Quantitative Finance

This paper solves the problem of optimal dynamic consumption, investment, and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect Gompertz' law and investment opportunities are constant. Healthcare slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. Optimal consumption and healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Healthcare spending steadily increases with age, both in absolute terms and relative to total spending. The optimal stochastic control problem reduces to a nonlinear ordinary differential equation with a unique solution, which has an explicit expression in the old-age limit. The main results are obtained through a novel version of Perron's method.

In this paper we find tight sufficient conditions for the continuity of the value of the utility maximization problem from terminal wealth with respect to the convergence in distribution of the underlying processes. We also establish a weak convergence result for the terminal wealths of the optimal portfolios. Finally, we apply our results to the computation of the minimal expected shortfall (shortfall risk) in the Heston model by building an appropriate lattice approximation.

In this paper, we propose a new class of optimization problems, which maximize the terminal wealth and accumulated consumption utility subject to a mean variance criterion controlling the final risk of the portfolio. The multiple-objective optimization problem is firstly transformed into a single-objective one by introducing the concept of overall "happiness" of an investor defined as the aggregation of the terminal wealth under the mean-variance criterion and the expected accumulated utility, and then solved under a game theoretic framework. We have managed to maintain analytical tractability; the closed-form solutions found for a set of special utility functions enable us to discuss some interesting optimal investment strategies that have not been revealed before in literature.

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.

Seeking robustness of risk among different assets, risk-budgeting portfolio selections have become popular in the last decade. Aiming at generalizing risk budgeting method from single-period case to the continuous-time, we characterize the risk contributions and marginal risk contributions on different assets as measurable processes, when terminal variance of wealth is recognized as the risk measure. Meanwhile this specified risk contribution has a aggregation property, namely that total risk can be represented as the aggregation of risk contributions among assets and (t,ω) . Subsequently, risk budgeting problem that how to obtain the policy with given risk budget in continuous-time case, is also explored which actually is a stochastic convex optimization problem parametrized by given risk budget. Moreover single-period risk budgeting policy is related to the projected risk budget in continuous-time case. Based on neural networks, numerical methods are given in order to get the policy with a specified budget process.

We examine Kreps' (2019) conjecture that optimal expected utility in the classic Black--Scholes--Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete-time economies that "approach" the BSM economy in a natural sense: The n th discrete-time economy is generated by a scaled n -step random walk, based on an unscaled random variable ζ with mean zero, variance one, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that E[ ζ 3 ]>0.

This paper investigates the large-time asymptotic behavior of the sensitivities of cash flows. In quantitative finance, the price of a cash flow is expressed in terms of a pricing operator of a Markov diffusion process. We study the extent to which the pricing operator is affected by small changes of the underlying Markov diffusion. The main idea is a partial differential equation (PDE) representation of the pricing operator by incorporating the Hansen--Scheinkman decomposition method. The sensitivities of the cash flows and their large-time convergence rates can be represented via simple expressions in terms of eigenvalues and eigenfunctions of the pricing operator. Furthermore, compared to the work of Park (Finance Stoch. 4:773-825, 2018), more detailed convergence rates are provided. In addition, we discuss the application of our results to three practical problems: utility maximization, entropic risk measures, and bond prices. Finally, as examples, explicit results for several market models such as the Cox--Ingersoll--Ross (CIR) model, 3/2 model and constant elasticity of variance (CEV) model are presented.

Credit Value Adjustment (CVA) is the difference between the value of the default-free and credit-risky derivative portfolio, which can be regarded as the cost of the credit hedge. Default probabilities are therefore needed, as input parameters to the valuation. When liquid CDS are available, then implied probabilities of default can be derived and used. However, in small markets, like the Nordic region of Europe, there are practically no CDS to use. We study the following problem: given that no liquid contracts written on the default event are available, choose a model for the default time and estimate the model parameters. We use the minimum variance hedge to show that we should use the real-world probabilities, first in a discrete time setting and later in the continuous time setting. We also argue that this approach should fulfil the requirements of IFRS 13, which means it could be used in accounting as well. We also present a method that can be used to estimate the real-world probabilities of default, making maximal use of market information (IFRS requirement).

Optimization methods are used to determine equilibria of investment in cryptocurrencies. The basic assumptions involve existence of a core group (the "wealthy") that fears the loss of substantial assets through government seizure. Speculators constitute another group that tends to introduce volatility and risk for the wealthy. The wealthy must divide their assets between the home currency and the cryptocurrency, while the government decides on the probability of seizing a fraction the assets of this group. Under the assumption that each group exhibits risk aversion through a utility function, we establish the existence and uniqueness of Nash equilibrium. Also examined is the more realistic optimization problem in which the government policy cannot be reversed, while the wealthy can adjust their allocation in reaction to the government's designation of probability. The methodology leads to an understanding the equilibrium market capitalization of cryptocurrencies.

Can deep reinforcement learning algorithms be exploited as solvers for optimal trading strategies? The aim of this work is to test reinforcement learning algorithms on conceptually simple, but mathematically non-trivial, trading environments. The environments are chosen such that an optimal or close-to-optimal trading strategy is known. We study the deep deterministic policy gradient algorithm and show that such a reinforcement learning agent can successfully recover the essential features of the optimal trading strategies and achieve close-to-optimal rewards.