scholarly journals On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2930
Author(s):  
Siow Woon Jeng ◽  
Adem Kiliçman
Keyword(s):  
Monte Carlo ◽  
Stock Price ◽  
Variance Reduction ◽  
Volterra Equation ◽  
Price Volatility ◽  
Multilevel Methods ◽  
Heston Model ◽  
Control Variate ◽  
Stochastic Volterra Equation ◽  
And Control

The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient error[|Vt−Vtn|p] is finitely bounded by an exponential function of t. Furthermore, the weak error |error[Vt−Vtn]| and convergence for the stochastic Volterra equation are proven at the rate of O(n−H). In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerical experiments indicate that the proposed method is capable of achieving a substantial cost-adjusted variance reduction up to 17 times, and it is better than its predecessor individual methods in terms of cost-adjusted performance. Due to the cost-adjusted basis for our numerical experiment, the result also indicates a high possibility of potential use in practice.

scholarly journals Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities

2019 ◽  
Vol 11 (3) ◽  
pp. 815 ◽  
Author(s):  
Yijuan Liang ◽  
Xiuchuan Xu
Keyword(s):  
Monte Carlo ◽  
Variance Reduction ◽  
Convergence Rates ◽  
Option Price ◽  
Correlation Coefficients ◽  
Control Variate ◽  
Stochastic Volatilities ◽  
Conditional Monte Carlo ◽  
Mc Simulation ◽  
Martingale Representation Theorem

Pricing multi-asset options has always been one of the key problems in financial engineering because of their high dimensionality and the low convergence rates of pricing algorithms. This paper studies a method to accelerate Monte Carlo (MC) simulations for pricing multi-asset options with stochastic volatilities. First, a conditional Monte Carlo (CMC) pricing formula is constructed to reduce the dimension and variance of the MC simulation. Then, an efficient martingale control variate (CV), based on the martingale representation theorem, is designed by selecting volatility parameters in the approximated option price for further variance reduction. Numerical tests illustrated the sensitivity of the CMC method to correlation coefficients and the effectiveness and robustness of our martingale CV method. The idea in this paper is also applicable for the valuation of other derivatives with stochastic volatility.

scholarly journals Examining the Efficiency of American Put Option Pricing by Monte Carlo Methods with Variance Reduction

2018 ◽  
Vol 10 (2) ◽  
pp. 10
Author(s):  
George Chang
Keyword(s):  
Monte Carlo ◽  
Option Pricing ◽  
Variance Reduction ◽  
American Options ◽  
Balance Sheet ◽  
Put Options ◽  
Control Variate ◽  
American Put Option ◽  
Put Option ◽  
American Put

We apply the Monte Carlo simulation algorithm developed by Broadie and Glasserman (1997) and the control variate technique first introduced to asset pricing via simulation by Boyle (1977) to examine the efficiency of American put option pricing via this combined method. The importance and effectiveness of variance reduction is clearly demonstrated in our simulation results. We also found that the control variates technique does not work as well for deep-in-the-money American put options. This is because deep-in-the-money American options are more likely to be exercised early, thus the value of the American options are less in line (or less correlated) with those of their European counterparts. the same FPESS can also be observed when investigators partition large datasets into smaller datasets to address a variety of auditing questions. In this study, we fill the empirical gap in the literature by investigating the sensitivity of the FPESS to partitioned datasets. We randomly selected 16 balance-sheet datasets from: China Stock Market Financial Statements Database™, that tested to be Benford Conforming noted as RBCD. We then explore how partitioning these datasets affects the FPESS by repeated randomly sampling: first 10% of the RBCD and then selecting 250 observations from the RBCD. This created two partitioned groups of 160 datasets each. The Statistical profile observed was: For the RBCD there were no indications of Non-Conformity; for the 10%-Sample there were no overall indications that Extended Procedures would be warranted; and for the 250-Sample there were a number of indications that the dataset was Non-Conforming. This demonstrated clearly that small datasets are indeed likely to create the FPESS. We offer a discussion of these results with implications for audits in the Big-Data context where the audit In-charge would find it necessary to partition the datasets of the client. 

