11) X i+1 X i Christopher Ting QF 604 Week 2 April 1, 2017 33/59

No Risk-Free Arbitrage For three strike prices X a, X b, and X c such that X a < X b < X c the conditions necessary for the absence of arbitrage are (I) Price monotonicity (II) Gradient bounds (III) Convexity p a p b ; c b c a. (12) 0 p b p a X b X a 1 ; 1 c b c a X b X a 0. (13) p b p a X b X a p c p b X c X b ; c b c a X b X a c c c b X c X b. (14) Christopher Ting QF 604 Week 2 April 1, 2017 34/59

Interpolation of Prices Satisfying the Three Constraints OptionMetrics IvyDB US database Christopher Ting QF 604 Week 2 April 1, 2017 35/59

Volatility Index To obtain the annualized volatility index σ for a fixed time horizon or constant maturity T, we interpolate the model-free variances σat 2 a and σb 2T b with T a < T < T b, where T a is strictly smaller than T b. At time 0, following the standard practice (see CBOE (2009)), the model-free volatility index σ is obtained by linear interpolation as follows: σ 2 T = σat 2 T b T a + σb 2 T b T T T T a b. (15) a T b T a The Actual/365 day-count convention is used to annualize the variance. Based on the UNG ETF example, we have T a = 22/365 and T b = 50/365. We obtain σ a = 20.81% and σ b = 24.20%. For 30-day constant maturity, i.e., T = 30/365, applying Equation (15) results in a model-free natural gas volatility index of 22.49% for August 30, 2013. Christopher Ting QF 604 Week 2 April 1, 2017 36/59

KBW Nasdaq Bank Index Christopher Ting QF 604 Week 2 April 1, 2017 37/59

BIX Christopher Ting QF 604 Week 2 April 1, 2017 38/59

No Dividend When a stock does not pay dividend before option maturity, American call option C(X, T ) does not have early exercise premium. So it has the same price as a corresponding call option c(x, T ). What about American put P (X, T )? Let E(X, T ) be the early exercise premium and p(x, T ) the corresponding European option component of P (X, T ). In other words, P (X, T ) = p(x, T ) + E(X, T ). Applying put-call parity, i.e., c(x, T ) (P (X, T ) E(X, T )) = S 0 Xe rt, The only unknown is E(X, T ). We calculate its value from the observed prices: E(X, T ) = S 0 Xe rt ( c(x, T ) P (X, T ) ). Christopher Ting QF 604 Week 2 April 1, 2017 39/59

ETFs of Four Important Commodities SPDR Gold Trust (GLD) Gold bullion ishares Silver Trust (SLV) Silver bars United States Oil Fund (USO) Near month contract to expire on light, sweet crude oil delivered to Cushing, Oklahoma, United States Natural Gas Fund (UNG) Near month contract to expire on natural gas delivered at the Henry Hub, Louisiana These ETFs do not pay dividend. Christopher Ting QF 604 Week 2 April 1, 2017 40/59

GLD and Its Volatility Index Christopher Ting QF 604 Week 2 April 1, 2017 41/59

SLV and Its Volatility Index Christopher Ting QF 604 Week 2 April 1, 2017 42/59

UNG and Its Volatility Index Christopher Ting QF 604 Week 2 April 1, 2017 43/59

USO and Its Volatility Index Christopher Ting QF 604 Week 2 April 1, 2017 44/59

Correlation Table: Correlation Table Correlation between the simple return on the volatility index and the underlying ETF s simple return. Underlying ETF Gold Silver Natural Gas Oil Volatility Index GLD SLV UNG USO Correlation 21.35% 32.44% 15.56% 40.25% Christopher Ting QF 604 Week 2 April 1, 2017 45/59

Early Exercise Premium Adjustment To examine the sensitivity and impact of early exercise premium on the model-free volatility, we need to have a common benchmark for the three early exercise premium adjustment methods on the same basis for meaningful comparison. We first calculate the model-free volatility index σ o by treating the American options as if they are European by ignoring the need to subtract the early exercise premium. We then calculate the model-free volatility indexes σ a using each of the following methods to account for the early exercise premium: 1 put-call parity method proposed in this paper 2 binomial tree model 3 Barone-Adesi and Whaley (BAW) approximation Define the distance from the benchmark σ o : D := σ o σ a Christopher Ting QF 604 Week 2 April 1, 2017 46/59

