Numerical analysis of the multilevel Milstein discretisation Mike Giles Kristian Debrabant (Univ. of Southern Denmark) Andreas R¨ oßler (TU Darmstadt) mike.giles@maths.ox.ac.uk Oxford-Man Institute of Quantitative Finance Mathematical Institute, University of Oxford Multilevel numerical analysis – p. 1
Multilevel Monte Carlo Given a scalar SDE driven by a Brownian diffusion d S ( t ) = a ( S, t ) d t + b ( S, t ) d W ( t ) , 0 < t < T to estimate E [ P ] where the path-dependent payoff P can be P ℓ using 2 ℓ uniform timesteps, we use approximated by � L � E [ � P L ] = E [ � E [ � P ℓ − � P 0 ] + P ℓ − 1 ] . ℓ =1 E [ � P ℓ − � P ℓ − 1 ] is estimated using N ℓ simulations with same W ( t ) for both � P ℓ and � P ℓ − 1 , � � N ℓ � P ( i ) P ( i ) � � ℓ − � Y ℓ = N − 1 ℓ ℓ − 1 i =1 Multilevel numerical analysis – p. 2
MLMC Theorem Theorem: Let P be a functional of the solution of an SDE, and � P ℓ the discrete approximation using a timestep h ℓ = 2 − ℓ T . If there exist independent estimators � Y ℓ based on N ℓ Monte Carlo samples, with computational complexity (cost) C ℓ , and positive constants α ≥ 1 2 , β, c 1 , c 2 , c 3 such that � � � � � E [ � � ≤ c 1 h α i) P ℓ − P ] ℓ � E [ � P 0 ] , ℓ = 0 ii) E [ � Y ℓ ] = E [ � P ℓ − � P ℓ − 1 ] , ℓ > 0 h β iii) V [ � Y ℓ ] ≤ c 2 N − 1 ℓ ℓ iv) C ℓ ≤ c 3 N ℓ h − 1 ℓ Multilevel numerical analysis – p. 3
MLMC Theorem then there exists a positive constant c 4 such that for any ε<e − 1 there are values L and N ℓ for which the multilevel estimator L � � � Y = Y ℓ , ℓ =0 �� � 2 � � has Mean Square Error MSE ≡ E < ε 2 Y − E [ P ] with a computational complexity C with bound c 4 ε − 2 , β > 1 , c 4 ε − 2 (log ε ) 2 , C ≤ β = 1 , c 4 ε − 2 − (1 − β ) /α , 0 < β < 1 . Multilevel numerical analysis – p. 4
Numerical Analysis If P is a Lipschitz function of S ( T ) , value of underlying path simulation at a fixed time, the strong convergence property � � S N − S ( T )) 2 �� 1 / 2 ( � = O ( h γ ) E P ℓ − P ] = O ( h 2 γ implies that V [ � ℓ ) and hence P ℓ − 1 ] = O ( h 2 γ V ℓ ≡ V [ � P ℓ − � ℓ ) . Therefore β =1 for Euler-Maruyama discretisation, and β =2 for the Milstein discretisation. However, in general, good strong convergence is neither necessary nor sufficient for good convergence for V ℓ . Multilevel numerical analysis – p. 5
Numerics and Analysis Euler Milstein option numerics analysis numerics analysis O ( h 2 ) O ( h 2 ) Lipschitz O ( h ) O ( h ) O ( h 2 ) O ( h 2 ) Asian O ( h ) O ( h ) O ( h 2 ) o ( h 2 − δ ) lookback O ( h ) O ( h ) O ( h 1 / 2 ) o ( h 1 / 2 − δ ) O ( h 3 / 2 ) o ( h 3 / 2 − δ ) barrier O ( h 1 / 2 log h ) O ( h 1 / 2 ) O ( h 3 / 2 ) o ( h 3 / 2 − δ ) digital Table: V ℓ convergence observed numerically (for GBM) and proved analytically (for more general SDEs) for both the Euler and Milstein discretisations. δ can be any strictly positive constant. Multilevel numerical analysis – p. 6
Numerical Analysis Analysis for Euler discretisations: lookback and barrier options: Giles, Higham & Mao ( Finance & Stochastics, 2009 ) lookback analysis follows from strong convergence barrier analysis shows dominant contribution comes from paths which are near the barrier; uses asymptotic analysis, first proving that “extreme” paths have negligible contribution similar analysis for digital options gives O ( h 1 / 2 − δ ) bound instead of O ( h 1 / 2 log h ) digital options: Avikainen ( Finance & Stochastics, 2009 ) method of analysis is quite different Multilevel numerical analysis – p. 7
Numerical Analysis Analysis for Milstein discretisations: builds on approach in paper with Higham and Mao key idea is to use boundedness of all moments to bound the contribution to V ℓ from “extreme” paths | ∆ W n | > h 1 / 2 − δ for some δ > 0 ) (e.g. for which max n uses asymptotic analysis to bound the contribution from paths which are not “extreme” Multilevel numerical analysis – p. 8
Milstein Scheme MLMC Theorem allows different approximations on the coarse and fine levels: � � N ℓ � P f � � ℓ ( ω ( n ) ) − � Y ℓ = N − 1 P c ℓ − 1 ( ω ( n ) ) ℓ n =1 The telescoping sum still works provided � � � � P f � � P c = E E . ℓ ℓ The key is to exploit this freedom to reduce the variance � � P f � ℓ − � P c V . ℓ − 1 Multilevel numerical analysis – p. 9
Milstein Scheme Fine path Brownian interpolation: within each timestep, model the behaviour as simple Brownian motion (constant drift and volatility) conditional on two end-points S f � � n + λ ( t )( � n +1 − � S f ( t ) S f S f = n ) � � + b n W ( t ) − W n − λ ( t )( W n +1 − W n ) , t − t n where λ ( t ) = . t n +1 − t n There then exist analytic results for the distribution of the min/max/average over each timestep, and probability of crossing a barrier. Multilevel numerical analysis – p. 10
Milstein Scheme Coarse path Brownian interpolation: exactly the same, but with double the timestep, so for even n S c ( t ) � S c � n + λ ( t )( � S c n +2 − � S c = n ) � � + b n W ( t ) − W n − λ ( t )( W n +2 − W n ) , t − t n where λ ( t ) = . Hence, in particular, t n +2 − t n � n +1 ≡ � 2 ( � n + � S c S c ( t n +1 ) S c S c 1 = n +2 ) � � W n +1 − 1 + b n 2 ( W n + W n +2 ) , Multilevel numerical analysis – p. 11
Milstein Scheme Theorem: Under standard conditions, � � � � m � � �� = O (( h log h ) m ) , sup S ( t ) − S ( t ) E � [0 ,T ] � � � m � � � �� = O ( h m ) , sup S ( t ) − S ( t ) E � [0 ,T ] � 2 �� T � = O ( h 2 ) . S ( t ) − S ( t ) d t E 0 Multilevel numerical analysis – p. 12
Milstein Scheme The variance convergence for the Asian option comes directly from this. Will now outline the analysis for the lookback option – the barrier is similar but more complicated. The digital option is based on a Brownian extrapolation from one timestep before the end – the analysis is similar. The analysis for the lookback, barrier and digital options uses the idea of “extreme” paths which are highly improbable – the variance comes mainly from non-extreme paths for which one can use asymptotic analysis. Multilevel numerical analysis – p. 13
Extreme Paths Lemma: If X ℓ is a random variable on level l , and E [ | X ℓ | m ] ≤ C m is uniformly bounded, then, for any δ > 0 , ℓ ] = o ( h p P [ | X ℓ | > h − δ ℓ ) , ∀ p > 0 . Proof: Markov inequality P [ | X ℓ | m > h − mδ ] < h − mδ E [ | X ℓ | m ] . ℓ ℓ Lemma: If Y ℓ is a random variable on level ℓ , E [ Y 2 ℓ ] is uniformly bounded, and the indicator function 1 E ℓ satisfies E [ 1 E ℓ ] = o ( h p ℓ ) , ∀ p > 0 then E [ | Y ℓ | 1 E ℓ ] = o ( h p ℓ ) , ∀ p > 0 . � E [ Y 2 Proof: Hölder inequality E [ | Y ℓ | 1 E ℓ ] ≤ ℓ ] E [ 1 E ℓ ] Multilevel numerical analysis – p. 14
Extreme Paths Theorem: For any γ > 0 , the probability that W ( t ) , its increments ∆ W n and the corresponding SDE solution S ( t ) S f and approximations � n and � S c n satisfy any of the following “extreme” conditions � � max( | S ( nh ) | , | � S f n | , | � S c h − γ max n | > n � � max( | S ( nh ) − � n | , | S ( nh ) − � n | , | � n − � S c S f S f S c h 1 − γ max n | ) > n h 1 / 2 − γ max | ∆ W n | > n is o ( h p ) for all p> 0 . Multilevel numerical analysis – p. 15
Non-extreme paths Furthermore, there exist constants c 1 , c 2 , c 3 , c 4 such that if none of these conditions is satisfied, and γ < 1 2 , then S f | � n − � S f c 1 h 1 / 2 − 2 γ max n − 1 | ≤ n n − b f | b f c 2 h 1 / 2 − 2 γ max n − 1 | ≤ n � � | b f n | + | b c c 3 h − γ max n | ≤ n | b f n − b c c 4 h 1 / 2 − 2 γ max n | ≤ n where b c n is defined to equal b c n − 1 if n is odd. Multilevel numerical analysis – p. 16
Lookback Option Lookback options are a Lipschitz function of the minimum over the whole simulation path. For the fine path, the minimum over one timestep is � � �� � 2 � � 2 S f S f S f S f b f � � n + � � n +1 − � n,min = 1 S f n +1 − − 2 h ℓ log U n n n 2 where U m is a (0 , 1] uniform random variable. For the coarse path, define � S c n for odd n using conditional Brownian interpolation, then use the same expression for the minimum with same U n – this doesn’t change the distribution of the computed minimum over the coarse timestep, so the telescoping sum is OK. Multilevel numerical analysis – p. 17
Lookback Option E [( � P ℓ − � P ℓ − 1 ) 4 ] is bounded and therefore extreme paths have negligible contribution to E [( � P ℓ − � P ℓ − 1 ) 2 ] . For non-extreme paths, tedious asymptotic analysis gives � � � � � � � � S f S f �� min − � S c �� n,min − � S c ≤ max � � min n,min n o ( h 1 − 5 γ/ 2 = ) ℓ and hence E [( � P ℓ − � P ℓ − 1 ) 2 ] = o ( h 2 − δ ) for any δ > 0 . ℓ Multilevel numerical analysis – p. 18
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