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 Gatheral’s 60th Birthday Conference Courant Institute, New York, 14 October 2017 Joint work with Ryan McCrickerd Imperial Network of Excellence in Probabilistic F Ω Methods and Modelling P
Pricing under rough volatility Simulating rough volatility Variance reduction methods Volatility is (still) rough — 3rd anniversary! Standard Brownian motion 0.2 W t −0.2 −0.6 0.0 0.2 0.4 0.6 0.8 1.0 t Daily realised variance of CAC 40 index Source: Oxford−Man Realized Library ''Volatility is Rough'' posted on arXiv −6 log ( RV t ) −8 −12 2008−01−01 2010−01−01 2012−01−01 2014−01−01 2016−01−01 2018−01−01 t Fractional Brownian motion with Hurst index 0.15 1.0 0.0 Z t −1.5 0.0 0.2 0.4 0.6 0.8 1.0 t 2 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Pricing under rough volatility Simulating rough volatility Variance reduction methods 3 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods The rough Bergomi model The rough Bergomi model (Bayer, Friz, and Gatheral, 2016) is a non-Markovian extension of the variance curve model of Bergomi (2005). 4 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods The rough Bergomi model 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 dS t � ρ dW s + � � � V t 1 − ρ 2 dW ⊥ = , s S t � ������������������������ �� ������������������������ � = : B s 4 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods The rough Bergomi model 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 dS t � ρ dW s + � � � V t 1 − ρ 2 dW ⊥ = , s S t � ������������������������ �� ������������������������ � = : B s where W and W ⊥ are independent Brownians and ρ ∈ [ − 1 , 1 ] . 4 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods The rough Bergomi model 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 dS t � ρ dW s + � � � V t 1 − ρ 2 dW ⊥ = , s S t � ������������������������ �� ������������������������ � = : B s where W and W ⊥ are independent Brownians and ρ ∈ [ − 1 , 1 ] . The spot variance V t is a product V t = ξ 0 ( t ) E ( η W α ) t of • the forward variance curve t �→ ξ 0 ( t ) , known at time 0, t ] � of • the Wick exponential E ( η W α ) t = exp � η W α t − 1 2 Var [ η W α a parameter η > 0 times a Gaussian random variable W α t . 4 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods The rough Bergomi model (cont.) The random variable W α t follows the Gaussian Riemann– Liouville process � t √ W α 2 α + 1 ( t − s ) α dW s , t ≥ 0 , t = 0 where the parameter α ∈ ( − 1 2 , 0 ) controls the roughness of paths. 5 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods The rough Bergomi model (cont.) The random variable W α t follows the Gaussian Riemann– Liouville process � t √ W α 2 α + 1 ( t − s ) α dW s , t ≥ 0 , t = 0 where the parameter α ∈ ( − 1 2 , 0 ) controls the roughness of paths. The paths of W α have Hölder regularity α + 1 2 and locally look like the paths of a fractional Brownian motion with H = α + 1 2 . 5 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Example: rough Bergomi paths α = 0 α = − 0 . 43 3 3 2 2 1 1 W α W α t 0 t 0 1 1 2 2 3 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t ρ = − 0 . 9 , α = − 0 . 43 ρ = 0 , α = − 0 . 43 1.2 1.2 1.1 1.1 S t 1.0 S t 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t 6 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Intuition on the parameters The rough Bergomi model has three time-homogeneous parameters, α , η , and ρ , with the following interpretations in terms of the implied volatility surface: • η — smile, • ρ — skew, • α — near-maturity explosion (of smile and skew). 7 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Example: rough Bergomi smiles ρ = − 0 . 9 , α = − 0 . 43 ρ = 0 , α = − 0 . 43 0.35 0.28 1 D 0.30 1 W 0.26 1 M σ BS ( k, t ) σ BS ( k, t ) 3 M 0.25 6 M 0.24 1 Y 0.20 0.22 0.15 0.10 0.20 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 k k ρ = − 0 . 9 , α = 0 ρ = 0 , α = 0 0.35 0.28 0.30 0.26 σ BS ( k, t ) σ BS ( k, t ) 0.25 0.24 0.20 0.22 0.15 0.10 0.20 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 k k 8 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Example: rough Bergomi smiles ρ = − 0 . 9 , α = − 0 . 43 ρ = 0 , α = − 0 . 43 0.35 0.28 1 D 0.30 1 W 0.26 1 M σ BS ( k, t ) σ BS ( k, t ) 3 M 0.25 6 M 0.24 1 Y 0.20 0.22 0.15 0.10 0.20 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 k k ρ = − 0 . 9 , α = − 0 . 43 ρ = 0 , α = − 0 . 43 0.35 0.28 1 D 1 W 0.30 0.26 1 M σ BS (∆ , t ) σ BS (∆ , t ) 3 M 0.25 6 M 0.24 1 Y 0.20 0.22 0.15 0.10 0.20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ∆ ∆ 8 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Pricing under rough volatility Simulating rough volatility Variance reduction methods 9 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Pricing by Monte Carlo The introduction of the rough Bergomi model has launched a quest for efficient pricing methods for the model. 10 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Pricing by Monte Carlo 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. 10 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Pricing by Monte Carlo 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 Pricing by Monte Carlo 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. 10 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Pricing by Monte Carlo 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: • low bias, to avoid systematic error, • low variance, such that good accuracy can be achieved in reasonable runtime. 10 / 29
Pricing under rough volatility Simulating rough volatility Variance reduction methods Simulating the rough Bergomi model The rough Bergomi model has a simple stochastic structure — all randomness comes from a bivariate Gaussian process ( B , W α ) , while S can be approximated by Riemann sums. 11 / 29
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