Implied Volatility of Leveraged ETF Options

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1 IEOR Dept. Columbia University joint work with Ronnie Sircar (Princeton) Cornell Financial Engineering Seminar Feb. 6, / 37

2 LETFs and Their Options Leveraged Exchange Traded Funds (LETFs) promise a fixed leverage ratio with respect to a given underlying asset or index. Most typical leverage ratios are: (long) 1,2,3, and (short) 1, 2, 3. Significant rise in the use of (L)ETFs and their options in the past decade. In 212, total notional on ETF options is $4-5 bil, as compared to $9 bil for S&P 5 index options. ETF options across 4 asset classes: equity, debt, commodity, and currency. 85% of the total ETF options volume is traded among only four ETFs: SPY (S&P 5), IWM (Russell 2), QQQ (Nasdaq 1) and GLD (gold). 2 / 37

3 Empirical LEFT Prices.15 SSO SDS.1 Cumulative Return /28/21 2/16/211 4/7/211 5/27/211 7/16/211 9/4/211 1/24/211 Figure : SSO and SDS cumulative returns from Dec 21 to Nov 211. Observe that both SSO and SDS can give negative returns simultaneously over several periods in time. 3 / 37

4 return of SSO return of SDS times return of SPY times return of SPY return of SSO return of SDS times return of SPY times return of SPY return of SSO return of SDS times return of SPY times return of SPY Figure : 1-day (left), 2-week (mid) and 2-month (right) returns of SPY against SSO (top) and SDS (bottom), in logarithmic scale. We considered 1-day, 2-week and 2-month rolling periods from Sept 29, 21 to Sept 3, / 37

5 return of UPRO times return of SPY return of UPRO times return of SPY return of UPRO times return of SPY.4.4 return of SPXU.1 return of SPXU return of SPXU times return of SPY times return of SPY times return of SPY Figure : 1-day (left), 2-week (mid) and 2-month (right) returns of SPY against UPRO (top) and SPXU (bottom), in logarithmic scale. We considered 1-day, 2-week and 2-month rolling periods from Sept 29, 21 to Sept 3, / 37

6 LETF Price Dynamics in the Black-Scholes Model Under the unique risk-neutral measure P, the reference asset price follows: dx t X t = rdt+σdw, with constant interest rate r and constant volatility σ. A long LETF L on X with leverage ratio β 1 is constructed by investing the amount βlt (β times the fund value) in X, borrowing the amount (β 1)Lt at the risk-free rate r, expense charge at rate c. For a short (β 1) LETF, $ β L t is shorted on X, and $(1 β)l t is kept in the money market account. The LETF price dynamics: dl t L t = β ( dxt X t ) ((β 1)r+c)dt = (r c)dt+βσdw t. 6 / 37

7 LETF Option Prices (Black-Scholes) The LETF value L can be written in terms of X: L t L = ( Xt X ) β e (r(β 1)+c)t β(β 1) 2 σ 2t. Over longer times, the volatility will lead to significant attrition in fund value, even if the underlying is performing well. The no-arbitrage price a European call option on L is: C (β) BS (t,l;k,t) = e r(t t) IE {(L T K) + L t = L} = C BS (t,l;k,t,r,c, β σ), where C BS (t,l;k,t,r,c,σ) is the Black-Scholes formula for a call. 7 / 37

8 atility (IV) of LETF Options Given the market price C obs of a call on L, the implied volatility is given by I (β) (K,T) = (C (β) BS ) 1 (C obs ) = 1 β C 1 BS (Cobs ). We normalize by the β 1 factor in our definition of implied volatility so that they remain on the same scale. Proposition The slope of the implied volatility curve admits the bound: e (r c)(t t) β L T t (1 N(d (β) 2 )) N (d (β) 1 ) I(β) (K) K e (r c)(t t) β L T t where d (β) 2 = d (β) 1 β I (β) (K) T t, with σ = I β (K). N(d (β) 2 ) N (d (β) 1 ), 8 / 37

9 Empirical atilities SPX, SPY Figure : SPX (blue cross) and SPY (red circles) implied volatilities on Sept 1, 21 for different maturities ( (from ) 17 to 472 days) plotted against log-moneyness: = log strike. (L)ETF price 9 / 37

