Escape Velocity backed Avalanche Predictor- Neural Evidence from Nifty

The concept of escape velocity has been extended from physics to stochastic finance and used as an avalanche predictor. Escape velocity being an extreme event serves as a perfect proxy of this stochastic finance event. This study identifies the propensity of the capital market to explode on rare occasions, which could be termed as avalanche. The frequency of such movement (both up and down) may not be high; however, the amplitude will be significantly high. The underlying for the study is Nifty, bellwether Indian bourse. Escape velocity has been calculated for Nifty on a daily basis for 17 years and prediction modelling has been constructed applying artificial neural networks (ANN) and multiple adaptive regression splines (MARS) simultaneously. Results indicate queer coupling of US events and Nifty apart from the evident behavioural traces. This research work is aimed at providing an implicit form of avalanche predictor from a distinctly different reference point.

Introduction to Stopping Time in Stochastic Finance Theory

Summary We start with the definition of stopping time according to [4], p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in [6], pp.37–38). Finally we give an example with some given filtration. Stopping time is very important for stochastic finance. A stopping time is the moment, where a certain event occurs ([7], p.372) and can be used together with stochastic processes ([4], p.283). Look at the following example: we install a function ST: {1,2,3,4} → {0, 1, 2} ∪ {+∞}, we define: a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2. b. The set {0,1,2} consists of time points: 0=now,1=tomorrow,2=the day after tomorrow. We can prove: c. {w, where w is Element of Ω: ST.w=0}=∅ & {w, where w is Element of Ω: ST.w=1}={1,2} & {w, where w is Element of Ω: ST.w=2}={3,4} and ST is a stopping time. We use a function Filt as Filtration of {0,1,2}, Σ where Filt(0)=Ωnow, Filt(1)=Ωfut1 and Filt(2)=Ωfut2. From a., b. and c. we know that: d. {w, where w is Element of Ω: ST.w=0} in Ωnow and {w, where w is Element of Ω: ST.w=1} in Ωfut1 and {w, where w is Element of Ω: ST.w=2} in Ωfut2. The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also [7], p.371. Suppose we have ST(1)=+∞, then this means that for 1 the corresponding event never occurs. As an interpretation for our installed functions consider the given adapted stochastic process in the article [5]. ST(1)=1 means, that the given element 1 in {1,2,3,4} is stopped in 1 (=tomorrow). That tells us, that we have to look at the value f2(1) which is equal to 80. The same argumentation can be applied for the element 2 in {1,2,3,4}. ST(3)=2 means, that the given element 3 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(3) which is equal to 100. ST(4)=2 means, that the given element 4 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(4) which is equal to 120. In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [7], p.372).

Boundary Conditions of Options: A Demonstration Based on the Stochastic Discount

The aim of this paper is to provide a proof of the generally accepted boundary conditions of (call and put) financial options from a novel point of view. To do this, we will use an auxiliary discounting function which will be defined in this work. However, the financial options are derivative instruments whose function is risk hedging in contexts of uncertainty, whereby the employed discount function will be necessarily stochastic. More specifically, we will apply the classic properties of the magnitude “discount” to the so-defined discount function to obtain, in a natural way, the noteworthy boundary conditions of financial options. It is well-known that financial options (belonging to the field of stochastic finance) have been studied without any relation with the magnitude “discount” (more characteristic of the classic Financial Mathematics). Consequently, the principal contribution of this work is the construction of a stochastic discount function as a bridge connecting its associated discount and the financial options, being demonstrated that their properties can be mutually derived.

Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance

Summary We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature [10] (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct the Borel sets [16] (pp. 9-10). Literature [16] (pp. 9-10) and [11] (pp. 11-12) gives an overview, that there exists some other sets for this construction. Last we define special functions as random variables for stochastic finance in discrete time. The relevant functions are implemented in the article [15], see [9] (p. 4). The aim is to construct events and random variables, which can easily be used with a probability measure. See as an example theorems (10) and (14) in [20]. Then the formalization is more similar to the presentation used in the book [9]. As a background, further literatures is [3] (pp. 9-12), [13] (pp. 17-20), and [8] (pp.32-35).