Results for GLD days to maturity # put-call parity binomial tree model Barone-Adesi & Whaley σ a D std(d) σ a D std(d) σ a D std(d) 7 T 30 1935 20.73 0.6186 1.0272 21.52 0.1084 0.5684 21.51 0.1071 0.5669 31 T 60 1698 21.98 0.6205 0.7992 22.65 0.0766 0.5321 22.65 0.0755 0.5318 61 T 90 1521 22.66 0.6180 0.6046 23.31 0.0713 0.2850 23.31 0.0692 0.2806 91 T 120 1221 22.15 0.5454 0.5368 22.69 0.0633 0.2491 22.70 0.0619 0.2490 121 T 150 908 22.86 0.5344 0.5832 23.35 0.0613 0.3306 23.35 0.0601 0.3295 151 T 180 846 23.16 0.4819 0.4748 23.63 0.0388 0.2361 23.63 0.0384 0.2362 181 T 210 790 23.72 0.4761 0.4962 24.17 0.0492 0.1801 24.17 0.0481 0.1800 211 T 240 859 23.84 0.5212 0.4806 24.31 0.0612 0.1963 24.31 0.0609 0.1967 241 T 270 816 23.86 0.4659 0.4642 24.28 0.0644 0.2057 24.28 0.0643 0.2058 271 T 300 677 23.00 0.4254 0.4387 23.37 0.0538 0.1929 23.37 0.0539 0.1929 301 T 330 724 23.23 0.4713 0.5218 23.64 0.0604 0.2039 23.64 0.0608 0.2045 331 T 360 707 23.53 0.4746 1.1385 23.90 0.0871 1.0962 23.90 0.0874 1.0962 361 T 390 369 24.36 0.5706 0.4892 24.85 0.0712 0.2718 24.84 0.0718 0.2727 391 T 420 168 26.48 0.6171 0.4546 27.04 0.0545 0.1245 27.04 0.0582 0.1307 421 T 450 131 27.92 0.6365 0.5085 28.47 0.0848 0.1598 28.46 0.0926 0.1685 451 T 480 131 29.34 0.5524 0.4398 29.85 0.0554 0.0682 29.84 0.0631 0.0803 481 T 510 126 27.93 0.6780 0.5426 28.55 0.0601 0.0800 28.54 0.0674 0.0870 511 T 540 147 26.43 0.7941 0.6014 27.10 0.1204 0.1839 27.09 0.1295 0.1909 7 T 540 13774 22.90 0.5511 0.6966 23.45 0.0711 0.4254 23.45 0.0706 0.4248 Christopher Ting QF 604 Week 2 April 1, 2017 47/59

Results for SLV days to maturity # put-call parity binomial tree model Barone-Adesi & Whaley σ a D std(d) σ a D std(d) σ a D std(d) 7 T 30 1532 36.63 2.4268 3.8516 35.44 3.7278 5.0259 35.44 3.7265 5.0243 31 T 60 1273 37.43 2.1373 2.6729 37.53 2.1606 3.1264 37.54 2.1589 3.1256 61 T 90 771 37.10 1.9411 2.1499 37.91 1.2617 1.8683 37.92 1.2593 1.8685 91 T 120 728 37.35 1.8509 2.3624 38.18 1.1051 2.1780 38.19 1.1024 2.1782 121 T 150 436 38.07 1.6076 1.4562 39.03 0.8879 1.0962 39.03 0.8861 1.0961 151 T 180 428 37.70 1.4262 1.2359 38.60 0.6936 1.0256 38.60 0.6920 1.0256 181 T 210 379 38.41 1.2416 1.6796 39.16 0.5840 0.7798 39.16 0.5818 0.7804 211 T 240 410 38.09 1.3096 1.1555 39.06 0.4388 0.7739 39.06 0.4382 0.7743 241 T 270 263 36.89 1.4495 1.2809 38.08 0.2931 0.4655 38.08 0.2924 0.4659 271 T 300 238 37.28 1.4380 1.2022 38.54 0.1739 0.3163 38.54 0.1741 0.3181 301 T 330 141 37.35 1.5518 1.2635 38.84 0.1762 0.3362 38.83 0.1779 0.3360 331 T 360 109 36.21 1.2003 1.1451 37.27 0.1520 0.2415 37.26 0.1529 0.2418 361 T 390 100 36.94 1.2557 1.3035 38.03 0.2150 0.3774 38.02 0.2182 0.3772 391 T 420 97 39.03 1.6441 1.4663 40.51 0.2130 0.3783 40.51 0.2145 0.3792 421 T 450 101 40.04 1.5642 1.2649 41.48 0.2252 0.3880 41.48 0.2293 0.3878 451 T 480 109 41.24 1.9607 1.3573 42.87 0.5219 0.8170 42.87 0.5273 0.8170 481 T 510 106 40.84 1.6723 1.4710 42.03 0.4287 0.5331 42.03 0.4324 0.5335 511 T 540 126 39.58 1.5696 1.4655 40.55 0.5576 0.7114 40.55 0.5627 0.7120 7 T 540 7347 37.51 1.8656 2.5066 37.89 1.5954 3.0989 37.89 1.5943 3.0980 Christopher Ting QF 604 Week 2 April 1, 2017 48/59

