Hi all, good morning In my knowledge there is basically two types of model for daily closing price of an asset.Model-1Assumption : log of daily closing price follows random walk. therefore,ln(P[t+1]) given P[t] ~ N(ln(P[t]), sigma^2) Note : Instead of modeling ln(P), I can model P itself using the same configuration. But then I need to acknowledge there is a positive probability that Price can be negative, therefore I modeled ln(P).hence : ln(P[t+1]/P[t]) given P[t] ~ N(0, sigma^2).Model-2This model is based on following differential equation :d(P[t]) = mu*P[t]&dt + sigma*P[t]*d(W[t]) W[t] is assumed Weiner process i.e. d(W[t]) ~ N(0,dt)Closed from solution of that is :d(ln(P[t])) = (mu-0.5*sigma^2)*dt + sigma*d(W[t]) Hence : d(ln(P[t])) ~ N((mu-0.5*sigma^2)*dt , sigma^2*dt)If I take dt=1 then,ln(P[t+1]/P[t]) ~ N((mu-0.5*sigma^2), sigma^2)Hence E[ln(P[t+1]/P[t])] = mu-0.5*sigma^2 = -0.5*sigma^2 [if mu=0]However for model-1, E[ln(P[t+1]/P[t])] = 0My question is which model is correct? Is there any procedure to link those two model?Your help will be highly appreciated.Regards,

both can be correct depending under which probability measures you are working. however, if you have the two models under the same probability measure, then i will go for model 2.

ut, estimation of expected value of next day's return is completely different . Then how can I convince my client on my estimation?

Hi Buddies, still i donot understand. can anyone give me better idea in understanding that?

Under the risk neutral probability, which is the one under which you will (most likely) be pricing your derivativesmodel number 2 is correct, because model number 1 is not a martingale. since you are working under the risk neutral probability your drift (mu) should be equal to the risk free interest rate, which you have as an example set to zero in the derivation of the above models. E(exp(X)) when X is N(0,sigma^2) =! 0 and that's why you are using (mu-0.5*sigma^2) as driftI hope this helps.../J

Ok then u mean that model-2 is correct. Now come to the estimation of VaR. suppose i have invested in only 1 stock. Then I can estimate the VaR as follows:Return : P[t+1]-P[t] = P[t]* (P[t+1]/P[t] - 1) = (almost equal, provided there is not so much change in P[t+1] from P[t]) = P[t]* ln(P[t+1]/P[t]) Hence 5-th percentile of the distribution of P[t+1]-P[t] is :E(P[t+1]-P[t]) - 1.64*sd(P[t+1]-P[t]) = P[t]*{E( ln(P[t+1]/P[t] ) - 1.64*sd( ln(P[t+1]/P[t] )}.If I go with model-1 then I get VaR estimate as : -1.64*P[t]*sigma. However it will not be same for model-2Almost all the books including Riskmetrics give the 1st estimate as VaR.Does it mean that all are wrong?

Is there any better explanation?

In the continuous time model you have thatln(P[t+1]/P[t]) ~ N((m-0.5*s^2), s^2)In empirical work you can treat m-0.5*s^2 as a new constant, and writeln(P[t+1]/P[t]) ~ N(c, s^2)This is how you would estimate such a model from a time series, you would estimate c and s.The drift c is typically very small compared to the magnitude of the the volatility, and in many VaR calculations it is set to zero. This happens especially when the horizon is relatively short (since c grows with the horizon, while s grows with the sqrt(horizon) ). Also, while the volatility s is estimated quite accurately, the historical drift c is not very reliable as an estimator of the future c and omitting it might be again more robust.You can simulate samples of reasonable size (or bootstrap market returns) to see how it works.hope that it helps,Kyriakos