Solution Behavior of Heston Model Using Impression Matrix Norm

Abstract

We are interested in the behavior of solutions for several stochastic differential equations such as Heston model. We focus on the numerical solutions via Milstein method for different stock market conditions. We examine the advantages and limitations of the model. Moreover, we introduce 3-dimensional matrix norms. Furthermore, we define market impression matrix norm as an application to the 3-dimensional matrix norms. Later, we perform simulations for various parameters.

Appendix

1.1 Milstein Method

Definition.

Let $\overline{y}_{N}$ be the numerical approximation to y(t _N ) after N steps with constant stepsize $h = (t_{N} - t_{0}/N)$; then $\overline{y}$ is said to converge strongly to y with order p if $\exists C > 0$ (independent of h) and δ > 0 such that

$$\displaystyle{E(\|\overline{y}_{N} - y(t_{N})\|) \leq Ch^{p},\ h \in (0,\delta ).}$$

Lets consider the following SDEs:

$$\displaystyle{ dy = f(t,y)dt + g(t,y)dW,\ y(0) = y_{0}. }$$

(1)

Milstein method has the following form for equation (1):

$$\displaystyle\begin{array}{rcl} \Delta y_{i}& =& f(t_{i},y_{i})\Delta t_{i} + g(t_{i},y_{i})\Delta W_{i} + \frac{1} {2}g(t_{i},y_{i})\frac{\partial g} {\partial y}(t_{i},y_{i})(\Delta W_{i}^{2} - \Delta t_{ i}) {}\\ \Delta t_{i}& =& t_{i+1} - t_{i} {}\\ \Delta W_{i}& =& W_{i+1} - W_{i} {}\\ \end{array}$$

Milstein method has strong order 1 for solving SDEs. Also Brownian motion $\Delta W_{i}$ can be modeled as $\Delta W_{i} = z_{i}\sqrt{\Delta t_{i}}$ where z _i is chosen from N(0,1) standard normally random variable (see [3]).

1.2 Heston Model

In Heston’s stochastic volatility model the asset price process S _t and the variance process v _t : = σ _t ² solve the following two-dimensional stochastic differential equation (see [1]):

$$\displaystyle\begin{array}{rcl} dS_{t}& =& (r - q)S_{t}dt + \sqrt{v_{t}}S_{t}dW_{1}(t) {}\\ dv_{t}& =& \kappa (\theta -v_{t})dt +\xi \sqrt{v_{t}}dW_{2}(t) {}\\ \end{array}$$

At the Black-Scholes-Merton (BSM) model (see [7]), the volatility σ was assumed to be constant. The main difference between BSM and Heston model is volatility behavior. It is stochastic and it satisfies mean-reverting property with a mean-reverting drift at the Heston model. The W ₁andW ₂ represent Brownian motions of asset price process and the variance process is correlated with correlation coefficient ρ ∈ [−1, 1]. Here ξ > 0 is the volatility parameter of the variance process, r ≥ 0 is the risk-free interest rate, q ≥ 0 is the dividend yield, κ > 0 is the rate of mean reversion, and θ > 0 is the long-run variance level (see [1]). Stochastic volatility model of Heston (1993) is frequently used. Heston’s model is derived from the CIR model of Cox, Ingersoll, and Ross (1985) for interest rates (see [8]). We choose the parameters as they satisfy the Feller condition 2κ θ ≥ ξ ² at our simulations so that non-negativity of volatility can be guaranteed (see [9]).