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Register a new userRamsey Rule with Progressive Utility in Long Term Yield Curves Modeling
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Authors
Nicole El Karoui
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The purpose of this paper relies on the study of long term yield curves modeling. Inspired by the economic litterature, it provides a financial interpretation of the Ramsey rule that links discount rate and marginal utility of aggregate optimal consumption. For such a long maturity modelization, the possibility of adjusting preferences to new economic information is crucial. Thus, after recalling some important properties on progressive utility, this paper first provides an extension of the notion of a consistent progressive utility to a consistent pair of progressive utilities of investment and consumption. An optimality condition is that the utility from the wealth satisfies a second order SPDE of HJB type involving the Fenchel-Legendre transform of the utility from consumption. This SPDE is solved in order to give a full characterization of this class of consistent progressive pair of utilities. An application of this results is to revisit the classical backward optimization problem in the light of progressive utility theory, emphasizing intertemporal-consistency issue. Then we study the dynamics of the marginal utility yield curve, and give example with backward and progressive power utilities.
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The purpose of this paper relies on the study of long term affine yield curves modeling. It is inspired by the Ramsey rule of the economic literature, that links discount rate and marginal utility of aggregate optimal consumption. For such a long maturity modelization, the possibility of adjusting preferences to new economic information is crucial, justifying the use of progressive utility. This paper studies, in a framework with affine factors, the yield curve given from the Ramsey rule. It first characterizes consistent progressive utility of investment and consumption, given the optimal wealth and consumption processes. A special attention is paid to utilities associated with linear optimal processes with respect to their initial conditions, which is for example the case of power progressive utilities. Those utilities are the basis point to construct other progressive utilities generating non linear optimal processes but leading yet to still tractable computations. This is of particular interest to study the impact of initial wealth on yield curves.
This paper proposes a paradigm shift in the valuation of long term annuities, away from classical no-arbitrage valuation towards valuation under the real world probability measure. Furthermore, we apply this valuation method to two examples of annuity products, one having annual payments linked to a mortality index and the savings account and the other having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of real world valuation. In contrast to risk neutral valuation, which is a form of relative valuation, the long term average excess return of the equity market comes into play. Instead of the savings account, the numeraire portfolio is employed as the fundamental unit of value in the analysis. The numeraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked nonnegative portfolios supermartingales. The benchmarked real world value of a benchmarked contingent claim equals its real world conditional expectation. This yields the minimal possible value for its hedgeable part and minimizes the fluctuations for its benchmarked hedge error. Under classical assumptions, actuarial and risk neutral valuation emerge as special cases of the proposed real world valuation. In long term liability and asset valuation, the proposed real world valuation can lead to significantly lower values than suggested by classical approaches when an equivalent risk neutral probability measure does not exist.
In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black-Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. Moreover, these two methods solve the backward dynamic programming problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite this model is only bidimensional, the whole history of the process impacts on the price, and handling all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.
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