The continuous observation of the financial markets has identified some 'stylized facts' which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss-Wiener process (Brownian motion), characterized by stationary independent increments, by a fractional version. On the other hand, the CEV model addresses the Leverage effect and smile-skew phenomena, efficiently. In this paper, these two insights are merging and both the fractional and mixed-fractional extensions for the CEV model, are developed. Using the fractional versions of both the Itô's calculus and the Fokker-Planck equation, the transition probability density function of the asset price is obtained as the solution of a non-stationary Feller process with time-varying coefficients, getting an analytical valuation formula for a European Call option. Besides, the Greeks are computed and compared with the standard case.
Retail competition today can be described by three main features: i) oligopolistic competition, ii) multi-store settings, and iii) the presence of large economies of scale. In these markets, firms usually apply a centralized decisions making process in order to take full advantage of economies of scales, e.g. retail distribution centers. In this paper, we model and analyze the stability and chaos of retail competition considering all these issues. In particular, a dynamic multi-market Cournot-Nash equilibrium with global economies and diseconomies of scale model is developed. We confirm the non-intuitive hypothesis that retail multi-store competition is more unstable that traditional small business that cover the same demand. The main sources of stability are the scale parameter and the number of markets.
The Constant Elasticity of Variance (CEV) model significantly outperforms the Black-Scholes (BS) model in forecasting both prices and options. Furthermore, the CEV model has a marked advantage in capturing basic empirical regularities such as: heteroscedasticity, the leverage effect, and the volatility smile. In fact, the performance of the CEV model is comparable to most stochastic volatility models, but it is considerable easier to implement and calibrate. Nevertheless, the standard CEV model solution, using the non-central chi-square approach, still presents high computational times, specially when: i) the maturity is small, ii) the volatility is low, or iii) the elasticity of the variance tends to zero. In this paper, a new numerical method for computing the CEV model is developed. This new approach is based on the semiclassical approximation of Feynman's path integral. Our simulations show that the method is efficient and accurate compared to the standard CEV solution considering the pricing of European call options.
How to price and hedge claims on nontraded assets are becoming increasingly important matters in option pricing theory today. The most common practice to deal with these issues is to use another similar or 'closely related' asset or index which is traded, for hedging purposes. Implicitly, traders assume here that the higher the correlation between the traded and nontraded assets, the better the hedge is expected to perform. This raises the question as to how 'closely related' the assets really are. In this paper, the concept of twin assets is introduced, focusing the discussion precisely in what does it mean for two assets to be similar. Our findings point to the fact that, in order to have very similar assets, for example identical twins, high correlation measures are not enough. Specifically, two basic criteria of similarity are pointed out: i) the coefficient of variation of the assets and ii) the correlation between assets. From here, a method to measure the level of similarity between assets is proposed, and secondly, an option pricing model of twin assets is developed. The proposed model allows us to price an option of one nontraded asset using its twin asset, but this time knowing explicitly what levels of errors we are facing. Finally, some numerical illustrations show how twin assets behave depending upon their levels of similarities, and how their potential differences will traduce in MAPE (mean absolute percentage error) for the proposed option pricing model.
In this paper, we examine the interlinkages among firms through a financial network where crossholdings on both equity and debt are allowed. We relate mathematically the correlation among equities with the unconditional correlation of the assets, the values of their business assets and the sensitivity of the network, particularly the ∆-Greek. We noticed also this relation is independent of the Equities level. Besides, for the two-firms case, we analytically demonstrate that the equities correlation is always higher than the correlation of the assets; showing this issue by numerical illustrations. Finally, we study the relation between equity correlations and asset prices, where the model arrives to an increase in the former due to a fall in the assets.
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