discuss result from generating n path of stock price using exact discretization and euler scheme

What is Euler's discretisation?

The method utilised is known as Euler Discretisation. The volatility path will be discretised into constant-increment time steps of Δ t, with the updated volatility, ν i + 1 given as an explicit function of ν i:

How do you find the volatility path?

The volatility path will be discretised into constant-increment time steps of Δ t, with the updated volatility, ν i + 1 given as an explicit function of ν i: ν i + 1 = ν i + κ (θ − ν i) Δ t + ξ ν i Δ W i + 1 ν

How do you discretize the process in Equation 2?

The simplest way to discretize the process in Equation (2) is to use Euler dis- cretization. This is equivalent to approximating the integrals using the left- point rule. Hence the –rst integral is approximated as the product of the integrand at time t, and the integration range dt Z

What is the Black Scholes model of asset path?

The Black Scholes model uses a stochastic differential equation with a geometric Brownian motion to model the dynamics of the asset path. It is given by: Where I am using the notation of the Wikipedia Heston Model article.

Chapter 18. Simulating Stock Prices

In most of finance, especially in analysis of derivatives, we assume that asset prices are unpredictable and follow a geometric Brownian motion. Most people find it difficult to grasp exactly what this means, but having a good understanding of it is essentia! to do any work with derivatives.

Review of Theory and Concepts

We will explore geometric Brownian motion starting with some preliminaries and a simple model of stock prices. After we discuss geometric Brownian motion and its implications, we will discuss how to estimate the necessary parameters from historical data to simulate stock prices.

What is the issue when simulating SDEs?

An important issue that arises when simulating SDE's is the allocation of computational resources. In particular,we need to determine how many sample paths,n, to generate and how many time steps,m, to simulate on eachsample path. A smaller value ofmwill result in greater bias and numerical error, whereas a smaller value ofnwill result in greater statistical noise. Indeed numerical and statistical error were both discussed in Examples 1and 2 but we did not discuss the optimal tradeo between the two in those examples. That is the problem wenow discuss: how to choosenandmin an optimal manner given a xed computational budget.

What is the alternative to second order schemes?

An alternative to second order schemes is the Euler scheme with Richardson extrapolation. This is easy toimplement and often has superior performance to second order schemes, especially in high dimensions. As aresult, the Euler scheme with Richardson extrapolation is often considered to be a benchmark scheme forreducing discretization error. In order to simplify notation, we will write Xhbfor

Building a Pricing Model Simulation

Whether we are considering buying or selling a financial instrument, the decision can be aided by studying it both numerically and graphically. This data can help us judge the next likely move that the asset might make and the moves that are less likely.

Computing Historical Volatility in Excel

For this example, we will use the Excel function "= NORMSINV (RAND ())." With a basis from the normal distribution, this function computes a random number with a mean of zero and a standard deviation of one. To compute μ, simply average the yields using the function Ln (.): the log-normal distribution .

Mathematical Model

Euler Discretisation

• Given that the SDE for the asset path is now dependent (in a temporal manner) upon the solution of the second volatility SDE, it is necessary to simulate the volatility process first and then utilise this "volatility path" in order to simulate the asset path. In the case of the original Black Scholes SDE it is possible to use Ito's Lemma to directl...

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C++ Implementation

References