scholarly journals Introduction to Stochastic Finance: Random Variables and Arbitrage Theory

2018 ◽  
Vol 26 (1) ◽  
pp. 1-9
Author(s):  
Peter Jaeger
Keyword(s):  
Interest Rate ◽  
Probability Measure ◽  
Random Variables ◽  
Random Variable ◽  
Market Model ◽  
Borel Sets ◽  
Risk Neutral ◽  
Arbitrage Opportunity ◽  
The Interest Rate ◽  
Definition Of

Summary Using the Mizar system [1], [5], we start to show, that the Call-Option, the Put-Option and the Straddle (more generally defined as in the literature) are random variables ([4], p. 15), see (Def. 1) and (Def. 2). Next we construct and prove the simple random variables ([2], p. 14) in (Def. 8). In the third section, we introduce the definition of arbitrage opportunity, see (Def. 12). Next we show, that this definition can be characterized in a different way (Lemma 1.3. in [4], p. 5), see (17). In our formalization for Lemma 1.3 we make the assumption that ϕ is a sequence of real numbers (there are only finitely many valued of interest, the values of ϕ in Rd). For the definition of almost sure with probability 1 see p. 6 in [2]. Last we introduce the risk-neutral probability (Definition 1.4, p. 6 in [4]), here see (Def. 16). We give an example in real world: Suppose you have some assets like bonds (riskless assets). Then we can fix our price for these bonds with x for today and x · (1 + r) for tomorrow, r is the interest rate. So we simply assume, that in every possible market evolution of tomorrow we have a determinated value. Then every probability measure of Ωfut1 is a risk-neutral measure, see (21). This example shows the existence of some risk-neutral measure. If you find more than one of them, you can determine – with an additional conidition to the probability measures – whether a market model is arbitrage free or not (see Theorem 1.6. in [4], p. 6.) A short graph for (21): Suppose we have a portfolio with many (in this example infinitely many) assets. For asset d we have the price π(d) for today, and the price π(d) (1 + r) for tomorrow with some interest rate r > 0. Let G be a sequence of random variables on Ωfut1, Borel sets. So you have many functions fk : {1, 2, 3, 4}→ R with G(k) = fk and fk is a random variable of Ωfut1, Borel sets. For every fk we have fk(w) = π(k)·(1+r) for w {1, 2, 3, 4}. $$\matrix{ {Today} & {Tomorrow} \cr {{\rm{only}}\,{\rm{one}}\,{\rm{scenario}}} & {\left\{ {\matrix{ {w_{21} = \left\{ {1,2} \right\}} \hfill \cr {w_{22} = \left\{ {3,4} \right\}} \hfill \cr } } \right.} \cr {{\rm{for}}\,{\rm{all}}\,d \in N\,{\rm{holds}}\,\pi \left( d \right)} & {\left\{ {\matrix{ {f_d \left( w \right) = G\left( d \right)\left( w \right) = \pi \left( d \right) \cdot \left( {1 + r} \right),} \hfill \cr {w \in w_{21} \,or\,w \in w_{22} ,} \hfill \cr {r > 0\,{\rm{is}}\,{\rm{the}}\,{\rm{interest}}\,{\rm{rate}}.} \hfill \cr } } \right.} \cr }$$ Here, every probability measure of Ωfut1 is a risk-neutral measure.

Brazil: monetary policy and the neutral interest rate

2016 ◽  
Vol 43 (6) ◽  
pp. 966-979
Author(s):  
Cleomar Gomes da Silva ◽  
Rafael Cavalcanti de Araújo
Keyword(s):  
Monetary Policy ◽  
Interest Rate ◽  
Real Interest Rate ◽  
Design Methodology ◽  
Macroeconomic Model ◽  
Natural Rate ◽  
Content Type ◽  
Natural Rate Of Interest ◽  
The Interest Rate ◽  
Definition Of

Purpose The purpose of this paper is to analyze the conduct of monetary policy in Brazil and estimate the country’s neutral real interest rate. Design/methodology/approach The authors make use of a state-space macroeconomic model representation. Findings The period of analysis goes from 2003 up to the end of 2013 and the results show that the country’s natural rate of interest was around 4.2 percent in December 2013. Originality/value One of the main differences of this work is the inclusion of variables such as the real exchange rate and world interest rate. This is important because these variables play an important role in the definition of the interest rate and, consequently, in the definition of the neutral interest rate.

scholarly journals Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables

2010 ◽  
Vol 18 (4) ◽  
pp. 213-217
Author(s):  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Probability Measure ◽  
Cartesian Product ◽  
Random Variables ◽  
Random Variable ◽  
Final Part ◽  
Chebyshev’S Inequality ◽  
And Algebra ◽  
Discrete Spaces ◽  
Chebyshev's Inequality

Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables In this article we continue formalizing probability and randomness started in [13], where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables.

