Long-Term Risk: A Martingale Approach

Abstract

A. PROOFS FOR SECTION 3 WE FIRST RECALL SOME RESULTS about semimartingale topology originally introduced by Émery (1979) (see Czichowsky and Schweizer (2006), Kardaras (2013), andCuchiero andTeichmann (2015) for recent applications in mathematical finance). The semimartingale topology is stronger than the topology of uniform convergence in probability on compacts (ucp). In the latter case, the supremum in Eq. (2.1) is only taken over integrands in the form η t = 1 [0 s] (t) for every s > 0:The following inequality due to Burkholder is useful for proving convergence in the semimartingale topology in Theorem 3.1 (see Meyer (1972, Theorem 47, p. 50) for discrete martingales and Cuchiero and Teichmann (2015) for continuous martingales, where a proof is provided inside the proof of their Lemma 4.7).LEMMA A.1: For every martingale X and every predictable process η bounded by 1, |η t | ≤ 1, it holds that aP supfor all a ≥ 0 and t > 0.We will also use the following result (see Kardaras (2013, Proposition 2.10)).We will also make use of the following lemma.LEMMA A.3: Let (X n t ) t≥0 be a sequence of martingales such that X n t

Long-term factorization of affine pricing kernels

“…The following theorem, proved in Qin and Linetsky (2017), gives a sufficient condition that ensures convergence to the long bond in the semimartingale topology which is stronger than the ucp convergence in Definition 1 and convergence of Tforward measures to the long forward measure in total variation, which is stronger than the strong convergence in Definition 2 (we refer to Qin and Linetsky (2017) and the on-line appendix for proofs and details). Suppose that for each t > 0 the ratio of the F t -conditional expectation of the pricing kernel S T to its unconditional expectation converges to a positive limit in L 1 as T → ∞ (under P), i.e.…”

“…We now recall the definitions of the long bond and the long forward measure from Qin and Linetsky (2017). Definition 2.1.…”

“…Following Qin and Linetsky (2017), for each fixed T > 0 we define a self-financing trading strategy that rolls over investments in T -maturity zero-coupon bonds as follows. Fix T and consider a self-financing roll-over strategy that starts at time zero by investing one unit of account in 1/P T 0 units of the T -maturity zero-coupon bond.…”

“…This allows us to give a fully explicit treatment and illustrate dynamics of the long bond, the martingale component and the long-forward measure in affine models. In particular, we explicitly verify that when the Riccati ODE associated with the affine pricing kernel possesses a fixed point, the affine model satisfies the sufficient condition in Theorem 3.1 of Qin and Linetsky (2017) so that the long-term limit exists.…”

“…The permanent component is a martingale that accomplishes a change of probabilities to the long forward measure. Qin and Linetsky (2017) study the long-term factorization and the long forward measure in the general semimartingale setting.…”