Implied Volatility of Leveraged ETF Options: Consistency and Scaling

Transcription

1 Implied Volatility of Leveraged ETF Options: Consistency and Scaling Tim Leung Industrial Engineering & Operations Research Dept Columbia University tl2497 Risk USA Post-Conference Workshop on ETFs Oct 23, 2014 Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

2 LETFs and Their Options Recall that Leveraged Exchange Traded Funds (LETFs) promise a fixed multiple of the returns of an index/underlying asset. Most typical leverage ratios are: (long) 1,2,3, and (short) 1, 2, 3. (L)ETFs have also led to increased trading of options written on (L)ETFs. In 2012, total daily notional on (L)ETF options is $40-50 bil, as compared to $90 bil for S&P 500 index options. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

3 LETF Price Dynamics Suppose the reference asset price follows ds t S t = µ t dt+σ t dw t, along with a constant interest rate r. A long LETF L on X with leverage ratio β > 1 is constructed by investing the amount βlt (β times the fund value) in S, 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 S, and $(1 β)l t is kept in the money market account. The LETF price dynamics: ( ) dl t dst = β ((β 1)r+c)dt L t S t = (βµ t [(β 1)r+c])dt+βσ t dw t. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

4 LETF Option Price The LETF value L can be written in terms of S: L T L 0 = ( ST S 0 or equivalently in log-returns: ) β T e (r(β 1)+c)T β(β 1) 2 0 σ2 t dt, ( ) ( ) LT ST β(β 1) T log = βlog (r(β 1)+c)T σt 2 L 0 S 0 2 dt. 0 Hence, the realized volatility will lead to attrition in fund value. If σ is constant, then L is lognormal, and the no-arb price a European call option on L is C (β) BS (t,l;k,t) = e r(t t) E Q {(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. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

5 Implied Volatility (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 For every fixed T > 0, the slope of the IV curve admits the (model-free) bounds: 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 ), Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

6 The observed and B-S call prices must be decreasing in K: C obs K = C BS K (t,l;k,t,r,c, β I(β) (K)) = C BS K ( β I(β) (K))+ β C BS σ ( β I(β) (K)) I(β) (K) K 0. Since the B-S Vega CBS σ > 0, we rearrange to get I (β) (K) K 1 C BS / K β C BS / σ ( β I(β) (K)), where the partial derivatives are evaluated with volatility parameter ( β I (β) (K)). Similarly, put prices are increasing in K, implying I (β) (K) K 1 P BS / K β P BS / σ ( β I(β) (K)), where P BS is the B-S put price. Puts and calls with the same K and T have the same IV I (β) (K), hence the upper & lower bounds. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

7 Empirical Implied Volatilities SPX, SPY (+1) Implied Vol 0.4 Implied Vol 0.4 Implied Vol Implied Vol LM LM LM LM Implied Vol 0.4 Implied Vol 0.4 Implied Vol Implied Vol LM LM LM LM Figure : SPX (blue cross) and SPY (red circles) implied volatilities on Sept 1, 2010 for different maturities ( (from ) 17 to 472 days) plotted against log-moneyness: LM = log strike. (L)ETF price Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

8 Empirical Implied Volatilities SPY (+1), SSO (+2) SPY SSO SPY SSO 0.4 Implied Vol Implied Vol LM LM Figure : SPY (blue cross) and SSO (red circles) implied volatilities against log-moneyness. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

9 Empirical Implied Volatilities SPY (+1), SSO (+2) SPY SSO SPY SSO Implied Vol Implied Vol LM LM Figure : SPY (blue cross) and SSO (red circles) implied volatilities against log-moneyness. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

10 Empirical Implied Volatilities SPY (+1), SDS ( 2) Implied Vol Implied Vol SPY SDS 0.2 SPY SDS LM LM Figure : SPY (blue cross) and SDS (red circles) implied volatilities against log-moneyness (LM) for increasing maturities. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