scholarly journals Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

2008 ◽  
Vol 45 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Eric Carlen ◽  
Ester Gabetta ◽  
Eugenio Regazzini
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
Initial Energy ◽  
Infinite Variance ◽  
Borel Sets ◽  
Second Moment ◽  
Quantitative Estimates ◽  
Kac Equation ◽  
Kac’S Model

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure on R. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets of R). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variables x̃1, x̃2, …, where these random variables have an infinite second moment and zero mean. Then, with Tn := ∑j=1ηnλj,nx̃j, with max1 ≤ j ≤ ηnλj,n → 0 (as n → +∞), and ∑j=1ηnλj,n2 = 1, n = 1, 2, …, the classical central limit theorem suggests that T should in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

scholarly journals Random Variables and Product of Probability Spaces

2013 ◽  
Vol 21 (1) ◽  
pp. 33-39
Author(s):  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Borel Sets ◽  
The Real ◽  
Probability Spaces ◽  
Countably Infinite

Summary We have been working on the formalization of the probability and the randomness. In [15] and [16], we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of [15] and [16]. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on Σ, Borel sets and a real-valued random variable on Σ. Next, we formalize the product of countably infinite probability spaces.

scholarly journals Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

2008 ◽  
Vol 45 (01) ◽  
pp. 95-106 ◽  
Author(s):  
Eric Carlen ◽  
Ester Gabetta ◽  
Eugenio Regazzini
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
Initial Energy ◽  
Infinite Variance ◽  
Borel Sets ◽  
Second Moment ◽  
Quantitative Estimates ◽  
Kac Equation ◽  
Kac’S Model

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure onR. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets ofR). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variablesx̃1,x̃2, …, where these random variables have an infinite second moment and zero mean. Then, withTn:= ∑j=1ηnλj,nx̃j, with max1 ≤j≤ ηnλj,n→ 0 (asn→ +∞), and ∑j=1ηnλj,n2= 1,n= 1, 2, …, the classical central limit theorem suggests thatTshould in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

FINANCIAL MARKET MODEL WITH INFLUENTIAL INFORMED INVESTORS

2005 ◽  
Vol 08 (06) ◽  
pp. 693-716 ◽  
Author(s):  
AXEL GRORUD ◽  
MONIQUE PONTIER
Keyword(s):  
Brownian Motion ◽  
Interest Rate ◽  
Financial Market ◽  
Asset Prices ◽  
Optimization Problem ◽  
Statistical Tests ◽  
Market Model ◽  
Financial Model ◽  
Additional Information ◽  
The Interest Rate

We develop a financial model with an "influential informed" investor who has an additional information and influences asset prices by means of his strategy. The prices dynamics are supposed to be driven by a Brownian motion, the informed investor's strategies affect the risky asset trends and the interest rate. Our paper could be seen as an extension of Cuoco and Cvitanic's work [4] since, as these authors, we solve the informed influential investor's optimization problem. But our main result is the construction of statistical tests to detect if, observing asset prices and agent's strategies, this influential agent is or not an informed trader.

RISK-BASED CAPITAL FOR VARIABLE ANNUITY UNDER STOCHASTIC INTEREST RATE

2020 ◽  
Vol 50 (3) ◽  
pp. 959-999
Author(s):  
JinDong Wang ◽  
Wei Xu
Keyword(s):  
Monetary Policy ◽  
Interest Rate ◽  
Mutual Fund ◽  
Risk Measures ◽  
Market Model ◽  
Model Parameters ◽  
Joint Dynamics ◽  
Contract Attributes ◽  
The Interest Rate

AbstractInterest rate is one of the main risks for the liability of the variable annuity (VA) due to its long maturity. However, most existing studies on the risk measures of the VA assume a constant interest rate. In this paper, we propose an efficient two-dimensional willow tree method to compute the liability distribution of the VA with the joint dynamics of the mutual fund and interest rate. The risk measures can then be computed by the backward induction on the tree structure. We also analyze the sensitivity and impact on the risk measures with regard to the market model parameters, contract attributes, and monetary policy changes. It illustrates that the liability of the VA is determined by the long-term interest rate whose increment leads to a decrease in the liability. The positive correlation between the interest rate and mutual fund generates a fat-tailed liability distribution. Moreover, the monetary policy change has a bigger impact on the long-term VAs than the short-term contracts.

scholarly journals Modelling Real World Using Stochastic Processes and Filtration

2016 ◽  
Vol 24 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Peter Jaeger
Keyword(s):  
Stochastic Process ◽  
Real World ◽  
Random Variable ◽  
Call Option ◽  
Relevant Market ◽  
Real Random Variable ◽  
Time Points ◽  
Arbitrage Opportunity ◽  
Definition Of ◽  
The Given

