T urbocharging Monte Carlo pricing under rough volatility Mikko - - PowerPoint PPT Presentation
Pricing under rough volatility Simulating rough volatility Variance reduction methods T urbocharging Monte Carlo pricing under rough volatility Mikko Pakkanen Department of Mathematics, Imperial College London, UK Jim Gatherals 60th
Pricing under rough volatility Simulating rough volatility Variance reduction methods
Mikko Pakkanen
Department of Mathematics, Imperial College London, UK
Jim Gatheral’s 60th Birthday Conference Courant Institute, New York, 14 October 2017 Joint work with Ryan McCrickerd
Ω F P Imperial Network of Excellence in
Probabilistic
Methods and Modelling
Pricing under rough volatility Simulating rough volatility Variance reduction methods
0.0 0.2 0.4 0.6 0.8 1.0 −0.6 −0.2 0.2 t Wt Standard Brownian motion t log(RVt) 2008−01−01 2010−01−01 2012−01−01 2014−01−01 2016−01−01 2018−01−01 −12 −8 −6 Daily realised variance of CAC 40 index Source: Oxford−Man Realized Library ''Volatility is Rough'' posted on arXiv 0.0 0.2 0.4 0.6 0.8 1.0 −1.5 0.0 1.0 t Zt Fractional Brownian motion with Hurst index 0.15
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Pricing under rough volatility Simulating rough volatility Variance reduction methods
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model (Bayer, Friz, and Gatheral, 2016) is a non-Markovian extension of the variance curve model of Bergomi (2005).
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model (Bayer, Friz, and Gatheral, 2016) is a non-Markovian extension of the variance curve model of Bergomi (2005). This model, under a pricing measure, is given by dSt St =
ρdWs +
s
,
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model (Bayer, Friz, and Gatheral, 2016) is a non-Markovian extension of the variance curve model of Bergomi (2005). This model, under a pricing measure, is given by dSt St =
ρdWs +
s
, where W and W⊥ are independent Brownians and ρ ∈ [−1, 1].
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model (Bayer, Friz, and Gatheral, 2016) is a non-Markovian extension of the variance curve model of Bergomi (2005). This model, under a pricing measure, is given by dSt St =
ρdWs +
s
, where W and W⊥ are independent Brownians and ρ ∈ [−1, 1]. The spot variance Vt is a product Vt = ξ0(t)E(ηWα)t of
t − 1 2Var[ηWα t ] of
a parameter η > 0 times a Gaussian random variable Wα
t .
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The random variable Wα
t follows the Gaussian Riemann–
Liouville process Wα
t =
√ 2α + 1 t (t − s)αdWs, t ≥ 0, where the parameter α ∈ (− 1
2, 0) controls the roughness of
paths.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The random variable Wα
t follows the Gaussian Riemann–
Liouville process Wα
t =
√ 2α + 1 t (t − s)αdWs, t ≥ 0, where the parameter α ∈ (− 1
2, 0) controls the roughness of
paths. The paths of Wα have Hölder regularity α + 1
2 and locally look
like the paths of a fractional Brownian motion with H = α + 1
2.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
0.0 0.2 0.4 0.6 0.8 1.0
t
3 2 1 1 2 3
W α
t
α = 0
0.0 0.2 0.4 0.6 0.8 1.0
t
3 2 1 1 2 3
W α
t
α = − 0. 43
0.0 0.2 0.4 0.6 0.8 1.0
t
0.7 0.8 0.9 1.0 1.1 1.2
St ρ = − 0. 9, α = − 0. 43
0.0 0.2 0.4 0.6 0.8 1.0
t
0.7 0.8 0.9 1.0 1.1 1.2
St ρ = 0, α = − 0. 43
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model has three time-homogeneous parameters, α, η, and ρ, with the following interpretations in terms of the implied volatility surface:
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3
k
0.10 0.15 0.20 0.25 0.30 0.35
σBS(k, t) ρ = − 0. 9, α = − 0. 43
1D 1W 1M 3M 6M 1Y 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4
k
0.20 0.22 0.24 0.26 0.28
σBS(k, t) ρ = 0, α = − 0. 43
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3
k
0.10 0.15 0.20 0.25 0.30 0.35
σBS(k, t) ρ = − 0. 9, α = 0
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4
k
0.20 0.22 0.24 0.26 0.28
σBS(k, t) ρ = 0, α = 0
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3
k
0.10 0.15 0.20 0.25 0.30 0.35
σBS(k, t) ρ = − 0. 9, α = − 0. 43
1D 1W 1M 3M 6M 1Y 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4
k
0.20 0.22 0.24 0.26 0.28
σBS(k, t) ρ = 0, α = − 0. 43
0.0 0.2 0.4 0.6 0.8 1.0
∆
0.10 0.15 0.20 0.25 0.30 0.35
σBS(∆, t) ρ = − 0. 9, α = − 0. 43
1D 1W 1M 3M 6M 1Y 0.0 0.2 0.4 0.6 0.8 1.0
∆
0.20 0.22 0.24 0.26 0.28
σBS(∆, t) ρ = 0, α = − 0. 43
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Pricing under rough volatility Simulating rough volatility Variance reduction methods
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The introduction of the rough Bergomi model has launched a quest for efficient pricing methods for the model.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The introduction of the rough Bergomi model has launched a quest for efficient pricing methods for the model. The model is non-affine and non-Markovian so standard methods (PDEs, characteristic functions) seem inapplicable.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The introduction of the rough Bergomi model has launched a quest for efficient pricing methods for the model. The model is non-affine and non-Markovian so standard methods (PDEs, characteristic functions) seem inapplicable. Currently, the only operational pricing method for mere vanilla options is Monte Carlo.
10 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods
The introduction of the rough Bergomi model has launched a quest for efficient pricing methods for the model. The model is non-affine and non-Markovian so standard methods (PDEs, characteristic functions) seem inapplicable. Currently, the only operational pricing method for mere vanilla options is Monte Carlo. Thus it is worthwhile to try to optimise, “turbocharge”, Monte Carlo pricing as much as possible.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The introduction of the rough Bergomi model has launched a quest for efficient pricing methods for the model. The model is non-affine and non-Markovian so standard methods (PDEs, characteristic functions) seem inapplicable. Currently, the only operational pricing method for mere vanilla options is Monte Carlo. Thus it is worthwhile to try to optimise, “turbocharge”, Monte Carlo pricing as much as possible. In general, a good Monte Carlo pricer should have:
reasonable runtime.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model has a simple stochastic structure — all randomness comes from a bivariate Gaussian process (B, Wα), while S can be approximated by Riemann sums.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model has a simple stochastic structure — all randomness comes from a bivariate Gaussian process (B, Wα), while S can be approximated by Riemann sums. The covariance structure of (B, Wα) is not difficult to work out, so we could simulate exactly X := (B0, Wα
0), (B1/n, Wα 1/n), (B2/n, Wα 2/n), . . . , (B⌊nt⌋/n, Wα ⌊nt⌋/n)
by sampling from a 2⌊nt⌋-dimensional Gaussian distribution.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model has a simple stochastic structure — all randomness comes from a bivariate Gaussian process (B, Wα), while S can be approximated by Riemann sums. The covariance structure of (B, Wα) is not difficult to work out, so we could simulate exactly X := (B0, Wα
0), (B1/n, Wα 1/n), (B2/n, Wα 2/n), . . . , (B⌊nt⌋/n, Wα ⌊nt⌋/n)
by sampling from a 2⌊nt⌋-dimensional Gaussian distribution. The simulation is based on the Cholesky factorisation of the covariance matrix of X, which requires O(n3) flops.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The rough Bergomi model has a simple stochastic structure — all randomness comes from a bivariate Gaussian process (B, Wα), while S can be approximated by Riemann sums. The covariance structure of (B, Wα) is not difficult to work out, so we could simulate exactly X := (B0, Wα
0), (B1/n, Wα 1/n), (B2/n, Wα 2/n), . . . , (B⌊nt⌋/n, Wα ⌊nt⌋/n)
by sampling from a 2⌊nt⌋-dimensional Gaussian distribution. The simulation is based on the Cholesky factorisation of the covariance matrix of X, which requires O(n3) flops. The Cholesky factor needs to be computed only once, but subsequent realisations of X still take O(n2) flops.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Exact simulation being too expensive, we seek to approximate the process Wα.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Exact simulation being too expensive, we seek to approximate the process Wα. A naive approach would be to use forward Riemann sums Wα
i/n = i
n− k−1 n i n − k n
i
n−sαdWs ≈ i
k
n
α W i
n − k−1 n −W i n− k n
Wα,n
i/n .
