View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Spiral - Imperial College Digital Repository Particle Filtering-based Maximum Likelihood Estimation for Financial Parameter Estimation Jinzhe Yang Binghuan Lin Wayne Luk Terence Nahar Imperial College & SWIP Techila Technologies Ltd Imperial College SWIP London & Edinburgh, UK Tampere University of Technology London, UK London, UK [email protected] Tampere, Finland [email protected] [email protected] [email protected] Abstract—This paper presents a novel method for estimating result, which cannot meet runtime requirement in practice. In parameters of financial models with jump diffusions. It is a addition, high performance computing (HPC) technologies are Particle Filter based Maximum Likelihood Estimation process, still new to the finance industry. We provide a PF based MLE which uses particle streams to enable efficient evaluation of con- straints and weights. We also provide a CPU-FPGA collaborative solution to estimate parameters of jump diffusion models, and design for parameter estimation of Stochastic Volatility with we utilise HPC technology to speedup the estimation scheme Correlated and Contemporaneous Jumps model as a case study. so that it would be acceptable for practical use. The result is evaluated by comparing with a CPU and a cloud This paper presents the algorithm development, as well computing platform. We show 14 times speed up for the FPGA as the pipeline design solution in FPGA technology, of PF design compared with the CPU, and similar speedup but better convergence compared with an alternative parallelisation scheme based MLE for parameter estimation of Stochastic Volatility using Techila Middleware on a multi-CPU environment. with Correlated and Contemporaneous Jumps (SVCJ) [2]. The contributions of this paper are summarised as follows: NTRODUCTION I. I • We provide a PF-based MLE method that is parallelisable Particle Filter (PF), also known as Sequential Monte Carlo and robust. It can be a potential competitor for default method (SMC), is a computationally intensive state estimation estimation tools used for jump-diffusion models such as technique that is applied to solve dynamic problems involving MCMC. non-normality and non-linearity. Information about how the • We develop a collaborative CPU-FPGA platform for this method should evolve is contained in every single particle, method, making full use of both CPU and FPGA by and in order to acquire accurate estimation results, the number resampling on multi-threaded CPU and enabling efficient of particles is often large. Empirically, the more complex evaluation of constraints and weights on FPGA. the system, the larger the amount of particles. A common • We offer optimisations through algorithmic method concern is that the computational intensity for solving real- to minimise bottlenecks in complex computation, and world problems using PF is high, and time constraint limits though memory optimisation to reduce bandwidth over- the applicability of PF to fields with runtime requirement. A head. detailed discussion of PF methods in practice can be found in • We benchmark different parallelisation schemes of parti- [1]. cle filtering on different computing platforms. Parameter estimation is one of the most crucial topics in II. BACKGROUND the finance industry, especially for buy-side practitioners. The parameters are estimated from historical data and could be Introducing jump components of diffusion models used in used to predict future movements of the market. Because of finance is one of the main developments in financial modeling the lack of likelihood functions for jump diffusion models, literature during the past decade. This paper focuses on practitioners use Bayesian methods as default estimation tools, the affine jump diffusion model, which has good analytical such as Markov Chain Monte Carlo (MCMC) methods. This properties. On the one hand, this method allows the derivation paper explores an alternative estimation method, the maximum of semi-closed form for option pricing formulae that enable likelihood estimation (MLE), for financial parameter estima- fast option pricing. This feature is a necessary criterion of the tion. model to be acceptable by the financial industry. On the other Some attempts have been made for parameter estimation hand, this model captures large movements in the financial using PF, but they only cover models without jump diffusion. market by incorporating jump components. The particles are treated as the evaluation function for the In our approach, the joint dynamic (SVCJ) of log-price st MLE, so the likelihood function could also be provided by the and stochastic variance Vt is given by: PF. The restriction of PF is that it brings heavy computational Nt burden which might