Weak time-derivatives and no-arbitrage pricing

Abstract

We prove a risk-neutral pricing formula for a large class of semimartingale processes through a novel notion of weak time-differentiability that permits to differentiate adapted processes. In particular, the weak time-derivative isolates drifts of semimartingales and is null for martingales. Weak time-differentiability enables us to characterize no-arbitrage prices as solutions of differential equations, where interest rates play a key role. Finally, we reformulate the eigenvalue problem of Hansen and Scheinkman (Econometrica 77:177–234, 2009) by employing weak time-derivatives.

Appendix: A routine lemma

We state and prove a version of a standard result that is best suited for our purposes.

Lemma A.1

Let $f:[t,T]\to \mathbb{R}$ be a measurable function.

• (i) If $f$ is bounded, nonnegative, with compact support and $\int _{t}^{T} f(\tau )g(\tau )d\tau =0$ for any $g \in C_{c}((t,T))$, then $f=0$ a.e.

• (ii) If $\int _{t}^{T} f(\tau )g(\tau )d\tau =0$ for any $g \in C_{c}((t,T))$, then $f=0$ a.e.

Proof

(i) If $f$ is strictly positive on a set $A$ with positive measure, consider the indicator function $\mathbf{1}_{A}$ and a sequence $(U_{n})$ of continuous positive approximations of $\mathbf{1}_{A}$, obtained by convolution with a smooth positive kernel. As $U_{n}$ converges to $\mathbf{1}_{A}$ in $L^{2}$, $0 \leqslant \int _{t}^{T} f(\tau ) \mathbf{1}_{A}(\tau ) d\tau =\lim _{n} \int _{t}^{T} f( \tau ) U_{n}(\tau )d\tau =0$. Consequently, $f$ is null a.e.

(ii) Suppose that $f$ is positive with compact support. For any $N>0$, consider $f_{N}(s)=\min \{f(\tau ),N\}$. Then,

$$ 0 \leqslant \int _{t}^{T} f_{N}(\tau ) g(\tau )d\tau \leqslant \int _{t}^{T} f(\tau )g(\tau ) d\tau =0. $$

Therefore, each $f_{N}$ is null a.e. by (i) and so is $f$. □