Since Wα,n
i/n is a discrete convolution,
Wα,n
0 ,
Wα,n
1/n, . . . ,
Wα,n
⌊nt⌋/n can
be generated (using FFT) in O(n log n) flops.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Exact simulation being too expensive, we seek to approximate the process Wα. A naive approach would be to use forward Riemann sums Wα
i/n = i
n− k−1 n i n − k n
i
n−sαdWs ≈ i
k
n
α W i
n − k−1 n −W i n− k n
Wα,n
i/n .
Since Wα,n
i/n is a discrete convolution,
Wα,n
0 ,
Wα,n
1/n, . . . ,
Wα,n
⌊nt⌋/n can
be generated (using FFT) in O(n log n) flops. However:
integrand s → i
n − sα has a singularity.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The hybrid scheme of Bennedsen, Lunde, and Pakkanen (2017) fixes the deficiencies of forward Riemann sums by using:
i/n := κ
n − k−1 n i n− k n
i
n − sαdWs
+
i
bk
n
α W i
n − k−1 n − W i n− k n
, where bk ∈ [k − 1, k] \ {0} can be chosen (MSE) optimally.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The hybrid scheme of Bennedsen, Lunde, and Pakkanen (2017) fixes the deficiencies of forward Riemann sums by using:
i/n := κ
n − k−1 n i n− k n
i
n − sαdWs
+
i
bk
n
α W i
n − k−1 n − W i n− k n
, where bk ∈ [k − 1, k] \ {0} can be chosen (MSE) optimally. Usually κ = 1 suffices.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The hybrid scheme of Bennedsen, Lunde, and Pakkanen (2017) fixes the deficiencies of forward Riemann sums by using:
i/n := κ
n − k−1 n i n− k n
i
n − sαdWs
+
i
bk
n
α W i
n − k−1 n − W i n− k n
, where bk ∈ [k − 1, k] \ {0} can be chosen (MSE) optimally. Usually κ = 1 suffices. The variates Wα,n
0 ,
Wα,n
1/n, . . . ,
Wα,n
⌊nt⌋/n can be generated by
sampling ⌊nt⌋ iid draws from a κ + 1-dimensional Gaussian distribution and computing a discrete convolution.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The hybrid scheme of Bennedsen, Lunde, and Pakkanen (2017) fixes the deficiencies of forward Riemann sums by using:
i/n := κ
n − k−1 n i n− k n
i
n − sαdWs
+
i
bk
n
α W i
n − k−1 n − W i n− k n
, where bk ∈ [k − 1, k] \ {0} can be chosen (MSE) optimally. Usually κ = 1 suffices. The variates Wα,n
0 ,
Wα,n
1/n, . . . ,
Wα,n
⌊nt⌋/n can be generated by
sampling ⌊nt⌋ iid draws from a κ + 1-dimensional Gaussian distribution and computing a discrete convolution. Again, this requires only O(n log n) flops.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 x g(x)
n = 10 α = −0.4
approximation 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 x g(x)
n = 10 κ = 2 α = −0.4
approximation
Forward Riemann sums vs. the hybrid scheme
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
−0.4 −0.2 0.2 0.4 0.6 0.8 k IV (k, T ) T = 0.041
exact κ = 0 κ = 1 κ = 2
−0.5 0.5 0.1 0.2 0.3 0.4 k IV (k, T ) T = 1
exact κ = 0 κ = 1 κ = 2
Solid/patterned line: optimal bk Dashed line: bk = k. S0 ξ0(t) η α ρ n paths 1 0.2352 1.9 −0.43 −0.9 500 106
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Pricing under rough volatility Simulating rough volatility Variance reduction methods
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
While the hybrid scheme simulates the variance process V efficiently, ceteris paribus it does essentially nothing to the variance of the Monte Carlo pricer.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
While the hybrid scheme simulates the variance process V efficiently, ceteris paribus it does essentially nothing to the variance of the Monte Carlo pricer. Indeed there is scope for improving the efficiency of the pricer by deploying a “cocktail” of variance reduction methods.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
While the hybrid scheme simulates the variance process V efficiently, ceteris paribus it does essentially nothing to the variance of the Monte Carlo pricer. Indeed there is scope for improving the efficiency of the pricer by deploying a “cocktail” of variance reduction methods. To this end, we work with price estimators of the form ˆ Pn(k, t) := 1 n
n