Long-Time Trajectorial Large Deviations and Importance Sampling for Affine Stochastic Volatility Models

2021 ◽  
Vol 53 (1) ◽  
pp. 220-250
Author(s):  
Zorana Grbac ◽  
David Krief ◽  
Peter Tankov
Keyword(s):  
Monte Carlo ◽  
Stochastic Volatility ◽  
Variance Reduction ◽  
Large Deviation ◽  
Heston Model ◽  
Stochastic Volatility Models ◽  
Monte Carlo Computation ◽  
Volatility Models ◽  
Long Time ◽  
Path Dependent

AbstractWe establish a pathwise large deviation principle for affine stochastic volatility models introduced by Keller-Ressel (2011), and present an application to variance reduction for Monte Carlo computation of prices of path-dependent options in these models, extending the method developed by Genin and Tankov (2020) for exponential Lévy models. To this end, we apply an exponentially affine change of measure and use Varadhan’s lemma, in the fashion of Guasoni and Robertson (2008) and Robertson (2010), to approximate the problem of finding the measure that minimizes the variance of the Monte Carlo estimator. We test the method on the Heston model with and without jumps to demonstrate its numerical efficiency.

Green Simulation with Database Monte Carlo

2021 ◽  
Vol 31 (1) ◽  
pp. 1-26
Author(s):  
Mingbin Feng ◽  
Jeremy Staum
Keyword(s):  
Monte Carlo ◽  
Computational Efficiency ◽  
Numerical Experiments ◽  
Variance Reduction ◽  
Idle Time ◽  
General Strategy ◽  
Control Variate ◽  
Simulation Procedure ◽  
Computational Budget ◽  
Over Time

In a setting in which experiments are performed repeatedly with the same simulation model, green simulation means reusing outputs from previous experiments to answer the question currently being asked of the model. In this article, we address the setting in which experiments are run to answer questions quickly, with a time limit providing a fixed computational budget, and then idle time is available for further experimentation before the next question is asked. The general strategy is database Monte Carlo for green simulation: the output of experiments is stored in a database and used to improve the computational efficiency of future experiments. In this article, the database provides a quasi-control variate, which reduces the variance of the estimated mean response in a future experiment that has a fixed computational budget. We propose a particular green simulation procedure using quasi-control variates, addressing practical issues such as experiment design, and analyze its theoretical properties. We show that, under some conditions, the variance of the estimated mean response in an experiment with a fixed computational budget drops to zero over a sequence of repeated experiments, as more and more idle time is invested in creating databases. Our numerical experiments on the procedure show that using idle time to create databases of simulation output provides variance reduction immediately, and that the variance reduction grows over time in a way that is consistent with the convergence analysis.

scholarly journals Penentuan Harga Opsi Asia dengan Metode Monte Carlo

2017 ◽  
Vol 3 (1) ◽  
pp. 44-48
Author(s):  
Surya Amami Pramuditya
Keyword(s):  
Monte Carlo ◽  
Confidence Interval ◽  
Stock Price ◽  
Variance Reduction ◽  
Option Price ◽  
Numerical Procedure ◽  
Reduction Technique ◽  
Asian Options ◽  
Call Option ◽  
Strike Price

An option is a contract between a holder and a writer in which the writer grants the rights (not obligations) to the holder to buy or sell the assets of the writer at a certain price (strike price) at maturity time. Asian options are included in the dependent path option. This means that Asia's payoff option depends not only on the stock price at maturity time, but it is the average stock price during its maturity and symbolized A (average). Monte Carlo is basically used as a numerical procedure to estimate the expected value of pricing product derivatives. The techniques used are the standard Monte Carlo and variance reduction. The result obtained the Asia call option price and put for both techniques with 95% confidence interval. The variance reduction technique looks faster reducing 95% confidence interval than standard method.

