Sequential Monte Carlo Methods for Option Pricing

In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure

Multidimensional Quasi-Monte Carlo Malliavin Greeks

We investigate the use of Malliavin calculus in order to calculate the Greeks of multidimensional complex path-dependent options by simulation. For this purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin Calculus approach over different techniques that have been previously proposed. Indeed, these techniques may be computationally expensive and do not provide flexibility for variance reduction. In contrast, the Malliavin approach exhibits a higher flexibility by providing a class of functions that return the same expected value (the Greek) with different accuracies. This versatility for variance reduction is not possible without the use of the generalized integral by part formula of Malliavin Calculus. In the multidimensional context, we find convenient formulas that permit to improve the localization technique, introduced in Fourni\'e et al and reduce both the computational cost and the variance. Moreover, we show that the parameters employed for variance reduction can be obtained \textit{on the flight} in the simulation. We illustrate the efficiency of the proposed procedures, coupled with the enhanced version of Quasi-Monte Carlo simulations as discussed in Sabino, for the numerical estimation of the Deltas of call, digital Asian-style and Exotic basket options with a fixed and a floating strike price in a multidimensional Black-Scholes market.Comment: 22 pages, 6 figure

Monte Carlo evaluation of sensitivities in computational finance

In computational finance, Monte Carlo simulation is used to compute the correct prices for financial options. More important, however, is the ability to compute the so-called "Greeks'', the first and second order derivatives of the prices with respect to input parameters such as the current asset price, interest rate and level of volatility.\ud \ud This paper discusses the three main approaches to computing Greeks: finite difference, likelihood ratio method (LRM) and pathwise sensitivity calculation. The last of these has an adjoint implementation with a computational cost which is independent of the number of first derivatives to be calculated. We explain how the practical development of adjoint codes is greatly assisted by using Algorithmic Differentiation, and in particular discuss the performance achieved by the FADBAD++ software package which is based on templates and operator overloading within C++.\ud \ud The pathwise approach is not applicable when the financial payoff function is not differentiable, and even when the payoff is differentiable, the use of scripting in real-world implementations means it can be very difficult in practice to evaluate the derivative of very complex financial products. A new idea is presented to address these limitations by combining the adjoint pathwise approach for the stochastic path evolution with LRM for the payoff evaluation

Investigation into Vibrato Monte Carlo for the Computation of\ud Greeks of Discontinuous Payoffs

Monte Carlo simulation is a popular method in computational finance. Its basic theory is relatively simple, it is also quite easy to implement and allows nevertheless an efficient pricing of financial options, even in high-dimensional problems (basket options, interest rates products...).\ud \ud The pricing of options is just one use of Monte Carlo in finance. More important than the prices themselves are their sensitivities to input parameters (underlying asset value, interest rates, market volatility...). Indeed we need those sensitivities (also known as "Greeks") to hedge against market risk.\ud \ud In this paper, we will first recall classical approaches to the computation of Greeks through Monte Carlo simulation: finite differences, Likelihood Ratio method (LRM) and Pathwise Sensitivities (PwS). Each of those approaches has particular limitations in the case of options with discontinuous payoffs. We will expound those limitations and introduce a new hybrid method proposed by Prof. Mike Giles, the Vibrato Monte Carlo, which combines both Pathwise Sensitivity and Likelihood Ratio methods to get around their shortcomings.\ud \ud We will discuss the possible use of Vibrato Monte Carlo ideas for options with discontinuous payoffs. My personal contribution is an improvement to the standard Vibrato Monte Carlo yielding both computational savings and an improved accuracy. I will call it Allargando Vibrato Monte Carlo (AVMC). I then also extend the Vibrato Monte Carlo technique to discretely sampled path dependent options (digital option with discretely sampled barrier, lookback option with discretely sampled maximum)

Quasi-Monte Carlo methods for calculating derivatives sensitivities on the GPU

The calculation of option Greeks is vital for risk management. Traditional pathwise and finite-difference methods work poorly for higher-order Greeks and options with discontinuous payoff functions. The Quasi-Monte Carlo-based conditional pathwise method (QMC-CPW) for options Greeks allows the payoff function of options to be effectively smoothed, allowing for increased efficiency when calculating sensitivities. Also demonstrated in literature is the increased computational speed gained by applying GPUs to highly parallelisable finance problems such as calculating Greeks. We pair QMC-CPW with simulation on the GPU using the CUDA platform. We estimate the delta, vega and gamma Greeks of three exotic options: arithmetic Asian, binary Asian, and lookback. Not only are the benefits of QMC-CPW shown through variance reduction factors of up to $1.0 \times 10^{18}$, but the increased computational speed through usage of the GPU is shown as we achieve speedups over sequential CPU implementations of more than $200$x for our most accurate method.Comment: 26 pages, 12 figure

Malliavin calculus in finance

This article is an introduction to Malliavin Calculus for practitioners. We treat one specific application to the calculation of greeks in Finance. We consider also the kernel density method to compute greeks and an extension of the Vega index called the local vega index.Malliavin claculus, computational finance, Greeks, Monte Carlo methods, kernel density method

Adjoint methods for computing sensitivities in local volatility surfaces

In this paper we present the adjoint method of computing sensitivities of option prices with respect to nodes in the local volatility surface. We first introduce the concept of algorithmic differentiation and how it relates to\ud path-wise sensitivity computations within a Monte Carlo framework. We explain the two approaches available: forward mode and adjoint mode. We illustrate these concepts on the simple example of a model with a geometric Brownian motion driving the underlying price process, for which\ud we compute the Delta and Vega in forward and adjoint mode. We then go on to explain in full detail how to apply these ideas to a model where the underlying has a volatility term defined by a local volatility surface. We provide source codes for both the simple and the more complex case and\ud analyze numerical results to show the strengths of the adjoint approach

Numerical schemes and Monte Carlo techniques for Greeks in stochastic volatility models

The main objective of this thesis is to propose approximations to option sensitivities in stochastic volatility models. The first part explores sequential Monte Carlo techniques for approximating the latent state in a Hidden Markov Model. These techniques are applied to the computation of Greeks by adapting the likelihood ratio method. Convergence of the Greek estimates is proved and tracking of option prices is performed in a stochastic volatility model. The second part defines a class of approximate Greek weights and provides high-order approximations and justification for extrapolation techniques. Under certain regularity assumptions on the value function of the problem, Greek approximations are proved for a fully implementable Monte Carlo framework, using weak Taylor discretisation schemes. The variance and bias are studied for the Delta and Gamma, when using such discrete-time approximations. The final part of the thesis introduces a modified explicit Euler scheme for stochastic differential equations with non-Lipschitz continuous drift or diffusion; a strong rate of convergence is proved. The literature on discretisation techniques for stochastic differential equations has been motivational for the development of techniques preserving the explicitness of the algorithm. Stochastic differential equations in the mathematical finance literature, including the Cox-Ingersoll-Ross, the 3/2 and the Ait-Sahalia models can be discretised, with a strong rate of convergence proved, which is a requirement for multilevel Monte Carlo techniques.Open Acces