April 21st, 2015, 12:56 pm
Hello, currently I'm working on a stochastic volatility model with jumps. The dynamics are the following:[$] dV_t = \kappa_V (\bar{V} - V_t) dt + \sigma_V \sqrt{q_t} dW^V_t[$][$] dq_t = \kappa_q (\bar{q} - q_t) dt + \sigma_q \sqrt{q_t} dW^q_t + Z_q dN^q_t[$],where [$]dW^VdW^q = \rho [$], Z_q is a exponential distributed variable with mean [$] \mu_q [$] and [$] N^q [$] is a compound poisson process with intensity [$]l0[$].I'm interested in the moments of [$] V_t[$] . I calculate the first moment of [$] V_t[$] under the assumption that [$]\sigma_V \sqrt{q_t} dW^V_t[$] is a martingale and, hence, get the usual first moment of a CIR process: [$] \mathbb{E}V_t = V_0 e^{-\kappa_V t} + \bar{V}(1-e^{-\kappa_V t})[$]However, if I do a MC simulation for a set of model parameters and compare the analytic first moment with the simulated first moment the discripancy is hugh. For example:I get [$] \mathbb{E}V_t = 1 [$] analytically and [$] \mathbb{E}V_t = 2.0303[$] via MC simulation using the following parameters: kappa_V = 1.4515; V_bar = 1; sigma_V = 1.3394; kappa_q = 2.0224; q_bar = 1; sigma_q = 1.3394; l0 = 0.1073; mu_q = 16.4938; rho = 0.944; V_0 = 1;Where is my error? Could it be that the jump size mu_q so big that the assumption that [$]\sigma_V \sqrt{q_t} dW^V_t[$] is a martingale does not hold anymore?Do anyone know how to calculate the analytic moment of $V_t$ nonetheless?Grettings
Last edited by
Phunfactory on April 20th, 2015, 10:00 pm, edited 1 time in total.