V
vegan
Member
Hi
Core reading 53 of chapter 14 (Cameron Martin Girsanov theorem) calculates the expected value of the discounted asset price under Q as shown below.
\[E_q[e^{-rt}S_t | F_u] =e^{-rt}S_uE_Q[exp{(r-0.5sigma^{2})(t-u)+sigma(*Z_t - *Z_u)}]\]
\[e^{-rt}S_uE_Q[exp{(r-0.5sigma^{2})(t-u)+0.5sigma^{2}(t -u)}\]
Note: \[*Z_t\] denotes standard brownian motion under Q
Please could someone explain how \[sigma(*Z_t - *Z_u)\] from the first line, simplifies to \[0.5sigma^{2}(t -u)\] of the second line?
Thanks,
Core reading 53 of chapter 14 (Cameron Martin Girsanov theorem) calculates the expected value of the discounted asset price under Q as shown below.
\[E_q[e^{-rt}S_t | F_u] =e^{-rt}S_uE_Q[exp{(r-0.5sigma^{2})(t-u)+sigma(*Z_t - *Z_u)}]\]
\[e^{-rt}S_uE_Q[exp{(r-0.5sigma^{2})(t-u)+0.5sigma^{2}(t -u)}\]
Note: \[*Z_t\] denotes standard brownian motion under Q
Please could someone explain how \[sigma(*Z_t - *Z_u)\] from the first line, simplifies to \[0.5sigma^{2}(t -u)\] of the second line?
Thanks,