Results for UNG days to maturity # put-call parity binomial tree model Barone-Adesi & Whaley σ a D std(d) σ a D std(d) σ a D std(d) 7 T 30 926 45.86 3.4258 5.1764 45.19 4.3989 5.8875 45.19 4.3988 5.8895 31 T 60 1377 46.40 2.0768 2.9768 46.21 2.4487 3.3242 46.21 2.4486 3.3253 61 T 90 579 45.73 2.2414 2.9076 45.49 2.4645 3.6649 45.48 2.4740 3.6777 91 T 120 525 45.08 2.3169 3.9778 45.31 2.0443 3.9174 45.31 2.0439 3.9165 121 T 150 544 45.96 2.0246 2.4163 46.42 1.5892 2.7138 46.42 1.5905 2.7135 151 T 180 547 45.28 1.9402 2.2608 45.61 1.5803 2.4852 45.62 1.5780 2.4838 181 T 210 476 44.59 2.2884 2.8026 45.26 1.5614 2.7290 45.26 1.5650 2.7268 211 T 240 522 44.58 2.1949 2.4998 45.35 1.3681 2.5961 45.34 1.3721 2.5941 241 T 270 142 44.38 2.5960 2.6768 45.38 1.4564 2.3592 45.38 1.4584 2.3574 271 T 300 80 41.68 2.1431 2.1979 42.55 1.1769 1.0395 42.55 1.1765 1.0390 301 T 330 82 39.86 1.7372 2.2700 40.31 1.3286 1.5432 40.32 1.3238 1.5419 331 T 360 82 41.54 1.9589 2.1093 41.04 2.3165 1.7517 41.04 2.3147 1.7547 361 T 390 79 40.65 1.4594 1.9604 40.76 1.3149 1.1483 40.77 1.3083 1.1500 391 T 420 78 41.22 1.4860 1.9919 41.72 1.0503 1.0410 41.72 1.0512 1.0405 421 T 450 82 42.42 1.7640 2.5094 42.95 0.9533 0.9531 42.95 0.9550 0.9525 451 T 480 88 42.16 2.3511 2.4814 43.12 1.0141 0.9347 43.11 1.0166 0.9360 481 T 510 85 43.96 2.8328 2.8181 45.62 0.9287 0.8554 45.62 0.9303 0.8559 511 T 540 105 42.70 2.5946 3.0826 43.96 1.1651 1.4369 43.95 1.1693 1.4356 7 T 540 6399 45.15 2.3258 3.3113 45.29 2.2326 3.6638 45.29 2.2339 3.6652 Christopher Ting QF 604 Week 2 April 1, 2017 49/59

Results for USO days to maturity # put-call parity binomial tree model Barone-Adesi & Whaley σ a D std(d) σ a D std(d) σ a D std(d) 7 T 30 1469 35.61 2.2368 3.6300 36.85 1.3971 3.6547 36.85 1.3959 3.6551 31 T 60 1591 36.24 2.0718 1.8065 37.87 0.4987 1.3400 37.87 0.4972 1.3402 61 T 90 943 34.71 2.1229 1.3632 36.63 0.3157 0.6882 36.63 0.3150 0.6882 91 T 120 857 33.50 2.3755 1.4496 35.71 0.2494 0.4346 35.71 0.2492 0.4341 121 T 150 605 35.51 2.4383 1.7833 37.73 0.2446 0.6734 37.73 0.2467 0.6726 151 T 180 590 35.04 2.4516 1.7160 37.24 0.2106 0.4201 37.23 0.2139 0.4193 181 T 210 515 34.89 2.5587 1.6851 37.15 0.2115 0.4916 37.14 0.2169 0.4914 211 T 240 566 34.76 2.4466 1.7357 36.93 0.2284 0.5880 36.92 0.2345 0.5901 241 T 270 318 31.27 2.5282 1.6519 33.50 0.2231 0.3155 33.49 0.2253 0.3158 271 T 300 260 32.18 2.5354 1.7516 34.42 0.2561 0.6811 34.42 0.2585 0.6842 301 T 330 124 35.02 2.2831 2.6411 37.07 0.2831 0.6862 37.07 0.2863 0.6867 331 T 360 104 35.61 1.6873 1.3852 37.31 0.0912 0.1900 37.31 0.0919 0.1885 361 T 390 98 37.81 1.9864 1.6022 39.57 0.4147 1.3164 39.57 0.4172 1.3172 391 T 420 97 39.53 2.2059 1.3100 41.64 0.1315 0.3278 41.63 0.1393 0.3292 421 T 450 104 37.80 2.4890 1.6667 40.43 0.0497 0.0687 40.42 0.0562 0.0774 451 T 480 110 37.45 2.6577 2.1996 39.98 0.1159 0.2130 39.97 0.1251 0.2200 481 T 510 105 35.90 2.6957 2.5188 38.35 0.2486 0.6726 38.34 0.2617 0.6812 511 T 540 126 33.75 3.1846 2.9540 36.77 0.1712 0.2938 36.75 0.1857 0.3041 7 T 540 8582 35.09 2.3083 2.1859 37.01 0.4858 1.7339 37.01 0.4872 1.7337 Christopher Ting QF 604 Week 2 April 1, 2017 50/59

Variance Swap The model-free variance σ 2 MF is valued under the risk neutral measure Q: σ 2 MFT = E Q 0 [ T Therefore, σ 2 MF is the fixed leg of a variance swap. 0 ] σt 2 dt. (16) The floating leg of the m-day variance swap is computed using the spot prices S k from calendar day 0 up to calendar day m when the variance swap matures. Christopher Ting QF 604 Week 2 April 1, 2017 51/59

Variance Swap (Cont d) Denoting the end-of-day ETF price by S k, the annualized realized variance is defined as the average of the squared logarithmic returns: V = 252 N m N m k=1 [ ( )] 2 Sk ln. (17) S k 1 Here, the subscript k in the daily ETF price S k refers to the number of trading days from today. Variance swap payoff G = Notional Amount ( V σ 2 MF). Christopher Ting QF 604 Week 2 April 1, 2017 52/59

P&L of Using Different Variance Swap Rates Notional amount of G is $100. 30-day maturity Gold VIX 2 EEP µ std γ κ NW t stat 1 st 25 th 50 th 75 th 99 th CBOE -0.20 5.88 1.38 7.36-0.27-17.81-2.35-1.34 0.07 20.37 None 0.11 5.89 1.36 7.45 0.15-16.40-2.17-1.15 0.36 20.67 BT 0.18 5.88 1.36 7.47 0.24-16.39-2.06-1.09 0.40 20.82 BAW 0.18 5.88 1.36 7.47 0.24-16.39-2.06-1.08 0.40 20.82 PCP 0.48 5.80 1.52 7.41 0.65-15.22-1.89-0.91 0.60 20.93 µ: Average, γ: skewness, κ: kurtosis NW t stat: Newey-West (1987) adjusted t statistic Christopher Ting QF 604 Week 2 April 1, 2017 53/59

P&L of Using Different Variance Swap Rates (Cont d) Silver VIX 2 EEP µ std γ κ NW t stat 1 st 25 th 50 th 75 th 99 th CBOE -2.85 13.11 0.90 6.16-1.96-40.35-8.44-4.18-1.14 37.86 None -4.83 10.93 1.12 8.02-5.07-31.77-9.23-5.35-1.86 36.19 BT -2.97 10.16 1.37 9.25-3.38-29.45-7.45-4 -0.65 36.78 BAW -2.97 10.16 1.37 9.25-3.38-29.45-7.45-3.99-0.65 36.78 PCP -3.08 10.20 1.50 9.49-3.50-27.00-7.58-4.16-0.61 37.96 Christopher Ting QF 604 Week 2 April 1, 2017 54/59

