Pricing and hedging American options by Monte Carlo methods using a Malliavin calculus approach

Bally, Vlad; Caramellino, Lucia; Zanette, Antonino

doi:10.1515/mcma.2005.11.2.97

Cited by 14 publications

(31 citation statements)

References 15 publications

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“…3) where B is a d-dimensional uncorrelated Brownian motion and Σ i is the i-th row of the matrix Σ defined as a square root of the correlation matrix Γ, given by12 . .…”

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confidence: 99%

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

Goudenège

Molent

Zanette

2020

Quantitative Finance

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In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black-Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. Moreover, these two methods solve the backward dynamic programming problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite this model is only bidimensional, the whole history of the process impacts on the price, and handling all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.Pricing American options is clearly a crucial question of finance but also a challenging one since computing the optimal exercise strategy is not an evident task. This issue is even more exacting when the underling of the option is a multi-dimensional process, such as a baskets of d assets, since in this case the direct application of standard numerical schemes, such as finite difference or tree methods, is not possible because of the exponential growth of the calculation time and the required working memory.Common approaches in this field can be divided in four groups: techniques which rely on recombinant trees to discretize the underlyings (see [4], [11] and [24]), techniques which employ regression on a truncated basis of L 2 in order to compute the conditional expectations (see [28] and [32]), techniques which exploit Malliavin calculus to obtain representation formulas for the conditional expectation (see [1], [3], [9], and [27]) and techniques which make use of duality-based approaches for Bermudan option pricing (see [21], [26] and [31]). Recently, Machine Learning algorithms (Rasmussen and Williams [33]) and Deep Learning techniques (Nielsen [30]) have found great application in this sector of option pricing. Neural networks are used by Kohler et al. [25] to price American options based on several underlyings. Deep Learning techniques are nowadays widely used in solving large differential equations, which is intimately related to option pricing. In particular, Han et al. [20] introduce a Deep Learning-based approach that can handle general high-dimensional parabo...

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“…3) where B is a d-dimensional uncorrelated Brownian motion and Σ i is the i-th row of the matrix Σ defined as a square root of the correlation matrix Γ, given by12 . .…”

mentioning

confidence: 99%

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

Goudenège

Molent

Zanette

2020

Quantitative Finance

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“…Many works have been published about option pricing with stochastic volatility models. In the last few years, the importance of applying Malliavin calculus was demonstrated ( [1,2,6]). Malliavin calculus is a suitable tool to compute the value of the conditional expectation in order to resolve several problems in the field of financial mathematics, and in particular for the American options pricing problem.…”

Section: Introductionmentioning

confidence: 99%

“…The papers developed by Fournié et al [4,5] are considered as the background basis for Malliavin calculus in financial mathematics. In [2], Bally et al have developed a representation formula for the conditional expectation using Malliavin calculus in order to evaluate the American option for a constant volatility. Abbas-Turki and Lapeyre [1] have developed a new method to price American options under stochastic volatility.…”

Section: Introductionmentioning

confidence: 99%

Pricing American put options under stochastic volatility using the Malliavin derivative

Kharrat¹

2019

Rev. Un. Mat. Argentina

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“…In recent times, several studies on the computation of Greeks for American options have been reported. See Gobet [5] and Bally et al [6], for example. One can obtain Greeks by simply taking the differential of the option pricing function if the closed-form pricing formula is known.…”

Section: Introductionmentioning

confidence: 99%

“…Although Gobet [5], Bally et al [6] considered the Monte Carlo simulation an approach to compute Greeks for American options, the approach is mathematically difficult to understand. Therefore, the finite difference approach is still widely used in practical purposes to compute vega and rho.…”

Section: Introductionmentioning

confidence: 99%

Computation of Greeks Using Binomial Tree

Muroi¹,

Suda²

2017

JMF

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This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that Greeks for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithm of Greeks for American options using the binomial tree is also given in this article. There are three advantageous points to use binomial tree approach for the computation of Greeks. First, mathematics is much simpler than using the continuous time Malliavin calculus approach. Second, we can construct a simple algorithm to obtain the Greeks for American options. Third, this algorithm is very efficient because one can compute the price and Greeks (delta, gamma, vega, and rho) at once. In spite of its importance, only a few previous studies on the computation of Greeks for American options exist, because performing sensitivity analysis for the optimal stopping problem is difficult. We believe that our method will become one of the popular ways to compute Greeks for options.

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Pricing and hedging American options by Monte Carlo methods using a Malliavin calculus approach

Cited by 14 publications

References 15 publications

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

Pricing American put options under stochastic volatility using the Malliavin derivative

Computation of Greeks Using Binomial Tree

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