# Law of one price in continuous time

The law of one price (i.e. for assets $S^{(i)}$ and $S^{(j)}$, $S^{(i)}_T = S^{(j)}_T$ almost surely implies that $S^{(i)}_t = S^{(j)}_t$ almost surely for all $0 \leq t \leq T$) is known to hold in discrete time when there is no arbitrage.

However, my lecturer claims that this statement might not hold in continuous time. Can anyone give me an example for that?

• I dont think you are referring to assets, but only European claims. – emcor Nov 29 '14 at 11:50
• but that's how the lecturer defines it on the board! – Richard Nov 29 '14 at 11:54

Let $B_t=e^{\int_0^t r_s} ds$ be the money-market account value at time $t$, where $r_t$ is the short interest rate rate. Then both $\{\frac{S_t^i}{B_t}, \, t \ge 0\}$ and $\{\frac{S_t^j}{B_t}, \, t \ge 0\}$ are martingales. Therefore, for $0 \le t \le T$, \begin{align*} \frac{S_t^i}{B_t} &= E\left(\frac{S_T^i}{B_T} \mid \mathcal{F}_t \right)\\ &= E\left(\frac{S_T^j}{B_T} \mid \mathcal{F}_t \right)\\ &= \frac{S_t^j}{B_t}. \end{align*} That is, $S_t^i = S_t^j$.