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)