# Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d$- Brownian Motion (with $\mathcal F_T = \mathcal F$).

Also, consider a finantial market where the interest rate is nul, $r=0$, and the dynamics of the risky asset $S$ is given by $$S_t= S_0 + \int_0^t \mu_s ~ds +\int_0^t \sigma_s ~dW_s \quad , t \geq 0$$

where $t \in [0,T] \mapsto \mu_t$ and $t \in [0,T] \mapsto \sigma_t \geq 0$ are deterministic and continuous functions.

Show that:

1. If the absence of arbitrage opportunity hypothesis is verified, then $B:=\{t \in [0,T] : \sigma_t=0 \ \text{and} \ \mu_t \neq 0\}$ is a Lebesgue nul-measure set (ie, $\int_0^T\mathbf1_{t \in B} dt=0$).
2. $\nu_\sigma(O):= \int_0^T\mathbf1_{t \in O}\sigma_t ~dt$ dominates $\nu_\mu (O):= \int_0^T\mathbf1_{t \in O} \mu_t ~dt$, where $O$ is a borelian of $[0,T]$ ( ie, $\nu_\mu \ll \nu_\sigma$) and deduce from it that there is a measurable function $\lambda$ such that $\mu = \sigma \lambda$.
• Can you please tell from which book is this question? – Alexey Kalmykov Apr 8 '13 at 21:59
• @AlexeyKalmykov: I really don't know, but I'd like too if it makes part of a book. It was part of an exam question that I couldn't answer. If you find it somewher, please let me know. – Paul Apr 8 '13 at 22:08
• Since it's an exam question, what have you tried so far? – chrisaycock Apr 8 '13 at 23:43
• In your second question, should it be $1_{t \in O}$? – quasi Apr 9 '13 at 0:24
• For your second question, do you need to assume that $\sigma_t \geq 0$? – quasi Apr 9 '13 at 2:19

For the first one absurd reasoning allows you to construct an arbitrage (as r=0) by investing (or short selling according to the sign of $\mu$) at the time where $\sigma$ is null, or if you prefer as soon as $t$ is in $B$ (which is not a Lebesgue negligible set by hypothesis) which is absurd as no-arbitrage holds. The details that remain to be proved is that such a strategy is an admissible one.
For the first part as any borelian set such that $\nu_\sigma(O):= \int_0^T\mathbf1_{t \in O}\sigma_t dt$ doesn't dominate $\nu_\mu (O):= \int_0^T\mathbf1_{t \in O} \mu_t dt$ is included in $B$ and as $B$ is of null probability the conclusion holds true.