Let Q be the martingale measure with the money market account as the numeraire and Q^∗ denote the equivalent martingale measure where the asset price S_t is used as the numeraire. Suppose S_t follows the Geometric Brownian process with drift rate r and volatility σ under Q, where r is the riskless interest rate. By using (3.2.11), show that

where Z_T is Q-Brownian. Using the Girsanov Theorems, show that

then deduce that [see (3.3.12b)] 

Let U_t be another asset whose price dynamics under Q is governed by

where dZ^U_t dZ_t = ρdt and ρ is the correlation coefficient. Show that

is a Q^∗-Brownian process.

Since dZ^U_t and dZ_t are correlated with correlation coefficient ρ, we may write

Transcribed Image Text:

dQ* dQ FT ST -Ste = -rT =e=²²T+0ZT.