1 Introduction
Many praises have been used to describe Black-Scholes’ pricing theory, which is the most successful and widely used in the application. Although the elegant Black-Scholes’ theory is, there is an absolute disadvantage(Heston 1993). The strong assumption of constant volatility cannot express high peak and flat tail character of derivatives, and the relative between the spot return and the variance. Therefore, there are several theories try to extend and modify the original Black-Scholes’ model. The Heston stochastic volatility model is the most popular extension to the Black-Scholes’ model(Tse & Wan 2013).
The first scheme widely accepted is Euler-Maruyama Scheme. This scheme is efficient, easy to implement, and almost can be used to any stochastic differential equations. However, this model cannot handle a situation, if the Feller condition is not satisfied. Actually, the variance of a step can become negative, when the Feller condition is violated. There are several authors done great work to modify the Euler scheme to extend the situation which the model can be applied. Unfortunately, there is still significant biased when the Feller condition is violated(Andersen 2007). In another aspect, Broadie and Kaya present a method to exactly simulate from the Heston model. Their model based on acceptance-rejection sampling and introduce closed-form equations for the characteristics function. Despite elegant, the most serious problem of this model is the speed of application, which make this model almost has no practical implement. Based on Broadie and Kaya theory, Andersen proposes a means to efficiently approximate the variance. …show more content…
This method also is called quadratic exponential (QE method), which is be considered as one of the best algorithms for simulating from the Heston model. However, this method relies on huge time steps(Tse & Wan 2013). To solve this problem, Tse and Wan present a model with Inverse Gaussian approximation and IPZ scheme, which can reduce the number of time steps needed for pursuing same accurate as QE method. Recently, Begin et al raise a new method for sampling, which can imply model parameters of considered quite extreme. This paper focuses on reviewing the different procedures from simulation of Heston model and investigate the improvement raised by Begin et al and Tse and Wan. The rest session will be organized as follow: Section 2 will describe the main conclusions and several properties of Heston model. Section 3 will review some important simulation scheme from Heston model. And section 4 will conclude the opinions and shortages of this article and propose some issue need further study. 2 Definition and Basic Properties …show more content…
investors will not pay for risk premiums, and defined by following stochastic differential equations:
(1)
(2)
Where: S(t) represents the price process of an underlying assets. V(t) is the variance of the corresponding instantaneous returns. And the initial condition S(0), V(0) should be strictly positive. r is the risk-free return rate (non-negative constant). k is the speed of the mean reversion. θ is the mean value of the variance. σ is the volatility of the variance. These three are positive constants. Ws(t) and Wv(t) are Brownian motion variables, and ρ is the correlation coefficient between them. In some probability measure, we assume dWs(t) * dWv(t) = ρdt.
For computational convenient, it is general to employee logarithm to transform the asset price process S(t) into X(t) = Log (S(t)), and apply Itô’s Lemma(3) to equations (1).
(3)
We can get following stochastic differential equations:
Many praises have been used to describe Black-Scholes’ pricing theory, which is the most successful and widely used in the application. Although the elegant Black-Scholes’ theory is, there is an absolute disadvantage(Heston 1993). The strong assumption of constant volatility cannot express high peak and flat tail character of derivatives, and the relative between the spot return and the variance. Therefore, there are several theories try to extend and modify the original Black-Scholes’ model. The Heston stochastic volatility model is the most popular extension to the Black-Scholes’ model(Tse & Wan 2013).
The first scheme widely accepted is Euler-Maruyama Scheme. This scheme is efficient, easy to implement, and almost can be used to any stochastic differential equations. However, this model cannot handle a situation, if the Feller condition is not satisfied. Actually, the variance of a step can become negative, when the Feller condition is violated. There are several authors done great work to modify the Euler scheme to extend the situation which the model can be applied. Unfortunately, there is still significant biased when the Feller condition is violated(Andersen 2007). In another aspect, Broadie and Kaya present a method to exactly simulate from the Heston model. Their model based on acceptance-rejection sampling and introduce closed-form equations for the characteristics function. Despite elegant, the most serious problem of this model is the speed of application, which make this model almost has no practical implement. Based on Broadie and Kaya theory, Andersen proposes a means to efficiently approximate the variance. …show more content…
This method also is called quadratic exponential (QE method), which is be considered as one of the best algorithms for simulating from the Heston model. However, this method relies on huge time steps(Tse & Wan 2013). To solve this problem, Tse and Wan present a model with Inverse Gaussian approximation and IPZ scheme, which can reduce the number of time steps needed for pursuing same accurate as QE method. Recently, Begin et al raise a new method for sampling, which can imply model parameters of considered quite extreme. This paper focuses on reviewing the different procedures from simulation of Heston model and investigate the improvement raised by Begin et al and Tse and Wan. The rest session will be organized as follow: Section 2 will describe the main conclusions and several properties of Heston model. Section 3 will review some important simulation scheme from Heston model. And section 4 will conclude the opinions and shortages of this article and propose some issue need further study. 2 Definition and Basic Properties …show more content…
investors will not pay for risk premiums, and defined by following stochastic differential equations:
(1)
(2)
Where: S(t) represents the price process of an underlying assets. V(t) is the variance of the corresponding instantaneous returns. And the initial condition S(0), V(0) should be strictly positive. r is the risk-free return rate (non-negative constant). k is the speed of the mean reversion. θ is the mean value of the variance. σ is the volatility of the variance. These three are positive constants. Ws(t) and Wv(t) are Brownian motion variables, and ρ is the correlation coefficient between them. In some probability measure, we assume dWs(t) * dWv(t) = ρdt.
For computational convenient, it is general to employee logarithm to transform the asset price process S(t) into X(t) = Log (S(t)), and apply Itô’s Lemma(3) to equations (1).
(3)
We can get following stochastic differential equations: