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Q-measure

Under the risk-neutral measure $\mathbb{Q}$ all discounted traded assets should be martingales. Suppose $S$ and $B$ has the following $\mathbb{P}$-dynamics. \begin{eqnarray*} \text{d} B_t&=&r(t)B_t\text{d} t,~B_0=1,\\ \text{d} S_t&=&\text{diag}(S_t)\mu^\mathbb{P}(t,S_t)\text{d} t+\text{diag}(S_t)\sigma(t,S_t)\text{d} W^\mathbb{P}_t,~S_0=s \end{eqnarray*} Using a Girsanov measure change we should obtain a new dynamics with drift $\mu^\mathbb{Q}(t,S_t)=\mu^\mathbb{P}(t,S_t)-\sigma(t,S_t)g(t)$. One necessary condition for $S_i(t)/B(t)$ to be a $\mathbb{Q}$-martingale is that the drift part vanishes if we apply Ito's-formula to $S_i(t)/B(t)$. Suppose $S$ and $B$ has the following $\mathbb{Q}$-dynamics. \begin{eqnarray*} \text{d} B_t&=&r(t)B_t\text{d} t,~B_0=1,\\ \text{d} S_t&=&\text{diag}(S_t)\mu^\mathbb{Q}(t,S_t)\text{d} t+\text{diag}(S_t)\sigma(t,S_t)\text{d} W^\mathbb{Q}_t,~S_0=s \end{eqnarray*} Then we get \begin{eqnarray*} \text{d} \frac{S_i(t)}{B(t)}& =& \frac{S_i(t)}{B(t)}\mu_i^{\mathbb{Q}}(t,S_t)- \frac{S_i(t)}{B(t)}r(t)\text{d} t+\frac{S_i(t)}{B(t)}\sigma_i(t,S_t)\text{d} W^{\mathbb{Q}}(t),\\ &=&\frac{S_i(t)}{B(t)}(\mu_i^{\mathbb{Q}}(t,S_t)-r(t))\text{d} t+\frac{S_i(t)}{B(t)}\sigma^{\mathbb{Q}}_i(t,S_t)\text{d} W^{\mathbb{Q}}(t) \end{eqnarray*} The drift vanishes if and only if $\mu_i^{\mathbb{Q}}(t,S_t)=r(t)$! This means that the Girsanov kernel $g(t)$ should satisfy the following equation \[ \mu^\mathbb{P}(t,S_t)-\sigma(t,S_t)g(t)=1_n r(t),\] where $1_n$ is column vector of $n$ ones matching the dimension of $S$. If this equatuion has at least one solution which also satisfies the Novikov condition then the market is free of arbitrage. If the solution also is unique then the market is free of arbitrage and complete (see the fundamental theorems of asset pricing).

 

Questions: Magnus Wiktorsson
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