Abstract |
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Mathematical finance extensively relies on martingales. In some cases, they need to meet constraints. For instance, they can be required to evolve in a compact set; discounted zero-coupon bond prices are risk-neutral martingales in $[0,1]$ if interest rates are positive. So are survival probabilities $\mathbb{E}_t[1_{\{\tau>T\}}]$. These martingales are usually tackled via latent processes, but some constraints are typically relaxed in order to fit the initial term-structure (eg positivity in Hull-White or CIR++ models for default intensities). In that context, tools for direct modelling of such bounded processes could be valuable. Surprisingly, however, such approaches haven't received much attention so far. Filling this gap is the goal of this paper. We first review the conditions for a pure diffusion SDE to admit a (strong) solution, and for the latter to be a martingale. We then proceed with bounded (or conic) martingales. Martingales in $[0,1]$ can be obtained by solving pure diffusion SDEs whose diffusion coefficient has a specific form, eg $x(1-x)$ or the logistic curve. Unfortunately, explicit solutions to those are unavailable. On the other hand, martingales in $[a,b]$ can be obtained by mapping latent processes through smooth bijections $F$ with image $[a,b]$, provided that the associated drift is chosen according to the mapping's score. We study several mappings, and a unique solution is found in one case. Its variance, variance of increments and distribution are derived, a useful feature for tractability purposes. The solution's asymptotic distribution is proven to be typically singular (Bernoulli type). This is natural for floored/capped martingales. In particular, any quantile of the exponential martingale distribution tend to zero at some point. Interestingly, this proves that the maximum variance is attainable asymptotically. The existence of a unique solution is not always easy to establish for bounded martingale SDEs. We show how to choose the mapping for turning the SDE of a bounded martingale with separable diffusion coefficient of the form $h(t)g(x)$ to that of a drifted process with autonomous diffusion coefficient. We conclude with an example where the method is applied to survival probability modelling. In this setup, calibration to initial survival probability curve is trivial. Potential applications cover pricing of CDS options or CVA on CDS under weak collateral regimes, among others. |