10 Empirical atilities SPY, SSO SPY SSO SPY SSO Figure : SPY (blue cross) and SSO (red circles) implied volatilities against log-moneyness. 1 / 37

11 Empirical atilities SPY, SSO SPY SSO SPY SSO Figure : SPY (blue cross) and SSO (red circles) implied volatilities against log-moneyness. 11 / 37

12 Empirical atilities SPY, SSO SPY SSO Ratios SSO:SPY Figure : [Left] SPY (blue cross) and SSO (red circles) implied volatilities against log-moneyness. [Right] SSO:SPY implied volatility ratios for different maturities. 12 / 37

13 Empirical atilities SPY, SDS SPY SDS SPY SDS Figure : SPY (blue cross) and SDS (red circles) implied volatilities against log-moneyness () for increasing maturities. 13 / 37

14 Empirical atilities SPY, SDS SPY SDS SPY SDS Figure : SPY (blue cross) and SDS (red circles) implied volatilities against log-moneyness () for increasing maturities. 14 / 37

15 Empirical atilities SPY, SDS SPY SDS Ratios SDS:SPY Figure : [Left] SPY (blue cross) and SDS (red circles) implied volatilities against log-moneyness. [Right] SDS:SPY implied volatility ratios for different maturities. 15 / 37

16 Observations The most salient features of the empirical implied volatilities: IV skew for SSO appears to be flatter than that for SPY, IV skew is upward sloping for SDS, and downward sloping for SPY and SSO. Also, intuitively, a put on a long-letf and a call on a short-letf are both bearish, IVs should be higher for smaller (larger) for long (short) LETF. Traditionally, IV is used to compare option contracts across strikes & maturities. What about across leverage ratios? Which pair of LETF options should have comparable IV? 16 / 37

17 Observations The most salient features of the empirical implied volatilities: IV skew for SSO appears to be flatter than that for SPY, IV skew is upward sloping for SDS, and downward sloping for SPY and SSO. Also, intuitively, a put on a long-letf and a call on a short-letf are both bearish, IVs should be higher for smaller (larger) for long (short) LETF. Traditionally, IV is used to compare option contracts across strikes & maturities. What about across leverage ratios? Which pair of LETF options should have comparable IV? 16 / 37

18 Multiscale Stochastic Volatility Framework We assume that the reference index X, the LETF L, fast volatility factor Y and slow volatility factor Z are described by the system of SDEs: dx t = rx t dt+f(y t,z t )X t dw () t dl t = (r c)l t dt+βf(y t,z t )L t dw () t ( ) 1 dy t = dz t = ε α(y t) 1 η(y t )Λ 1 (Y t,z t ) ε ( δ l(z t ) ) δ g(z t )Λ 2 (Y t,z t ) dt+ 1 ε η(y t )dw (1) t dt+ δ g(z t )dw (2) t. Here, the standard IP -Brownian motions ( W (),W (1),W (2) ) are correlated: d W (),W (1) t = ρ 1 dt, d W (),W (2) t = ρ 2 dt, d W (1),W (2) t = ρ 12 dt, where ρ 1, ρ 2, ρ 12 < 1, and 1+2ρ 1 ρ 2 ρ 12 ρ 2 1 ρ 2 2 ρ 2 12 >. We call Λ 1 (y,z) and Λ 2 (y,z) the market prices of volatility risk. 17 / 37

19 ETF Option Price Approximation The no arbitrage price of a European option on X is given by P ε,δ (t,x t,y t,z t ) = IE { e r(t t) h(x T ) X t,y t,z t }. Instead of solving the high-dim PDE, we apply the perturbation theory, as discussed in Fouque et al. (211), to simplify and study the calibration problem. Proposition For fixed (t,x,y,z), the European option price P ε,δ (t,x,y,z) is approximated by P (t,x,z), where { ( P = PBS + τv δ +τv1 δ x )+ V ε ( 3 x σ x )} P BS x σ, and the order of accuracy is given by P ε,δ = P +O(εlog ε +δ). Here, PBS is the Black-Scholes call price with time-to-maturity τ = T t and the corrected volatility parameter σ. 18 / 37