(Xi − ˆ αnYi) − ˆ αnE[Y], where (X1, Y1), . . . , (Xn, Yn) are identical copies of a random vector (X, Y) and ˆ αn is a free parameter, to be defined shortly.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Our reference estimator, which we call the Base estimator, uses X := f(St) := (St − S0ek)+, k ≥ 0, (S0ek − St)+, k < 0, Y := 0.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Our reference estimator, which we call the Base estimator, uses X := f(St) := (St − S0ek)+, k ≥ 0, (S0ek − St)+, k < 0, Y := 0. This is really just the “naive” estimator, except that we price the out-of-the-money European call/put, which is less noisy, and derive implied volatility from its price.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Our reference estimator, which we call the Base estimator, uses X := f(St) := (St − S0ek)+, k ≥ 0, (S0ek − St)+, k < 0, Y := 0. This is really just the “naive” estimator, except that we price the out-of-the-money European call/put, which is less noisy, and derive implied volatility from its price. Without loss of generality, assume S0 = 1.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The well-known result of Romano and Touzi (1997) implies that E[f(St)] = E
t Vsds, SW
t , k
where BS is the appropriate Black–Scholes function, dSW
t /SW t = √Vt ρ dWt
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
The well-known result of Romano and Touzi (1997) implies that E[f(St)] = E
t Vsds, SW
t , k
where BS is the appropriate Black–Scholes function, dSW
t /SW t = √Vt ρ dWt
This suggests that we could use X := BS
t Vsds, SW
t , k
This method alone is rather effective in reducing variance when ρ ≈ 0, but its benefits evaporate as ρ → −1.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Inspired by the idea of Bergomi (2016) of using a timer option as a control variate, we choose Y := BS
ˆ Qn − t Vsds
t , k
where ˆ Qn is a free parameter, “variance budget”, to be chosen post simulation.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Inspired by the idea of Bergomi (2016) of using a timer option as a control variate, we choose Y := BS
ˆ Qn − t Vsds
t , k
where ˆ Qn is a free parameter, “variance budget”, to be chosen post simulation. By a martingale argument, E[Y] = BS(ρ2 ˆ Qn, 1, k)
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Our “turbocharged” Mixed estimator (McCrickerd and Pakkanen, 2017) is given by X := BS
t Vsds, SW
t , k
Y := BS
ˆ Qn − t Vsds
t , k
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Our “turbocharged” Mixed estimator (McCrickerd and Pakkanen, 2017) is given by X := BS
t Vsds, SW
t , k
Y := BS
ˆ Qn − t Vsds
t , k
We set, post simulation, ˆ αn := − n
i=1(Xi − Xi)(Yi − Yi)
n
i=1(Yi − Yi)2
, ˆ Qn := max
i=1,...,n
t Vsds
.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Our “turbocharged” Mixed estimator (McCrickerd and Pakkanen, 2017) is given by X := BS
t Vsds, SW
t , k
Y := BS
ˆ Qn − t Vsds
t , k
We set, post simulation, ˆ αn := − n
i=1(Xi − Xi)(Yi − Yi)
n
i=1(Yi − Yi)2
, ˆ Qn := max
i=1,...,n
t Vsds
. Additionally, we couple (X2i−1, Y2i−1) and (X2i, Y2i) for any i ≥ 1 by using antithetic pairs (B, W) and (−B, −W) as drivers.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
4 2 2 4 1 2 3
k = − 0. 18 σBS = 29. 6 B = − 0. 05 S = 1. 28
10 Delta Put
4 2 2 4
Base error : 100(ˆ σn
BS i − σBS). ρ = − 0. 9, n = 1000, τ = 114. 4ms, N = 1000, ψ 2 = 131. 4
k = 0. 00 σBS = 20. 6 B = 0. 01 S = 1. 24
ATM
4 2 2 4
k = 0. 10 σBS = 15. 8 B = − 0. 04 S = 0. 52
10 Delta Call
4 2 2 4 1 2 3
k = − 0. 15 σBS = 24. 2 B = − 0. 00 S = 0. 94
10 Delta Put
4 2 2 4
Base error : 100(ˆ σn
BS i − σBS). ρ = − 0. 0, n = 1000, τ = 114. 6ms, N = 1000, ψ 2 = 133. 6
k = 0. 00 σBS = 21. 7 B = − 0. 01 S = 1. 03
ATM
4 2 2 4
k = 0. 17 σBS = 24. 7 B = 0. 01 S = 1. 25
10 Delta Call
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
4 2 2 4 1 2 3
k = − 0. 18 σBS = 29. 6 B = 0. 00 S = 0. 55
10 Delta Put
4 2 2 4
Mixed error : 100(ˆ σn
BS i − σBS). ρ = − 0. 9, n = 1000, τ = 70. 6ms, N = 1000, ψ 2 = 10. 4
k = 0. 00 σBS = 20. 6 B = 0. 03 S = 0. 27
ATM
4 2 2 4