P&L of Using Different Variance Swap Rates (Cont d) Natural Gas VIX 2 EEP µ std γ κ NW t stat 1 st 25 th 50 th 75 th 99 th None -4.10 9.42 0.42 5.36-4.81-28.83-8.96-4.03-0.87 24.42 BT -1.80 9.10 0.87 5.52-2.19-22.74-6.50-2.27 0.86 26.59 BAW -1.80 9.10 0.87 5.52-2.19-22.74-6.51-2.27 0.86 26.59 PCP -1.81 9.44 0.67 5.05-2.12-24.30-6.40-2.29 1.25 27.00 Christopher Ting QF 604 Week 2 April 1, 2017 55/59

P&L of Using Different Variance Swap Rates (Cont d) Oil VIX 2 EEP µ std γ κ NW t stat 1 st 25 th 50 th 75 th 99 th CBOE -4.10 7.16 0.19 6.73-7.22-25.40-7.09-4.41-1.26 19.69 None -3.80 7.25 0.49 7.00-6.56-22.93-6.87-4.36-1.23 22.53 BT -3.24 6.99 0.67 7.03-5.79-21.58-6.41-3.93-0.82 22.64 BAW -3.24 6.99 0.67 7.03-5.79-21.58-6.41-3.93-0.81 22.62 PCP -2.25 7.13 1.02 7.63-3.91-19.31-5.44-3.11 0.03 25.16 Christopher Ting QF 604 Week 2 April 1, 2017 56/59

Term Structure of Variance Risk Premiums EEP µ std γ κ NW t stat 1 st 25 th 50 th 75 th 99 th Gold None -5.16 4.00-0.71 3.19-12.85-15.65-6.99-4.43-2.13 3.40 BT -5.07 3.96-0.71 3.24-12.73-15.52-6.91-4.30-2.10 3.52 BAW -5.06 3.96-0.71 3.24-12.72-15.48-6.90-4.29-2.10 3.53 PCP -4.82 3.97-0.62 3.20-12.09-14.72-6.71-4.16-2.00 3.89 Silver None -3.38 7.35-0.19 1.83-4.06-17.74-8.24-3.35 4.08 7.88 BT -3.08 7.32-0.21 1.87-3.71-17.64-7.97-3.12 4.19 8.00 BAW -3.07 7.32-0.21 1.87-3.70-17.63-7.96-3.11 4.20 8.01 PCP -2.19 6.84-0.19 1.89-2.84-15.91-6.98-1.73 4.63 8.16 Natural None -4.32 6.66 0.01 1.88-5.82-18.03-9.55-5.39 2.22 6.25 Gas BT -3.83 6.50-0.03 1.90-5.29-17.81-9.00-4.61 2.56 6.29 BAW -3.82 6.50-0.03 1.90-5.28-17.77-8.98-4.60 2.57 6.29 PCP -3.80 6.50 0.02 1.96-5.27-17.71-8.47-5.13 2.40 6.78 Crude None -5.09 7.80 0.11 3.52-6.54-22.58-9.46-3.90-1.21 15.17 Oil BT -4.89 7.76 0.16 3.63-6.32-22.43-9.06-3.78-1.08 15.38 BAW -4.87 7.76 0.17 3.63-6.29-22.37-9.04-3.78-1.08 15.41 PCP -3.33 8.24 0.49 4.45-4.07-22.42-7.03-3.20-0.39 19.70 Christopher Ting QF 604 Week 2 April 1, 2017 57/59

Concluding Remarks Existing methods are not fully model-free. Our proposed fully model-free method is better and robust Term structure of volatilities can be constructed with the fully model-free method. BIX, an index for measuring the systematic risk in the U.S. banking sector. Early exercise premiums are significant in the context of computing volatility indexes from American options. Variance risk premiums are both statistically and economically significant. Christopher Ting QF 604 Week 2 April 1, 2017 58/59

Acceptance of a Quantitative Finance Model In the end, a theory is accepted not because it is confirmed by conventional empirical tests, but because researchers persuade one another that the theory is correct and relevant. Fischer Black (1986) Christopher Ting QF 604 Week 2 April 1, 2017 59/59