11 Empirical Implied Volatilities SPY (+1), SDS ( 2) SPY SDS SPY SDS Implied Vol Implied Vol LM LM Figure : SPY (blue cross) and SDS (red circles) implied volatilities against log-moneyness (LM) for increasing maturities. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

12 Observations The most salient features of the empirical implied volatilities: IV skew for SSO (+2) is flatter than that for SPY, IV skew is downward sloping for long ETFs (e.g. SPY, SSO, UPRO), IV is skew is upward sloping for short ETFs (e.g. SDS, SPXU). Intuitively, a put on a long-letf and a call on a short-letf are both bearish, IVs should be higher for smaller (larger) LM for long (short) LETF. Traditionally, IV is used to compare option contracts across strikes & maturities. What about IVs across leverage ratios? Which pair of LETF options should have comparable IV? Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

13 Local-Stochastic Framework Assume zero interest rate, dividend rate, fee. Under the risk-neutral measure, we model the reference index S by the SDEs: S t = e Xt, (log-price) dx t = 1 2 σ2 (t,x t,y t )dt+σ(t,x t,y t )dwt x, (stoch. vol) dy t = c(t,x t,y t )dt+g(t,x t,y t )dw y t, d W x,w y t = ρ(t,x t,y t )dt. Then, the LETF price L follows L t = e Zt, dz t = 1 2 β2 σ 2 (t,x t,y t )dt+βσ(t,x t,y t )dw x t. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

14 Local-Stochastic Framework Stochastic Volatility: When σ and ρ are functions of (t,y) only, such as Heston, then options written on Z can be priced using (Y,Z) only. Local Volatility: if both σ and ρ are dependent on (t,x) only, such as CEV, then the ETF follows a local vol. model but not the LETF. Options on Z must be analyzed in conjunction with X. Local-Stochastic Volatility: If σ and/or ρ depend on (x, y), such as SABR, then to analyze options on Z, one must consider the triple (X,Y,Z). Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

15 LETF Option Price With a terminal payoff ϕ(z T ) = (e ZT e k ) + or (e k e ZT ) +, the LETF option price is given by the risk-neutral expectation u(t,x,y,z) = E[ϕ(Z T ) X t = x,y t = y,z t = z]. The price function u satisfies the Kolmogorov backward equation ( t +A(t))u = 0, u(t,x,y,z) = ϕ(z), where the operator A(t) is given by A(t) = a(t,x,y) (( 2 x x ) +β 2 ( 2 z z ) +2β x z ) with coefficients +b(t,x,y) 2 y +c(t,x,y) y +f(t,x,y)( x y +β y z ), a(t,x,y) = 1 2 σ2 (t,x,y), b(t,x,y) = 1 2 g2 (t,x,y), f(t,x,y) = g(t,x,y)σ(t,x,y)ρ(t,x,y). Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

16 Asymptotic Expansion of Option Price Expand the coefficients (a,b,c,f) of the operator A(t) as a Taylor series. For χ {a,b,c,f}, we write χ(t,x,y) = n=0k=0 χ n k,k (t) = n k x yχ(t, x,ȳ) k. (n k)!k! n χ n k,k (t)(x x) n k (y ȳ) k, Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

17 Asymptotic Expansion of Option Price By direct substitution, the operator A(t) can now be written as A(t) = A 0 (t)+b 1 (t), B 1 (t) = A n (t), n=1 where n A n (t) = (x x) n k (y ȳ) k A n k,k (t), k=0 A n k,k (t) = a n k,k (t) (( x 2 x) +β 2 ( z 2 ) ) z +2β x z +b n k,k (t) y 2 +c n k,k (t) y +f n k,k (t)( x y +β y z ), The price function now satisfies ( t +A 0 (t))u(t) = B 1 (t)u(t), u(t) = ϕ. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

