Multilevel Monte Carlo for exponential Lévy models

Finance and Stochastics, Sep 2017

We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and \(\alpha\)-stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments.

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Multilevel Monte Carlo for exponential Lévy models

Finance Stoch Multilevel Monte Carlo for exponential Lévy models Michael B. Giles 0 1 Yuan Xia 0 1 B Y. Xia 0 1 0 Mathematical Institute, Oxford University , Oxford , UK 1 JEL Classification C15 We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and α-stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments. Multilevel Monte Carlo; Exponential Lévy models; Asian options; Lookback options; Barrier options 1 Introduction Exponential Lévy models are based on the assumption that asset returns follow a Lévy process [25, 10]. The asset price follows St = S0 exp(Xt ) where X is an (m, σ, ν)-Lévy process, i.e., Xt = mt + σ Bt + t where m is a constant, B is a Brownian motion, J is the jump measure and ν is the Lévy measure (cf. [24, Theorem I.42]). Models with jumps give an intuitive explanation of implied volatility skews and smiles in the index option market and foreign exchange market ([10, Chap. 11]). The jump fear is mainly on the downside in the equity market which produces a premium for low-strike options; the jump risk is symmetric in the foreign exchange market so that the implied volatility has a smile shape. Chapter 7 in [ 10 ] shows that models building on pure jump processes can reproduce the stylised facts of asset returns, like heavy tails and the asymmetric distribution of increments. Since pure jump processes of finite activity without a diffusion component cannot generate a realistic path, it is natural to allow the jump activity to be infinite. In this work, we deal with infiniteactivity pure jump exponential Lévy models, in particular models driven by variance gamma (VG), normal inverse Gaussian (NIG) and α-stable processes which allow direct simulation of increments. We are interested in estimating the expected payoff value E[f (S)] in option pricing problems. In the case of European options, it is possible to directly sample the final value of the underlying Lévy process, but for Asian, lookback and barrier options, the option value depends on functionals of the Lévy process and so it is necessary to approximate those. In the case of a VG model with a lookback option, the convergence results in [ 13 ] show that to achieve an O( ) root mean square (RMS) error using a standard Monte Carlo method with a uniform timestep discretisation requires O( −2) paths, each with O( −1) timesteps, leading to a computational complexity of O( −3). In the case of a simple Brownian diffusion, Giles [ 16, 17 ] introduced a multilevel Monte Carlo (MLMC) method, reducing the computational complexity from O( −3) to O( −2) for a variety of payoffs. The objective of this paper is to investigate whether similar benefits can be obtained for exponential Lévy processes. Various researchers have investigated simulation methods for the running maximum of Lévy processes. Reference [ 15 ] develops an adaptive Monte Carlo method for functionals of killed Lévy processes with a controlled bias. Small-time asymptotic expansions of the exit probability are given with computable error bounds. For evaluating the exit probability when the barrier is close to the starting point of the process, this algorithm outperforms a uniform discretisation significantly. Reference [ 21 ] develops a novel Wiener–Hopf Monte Carlo method to generate the joint distribution of (XT , sup0≤t≤T Xt ) which is further extended to MLMC in [ 14 ], obtaining an RMS error with a computational complexity of O( −3) for Lévy processes with bounded variation, and O −4 for processes with infinite variation. The method currently cannot be directly applied to VG, NIG and α-stable processes. References [ 12, 11 ] adapt MLMC to Lévy-driven SDEs with payoffs which are Lipschitz with respect to the supremum norm. If the Lévy process does not incorporate a Brownian process, reference [11] obtains an O( −(6β)/(4−β)) upper bound on the worst case computational complexity, where β is the BG index which will be defined later. In contrast to those advanced techniques, we take the discretely monitored maximum based on a uniform timestep discretisation of the Lévy process as the approximation. The outline of the work is as follows. First we review the multilevel Monte Carlo method and present the three Lévy processes we consider in our numerical experiments. To prepare for the analysis of the multilevel variance of lookback and barrier, we bound the convergence rate of the discretely monitored running maximum for a large class of Lévy processes whose Lévy measures ha (...truncated)


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Michael B. Giles, Yuan Xia. Multilevel Monte Carlo for exponential Lévy models, Finance and Stochastics, 2017, pp. 995-1026, Volume 21, Issue 4, DOI: 10.1007/s00780-017-0341-7