k = 0. 10 σBS = 15. 8 B = − 0. 03 S = 0. 26
10 Delta Call
4 2 2 4 1 2 3
k = − 0. 15 σBS = 24. 2 B = 0. 04 S = 0. 26
10 Delta Put
4 2 2 4
Mixed error : 100(ˆ σn
BS i − σBS). ρ = − 0. 0, n = 1000, τ = 69. 8ms, N = 1000, ψ 2 = 3. 9
k = 0. 00 σBS = 21. 7 B = 0. 02 S = 0. 15
ATM
4 2 2 4
k = 0. 17 σBS = 24. 7 B = 0. 05 S = 0. 28
10 Delta Call
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Ultimately, the goal of this simulation methodology is to calibrate the rough Bergomi model to a volatility surface.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Ultimately, the goal of this simulation methodology is to calibrate the rough Bergomi model to a volatility surface. We conduct a simple experiment to demonstrate what difference using the Mixed estimator in calibration makes.
24 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods
Ultimately, the goal of this simulation methodology is to calibrate the rough Bergomi model to a volatility surface. We conduct a simple experiment to demonstrate what difference using the Mixed estimator in calibration makes. We calibrate the parameters η and ρ to a 3M rough Bergomi reference smile at 19 points, minimising RMSE using L-BFGS-B.
24 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods
Ultimately, the goal of this simulation methodology is to calibrate the rough Bergomi model to a volatility surface. We conduct a simple experiment to demonstrate what difference using the Mixed estimator in calibration makes. We calibrate the parameters η and ρ to a 3M rough Bergomi reference smile at 19 points, minimising RMSE using L-BFGS-B. We initialise the solver at the true parameter values and let it run for 700 milliseconds.
24 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods
Ultimately, the goal of this simulation methodology is to calibrate the rough Bergomi model to a volatility surface. We conduct a simple experiment to demonstrate what difference using the Mixed estimator in calibration makes. We calibrate the parameters η and ρ to a 3M rough Bergomi reference smile at 19 points, minimising RMSE using L-BFGS-B. We initialise the solver at the true parameter values and let it run for 700 milliseconds. Throughout the experiment, we use n = 1 000.
24 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
ρ
1.0 1.5 2.0 2.5 3.0
η Base targeting ( − 0. 9, 1. 9)
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4
ρ
1.0 1.5 2.0 2.5 3.0
η Base targeting (0, 1. 9)
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
ρ
1.0 1.5 2.0 2.5 3.0
η Mixed targeting ( − 0. 9, 1. 9)
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4
ρ
1.0 1.5 2.0 2.5 3.0
η Mixed targeting (0, 1. 9)
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
scheme, but does not appear to effective in reducing variance in this setting (Mamallan, 2017).
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
scheme, but does not appear to effective in reducing variance in this setting (Mamallan, 2017).
need to be tuned strike by strike.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
scheme, but does not appear to effective in reducing variance in this setting (Mamallan, 2017).
need to be tuned strike by strike.
and useful here, albeit the speed-up appears not to be dramatic.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Brownian semistationary processes. Finance Stoch. 21(4) 931–965.
Raton.
with comparison to the Heston model. Unpublished MSci dissertation, Imperial College London.
pricing for the rough Bergomi model. Preprint: http://arxiv.org/abs/1708.02563
completeness in a stochastic volatility model. Math. Finance 7(4), 399–412.
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
Python implementation of turbocharged pricing along with a Jupyter notebook are available from: https://github.com/ryanmccrickerd/roughbergomi
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Pricing under rough volatility Simulating rough volatility Variance reduction methods
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