18 Asymptotic Expansion of Option Price We define ū N our Nth order approximation of u by R ū N = N u n, n=0 where the zeroth order term is ( ) 1 (ζ m(t,t)) 2 u 0 (t) = dζ 2πs2 (t,t) exp 2s 2 ϕ(ζ), (t,t) T T m(t,t) = z β 2 dt 1 a 0,0 (t 1 ), s 2 (t,t) = 2β 2 dt 1 a 0,0 (t 1 ). t t Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

19 Asymptotic Expansion of Option Price And the higher order terms are given by where L n (t,t) = n k=1 T t dt 1 T u n (t) = L n (t,t)u 0 (t), t 1 dt 2 T t k 1 dt k i I n,k G i1 (t,t 1 )G i2 (t,t 2 ) G ik (t,t k ), with G n (t,t i ) := M x (t,t i ) := x+ M y (t,t i ) := y + n (M x (t,t i ) x) n k (M y (t,t i ) ȳ) k A n k,k (t i ) k=0 ti t ti t ) ds (a 0,0 (s)(2 x +2β z 1)+f 0,0 (s) y, ( ) ds f 0,0 (s)( x +β z )+2b 0,0 (s) y +c 0,0 (s), I n,k = {i = (i 1,i 2,,i k ) N k : i 1 +i 2 + +i k = n}. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

20 Implied Volatility Expansion The Black-Scholes Call price u BS : R + R + is given by u BS (σ) := e z N(d + (σ)) e k N(d (σ)), d ± (σ) := 1 σ τ where τ = T t, and N is the standard normal CDF. ( ) z k ± σ2 τ, 2 For fixed (t,t,z,k), the implied volatility corresponding to a call price u ((e z e k ) +,e z ) is defined as the unique strictly positive real solution σ of the equation We consider an expansion of the implied volatility u BS (σ) = u. (1) σ = η, η := σ 0 + σ n. n=1 Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

21 Implied Volatility Expansion Recall that the price expansion is of the form: u = u BS (σ 0 )+ n=1 u n. On the other hand, one expands u BS (σ) as a Taylor series about the point σ 0 u BS (σ) = u BS (σ 0 +η) = u BS (σ 0 )+η σ u BS (σ 0 )+ 1 2! η2 2 σu BS (σ 0 )+ 1 3! η3 3 σu BS (σ 0 )+... In turn, one can solve iteratively for every term of (σ n ) n 1. We have a general expression for the nth term. The first two terms are u 1 σ 1 = σ u BS (σ 0 ), σ 2 = u σ2 1 σu 2 BS (σ 0 ) σ u BS, (σ 0 ) which can be simplified using the explicit expressions of u BS and u BS σ. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

22 Implied Volatility Expansion In the time-homogeneous LSV setting, we write down up to the 1st order terms: σ 0 = β 2a 0,0, σ 1 = σ 1,0 +σ 0,1, where ( ) ( ) βa1,0 1 (1st order term) σ 1,0 = λ+ 2σ 0 4 ( 1+β)σ 0a 1,0 τ, ( β 3 ) ( a 0,1 f 0,0 β 2 ) a 0,1 (2c 0,0 +βf 0,0 ) (1st order term) σ 0,1 = 2σ0 3 λ+ τ, 4σ 0 (log-moneyness) λ =k z, (time-to-maturity) τ = T t. These expressions show explicitly the non-trivial dependence of IV on the leverage ratio β. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

23 Implied Volatility Scaling Let σ Z (τ,λ) (resp. σ X (τ,λ)) be the implied volatility of a call written on the LETF Z (resp. X). From the above IV expressions, we have LETF : σ Z β ( ) a 1,0 2a 0,0 + β 2 + a 0,1f 0,0 λ 2a 0,0 2(2a 0,0 ) 3/2 β +O(τ), ETF : σ X ( ) a 1,0 2a 0, a 0,1f 0,0 λ+o(τ). 2a 0,0 2(2a 0,0 ) 3/2 Two effects of β: the vertical axis of σ Z is scaled by a factor of β. Second, the horizontal axis is scaled by 1/β. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

