Hi all,Suppose I have two correlated GBMs dA = A*mu_A*dt + rho*A*sig_A*dWadB = B*mu_B*dt + rho+B*sig_B*dWb + sqrt(1-rho^2)*B*sig_B*dWbWhat is the formula for Cov(A,B) as a function of a general time t? i.e Cov(A(t),B(t)), and how do you obtain it?Can you just integrate the above to get ln A = N[(mu_A - 0.5*sig_A^2)t, rho^2*sig_A^2*t], ln B = N[(mu_B - 0.5*sig_B^2)t, rho^2*sig_B^2*t + (1-rho^2)*sig_B^2*t], then use the PDFs to calculate Cov(A,B) = E(AB) - E(A)E(B)?Thanks

rho * sigA * sigB

I doubt that's right. Start with A=B, i.e. you are calculating the variance of the lognormal distribution.This can be done with the pdf, switching the integration dummy to x = log A(t).Now do the case you're interested in. The pdf is a bivariate log-normal. Switching to normals again should leave a tractable integral.

This is the (instantaneous) covariance of the log processes. Covar (A,B) is more complicated. It will be somthing like You get if for example by using the known solution for the GBM sde. Then you calculate expectation of A*B and then you substract from this the product of the expectations. Sure you know it. But i wanted to train my latex skills a bit. Cheers.

i meant to answer daveangel but saw Alan already responded.Also i think i missed a t at the mu. And the plus should be a minus i am afraid. But that should then be it hopefully. If not i am sure sombody will post the right thing

Related to this computation, I have suggested before on this forum that every quant should memorize the formula for int e^{h' . x - 0.5 x' . C^(-1) . x} dx/N, with x and h being n-vectors, primes denoting transpose, C=variance-cov matrix, and N is the normalization thatmakes the integral=1 when h=0.

QuoteOriginally posted by: AlanI doubt that's right. Start with A=B, i.e. you are calculating the variance of the lognormal distribution.This can be done with the pdf, switching the integration dummy to x = log A(t).Now do the case you're interested in. The pdf is a bivariate log-normal. Switching to normals again should leave a tractable integral.why do you think its not right ? did I miss something ? whats the covariance of the returns of A and B ?

I believe daveangel has cov(dA,dB) while velocidex, berndL and alan are referring to cov(A,B)

thanks - but why would the cov(A,B) be of interest ?

I am trying to demonstrate (see also this thread: http://www.wilmott.com/messageview.cfm? ... adid=91814 ) something about the CAPM explicitly. The CAPM in its "certainty equivalent" form makes a statement about the covariance of a cash flow with the market return. If I assume the cash flow in each period and the market return in each period follow GBM, then I want to be able to calculate the covariance between the cash flow and the market return (this is different from the covariance of the cash flow returns and the instantaneous market return).

so you want to figure out what the covariance of multi-periods or time varying returns would be ?

The problem is this: assuming that a cash flow follows GBM, and the cash flow is taken to be a payoff at certain intervals, then to calculate the value of that cash flow using the certainty equivalent form of the CAPM.I'm not talking about the cash flow as a type of flux, but rather just assuming that a cash flow is whatever the value of the GBM process is at t=1,2,3,...The certainty equivalent form of the CAPM requires as inputs: a) the risk free rate, b) the expected value of the cash flow, c) the market price of risk, and d) the covariance between the cash flow and the market return. a and b are easy, c is well known, d is the remaining challenge. If you assume that the market follows GBM as well, then you need the covariance between the value of the cash flow (= the value of the first GBM at time t), and the market return (which follows directly from the second GBM at time t). Hence you need the actual covariance of these, not the covariance of the log returns of the GBM process (which is trivially rho*sigA*sigB).

i am no expert no CAPM but I am doubtful that it would deal with levels of prices rather than changes in prices i.e. returns.