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snvk4u
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USing monte carlo to find Variance Covariance Matrix for Underlying asset and option

June 16th, 2006, 9:42 am

Hi ,Sorry if anyone will get confused seeing my last threadMy real concern is how to find CALCULATE VAR COVAR MATRIX FOR A PORTFOLIO MADE OF UNDERLYING ASSET (EXAMPLE . FUTURE) AND DERIVATIVES (SUCH AS OPTIONS)CAN SOMEONE TELL ME THE METHODOLOGY AND STEPS INVOLVED AND CAN RECOMMENED SOME LITERATURE TOO.THANKS
 
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Alan
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USing monte carlo to find Variance Covariance Matrix for Underlying asset and option

June 16th, 2006, 2:27 pm

In principle, you can measure it historically from the price history, but Isuspect you want to predict it for the future. To do that, you need a modelof the joint movement of all the underlying security prices. What is your model?regards, p.s. If you don't have a model, thenSTEP1: Get a model.How sophisticated and realistic it needs to be depends on your application.The basic model, which may or may not be suitable is: joint-lognormal
Last edited by Alan on June 15th, 2006, 10:00 pm, edited 1 time in total.
 
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snvk4u
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USing monte carlo to find Variance Covariance Matrix for Underlying asset and option

June 17th, 2006, 3:35 am

well at the moment yes i am following log normal approach. I have taken the returns to be normally distributed with mean zero.In my portfolio I have two assets i.e. two futures assets X and Y. and I have one option on asset X.problem I am facing is how to calculate var. covarianc matrix. since the option derives from the underlying asset. And taking option prices wud be wrong.any views?
 
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Alan
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USing monte carlo to find Variance Covariance Matrix for Underlying asset and option

June 17th, 2006, 5:13 pm

Ok, I understand the problem with the option.But before we can deal with that, I think thenext step is to decide on a time horizon for the returns.There are two natural choices: (i) the instantaneous variance-cov. matrix, or(ii) the variance-cov. matrix for returns from today until the option expires (assuming the future matures no earlier). Other choices are possible, too.What is your time horizon?
Last edited by Alan on June 16th, 2006, 10:00 pm, edited 1 time in total.
 
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snvk4u
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USing monte carlo to find Variance Covariance Matrix for Underlying asset and option

June 19th, 2006, 8:19 am

I am generating var/covar for next day. I have taken time horizon to be next day. I am predicting the next day option price to arrive at the Var/covar matrix to find the next day Value at Risk.I have calculated VaR through my approach. If you are interested I can send you the file so you can analyse. I cant upload it since it is not letting me. I can send to your email if you provide me the same.thanks.
 
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Alan
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USing monte carlo to find Variance Covariance Matrix for Underlying asset and option

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,