24 Implied Volatility Scaling This motivates us to introduce σ (β) X which we define as σ (β) X (τ,λ) := β σ X(τ,λ/β), They offer two ways to link the IV surfaces σ X and σ Z. and σ(β) Z, the scaled implied volatilities, σ (β) 1 Z (τ,λ) := β σ Z(τ,βλ). Viewed one way, the ETF IV σ X (τ,λ) should roughly coincide with the LETF implied volatility 1 β σ Z(τ,βλ). Conversely, the LETF implied volatility σ Z (τ,λ) should be close to the ETF implied volatility β σ X (τ,λ/β). In summary, we have σ Z (τ,λ) σ (β) X (τ,λ), σ X(τ,λ) σ (β) Z (τ,λ). Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

25 Empirical IV Scaling SPY 0.15 SSO SDS SPY SSO SDS Figure : Left: Empirical IV σ Z(τ,λ) plotted as a function of log-moneyness λ for SPY (red, β = +1), SSO (purple, β = +2), and SDS (blue, β = 2) on August 15, 2013 with τ = 155 days to maturity. Note that the implied volatility of SDS is increasing in the LETF log-moneyness. Right: Using the same data, the scaled LETF implied volatilities σ (β) Z (τ,λ) nearly coincide. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

26 Predicting from SPY IVs to LETF IVs UPRO Prediction Implied Vol Implied Vol SPXU Prediction LM LM Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

27 Alternative IV Scaling In a general stochastic volatility model, the log LETF price is ( ) ( ) LT XT β(β 1) T log = βlog (r(β 1)+c)T σ L 0 X 0 2 tdt. 2 0 Key idea: Condition on that the terminal LM log( XT LM (1). X 0 ) equal to constant Then, the best estimate of the β-letf s LM is given by the cond l expectation: LM (β) := βlm (1) (r(β 1)+c)T { β(β 1) T ( ) } E σt 2 2 dt XT log = LM (1). 0 X 0 Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

28 Connecting Log-moneyness Assuming constant σ as in the B-S model, we have the formula: LM (β) = βlm (1) (r(β 1)+c)T β(β 1) σ 2 T. (2) 2 Hence, the β-letf log-moneyness LM (β) is expressed as an affine function of the unleveraged ETF log-moneyness LM (1), reflecting the role of β. To link the LM pair (LM (β),lm (ˆβ) ) different leverage ratios (β, ˆβ), we have LM (β) = βˆβ ( LM (ˆβ) +(r(ˆβ 1)+ĉ)T + ˆβ(ˆβ ) 1) σ 2 T 2 β(β 1) (r(β 1)+c)T σ 2 T. 2 Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

29 Empirical IV Comparison via Moneyness Scaling SPY SSO SPY SSO Implied Vol Implied Vol LM LM Figure : SPY (blue cross) and LETF (red circles) implied volatilities after moneyness scaling on Sept 1, 2010 with 108 days to maturity, plotted against log-moneyness of SPY options. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

30 Example: CEV Model The CEV model (in log prices) is described by dx t = 1 2 δ2 e 2(γ 1)Xt dt+δe (γ 1)Xt dwt x, X 0 = x := logs 0. dz t = 1 2 β2 δ 2 e 2(γ 1)Xt dt+βδe (γ 1)Xt dwt x, Z 0 = z := logl 0, with γ 1. The generator of (X,Z) is given by A = 1 2 δ2 e 2(γ 1)x( ( x 2 x )+β 2 ( z 2 ) z )+2β x z. a(x,y) = 1 2 δ2 e 2(γ 1)x, b(x,y) = 0, c(x,y) = 0, f(x,y) = 0. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

