International Journal of Theoretical and Applied Finance

This paper characterizes the long term social discount rate (SDR) in terms of indices of variation and identifies a class of discount functions that assign weight to the distant future in terms of the asymptotes of their hazard rates. Let A(t) be a discount function supported on [0, +∞[, whose instantaneous discount rate is δ(t) = -(ln A(t))′. In this paper, we study the limiting behavior of δ(t) and a criterion to relate this limit to so-called long and heavy-tailed discount functions. Moreover, we study the discount functions of regular variation and, more specifically, the rapidly and the slowly-varying (singular) discount functions. In this way, the hyperbolic discounting exhibits long (and then heavy) tailedness, and it is intermediate between rapidly and the slowly-varying discount functions, which makes it a suitable function for long term discounting processes.

We study optimal investment in an asset subject to risk of default for investors that rely on different levels of information. The price dynamics can include noises both from a Wiener process and a Poisson random measure with infinite activity. The default events are modeled via a counting process in line with large part of the literature in credit risk. In order to deal with both cases of inside and partial information we consider the framework of the anticipating calculus of forward integration. This does not require a priori assumptions typical of the framework of enlargement of filtrations. We find necessary and sufficient conditions for the existence of a locally maximizing portfolio of the expected utility at terminal time. We consider a large class of utility functions. In addition we show that the existence of the solution implies the semi-martingale property of the noises driving the stock. Some discussion on unicity of the maxima is included.

In a Constant Maturity Treasury (CMT) swap the exotic leg pays, for a given tenor, the yield-to-maturity computed out of a reference bond curve. This paper introduces a theoretical framework for the modelling of CMT that takes into account default risk of bond issuer. As an application, we obtain, under simple but standard assumptions, analytical convexity corrections for some fundamental payoffs contingent on the CMT.

American options in a multi-asset market model with proportional transaction costs are studied in the case when the holder of an option is able to exercise it gradually at a so-called mixed (randomized) stopping time. The introduction of gradual exercise leads to tighter bounds on the option price when compared to the case studied in the existing literature, where the standard assumption is that the option can only be exercised instantly at an ordinary stopping time. Algorithmic constructions for the bid and ask prices and the associated superhedging strategies and optimal mixed stopping times for an American option with gradual exercise are developed and implemented, and dual representations are established.

Implied volatility is a key value in financial mathematics. We discuss some of the pros and cons of the standard ways to compute this quantity, i.e. numerical inversion of the well-known Black–Scholes formula or asymptotic expansion approximations, and propose a new way to directly calculate the implied variance in a local volatility framework based on the solution of a quasilinear degenerate parabolic partial differential equation. Since the numerical solution of this equation may lead to large nonlinear systems of equations and thus high computation times compared to the classical approaches, we apply model order reduction techniques to achieve computational efficiency. Our method of choice for the derivation of a reduced-order model (ROM) will be proper orthogonal decomposition (POD). This strategy is additionally combined with the discrete empirical interpolation method (DEIM) to deal with the nonlinear terms. Numerical results prove the quality of our approach compared to other methods.

This paper demonstrates the simple incorporation of any shape of risk aversion into an asset allocation framework. Indeed, the relevant literature about risk aversion shows mixed evidence regarding the shape of this important but subjective variable. Our setting builds on, and can be compared with, the well-known constant relative risk aversion (CRRA) framework and mostly preserves the tractability of the affine-CRRA framework. Our numerical analysis exhibits some link between measures of risk aversions and empirical studies of asset allocation.

We propose a framework for analyzing the credit risk of secured loans with maximum loan-to-value covenants. Here, we do not assume that the collateral can be liquidated as soon as the maximum loan-to-value is breached. Closed-form solutions for the expected loss are obtained for nonrevolving loans. In the revolving case, we introduce a minimization problem with an objective function parameterized by a risk reluctance coefficient, capturing the trade-off between minimizing the expected loss incurred in the event of liquidation and maximizing the interest gain. Using stochastic control techniques, we derive the partial integro-differential equation satisfied by the value function, and solve it numerically with a finite difference scheme. The experimental results and their comparison with a standard loan-to-value-based lending policy suggest that stricter lending decisions would benefit the lender.