June 19th, 2006, 12:50 pm
For a next day horizon, I would first try the "instantaneous var-cov" approximation.Let's take a porfolio of stocks S_i and options C_i, where i indexes the underlying security (so C_i an an option on S_i).You can treat S_i as a future if you like.For stocks and options, the short time horizon evolution in your log-normal model can be described by:dS_i = (... ) dt + sig_i S_i dB_i => dS_i/S_i = (... ) dt + sig_i dB_idC_i = (... ) dt + h_i sig_i S_i dB_i => dC_i/C_i = (...) dt + (h_i sig_i S_i/C_i) dB_i Here B_i(t) is a Brownian motion; they are correlated so dB_i dB_j = rho_ij dt.The (...) terms don't matter for this analysis.And h_i = (partial C_i)/(partial S_i) is the hedge ratio or delta, which you get from the Black-Scholes formula,using (today's) implied volatility.So, to get the var-cov of the returns and answer your original question, call that matrix Cov_ij, For stock i and stock j, take: Cov_ij = sig_i sig_j rho_ij dtFor stock i and option j, take: Cov_ij = sig_i (sig_j h_j S_j/C_j) rho_ij dtFor option i and option j, take: Cov_ij = (sig_i h_i S_i/C_i) (sig_j h_j S_j/C_j) rho_ij dtHere dt = 1/252 years, and the sig_i are the std. devs of log-returns on an annualized basis (theBlack-Scholes' implied volatilities).I don't need to see your file, but hopethis helps,