be infeasible for traditional platforms. It p s X s dst = (µ − 1=2Vt)dt + VtdWt + d( Zj ) (1) can take a few days or even up to weeks to calculate the j=1 N p Xt We extend the SIR algorithm to become the auxiliary dV = κ(θ − V )dt + σ V dW v + d( Zv) (2) t t t t j particle filter (APF) to handle such problem. In contrast to j=1 a standard SIR particle filter, where the sampling of Vt+1 is where Nt ∼ poi(λdt) is a Poisson distributed random number blind of st+1, APF samples Vt+1 from p(Vt+1jVt; st+1), and s v that describes the number of jumps, Zj = µs + ρsZj + σsj, using auxiliary variables as an approximation when the exact v j ∼ N(0; 1) (normal distribution) and Zj ∼ exp(µv) distribution of p(Vt+1jVt; st + 1) is not available. (exponential distribution) are random numbers that describe Based on [6] and [7], we extend the algorithm to the SVCJ the jump size in log-price process and variance process, model. respectively. The set of parameters from the SVCJ model that we estimate Algorithm 1 APF are: µ, µv, µy, κ, θ, σv, σy, ρ, ρj and λ, where for t=1 to T do • µ: long term mean of the return for n=1 to N do • µv: rate of exponential jump size in volatility Generate auxiliary variable of volatility as the expected • µ : mean of jump size in return y integrated volatility: v^t = Vt−1 + κ(θ − Vt−1) + • κ: rate of mean reversion 1 ρσv(yt−1 − µ + 2 Vt−1 − Jt−1) + Jt−1µv • θ: long term variance Compute first stage weights wt / p(stjst−1; v^t) • σ : volatility of volatility v Resample Vt−1, v^ according to the first stage weights. • σy: volatility of jump size in return for n=1 to N do • ρ: correlation between volatility process and return pro- Propagating the current particles by generating: cess Number of jumps J, • ρ : correlation between jumps in return and in volatility j Jump sizes in return Zy • λ: jump intensity Jump sizes in volatility Zv using v^t; Some of the parameters above are not provided in formula 1 Generating volatility from p(Vtjst; st−1; J; Zv;Zy) and 2, because they appear when the model is discretised: it is Calculate second stage reweighting π as in standard transformed into functions of finite set of state variables with auxiliary particle filter using importance sampling rule. known joint distribution. It is a generic model representing Resample V , J, Zv, Zy jump-diffusion models: • When λ = 0 (SV), it is a Heston Model [3]. • When µv = 0 (SVJ), it is a Bates Model [4]. According to [8], the likelihood function can be derived as: In other words, the model we estimate is more general than T Y several widely used financial models. Following the literature, LIK =p ^(s1) p^(stjst−1) (3) s v we assume two Brownian motions W and W are correlated t=2 with correlation coefficient ρ. This enables the model to d capture the well-known leverage effect in financial time series. As proved in [9], the likelihood estimator LN is an unbiased estimator of the likelihood Ld of the state space model (Euler III. ALGORITHM DEVELOPMENT discretisation of the original jump-diffusion process). Estimation of diffusion models in finance has long been Furthermore, the likelihood of the discrete process con- a challenge due to the existence of latent states and the verges to the likelihood of the diffusion process. Denote the discreteness of observable data. Mixing a jump process and set of model parameters by θ. As ∆t ! 0, the original process naturally introduces more difficulties by providing more flexibility to the model. Ld(θ) ! Lc(θ) (4) Maximum likelihood estimation (MLE) is a widely used statistical tool to estimate the parameters of a model. By where Lc is the likelihood of the original jump diffusion maximising the likelihood function, the estimation procedure process. fits the model to the observed data. We use stock prices as a Since we can get an estimation of likelihood as a byproduct set of the observations and variance as the latent states. of the particle filtering procedure, it is straightforward to con- We first develop a Sequential Importance Resampling (SIR) struct the particle filtering-based MLE by taking the estimator strategy for our parameter estimation. It is a standard strategy based on particle filtering as the objective function of an for models with/without jumps. However, the result accuracy optimisation algorithm. To summarise the MLE procedure: for are different between the two kinds of models. For models each iteration of MLE, we input a set of estimated parameters, without jumps, such as the Heston Model, it could provide ac- the PF outputs all corresponding results and evaluates the curate parameter sets. Nevertheless, when it is applied to jump- likelihood function, the optimiser refers to the likelihood and i diffusion models, where we take distribution q(Vt+1jVt ; st+1) generates another set of parameters.