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 …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 …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 …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 …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: