On VIX Futures in the rough Bergomi model

Transcription

1 On VIX Futures in the rough Bergomi model Oberwolfach Research Institute for Mathematics, February 28, 2017 joint work with Antoine Jacquier and Claude Martini

2 Contents VIX future dynamics under rbergomi Forward variance curve dynamics in rbergomi Upper and lower bounds for VIX futures Simulation scheme for VIX under rbergomi and log-normal approximations VIX futures calibration S&P 500 options calibration

3 rbergomi setting (Bayer, Friz and Gatheral): ds t = S t r t dt + S t Vt dw t, S 0 > 0 V t = ξ 0 (t)e (2νC H V t ), ν, ξ 0 ( ) > 0 V t = t dz u 0 (t u) 1/2 H dz t dw t = ρdt, ρ (0, 1)

4 rbergomi setting (Bayer, Friz and Gatheral): ds t = S t r t dt + S t Vt dw t, S 0 > 0 V t = ξ 0 (t)e (2νC H V t ), ν, ξ 0 ( ) > 0 V t = t dz u 0 (t u) 1/2 H dz t dw t = ρdt, ρ (0, 1) Focus on VIX [ VIX 2 ] E Q 1 0 V tds

5 rbergomi setting (Bayer, Friz and Gatheral): ds t = S t r t dt + S t Vt dw t, S 0 > 0 V t = ξ 0 (t)e (2νC H V t ), ν, ξ 0 ( ) > 0 V t = t dz u 0 (t u) 1/2 H dz t dw t = ρdt, ρ (0, 1) Focus on VIX [ VIX 2 ] E Q 1 0 V tds where = 30 days V T := E Q [ 1 ] [ T + E T Q [V t F T ] ds = E Q 1 ] T + ξ T T (s)ds

6 Proposition 1 VIX futures dynamics under rbergomi are given by: VIX 2 T = 1 T + T ξ 0 (t)η T (t) exp ( T η T (t) = exp 2νC H 0 ) dz u (t u) 1/2 H ( ν 2 C 2 H H ( (t T ) 2H t 2H)) dt,

7 Proposition 1 VIX futures dynamics under rbergomi are given by: VIX 2 T = 1 T + T ξ 0 (t)η T (t) exp ( T η T (t) = exp 2νC H 0 Corollary ) dz u (t u) 1/2 H ( ν 2 C 2 H H ( (t T ) 2H t 2H)) dt, In the rbergomi model the forward variance Curve, ξ T (t) follows log-normal dynamics, T ξ T (t) = ξ 0 (t)e (2νC H 0 ) dz u (t u) 1/2 H

8 Theorem (Upper and lower bounds for VIX futures) The following bounds hold for VIX Futures: 1 T + T ( ν 2 C 2 [ H ξ0 (t) exp (t T ) 2H t 2H]) { 1 dt V T 4H T + T ξ 0 (s)ds} 1/2.

9 VIX futures price VIX futures scenario 1 Monte-Carlo Lower bound Upper bound VIX futures scenario 2 VIX futures price Monte-Carlo Lower bound Upper bound

10 Simulation and approximations where { VIX 2 T = 1 η T (t) T + T = exp ( 2νC H V T t V T t = T 0 ( ν ξ 0 (t)η T (t) exp 2 C 2 H H ) ( (t T ) 2H t 2H)) dt, dz u, t [T, T + ] (t u) 1/2 H

11 Simulation and approximations where { VIX 2 T = 1 η T (t) T + T = exp ( 2νC H V T t V T t = Two simulation approaches T 0 ( ν ξ 0 (t)η T (t) exp 2 C 2 H H ) ( (t T ) 2H t 2H)) dt, dz u, t [T, T + ] (t u) 1/2 H

12 Simulation and approximations where { VIX 2 T = 1 η T (t) T + T = exp ( 2νC H V T t V T t = Two simulation approaches T 0 ( ν ξ 0 (t)η T (t) exp 2 C 2 H H ) ( (t T ) 2H t 2H)) dt, dz u, t [T, T + ] (t u) 1/2 H Hybrid Scheme (Bennedsen, Lunde and Pakkanen) + Forward Euler Truncated Cholesky (8 components)

13 Algorithm (VIX simulation: Hybrid Scheme + Forward Euler) Fix a grid T = {t i } i=0,...,nt and κ Simulate the Volterra process (V t) t [0,T ] using the hybrid scheme, yielding a sample of the random variable V T T = V T ; 2. extract the path of the Brownian motion Z driving the Volterra process. Z ti = Z ti 1 + n H 1/2 (V(t i ) V(t i 1 )), for i = 1,..., κ, Z ti = Z ti 1 + Z i 1, for i > κ; 3. fix a grid T = {τ j } j=0,...,n on [T, T + ] and approximate the continuous-time process V T by the discrete-time version ṼT defined via the following forward Euler scheme: n T Ṽτ T 0 := VT T and Ṽτ T Z ti Z ti 1 j :=, for j = 1,..., N; (τ i=1 j t i 1 ) H+1/2 4. compute the VIX process via numerical integration, for example using a composite trapezoidal rule: N 1 1 QT 2,τ + Q 2 1/2 ) j T,τ j+1 VIX T (τ j τ j 1 ), where Q 2 2 T,τ j := ξ 0 (τ j )E (2vC H Ṽ T τj. j=0

14 Algorithm (VIX simulation: Truncated Cholesky) Fix a grid T = {τ j } j=0,...,n on [T, T + ], (i) compute the covariance matrix of (Vτ T j ) i=j,...,8 which is available is close-form (ii) generate {Vτ T j } j=1,...,n by appropriately correlating and rescaling Vτ T j = V(Vτ T j ) ρ(v T τ j 1, V T τ j )V T τ j 1 V(V T τ j 1 ) + 1 ρ(vτ T j 1, Vτ T j ) 2 N (0, 1), for j = 9,..., N; (iii) compute the VIX via numerical integration as in the previous Algorithm.

15 VIX futures price Simulation and approximations VIX futures simulation methods Hybrid+Forward Euler Truncated Cholesky Time Monte-Carlo standard deviations (10log-scale) Computational time Hybrid+forward Euler Truncated Cholesky Hybrid+forward Euler Truncated Cholesky ξ H 0.07 ν κ 2 Standard deviation Seconds Simulations Simulations

16 Simulation and approximations Inspired by Bayer, Friz and Gatheral we consider a log-normal approximation such that T + VIX 2 T = E [V t F T ] dt = T log VIX 2 T N(µ, σ2 ) T + T ξ T (t)dt

17 Simulation and approximations Approach 1: Exactly match first and second moments of VIX 2 using Proposition 1

18 Simulation and approximations Approach 1: Exactly match first and second moments of VIX 2 using Proposition 1 E( VIX 2 T ) = [ E ( VIX 2 T ) 2] = T + T ξ 0(t)dt [T,T + ] 2 ξ 0(u)ξ 0(t) exp { ν 2 C 2 H H [ (u T ) 2H + (t T ) 2H u 2H t 2H]} e Θu,t dudt.

19 Simulation and approximations Approach 1: Exactly match first and second moments of VIX 2 using Proposition 1 E( VIX 2 T ) = [ E ( VIX 2 T ) 2] = T + T ξ 0(t)dt [T,T + ] 2 ξ 0(u)ξ 0(t) exp { ν 2 C 2 H H [ (u T ) 2H + (t T ) 2H u 2H t 2H]} e Θu,t dudt. Approach 2: Use the approximation by Bayer, Friz and Gatheral

20 Simulation and approximations Approach 1: Exactly match first and second moments of VIX 2 using Proposition 1 E( VIX 2 T ) = [ E ( VIX 2 T ) 2] = T + T ξ 0(t)dt [T,T + ] 2 ξ 0(u)ξ 0(t) exp { ν 2 C 2 H H [ (u T ) 2H + (t T ) 2H u 2H t 2H]} e Θu,t dudt. Approach 2: Use the approximation by Bayer, Friz and Gatheral E( VIX 2 T ) = T + T ξ 0(t)dt, [ ] V log( VIX 2 4ν 2 CH 2 T ) 2 (H + 1/2) 2 T 0 ( (T s + ) H+1/2 (T s) H+1/2) 2 ds

21 Simulation and approximations Approach 1: Exactly match first and second moments of VIX 2 using Proposition 1 E( VIX 2 T ) = [ E ( VIX 2 T ) 2] = T + T ξ 0(t)dt [T,T + ] 2 ξ 0(u)ξ 0(t) exp { ν 2 C 2 H H [ (u T ) 2H + (t T ) 2H u 2H t 2H]} e Θu,t dudt. Approach 2: Use the approximation by Bayer, Friz and Gatheral E( VIX 2 T ) = T + T ξ 0(t)dt, [ ] V log( VIX 2 4ν 2 CH 2 T ) 2 (H + 1/2) 2 T 0 ( (T s + ) H+1/2 (T s) H+1/2) 2 ds Either way we obtain a semi-explicit formula for VIX futures, computing E[ VIX 2 T )]

