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We normally assumed that stocks' prices follow geometric brownian motion which is
(dS_{i,t}=\mu_i S_{i,t} dt + \sigma_i S_{i,t} dW_t)
note that
(S_{i,t} =) price of the ith stock at time t
Moreover, I see lots of papers or some textbooks assume that stock index also follows geometric brownian motion.
Hence
(dI=\mu I dt + \sigma I dW_t)
Why can we assume that index also follows geometric brownian motion
Because sometimes to compute index we have to readjust the base value like S&P500 to maintain its continuity. This complicate things.
The question is that if there is the prove of index being geometric brownian motion.
Lots of thanks..
---------- Post added at 05:02 AM ---------- Previous post was at 04:44 AM ----------
Base Adjustment
In order to keep the S&P 500 Index comparable across time, the index needs to take into account corporate actions such as stock splits, share issuance, dividends and restructuring events (such as merger or spinoffs). Additionally, in order to keep the Index reflective of American stocks, the constituent stocks need to be changed from time to time.
To prevent the value of the Index from changing merely as a result of corporate financial actions, all such actions affecting the market value of the Index require a Divisor adjustment. Also, when a company is dropped and replaced by another with a different market capitalization, the divisor needs to be adjusted in such a way that the value of the S&P 500 Index remains constant. All Divisor adjustments are made after the close of trading and after the calculation of the closing value of the S&P 500 Index.
From http://en.wikipedia.org/wiki/S%26P_500#Index_maintenance
(dS_{i,t}=\mu_i S_{i,t} dt + \sigma_i S_{i,t} dW_t)
note that
(S_{i,t} =) price of the ith stock at time t
Moreover, I see lots of papers or some textbooks assume that stock index also follows geometric brownian motion.
Hence
(dI=\mu I dt + \sigma I dW_t)
Why can we assume that index also follows geometric brownian motion
Because sometimes to compute index we have to readjust the base value like S&P500 to maintain its continuity. This complicate things.
The question is that if there is the prove of index being geometric brownian motion.
Lots of thanks..
---------- Post added at 05:02 AM ---------- Previous post was at 04:44 AM ----------
Base Adjustment
In order to keep the S&P 500 Index comparable across time, the index needs to take into account corporate actions such as stock splits, share issuance, dividends and restructuring events (such as merger or spinoffs). Additionally, in order to keep the Index reflective of American stocks, the constituent stocks need to be changed from time to time.
To prevent the value of the Index from changing merely as a result of corporate financial actions, all such actions affecting the market value of the Index require a Divisor adjustment. Also, when a company is dropped and replaced by another with a different market capitalization, the divisor needs to be adjusted in such a way that the value of the S&P 500 Index remains constant. All Divisor adjustments are made after the close of trading and after the calculation of the closing value of the S&P 500 Index.
From http://en.wikipedia.org/wiki/S%26P_500#Index_maintenance