Effects of measurement error on Monte Carlo integration estimators of tree volume: sample diameters measured at random upper-stem heights

2015 ◽  
Vol 45 (4) ◽  
pp. 471-479 ◽  
Author(s):  
Thomas B. Lynch
Keyword(s):  
Monte Carlo ◽  
Measurement Error ◽  
Importance Sampling ◽  
Sampling Error ◽  
Monte Carlo Integration ◽  
Stem Diameter ◽  
Tree Level ◽  
Stem Volume ◽  
Control Variate ◽  
And Control

A study of the effects of measurement error was conducted on importance sampling and control variate sampling estimators of tree stem volume in which sample diameters are measured at randomly selected upper-stem heights. It was found that these estimators were unbiased in the presence of additive mean zero and multiplicative mean one measurement error applied to random samples of upper-stem diameter squared. However, biases due to measurement error are present if additive or multiplicative error is applied to upper-stem diameter rather than to upper-stem diameter squared. This is significant, as it appears that most of the previous studies on the magnitude of upper-stem diameter measurement error implicitly assume that the mean error is centered around the diameter rather than about the square of the diameter. Application of typical upper-stem measurement error obtained from previous studies to bias formulae derived here indicates that the bias could be a concern for small trees and with additive measurement error within ranges found in previous studies. Formulae for the variances of importance sampling and control variate sampling are derived, which include the contribution of both measurement error and sampling error. Results from previous studies of Monte Carlo integration estimator sampling error are combined with results from studies of upper-stem measurement error to obtain estimates of the typical magnitude of the contribution of measurement error to total estimator variability. Increases in upper-stem sample size may be warranted due to the impact of measurement error if precise estimates of stem volume at the individual-tree level are desired.

Penentuan Harga Opsi Dengan Volatilitas Stokastik Menggunakan Metode Monte Carlo

2021 ◽  
Vol 3 (1) ◽  
pp. 80-92
Author(s):  
Chalimatusadiah Chalimatusadiah ◽  
Donny Citra Lesmana ◽  
Retno Budiarti
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Stochastic Volatility ◽  
Stock Price ◽  
Option Price ◽  
Least Square ◽  
Heston Model ◽  
Strike Price ◽  
Ordinary Least Square ◽  
The Monte Carlo Method

ABSTRAKHal yang utama dalam perdagangan opsi adalah penentuan harga jual opsi yang optimal. Namun pada kenyataan sebenarnya fluktuasi harga aset yang terjadi di pasar menandakan bahwa volatilitas dari harga aset tidaklah konstan, hal ini menyebabkan investor mengalami kesulitan dalam menentukan harga opsi yang optimal. Artikel ini membahas tentang penentuan harga opsi tipe Eropa yang optimal dengan volatilitas stokastik menggunakan metode Monte Carlo dan pengaruh harga saham awal, harga strike, dan waktu jatuh tempo terhadap harga opsi Eropa. Adapun model volatilitas stokastik yang digunakan dalam penelitian ini adalah model Heston, yang mengasumsikan bahwa proses harga saham (St) mengikuti distribusi log-normal, dan proses volatilitas saham (Vt) mengikuti Proses Cox-Ingersoll-Ross. Hal pertama yang dilakukan dalam penelitian ini adalah mengestimasi parameter model Heston untuk mendapatkan harga saham dengan menggunakan metode ordinary least square dan metode numerik Euler-Maruyama. Langkah kedua adalah melakukan estimasi harga saham untuk mendapatkan harga opsi tipe Eropa menggunakan metode Monte Carlo. Hasil dari penelitian ini menunjukkan bahwa penggunaan metode Monte Carlo dalam penentuan harga opsi tipe Eropa dengan volatilitas stokastik model Heston menghasilkan solusi yang cukup baik karena memiliki nilai error yang kecil dan akan konvergen ke solusi eksaknya dengan semakin banyak simulasi. Selain itu, simulasi Monte Carlo memberikan kesimpulan bahwa parameter harga strike, harga saham awal dan waktu jatuh tempo memiliki pengaruh terhadap harga opsi yang konsisten dengan teori harga opsi. ABSTRACTWhat is important in options trading is determining the optimal selling price. However, in real market conditions, fluctuations in asset prices that occur in the market indicate that the volatility of asset prices is not constant, this causes investors to experience difficulty in determining the optimal option price. This article discusses the optimal determination of the European type option price with stochastic volatility using the Monte Carlo method and the effect of the initial stock price, strike price, and expiration date on European option prices. The stochastic volatility model used in this study is the Heston model, which assumes that the stock price process (S) follows the normal log distribution, and the stock volatility process (V) follows the Ingersoll-Ross Cox Process. The first thing to do in this study is to estimate the parameters of the Heston model to get stock prices using the ordinary least square method and the Euler-Maruyama numerical method. The second step is to estimate the share price to get the European type option price using a Monte Carlo Simulation. This study indicates that using the Monte Carlo method in determining the price of European type options with the Heston model of stochastic volatility produces a fairly good solution because it has a small error value and will converge to the exact solution with more simulations. Also, the Monte Carlo simulation concludes that the parameters of the strike price, initial stock price, and maturity date influence the option price, which is consistent with the option price theory.

