Implied Volatilities

Transcription

1 Implied Volatilities Christopher Ting Christopher Ting : christopherting@smu.edu.sg : : LKCSB 5036 April 1, 2017 Christopher Ting QF 604 Week 2 April 1, /59

2 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, /59

3 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, /59

4 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, /59

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

6 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, /59

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

8 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, /59

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

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

11 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, /59

12 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, /59

13 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, /59

14 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, /59

15 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, /59

16 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, /59

17 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, /59

18 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, /59

19 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, /59

20 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, /59

21 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, /59

22 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, /59

23 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, /59

24 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. The formula is beautiful! Christopher Ting QF 604 Week 2 April 1, /59

25 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, /59

26 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, /59

27 Calls Puts Bid Ask Strike Bid Ask 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, /59

28 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, /59

29 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, /59

30 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, /59

31 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, /59

32 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 ln s k 4 1 ). X k X k+1 X k (10) Christopher Ting QF 604 Week 2 April 1, /59

33 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 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, /59

34 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, /59

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

36 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, Christopher Ting QF 604 Week 2 April 1, /59

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

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

39 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, /59

40 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, /59

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

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

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

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

45 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, /59

46 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, /59

47 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 T T T T T T T T T T T T T T T T T T Christopher Ting QF 604 Week 2 April 1, /59

48 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 T T T T T T T T T T T T T T T T T T Christopher Ting QF 604 Week 2 April 1, /59

49 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 T T T T T T T T T T T T T T T T T T Christopher Ting QF 604 Week 2 April 1, /59

50 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 T T T T T T T T T T T T T T T T T T Christopher Ting QF 604 Week 2 April 1, /59

51 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, /59

52 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, /59

53 P&L of Using Different Variance Swap Rates Notional amount of G is $ day maturity Gold VIX 2 EEP µ std γ κ NW t stat 1 st 25 th 50 th 75 th 99 th CBOE None BT BAW PCP µ: Average, γ: skewness, κ: kurtosis NW t stat: Newey-West (1987) adjusted t statistic Christopher Ting QF 604 Week 2 April 1, /59

54 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 None BT BAW PCP Christopher Ting QF 604 Week 2 April 1, /59

55 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 BT BAW PCP Christopher Ting QF 604 Week 2 April 1, /59

56 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 None BT BAW PCP Christopher Ting QF 604 Week 2 April 1, /59

57 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 BT BAW PCP Silver None BT BAW PCP Natural None Gas BT BAW PCP Crude None Oil BT BAW PCP Christopher Ting QF 604 Week 2 April 1, /59

58 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, /59

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, /59