Question

Review Questions 1. One popular concept on stock market price is called "martingale pricing", which is relevant to "rational expectations" or "efficient market hypothesis" in economic theories. A rough statement on this is that, a (log of) stock market price pt (e.g., S&P 500 index) is supposed to reflect all the relevant information available up to time t, i.e., E [pt|Ft-1] = pt-1, therefore pt = log Pt is a martingale. (a) Under the martingale pricing, show that continuously compounded (cc) return for stock market price is an martingale difference sequence (mds). (b) In what follows, we assume cc return rt is a covariance stationary process. Prove the following statements: i. If rt ~ iid (0, ?²) (or independent white noise), then rt ~ mds (0, ?²). ii. If rt ~ mds (0, ?²), then rt ~ WN (0, ?²) (or weak white noise). (c) Prove the following statements: i. If {rt} is iid, then it is strictly stationary. ii. If {rt} is strictly stationary and E[r?²] < ?, then it is covariance stationary.

          Review Questions
1. One popular concept on stock market price is called "martingale pricing", which is relevant to "rational expectations" or "efficient market hypothesis" in economic theories. A rough statement on this is that, a (log of) stock market price pt (e.g., S&P 500 index) is supposed to reflect all the relevant information available up to time t, i.e.,
E [pt|Ft-1] = pt-1,
therefore pt = log Pt is a martingale.
(a) Under the martingale pricing, show that continuously compounded (cc) return for stock market price is an martingale difference sequence (mds).
(b) In what follows, we assume cc return rt is a covariance stationary process. Prove the following statements:
i. If rt ~ iid (0, ?²) (or independent white noise), then rt ~ mds (0, ?²).
ii. If rt ~ mds (0, ?²), then rt ~ WN (0, ?²) (or weak white noise).
(c) Prove the following statements:
i. If {rt} is iid, then it is strictly stationary.
ii. If {rt} is strictly stationary and E[r?²] < ?, then it is covariance stationary.

Review Questions
1. One popular concept on stock market price is called "martingale pricing", which is relevant to "rational expectations" or "efficient market hypothesis" in economic theories. A rough statement on this is that, a (log of) stock market price pt (e.g., S P 500 index) is supposed to reflect all the relevant information available up to time t, i.e.,
E [pt|Ft-1] = pt-1,
therefore pt = log Pt is a martingale.
(a) Under the martingale pricing, show that continuously compounded (cc) return for stock market price is an martingale difference sequence (mds).
(b) In what follows, we assume cc return rt is a covariance stationary process. Prove the following statements:
i. If rt iid (0, ?²) (or independent white noise), then rt mds (0, ?²).
ii. If rt mds (0, ?²), then rt WN (0, ?²) (or weak white noise).
(c) Prove the following statements:
i. If rt is iid, then it is strictly stationary.
ii. If rt is strictly stationary and E[r?²] < ?, then it is covariance stationary.

Added by Renee J.

Question

Please give Ace some feedback