EXACT SIMULATION OF THE 3/2 MODEL

2012 ◽  
Vol 15 (05) ◽  
pp. 1250032 ◽  
Author(s):  
JAN BALDEAUX
Keyword(s):  
Monte Carlo ◽  
Stock Price ◽  
Variance Reduction ◽  
Lie Symmetry Analysis ◽  
Exact Simulation ◽  
Quasi Monte Carlo ◽  
Monte Carlo Techniques ◽  
Price Process ◽  
Reduction Techniques ◽  
Conditional Monte Carlo

This paper discusses the exact simulation of the stock price process underlying the 3/2 model. Using a result derived by Craddock and Lennox using Lie Symmetry Analysis, we adapt the Broadie-Kaya algorithm for the simulation of affine processes to the 3/2 model. We also discuss variance reduction techniques and find that conditional Monte Carlo techniques combined with quasi-Monte Carlo point sets result in significant variance reductions.

scholarly journals Automatic control variates for option pricing using neural networks

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zineb El Filali Ech-Chafiq ◽  
Jérôme Lelong ◽  
Adil Reghai
Keyword(s):  
Neural Networks ◽  
Monte Carlo ◽  
Automatic Control ◽  
Variance Reduction ◽  
Gaussian Quadrature ◽  
Payoff Function ◽  
High Dimensional ◽  
Control Variates ◽  
Control Variate ◽  
Reduction Techniques

Abstract Many pricing problems boil down to the computation of a high-dimensional integral, which is usually estimated using Monte Carlo. In fact, the accuracy of a Monte Carlo estimator with M simulations is given by σ M {\frac{\sigma}{\sqrt{M}}} . Meaning that its convergence is immune to the dimension of the problem. However, this convergence can be relatively slow depending on the variance σ of the function to be integrated. To resolve such a problem, one would perform some variance reduction techniques such as importance sampling, stratification, or control variates. In this paper, we will study two approaches for improving the convergence of Monte Carlo using Neural Networks. The first approach relies on the fact that many high-dimensional financial problems are of low effective dimensions. We expose a method to reduce the dimension of such problems in order to keep only the necessary variables. The integration can then be done using fast numerical integration techniques such as Gaussian quadrature. The second approach consists in building an automatic control variate using neural networks. We learn the function to be integrated (which incorporates the diffusion model plus the payoff function) in order to build a network that is highly correlated to it. As the network that we use can be integrated exactly, we can use it as a control variate.

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