31 Example: CEV Model The first 3 terms in the IV expansion are σ 0 = β e 2x(γ 1) δ 2, σ 1 = τ(β 1)(γ 1)σ3 0 4β 2 + (γ 1)σ 0 (k z), 2β σ 2 = τ(γ ( 1)2 σ0 3 4β 2 +t(13+2β( 13+6β))σ0) 2 96β 4 + 7τ(β 1)(γ 1)2 σ β 3 (k z)+ (γ 1)2 σ 0 12β 2 (k z) 2, The factor (γ 1) appears in every term of these expressions. If γ = 1, σ 0 = β δ and σ 1 = σ 2 = σ 3 = 0. The higher order terms also vanish since a(x,y) = 1 2 δ2 in this case. Hence, just as in the B-S case, the IV expansion becomes flat. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

32 IV Comparison Figure : Exact (solid computed by Fourier inversion) and approximate (dashed) scaled IV σ (β) Z (τ,λ) under CEV, plotted against λ. For comparison, we also plot the ETF s exact CEV IV σ X(τ,λ) (dotted). Parameters: δ = 0.2, γ = 0.75, x = 0. Left: β = +2. Right: β = 2. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

33 IV Comparison Figure : Exact (solid computed by Fourier inversion) and approximate (dashed) scaled IV σ (β) Z (τ,λ) under CEV, plotted against λ. For comparison, we also plot the ETF s exact CEV IV σ X(τ,λ) (dotted). Parameters: δ = 0.2, γ = 0.75, x = 0. Left: β = +2. Right: β = 2. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

34 Example: SABR Model The SABR model (in log prices) is described by dx t = 1 dt+e 2 e2yt+2(γ 1)Xt Yt+(γ 1)Xt dwt x, dy t = 1 2 δ2 dt+δdw y t, dz t = 1 2 β2 e 2Yt+2(γ 1)Xt dt+βe Yt+(γ 1)Xt dwt x, d W x,w y t = ρdt. The generator of (X,Y,Z) is given by A = 1 2 e2y+2(γ 1)x( ( x 2 x)+β 2 ( z 2 ) z)+2β x y 1 2 δ2 y δ2 y 2 +ρδey+(γ 1)x ( x y +β y z ). a(x,y) = 1 2 e2y+2(γ 1)x, b = 1 2 δ2, c = 1 2 δ2, f(x,y) = ρδe y+(γ 1)x. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

35 Example: SABR Model The first 3 terms in the IV expansion are σ 0 = β e 2y+2(γ 1)x, where σ 1 = σ 1,0 +σ 0,1, σ 2 = σ 2,0 +σ 1,1 +σ 0,2. σ 1,0 = τ( 1+β)(γ 1)σ3 0 4β 2 + (γ 1)σ 0 (k z), 2β σ 0,1 = 1 4 τδσ 0(δ ρsign[β]σ 0 )+ 1 δρsign[β](k z). 2 Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

36 Example: SABR Model σ 2,0 = τ(γ ( 1)2 σ0 3 4β 2 +τ(13+2β( 13+6β))σ0) 2 96β 4 + 7τ( 1+β)(γ 1)2 σ β 3 (k z)+ (γ 1)2 σ 0 12β 2 (k z) 2, σ 1,1 = τ(γ 1)δσ2 0 (12ρ β +τσ 0 ( 9( 1+β)δ +( 11+10β)ρsign[β]σ 0 )) 48β ) 2 τ(γ 1)δσ 0 (3δ + 5(1 2β)ρσ0 β (k z), 24β σ 0,2 = 1 ( 96 τδ2 σ τδ 2 12ρ 2 ( ( +2τσ 0 7δρsign[β]+ 2+6ρ 2 ) )) σ τδ2 ρ(δsign[β] 3ρσ 0 )+ δ2 ( 2 3ρ 2 ) 12σ 0 (k z) 2, Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

37 IV Comparison Figure : Exact (solid computed by Fourier inversion) and approximate (dashed) scaled IV σ (β) Z (τ,λ) under SABR, plotted against λ. For comparison, we also plot the ETF s exact SABR IV σ X(τ,λ) (dotted). Parameters: δ = 0.5, γ = 0.5, ρ = 0.0 x = 0, y = 1.5. Left: β = +2. Right: β = 2. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