20 ETF atility Approximation Using the price approximation, we can derive the 1st-order approximation for the IV. To do so, we define the variable Log-Moneyness to Maturity Ratio by Proposition MR = log(k/x). τ The 1st-order approximation for the IV is given by I = b +τb δ + ( a ε +τa δ) MR+O(εlog ε +δ), where (b,b δ,a ε,a δ ) are defined by b = σ + V ε 3 2σ b δ = V δ + V 1 δ 2 ( 1 2r σ 2 ( 1 2r σ 2 ) ), a ε = V 3 ε σ 3,, a δ = V δ 1 σ / 37

21 LETF Options Price Approximation The no-arb. price of a European LETF option on L is given by P ε,δ β (t,l t,y t,z t ) = IE { e r(t t) h(l T ) L t,y t,z t }. Goal: examine the role of β in the coeff. of LETF option price asymptotics. Proposition For fixed (t,x,y,z), the LETF option price P ε,δ β (t,x,y,z) is approximated by where P β = P,β BS + { P ε,δ β = P β +O(εlog ε +δ), τv δ,β +τv δ 1,β Here the β-dependent group market parameters are: ( x )+ V ε ( 3,β x σβ x ) } P,β BS x σ. V δ,β = β V δ, V δ 1,β = β β V δ 1, V ε 3,β = β 3 V ε 3, σ β = β σ. 2 / 37

22 LETF atility Approximation The IV of an LETF option I (β) is defined by I (β) = 1 β P 1 BS (P β). How does β impact the 1st-order approximation of the IV? Proposition The 1st-order approximation of I (β) is given by I (β) b β +τbδ β +( a ε β +τaδ β) MR, where the skew slopes are given in terms of the unleveraged ETF skew slopes by a ε β = 1 β aε, a δ β = 1 β aδ, and where the level parameters (b β,bδ β ) are b β = σ + βv 3 ε 2σ ( 1 2r ), b δ β 2 σ 2 β = V δ + βv 1 δ 2 ( 1 2r ). β 2 σ 2 21 / 37

23 Predicting from SPY IVs to LETF IVs SSO Prediction SDS Prediction UPRO Prediction SPXU Prediction / 37

24 Predicting from SPY IVs to LETF IVs SSO Prediction SDS Prediction UPRO Prediction SPXU Prediction / 37

25 Discrepancy of Calibrated IV Slope & Intercept, 9/9 9/1 3 SSO Slope Relative Error 3 SSO Intercept Relative Error SDS Slope Relative Error 4 SDS Intercept Relative Error UPRO Slope Relative Error 3 UPRO Intercept Relative Error SPXU Slope Relative Error 4 SPXU Intercept Relative Error / 37

26 IV Approximation for All Maturities.55 Fitted Surface and Data SSO.36 Fitted Surface and Data SDS atility atility MR MR Figure : LETF options IVs and their calibrated first-order approximations for all available maturities on August 23, / 37

27 IV Approximation for All Maturities.4 Fitted Surface and Data UPRO.32 Fitted Surface and Data SPXU atility.3 5 atility MR MR Figure : LETF options IVs and their calibrated first-order approximations for all available maturities on August 23, / 37

28 Examining the Slope Relationships We calibrate daily the skew parameters (a ε β,aδ β ) for β {1,±2,±3}, and compute the ratio R ij := aε β i +τa δ β i a ε β j +τa δ β j, i,j {1,±2,±3}, i j, for the 1 pairwise combinations. In theory, we should have R ij = β j /β i. Anticipating this might not hold precisely in the data, we look for any systematic deviation from these relationships. For each leverage, we calculate estimates of each β i as if β j were correct, and then average over the four estimates for each β i. That is, for each daily observation R ij, we compute the estimates β j i := β i R ij, βi j := β j /R ij, and take average to get β i = 1 β i j. 4 j i 27 / 37

29 Daily implied β 1 for SPY Day number Implied SSO β Implied SPY β Implied SDS β Implied UPRO β Implied SPXU β Figure : Implied β: Histograms of β i from 9/9-9/1 LETF IVs. 28 / 37