22 VIX futures price Simulation and approximations Log-normal benchmarking Our approach Bayer, Friz & Gatheral Evolution of variance Variance Our approach Bayer, Friz & Gatheral

23 Price Simulation and approximations VIX Futures scenario 1 Log-normal approximation Truncated Cholesky VIX Futures scenario 2 Price Log-normal approximation Truncated Cholesky Difference Difference

24 VIX futures calibration In order to obtain ξ 0 ( ) we fit a essvi parametrisation to the available S&P 500 option data and numerically differentiate using the closed-form formula for variance swaps in essvi We minimise a least squares criterion, i.e. L F (ν, H) := N (V Ti F i ) 2, i=1 which we minimise over (ν, H). Here (F i ) i=1,...,n are the observed Futures prices on the time grid T 1 <... < T N, ( ) V Ti = 1/2 Ti + T i ξ 0 (t)dt exp σ2 i 8, σ 2 ( = (T s + ) H+1/2 (T s) H+1/2) 2 ds 4ν2 CH 2 T 2 (H+1/2) 2 0

25 Results: essvi fit 0.45 S&P 500 data at maturity S&P 500 data at maturity Implied Vol Ask Bid 0.15 essvi Log-moneyness S&P 500 data at maturity Implied Vol Ask 0.1 Bid essvi Log-moneyness S&P 500 data at maturity Implied Vol Ask 0.1 Bid essvi Log-moneyness and ρ(θ) = (A C) exp( Bθ) + C Implied Vol Ask 0.15 Bid 0.10 essvi Log-moneyness Figure 1: essvi calibration results on the 4th of December 2015, using ϕ(θ) = ηθ λ (1 + θ) λ 1

26 VIX futures price VIX futures term structure Log-normal approx. Observed VIX futures term structure Results: VIX futures Figure 2: Fitted VIX future prices vs. observed futures prices on the 4th of December 2015 H ν

27 VIX futures price VIX futures term structure Results: VIX futures Log-normal approx. Observed VIX futures term structure Figure 3: Fitted VIX future prices vs. observed futures prices on the 22nd of February 2016 H ν

28 VIX futures price VIX futures term structure Log-normal approx. Observed VIX futures term structure Results: VIX futures Figure 4: Fitted VIX future prices vs. observed futures prices on the 4th of January 2016 H ν

29 From VIX futures to S&P 500 options Algorithm (Calibration algorithm for SPX options via VIX Futures) (i) Calibrate H and ξ 0 using the VIX Futures; (ii) compute M paths of the Volterra process, {V (u) } M u=1, extract the BM {Z (u) } M u=1 driving each process and draw independent BM {Z (u) } M u=1; (iii) evaluate the Call prices in each calibration step: ( ) V (u) t = ξ 0(t)E 2νC H V (u) t, u = 1,..., M, W (u) = ρz (u) + 1 ρ 2 Z (u), u = 1,..., M, ( ) = S (u) t + S (u) t v (u) t W (u) t+ W (u) t, u = 1,..., M; S (u) t+ (iv) compute the Call price for each available maturity {T 1,..., T N } and strikes {K (1),..., K (L) } (v) minimise over (ν, ρ) the selected objective function L C (ν, ρ).

30 S&P 500 options calibration Implied Vol Implied Vol Implied Vol S&P 500 data at maturity Ask 0.15 Bid 0.10 rbergomi Log-moneyness S&P 500 data at maturity Ask 0.2 Bid 0.1 rbergomi Log-moneyness S&P 500 data at maturity Ask 0.2 Bid 0.1 rbergomi Log-moneyness Implied Vol Implied Vol Implied Vol S&P 500 data at maturity Ask 0.2 Bid 0.1 rbergomi Log-moneyness S&P 500 data at maturity Ask 0.2 Bid 0.1 rbergomi Log-moneyness S&P 500 data at maturity Ask 0.2 Bid 0.1 rbergomi Log-moneyness ρ ν 1.190

31 References C. Bayer, P. Friz and J. Gatheral.Pricing under rough volatility. Quantitative Finance: 1-18, M. Bennedsen, A. Lunde and M. S. Pakkanen. Hybrid scheme for Brownian semistationary processes.arxiv: J. Gatheral, T. Jaisson and M. Rosenbaum. Volatility is rough. arxiv: , J. Gatheral and A. Jacquier. Arbitrage-free SVI volatility surfaces. Quantitative Finance, 14(1): 59-71, 2014.

32 On VIX Futures in the rough Bergomi model Oberwolfach Research Institute for Mathematics, February 28, 2017 joint work with Antoine Jacquier and Claude Martini