Markov switching multifractal

The MSM model can be specified in both discrete time and continuous time. Discrete time Let P_t denote the price of a financial asset, and let r_t = \ln (P_t / P_{t-1}) denote the return over two consecutive periods. In MSM, returns are specified as : r_t = \mu + \bar{\sigma}(M_{1,t}M_{2,t}...M_{\bar{k},t})^{1/2}\epsilon_t, where \mu and \sigma are constants and {\epsilon_t} are independent standard Gaussians. Volatility is driven by the first-order latent Markov state vector: : M_t = (M_{1,t}M_{2,t}\dots M_{\bar{k},t}) \in R_+^\bar{k}. Given the volatility state M_t, the next-period multiplier M_{k,t+1} is drawn from a fixed distribution with probability \gamma_k, and is otherwise left unchanged. : The transition probabilities are specified by : \gamma_k = 1 - (1 - \gamma_1)^{(b^{k-1})} . The sequence \gamma_k is approximately geometric \gamma_k \approx \gamma_1b^{k-1} at low frequency. The marginal distribution has a unit mean, has a positive support, and is independent of . Binomial MSM In empirical applications, the distribution is often a discrete distribution that can take the values m_0 or 2-m_0 with equal probability. The return process r_t is then specified by the parameters \theta = (m_0,\mu,\bar{\sigma},b,\gamma_1). Note that the number of parameters is the same for all \bar{k}>1. Continuous time MSM is similarly defined in continuous time. The price process follows the diffusion: : \frac{dP_t}{P_t} = \mu dt + \sigma(M_t)\,dW_t, where \sigma(M_t) = \bar{\sigma}(M_{1,t}\dots M_{\bar{k},t})^{1/2}, W_t is a standard Brownian motion, and \mu and \bar{\sigma} are constants. Each component follows the dynamics: : The intensities vary geometrically with : :\gamma_k = \gamma_1b^{k-1}. When the number of components \bar{k} goes to infinity, continuous-time MSM converges to a multifractal diffusion, whose sample paths take a continuum of local Hölder exponents on any finite time interval. == Inference and closed-form likelihood ==

When M has a discrete distribution, the Markov state vector M_t takes finitely many values m^1,...,m^d \in R_+^{\bar{k}}. For instance, there are d = 2^{\bar{k}} possible states in binomial MSM. The Markov dynamics are characterized by the transition matrix A = (a_{i,j})_{1\leq i,j\leq d} with components a_{i,j} = P\left(M_{t+1} = m^j| M_t = m^i\right). Conditional on the volatility state, the return r_t has Gaussian density : f( r_t | M_t = m^i) = \frac{1} {\sqrt{2\pi\sigma^2(m^i)}}\exp\left[-\frac{(r_t-\mu)^2}{2\sigma^2(m^i)}\right] . Conditional distribution Closed-form Likelihood The log likelihood function has the following analytical expression: :\ln L(r_1,\dots,r_T;\theta) = \sum_{t=1}^{T}\ln[\omega(r_t).(\Pi_{t-1}A)]. Maximum likelihood provides reasonably precise estimates in finite samples. or simulated likelihood via a particle filter. == Forecasting ==

Given r_1,\dots,r_t, the conditional distribution of the latent state vector at date t+n is given by: :\hat{\Pi}_{t,n} = \Pi_tA^n.\, MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample. Calvet and Fisher and Fractionally Integrated GARCH. Lux obtains similar results using linear predictions. == Applications ==

Multiple assets and value-at-risk Extensions of MSM to multiple assets provide reliable estimates of the value-at-risk in a portfolio of securities. == Related approaches ==

MSM is a stochastic volatility model with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D. Hamilton. MSM is closely related to the Multifractal Model of Asset Returns. MSM improves on the MMAR's combinatorial construction by randomizing arrival times, guaranteeing a strictly stationary process. MSM provides a pure regime-switching formulation of multifractal measures, which were pioneered by Benoit Mandelbrot. == See also ==