30 Average Implied β s from IVs SPY SSO SDS UPRO SPXU β mean β standard dev Table : The mean and standard deviation of estimated β over 9/9 9/1. SPY, SSO and, surprisingly, SPXU implied volatilities are consistent with their leverage ratios of 1, 2 and 3 respectively. IV skews from SDS (β = 2) systematically overestimate the magnitude of leverage ratio, as if the LETF was more short than it is actually supposed to be. Skews from UPRO (β = 3) systematically underestimate the leverage ratio, as if the LETF was not so ultra-leveraged. 29 / 37

31 Moneyness Scaling To link the IVs between ETF and LETF options, we introduce the method of moneyness scaling. In a general stochastic volatility model, we can write the log LETF price as ( ) ( ) LT XT β(β 1) T log = βlog (r(β 1)+c)T σ L X 2 tdt. 2 Conditioning on that the terminal (random) log-moneyness log( XT X ) to constant (1), we compute the cond l expectation (best estimate) of the β-letf log-moneyness. This leads us to define (β) := β (1) (r(β 1)+c)T β(β 1) 2 equal { T ( ) } IE σt 2 dt XT log = (1). X 3 / 37

32 Moneyness Scaling To link the IVs between ETF and LETF options, we introduce the method of moneyness scaling. In a general stochastic volatility model, we can write the log LETF price as ( ) ( ) LT XT β(β 1) T log = βlog (r(β 1)+c)T σ L X 2 tdt. 2 Conditioning on that the terminal (random) log-moneyness log( XT X ) to constant (1), we compute the cond l expectation (best estimate) of the β-letf log-moneyness. This leads us to define (β) := β (1) (r(β 1)+c)T β(β 1) 2 equal { T ( ) } IE σt 2 dt XT log = (1). X 3 / 37

33 Connecting Log-moneyness Assuming constant σ as in the B-S model, we have the formula: (β) = β (1) (r(β 1)+c)T β(β 1) σ 2 T. (1) 2 Hence, the β-letf log-moneyness (β) is expressed as an affine function of the unleveraged ETF log-moneyness (1), reflecting the role of β. The moneyness scaling formula can be interpreted via Delta matching. Proposition Under the B-S model, an ETF call with (1) and a β-letf call with (β) have the same Delta if and only if (1) holds. 31 / 37

34 Simulated IV Comparison via Moneyness Scaling.45.4 ETF LEFT (β=2) LEFT (β=3).45.4 ETF LEFT (β=2) LEFT (β=3) atility atility Log-Moneyness Log-Moneyness Figure : [Left] As β increases from 1 to 3, the IV skew becomes visibly flatter. [Right] After moneyness scaling, the IVs are significantly closer. 32 / 37

35 Empirical IV Comparison via Moneyness Scaling SPY SSO SPY SDS Figure : SPY (blue cross) and LETF (red circles) implied volatilities after moneyness scaling on Sept 1, 21 with 18 days to maturity, plotted against log-moneyness of SPY options. 33 / 37

36 Empirical IV Comparison via Moneyness Scaling SPY UPRO SPY SPXU Figure : SPY (blue cross) and LETF (red circles) implied volatilities after moneyness scaling on Sept 1, 21 with 18 days to maturity, plotted against log-moneyness of SPY options. 34 / 37

37 Concluding Remarks We have discussed a stochastic volatility framework to understand the inter-connectedness of LETF options. The method of moneyness scaling enhances the comparison of IVs with different leverage ratios. The connection allows us to use the richer unleveraged index/etf option data to shed light on the less liquid LETF options market. Related problems: long or short time-to-expiration, liquidity, tracking errors, and feedback effect created by ETFs and LETFs. 35 / 37

38 Appendix 36 / 37

39 LETF Options under Local Volatility Models Assume X follows the local volatility model: Then the β-letf L follows dx t X t = rdt+σ(t,x t )dw t. or equivalently, dl t L t = (r c)dt+βσ(t,x t )dw t. L t L = ( Xt X ) β e (r(β 1)+c)t β(β 1) 2 t σ(u,xu)2 du. The realized variance term t σ(u,x u) 2 du means that X t and L t do not have one-to-one correspondence. L no longer follows a local volatility model. 37 / 37

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