# The Euler-Maruyama approximation for the asset price in the mean-reverting-theta stochastic volatility model

Research output: Contribution to journalArticle

6 Citations (Scopus)

### Abstract

Stochastic differential equations (SDEs) have been used to model an asset price and its volatility in finance. Lewis (2000)  developed the mean-reverting-theta processes which can not only model the volatility but also the asset price. In this paper, we will consider the following mean-reverting-theta stochastic volatility model
dV(t)=α2(μ2−V(t))dt+σ2V(t)βdw2(t).
We will first develop a technique to prove the non-negativity of solutions to the model. We will then show that the EM numerical solutions will converge to the true solution in probability. We will also show that the EM solutions can be used to compute some financial quantities related to the SDE model including the option value, for example.
Original language English 2209-2223 15 Computers and Mathematics with Applications 64 7 https://doi.org/10.1016/j.camwa.2012.01.037 Published - Oct 2012

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Stochastic Volatility Model
Stochastic models
Euler
Approximation
Volatility
Stochastic Equations
Differential equations
Differential equation
Stochastic Volatility
Nonnegativity
Finance
Model
Numerical Solution
Converge

### Keywords

• Euler-Maruyama method
• stochastic differential equation
• Brownian motion
• option value

### Cite this

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title = "The Euler-Maruyama approximation for the asset price in the mean-reverting-theta stochastic volatility model",
abstract = "Stochastic differential equations (SDEs) have been used to model an asset price and its volatility in finance. Lewis (2000)  developed the mean-reverting-theta processes which can not only model the volatility but also the asset price. In this paper, we will consider the following mean-reverting-theta stochastic volatility modeldV(t)=α2(μ2−V(t))dt+σ2V(t)βdw2(t).We will first develop a technique to prove the non-negativity of solutions to the model. We will then show that the EM numerical solutions will converge to the true solution in probability. We will also show that the EM solutions can be used to compute some financial quantities related to the SDE model including the option value, for example.",
keywords = "Euler-Maruyama method, stochastic differential equation, Brownian motion, option value",
author = "Chaminda Baduraliya and Xuerong Mao",
year = "2012",
month = "10",
doi = "10.1016/j.camwa.2012.01.037",
language = "English",
volume = "64",
pages = "2209--2223",
journal = "Computers and Mathematics with Applications",
issn = "0898-1221",
number = "7",

}

In: Computers and Mathematics with Applications, Vol. 64, No. 7, 10.2012, p. 2209-2223.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The Euler-Maruyama approximation for the asset price in the mean-reverting-theta stochastic volatility model

AU - Mao, Xuerong

PY - 2012/10

Y1 - 2012/10

N2 - Stochastic differential equations (SDEs) have been used to model an asset price and its volatility in finance. Lewis (2000)  developed the mean-reverting-theta processes which can not only model the volatility but also the asset price. In this paper, we will consider the following mean-reverting-theta stochastic volatility modeldV(t)=α2(μ2−V(t))dt+σ2V(t)βdw2(t).We will first develop a technique to prove the non-negativity of solutions to the model. We will then show that the EM numerical solutions will converge to the true solution in probability. We will also show that the EM solutions can be used to compute some financial quantities related to the SDE model including the option value, for example.

AB - Stochastic differential equations (SDEs) have been used to model an asset price and its volatility in finance. Lewis (2000)  developed the mean-reverting-theta processes which can not only model the volatility but also the asset price. In this paper, we will consider the following mean-reverting-theta stochastic volatility modeldV(t)=α2(μ2−V(t))dt+σ2V(t)βdw2(t).We will first develop a technique to prove the non-negativity of solutions to the model. We will then show that the EM numerical solutions will converge to the true solution in probability. We will also show that the EM solutions can be used to compute some financial quantities related to the SDE model including the option value, for example.

KW - Euler-Maruyama method

KW - stochastic differential equation

KW - Brownian motion

KW - option value

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U2 - 10.1016/j.camwa.2012.01.037

DO - 10.1016/j.camwa.2012.01.037

M3 - Article

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EP - 2223

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

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ER -