Summary First we give an implementation in Mizar [2] basic important definitions of stochastic finance, i.e. filtration ([9], pp. 183 and 185), adapted stochastic process ([9], p. 185) and predictable stochastic process ([6], p. 224). Second we give some concrete formalization and verification to real world examples. In article [8] we started to define random variables for a similar presentation to the book [6]. Here we continue this study. Next we define the stochastic process. For further definitions based on stochastic process we implement the definition of filtration. To get a better understanding we give a real world example and connect the statements to the theorems. Other similar examples are given in [10], pp. 143-159 and in [12], pp. 110-124. First we introduce sets which give informations referring to today (Ωnow, Def.6), tomorrow (Ωfut1 , Def.7) and the day after tomorrow (Ωfut2 , Def.8). We give an overview for some events in the σ-algebras Ωnow, Ωfut1 and Ωfut2, see theorems (22) and (23). The given events are necessary for creating our next functions. The implementations take the form of: Ωnow ⊂ Ωfut1 ⊂ Ωfut2 see theorem (24). This tells us growing informations from now to the future 1=now, 2=tomorrow, 3=the day after tomorrow. We install functions f : {1, 2, 3, 4} → ℝ as following: f1 : x → 100, ∀x ∈ dom f, see theorem (36), f2 : x → 80, for x = 1 or x = 2 and f2 : x → 120, for x = 3 or x = 4, see theorem (37), f3 : x → 60, for x = 1, f3 : x → 80, for x = 2 and f3 : x → 100, for x = 3, f3 : x → 120, for x = 4 see theorem (38). These functions are real random variable: f1 over Ωnow, f2 over Ωfut1, f3 over Ωfut2, see theorems (46), (43) and (40). We can prove that these functions can be used for giving an example for an adapted stochastic process. See theorem (49). We want to give an interpretation to these functions: suppose you have an equity A which has now (= w1) the value 100. Tomorrow A changes depending which scenario occurs − e.g. another marketing strategy. In scenario 1 (= w11) it has the value 80, in scenario 2 (= w12) it has the value 120. The day after tomorrow A changes again. In scenario 1 (= w111) it has the value 60, in scenario 2 (= w112) the value 80, in scenario 3 (= w121) the value 100 and in scenario 4 (= w122) it has the value 120. For a visualization refer to the tree: The sets w1,w11,w12,w111,w112,w121,w122 which are subsets of {1, 2, 3, 4}, see (22), tell us which market scenario occurs. The functions tell us the values to the relevant market scenario: For a better understanding of the definition of the random variable and the relation to the functions refer to [7], p. 20. For the proof of certain sets as σ-fields refer to [7], pp. 10-11 and [9], pp. 1-2. This article is the next step to the arbitrage opportunity. If you use for example a simple probability measure, refer, for example to literature [3], pp. 28-34, [6], p. 6 and p. 232 you can calculate whether an arbitrage exists or not. Note, that the example given in literature [3] needs 8 instead of 4 informations as in our model. If we want to code the first 3 given time points into our model we would have the following graph, see theorems (47), (44) and (41): The function for the “Call-Option” is given in literature [3], p. 28. The function is realized in Def.5. As a background, more examples for using the definition of filtration are given in [9], pp. 185-188.

scholarly journals Retrospectives: From Usury To Interest

2007 ◽  
Vol 21 (1) ◽  
pp. 227-236 ◽  
Author(s):  
Joseph Persky
Keyword(s):  
Political Economy ◽  
Interest Rate ◽  
Middle Ages ◽  
Industrial Revolution ◽  
Adam Smith ◽  
Modern World ◽  
Jeremy Bentham ◽  
The Interest Rate ◽  
Definition Of ◽  
Total Welfare

Since the Middle Ages, each epoch has participated in the debate over the conditions in which lending should be prohibited as usury. While disagreements over the definition of usury remain, the debate came to its modern climax on the eve of the industrial revolution, in a well-known interchange between Jeremy Bentham and Adam Smith in the late 1780s. Smith, for all his faith in a system of natural liberty, proved unwilling to let the interest rate float. Bentham argued anything else must reduce total welfare. From a superficial perspective, the entire affair amounts to nothing more than a modest dispute between a failing master (Smith died in 1790) and an over-eager disciple. (Bentham acknowledged in the Defence that all he knew of political economy originated in Smith's works.) Yet the argument struck a fundamental chord. Gilbert K. Chesterton identified Bentham's essay on usury as the very beginning of the “modern world.” I tend to agree.

scholarly journals Jump Telegraph Processes and Financial Markets with Memory

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
Nikita Ratanov
Keyword(s):  
Interest Rate ◽  
Financial Markets ◽  
Financial Market ◽  
Stock Prices ◽  
Call Option ◽  
Market Models ◽  
New Class ◽  
Risk Neutral ◽  
The Interest Rate ◽  
With Memory

The paper develops a new class of financial market models. These models are based on generalized telegraph processes with alternating velocities and jumps occurring at switching velocities. The model under consideration is arbitrage-free and complete if the directions of jumps in stock prices are in a certain correspondence with their velocity and with the behaviour of the interest rate. A risk-neutral measure and arbitrage-free formulae for a standard call option are constructed. This model has some features of models with memory, but it is more simple.

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