38 IV Comparison Figure : Exact (solid computed by Fourier inversion) and approximate (dashed) scaled IV σ (β) Z (τ,λ) under SABR, plotted against λ. For comparison, we also plot the ETF s exact SABR IV σ X(τ,λ) (dotted). Parameters: δ = 0.5, γ = 0.5, ρ = 0.0 x = 0, y = 1.5. Left: β = +2. Right: β = 2. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

39 Example: Heston Model The Heston model (in log prices) is described by dx t = 1 dt+e 1 2 eyt 2 Yt dwt x, X 0 = x := logs 0, dy t = ( (κθ 1 2 δ2 )e Yt κ ) dt+δe 1 2 Yt dw y t, Y 0 = y := logv 0, dz t = β 21 dt+βe 1 2 eyt 2 Yt dwt x, Z 0 = z := logl 0, d W x,w y t = ρdt. The generator of (X,Y,Z) is given by A = 1 2 ey( ( x 2 x)+β 2 ( z 2 ) z)+2β x z + ( (κθ 1 2 δ2 )e y κ ) y δ2 e y y 2 +ρδ( x y +β x z ), a(x,y) = 1 2 ey, b(x,y) = 1 2 δ2 e y, c(x,y) = ( (κθ 1 2 δ2 )e y κ ), f(x,y) = ρδ. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

40 Example: Heston Model The first two terms in the IV expansion are σ 0 = β e y, σ 1 = τ ( β 2( δ 2 2θκ ) ) +( 2κ+βδρ)σ0 2 8σ 0 + βδρ 4σ 0 (k z). Note that when X has Heston dynamics with parameters (κ, θ, δ, ρ, y), then Z also admits Heston dynamics with parameters (κ Z,θ Z,δ Z,ρ Z,y Z ) = (κ,β 2 θ, β δ,sign(β)ρ,y +logβ 2 ). This can also be inferred from our IV expressions. Indeed, the coefficients dependence on β is present only in the terms β 2 θ, β δ,sign(β)ρ,y +logβ 2. For instance, we can write σ 0 = e y+logβ2 = e yz, and the coeff. of (k z) in σ 1 is βδρ/4σ 0 = β δsign(β)ρ/4σ 0 = δ Z ρ Z /4σ 0. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

41 IV Comparison Figure : Exact (solid computed by Fourier inversion) and approximate (dashed) scaled IV σ (β) Z (τ,λ) under Heston, plotted against λ. For comparison, we also plot the ETF s exact Heston IV σ X(τ,λ) (dotted). Parameters: κ = 1.15, θ = 0.04, δ = 0.2, ρ = 0.4, y = logθ, τ = Left: β = +2. Right: β = 2. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

42 IV comparison Figure : Exact (solid computed by Fourier inversion) and approximate (dashed) scaled IV σ (β) Z (τ,λ) under Heston, plotted against λ. For comparison, we also plot the ETF s exact Heston IV σ X(τ,λ) (dotted). Parameters: κ = 1.15, θ = 0.04, δ = 0.2, ρ = 0.4, y = logθ, τ = 0.5. Left: β = +2. Right: β = 2. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

43 Concluding Remarks We have discussed a stochastic volatility framework to understand the inter-connectedness of LETF options. Explicit price and IV expansions are provided for Heston, CEV, and SABR models. 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. Our procedure can be applied to identify IV discrepancies across LETF options markets. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44

44 References T. Leung and R. Sircar, 2014 Implied Volatility of Leveraged ETF Options Applied Mathematical Finance, forthcoming. T. Leung, M. Lorig and A. Pascucci, 2014 Leveraged ETF Implied Volatilities from ETF Dynamics Submitted for publication. Tim Leung (Columbia Univ.) Implied Volatility of Leveraged ETF Options Oct 23, / 44