Quasi-Monte Carlo Methods

Monte Carlo is a very flexible numerical method which can model and price complex instruments when other methods can not. But it has the strong disadvantage of being very time consuming.

The pricing of complex exotic products can take a few seconds when we have to simulate a high number of paths with numerous time steps to get enough accuracy. And if we need to generate scenario analysis or stress testing by changing input parameters, seconds can become minutes or hours…

Computation speed is quickly one key aspect when running Monte Carlo simulations.

Quasi-Monte Carlo simulations are variations of Monte Carlo simulations.

They use low-discrepancy deterministic sequences with better uniformity properties than purely random sequences providing more accurate and efficient estimates compared to traditional Monte Carlo simulations.

Halton or Sobol are two examples of such sequences.

In the slides below, we generate 256 points uniformly distributed with traditional Monte Carlo method. The distribution of these random numbers might exhibit clustering or gaps, which can lead to suboptimal coverage of the space and this can result in inaccuracies in the estimations.

Quasi-Monte Carlo simulations address this issue. They are using low-discrepancy deterministic sequences, which have better uniformity properties than purely random sequences reducing clustering and gaps with a better coverage of the space. In this example we generate 256 points with Sobol sequence. The number of point has to be a power of 2 with this method.

As the sequences are deterministic, it is possible to add some noise, scrambling the numbers to add some randomness.

Quasi-Monte Carlo allows more accurate and efficient estimates for low dimension problems, with a rate of convergence close to O(log(N)^k / N) for a problem of dimension k compared to O(1 / N^0.5) with Monte Carlo simulations.

For a problem of dimension 1 we would get roughly the same accuracy of the estimate with 10^4 quasi-Monte Carlo simulations than with 10^7  Monte Carlo simulations, which allows a huge gain in terms of computation time.

We illustrate this below, with the estimation of the number π by Monte Carlo simulations. We simulate points in a square which side length is two units. The probability that the point is in the disc inside the square which has a radius of 1 is π / 4.

As highlighted on the chart, which shows π estimations by increasing the number of simulations, we start to get a strong accuracy with close to 10000 simulations with Quasi-Monte Carlo. It would require 10M to get similar accuracy with traditional Monte Carlo.

Presentation

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List of Quant Next Courses

We summarize below quantitative finance training courses proposed by Quant Next. Courses are 100% digital, they are composed of many videos, quizzes, applications and tutorials in Python.

Complete training program:

Options, Pricing, and Risk Management Part I: introduction to derivatives, arbitrage free pricing, Black-Scholes model, option Greeks and risk management.

Options, Pricing, and Risk Management Part II: numerical methods for option pricing (Monte Carlo simulations, finite difference methods), replication and risk management of exotic options.

Options, Pricing, and Risk Management Part III:  modelling of the volatility surface,  parametric models with a focus on the SVI model, and stochastic volatility models with a focus on the Heston and the SABR models.

A la carte:

Monte Carlo Simulations for Option Pricing: introduction to Monte Carlo simulations, applications to price options, methods to accelerate computation speed (quasi-Monte Carlo, variance reduction, code optimisation).

Finite Difference Methods for Option Pricing: numerical solving of the Black-Scholes equation, focus on the three main methods: explicit, implicit and Crank-Nicolson.

Replication and Risk Management of Exotic Options: dynamic and static replication methods of exotic options with several concrete examples.

Volatility Surface Parameterization: the SVI Model: introduction on the modelling of the volatility surface implied by option prices, focus on the parametric methods, and particularly on the Stochastic Volatility Inspired (SVI) model and some of its extensions.

The SABR Model: deep dive on on the SABR (Stochastic Alpha Beta Rho) model, one popular stochastic volatility model developed to model the dynamic of the forward price and to price options.

The Heston Model for Option Pricing: deep dive on the Heston model, one of the most popular stochastic volatility model for the pricing of options.