Aziz Issaka

Bio: Aziz Issaka is an academic researcher from University of North Carolina at Charlotte. The author has contributed to research in topics: Asymptotic analysis & Asymptotic expansion. The author has an hindex of 3, co-authored 8 publications receiving 66 citations. Previous affiliations of Aziz Issaka include North Dakota State University & Central European University.

Analysis of variance based instruments for Ornstein–Uhlenbeck type models: swap and price index

Abstract: In this paper a couple of variance dependent instruments in the financial market are studied. Firstly, a number of aspects of the variance swap in connection to the Barndorff-Nielsen and Shephard model are studied. A partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated. Under appropriate assumptions for the first four cumulants of the driving subordinator, a Veceř-type theorem is proved. The bounds of the arbitrage-free variance swap price are also found. Finally, a price-weighted index modulated by market variance is introduced. The large-basket limit dynamics of the price index and the “error term” are derived. Empirical data driven numerical examples are provided in support of the proposed price index.

Feynman path integrals and asymptotic expansions for transition probability densities of some Lévy driven financial markets

Abstract: In this paper we implement the method of Feynman path integral for the analysis of option pricing for certain Levy process driven financial markets. For such markets, we find closed form solutions of transition probability density functions of option pricing in terms of various special functions. Asymptotic analysis of transition probability density functions is provided. We also find expressions for transition probability density functions in terms of various special functions for certain Levy process driven market where the interest rate is stochastic.

An asymptotic series related to Ramanujan’s expansion for the $$n$$ n th Harmonic number

Abstract: In this paper, we investigate certain asymptotic series for the tail of the Riemann Zeta function used by M. D. Hirschhorn to prove an asymptotic expansion of Ramanujan for the $$n$$ th Harmonic number. We give a general form of these series with an explicit formula for its coefficients and with a precise error term.

Variance swaps, volatility swaps, hedging and bounds under multi-factor Heston stochastic volatility model

Abstract: In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is th...

Feynman Path Integrals and Asymptotic Expansions for Transition Probability Densities of Some Levy Driven Financial Markets

Abstract: In this paper we implement the method of Feynman path integral for the analysis of option pricing for certain L'evy process driven financial markets. For such markets, we find closed form solutions of transition probability density functions of option pricing in terms of various special functions. Asymptotic analysis of transition probability density functions is provided. We also find expressions for transition probability density functions in terms of various special functions for certain L'evy process driven market where the interest rate is stochastic.

Lévy processes and infinitely divisible distributions

Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Barndorff-Nielsen and Shephard model: oil hedging with variance swap and option

Abstract: In this paper the Barndorff-Nielsen and Shephard (BN-S) model is implemented to find an optimal hedging strategy for the oil commodity from the Bakken, a new region of oil extraction that is benefiting from fracking technology. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The results indicate that oil can be optimally hedged with the use of a combination of options and variance swaps. Theoretical results related to the variance process are established and implemented for the analysis of the variance swap. In this paper we also determined the optimal amount of the underlying oil commodity that has to be held for minimizing the hedging error. The model and analysis are used to numerically analyze hedging decisions for managing price risk in Bakken oil commodities. From the numerical results, a number of important features of the usefulness of the Barndorff-Nielsen and Shephard model are illustrated.

Fractional Queues with Catastrophes and Their Transient Behaviour

Abstract: Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al in 2003, we define and study a fractional M/M/1 queue with catastrophes In particular, we focus our attention on the transient behaviour, in which the time-change plays a key role We first specify the conditions for the global uniqueness of solutions of the corresponding linear fractional differential problem Then, we provide an alternative expression for the transient distribution of the fractional M/M/1 model, the state probabilities for the fractional queue with catastrophes, the distributions of the busy period for fractional queues without and with catastrophes and, finally, the distribution of the time of the first occurrence of a catastrophe

Refinements of Barndorff-Nielsen and Shephard Model: An Analysis of Crude Oil Price with Machine Learning

Abstract: A commonly used stochastic model for derivative and commodity market analysis is the Barndorff-Nielsen and Shephard (BN–S) model. Though this model is very efficient and analytically tractable, it suffers from the absence of long range dependence and many other issues. For this paper, the analysis is restricted to crude oil price dynamics. A simple way of improving the BN–S model with the implementation of various machine learning algorithms is proposed. This refined BN–S model is more efficient and has fewer parameters than other models which are used in practice as improvements of the BN–S model. The procedure and the model show the application of data science for extracting a “deterministic component” out of processes that are usually considered to be completely stochastic. Empirical applications validate the efficacy of the proposed model for long range dependence.

Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging

Abstract: In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with $$H>1/2$$ , and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model.