A consumption-investment model with state-dependent lower bound constraint on consumption

This paper considers a life-time consumption-investment problem under the Black-Scholes framework, where the investors consumption rate is subject to a lower bound constraint that linearly depends on the investors wealth. Due to the state-dependent control constraint, the standard stochastic control theory cannot be directly applied to our problem. We overcome this obstacle by examining an equivalent problem that does not impose state-dependent control constraint. It is shown that the value function is a third-order continuously differentiable function by using differential equation approaches. The feedback form optimal consumption and investment strategies are given. According to our findings, if the investor is more concerned with long-term consumption than short-term consumption, then she should, regardless of her financial condition, always consume as few as possible; otherwise, her optimal consumption strategy is state-dependent: consuming optimally when her financial condition is good, and consuming at the lowest possible rate when her financial situation is bad.

Stability of the Weak Martingale Optimal Transport Problem

While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of Gozlan, Roberto, Samson and Tetali this corresponds to weak martingale optimal transport (WMOT). In this article we establish stability of WMOT which is important since financial data can give only imprecise information on the underlying marginals. As application, we deduce the stability of the superreplication bound for VIX futures as well as the stability of stretched Brownian motion and we derive a monotonicity principle for WMOT.

Option Pricing under Bayesian MS-VAR Process

In this paper, we have studied option pricing methods that are based on a Bayesian Markov-Switching Vector Autoregressive (MS-BVAR) process using a risk-neutral valuation approach. A BVAR process which is a special case of the Bayesian MS-VAR process is widely used to model inter-dependencies of economic variables and forecast economic variables. Here we assumed that a regime-switching process is generated by a homogeneous Markov process and for a normal system, a residual process follows a conditional heteroscedastic model. With a direct calculation and change of probability measure, for some frequently used options, we derive pricing formulas. An advantage of our model is it depends on economic variables and is easy to use compared to previous option pricing papers which depend on regime-switching.

A note on closed-form spread option valuation under log-normal models

In the papers Carmona and Durrleman [7] and Bjerksund and Stensland [1], closed form approximations for spread call option prices were studied under the log normal models. In this paper, we give an alternative closed form formula for the price of spread call options under the log-normal models also. Our formula can be seen as a generalization of the closed-form formula presented in Bjerksund and Stensland [1] as their formula can be obtained by selecting special parameter values to our formula. Numerical tests show that our formula performs better for certain range of model parameters than the closed-form formula presented in Bjerksund and Stensland [1].

An Alternative Approach to Evaluate American Options Price Using HJM Approach

Developments in finance industry and academic research has led to innovative financial products. This paper presents an alternative approach to price American options. Our approach utilizes famous cite{heath1992bond} (HJM) technique to calculate American option written on an asset. Originally, HJM forward modeling approach was introduced as an alternative approach to bond pricing in fixed income market. Since then, cite{schweizer2008term} and cite{carmona2008infinite} extended HJM forward modeling approach to equity market by capturing dynamic nature of volatility. They modeled the term structure of volatility, which is commonly observed in the market place as opposed to constant volatility assumption under Black - Scholes framework. Using this approach, we propose an alternative value function, a stopping criteria and a stopping time. We give an example of how to price American put option using proposed methodology.

A C`adl`ag Rough Path Foundation for Robust Finance

Using rough path theory, we provide a pathwise foundation for stochastic It^o integration, which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called Property (RIE) for c`adl`ag paths, which is shown to imply the existence of a c`adl`ag rough path and of quadratic variation in the sense of Follmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type, and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Covers universal portfolio are admissible integrands, and that Property (RIE) is satisfied by both (Young) semimartingales and typical price paths.

Security Investment Risk Analysis Using Coefficient of Variation: An Alternative to Mean-Variance Analysis

This manuscript presents the mathematical relationship between coefficient of variation (CV) and security investment risk, defined herein as the probability of occurrence of negative returns. The equation suggests that there exists a range of CV where risk is zero and that risk never crosses 50% for securities with positive returns. We also found that at least for stocks, there is a strong correlation between CV and stock performance when CV is derived from annual returns calculated for each month (as opposed to using, for example, only annual returns based on end-of-the-year closing prices). We found that a low nonnegative CV of up to ~ 1.0 (~ 15% risk) correlates well with strong and consistent stock performance. Beyond this CV, share price growth gradually shows plateaus and/or large peaks and valleys. The efficient frontier was also re-examined based on CV analysis, and it was found that the direct relationship between risk and return (e.g., high risk, high return) is only robust when the correlation of returns among the portfolio securities is sufficiently negative. At low negative to positive correlation, the efficient frontier hypothesis breaks down and risk analysis based on CV becomes an important consideration.

Three fundamental problems in risk modeling on big data: an information theory view

Since Claude Shannon founded Information Theory, information theory has widely fostered other scientific fields, such as statistics, artificial intelligence, biology, behavioral science, neuroscience, economics, and finance. Unfortunately, actuarial science has hardly benefited from information theory. So far, only one actuarial paper on information theory can be searched by academic search engines. Undoubtedly, information and risk, both as Uncertainty, are constrained by entropy law. Todays insurance big data era means more data and more information. It is unacceptable for risk management and actuarial science to ignore information theory. Therefore, this paper aims to exploit information theory to discover the performance limits of insurance big data systems and seek guidance for risk modeling and the development of actuarial pricing systems.

Positive Stochastic Collocation for the Collocated Local Volatility Model

This paper presents how to apply the stochastic collocation technique to assets that can not move below a boundary. It shows that the polynomial collocation towards a lognormal distribution does not work well. Then, the potentials issues of the related collocated local volatility model (CLV) are explored. Finally, a simple analytical expression for the Dupire local volatility derived from the option prices modelled by stochastic collocation is given.

An axiomatization of $Lambda$-quantiles

We give an axiomatic foundation to $Lambda$-quantiles, a family of generalized quantiles introduced by Frittelli et al. (2014) under the name of Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call locality, that means that any change in the distribution of the probability mass that arises entirely above or below the value of the $Lambda$-quantile does not modify its value. We compare with a related axiomatization of the usual quantiles given by Chambers (2009), based on the stronger property of ordinal covariance, that means that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of $Lambda$-quantiles, refining some of the results of Frittelli et al. (2014) and Burzoni et al. (2017) and showing that in the case of a nonincreasing $Lambda$ the properties of $Lambda$-quantiles closely resemble those of the usual quantiles.