Mean reversion and regime switching are well‐known features of commodity prices. Recent empirical research additionally documents the time variation of the mean reversion rate and volatility. This paper considers the option pricing framework for an underlying commodity price with mean reversion rate and volatility change according to a self‐exciting regime switching model. We offer empirical evidence for the proposed model and derive analytic pricing formulas for the European and barrier options. Numerical examples demonstrate the application and the ability of the proposed model in capturing volatility smile and regime‐switching in the mean reversion rate, simultaneously. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:107–131, 2017

Variance swap is a typical financial tool for managing volatility risk. In this paper, we evaluate different types of variance swaps under a threshold Ornstein-Uhlenbeck model, which exhibits both mean reversion and regime switching features in the underlying asset price. We derive the analytical solution for the joint moment generating function of log-asset prices at two distinct time points. This enables us to price various types of variance swaps analytically.regime-switching is originated by Hamilton ([10]). Although there are many possible models for regime-switching, the self-exciting threshold autoregressive (SETAR) model proposed by Tong ([11]) admits a tractable likelihood function for parameter estimation and is a popular regime-switching model alternative to the one used in Kim et al. ([3]). Tsay ([12]) points out that the SETAR model can capture jump phenomena observed in practice because of discontinuity at the threshold. Therefore, it can be efficiently estimated and empirically tested with real data. We offer empirical evidence for the SETAR model with historical crude-oil spot price.By considering the diffusion limit of the SETAR model, we propose a threshold Ornstein-Uhlenbeck (TOU) process {X t } t≥0 and derive the corresponding analytical formulas for various variance swaps. To this end, we have to examine the joint distribution of X T 1 and X T 2 at two fixed time points. To characterize the distributional property, we derive the joint moment-generating function (MGF) [e 1 X T 1 + 2 X T 2 |X t = x] in two steps. We find the closed-form expression for[e X T |X t = x] in the first step and apply it to the joint MGF in the second step. A PDE framework similar to Wong and Zhao ([13]) is applied to obtain the solution. Consequently, the analytical solution of the MGF brings us the pricing formulas of variance swaps in terms of Laplace inversion. We implement the solution using numerical methods proposed by Abate and Valkó ([14], [15]).The remainder of the paper is organized as follows. Section 2 introduces TOU model with empirical support. Section 3 states the two-step method for deriving the analytical solution of the joint MGF. Section 4 is the application of the joint MGF in variance swaps pricing. Section 5 numerically examines the accuracy of the closed-form solution of MGF and variance swaps price derived under TOU model. Section 6 concludes. The preliminaryTo formulate our problem in a PDE framework under a n-regime TOU model, we introduce Lemma 2.1 in the succeeding text, which is useful in many circumstances in this paper.With the conditions in (27) and (28) and the derivative property of Kummer's function M (a, b, z) and U(a, b, z), we deduce a linear equation system (17) for solving the coefficients c 21 and c 22 .

Life Insurance Retirement Plans (LIRPs) offer tax-deferred cash value accumulation, tax-free withdrawals (if properly structured), and a tax-free death benefit to beneficiaries. Thus, LIRPs share many of the tax advantages of other retirement savings vehicles but with less restrictive limitations on income and contributions. Opinions are mixed about the effectiveness of LIRPs; some financial advisers recommend them enthusiastically, while others are more skeptical. In this paper, we examine the potential of LIRPs to meet both income and bequest needs in retirement. We contrast retirement portfolios that include a LIRP with those that include only investment products with no life insurance. We consider different issue ages, face amounts, and withdrawal patterns. We simulate market scenarios and we demonstrate that portfolios that include LIRPs yield higher legacy potential and smaller income risk than those that exclude it. Thus, we conclude that the inclusion of a LIRP can improve financial outcomes in retirement.

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