Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate

We investigate a dividend maximization problem under stochastic interest rates with Ornstein-Uhlenbeck dynamics. This setup also takes negative rates into account. First a deterministic time is considered, where an explicit separating curve α(t) can be found to determine the optimal strategy at time t. In a second setting, we introduce a strategy-independent stopping time. The properties and behavior of these optimal control problems in both settings are analyzed in an analytical HJB-driven approach, and we also use backward stochastic differential equations.

An Investor’s Investment Plan with Stochastic Interest Rate under the CEV Model and the Ornstein-Uhlenbeck Process

The aim of this paper is to maximize an investor’s terminal wealth which exhibits constant relative risk aversion (CRRA). Considering the fluctuating nature of the stock market price, it is imperative for investors to study and develop an effective investment plan that considers the volatility of the stock market price and the fluctuation in interest rate. To achieve this, the optimal investment plan for an investor with logarithm utility under constant elasticity of variance (CEV) model in the presence of stochastic interest rate is considered. Also, a portfolio with one risk free asset and two risky assets is considered where the risk free interest rate follows the Ornstein-Uhlenbeck (O-U) process and the two risky assets follow the CEV process. Using the Legendre transformation and dual theory with asymptotic expansion technique, closed form solutions of the optimal investment plans are obtained. Furthermore, the impacts of some sensitive parameters on the optimal investment plans are analyzed numerically. We observed that the optimal investment plan for the three assets give a fluctuation effect, showing that the investor’s behaviour in his investment pattern changes at different time intervals due to some information available in the financial market such as the fluctuations in the risk free interest rate occasioned by the O-U process, appreciation rates of the risky assets prices and the volatility of the stock market price due to changes in the elasticity parameters. Also, the optimal investment plans for the risky assets are directly proportional to the elasticity parameters and inversely proportional to the risk free interest rate and does not depend on the risk averse coefficient.

A closed-form pricing formula for catastrophe equity options

In this article, we derive a closed-form pricing formula for catastrophe equity put options under a stochastic interest rate framework. A distinguishing feature of the proposed solution is its simplified form in contrast to several recently published formulae that require evaluating several layers of infinite sums of $n$ -fold convoluted distribution functions. As an application of the proposed formula, we consider two different frameworks and obtain the closed-form formula for the joint characteristic function of the asset price and the losses, which is the only required ingredient in our pricing formula. The prices obtained by the newly derived formula are compared with those obtained using Monte-Carlo simulations to show the accuracy of our formula.

PRICING OPTIONS UNDER STOCHASTIC INTEREST RATE AND THE FRASCA–FARINA PROCESS: A SIMPLE, EXPLICIT FORMULA

Assuming a stochastic interest rate, we introduce a simple formula for pricing European options. In doing so, we provide a complete closed-form formula that does not require any numerical/computational methods. Furthermore, the model and formula are far simpler than the previous models/formulas. Our formula is as simple as the classical Black–Scholes pricing formula. Moreover, it removes the theoretical limitation of the original Black–Scholes model without any added practical complexity.