A probabilistic approach for nancial IoT data

Abstract

The extraction of information from the Internet of Things (IoT) plays a fundamental role in many research elds. In this work we focus our attention on nancial data, used to describe self- nancing portfolios in a complete market. Here, the absence of the arbitrage principle, the existence and the uniqueness of no arbitrage price are valid. With these hypotheses we can resort to the Black-Scholes model in order to determine the expression of no arbitrage price. In this model, frictional costs are avoided. Moreover, selling and buying of every amount of the assets and short sellings are allowed. In other words, traders can sell amount of assets even if they do not own them. Finally, this model is composed by a risk-free and a Geometric Brownian motion risk assets.

Introduction

Analytical and Numerical Model

A portfolio (or strategy) is an integrable stochastic process rappresenting the shares of the assets and identi ed by its value function. In particular, selfnancing portfolios, are strategies where a change of its value depends only on a change in the values of assets. The following equation, known as BlackScholes equation [ 1 ], characterizes the self- nancing portfolios f depending both on present time and the risk asset: 2s2 2 8(t; s) 2 [0; T [ R+ where r 2 R+ is the free risk interest rate. An arbitrage strategy is a selfnancing portfolio ensuring a future positive also with a null initial value. More in detail, in a Black-Scholes market the no arbitrage price P0 of a derivative F (S) assumes the expression P0 = e rT E[F (ST )], where S is the solution of the stochastic di erential equation dSt = Stdt + StdWt with ; 2 R+ and Wt is a Brownian motion. Now we focus on a statistical approach (i.e., Monte Carlo method [ 2 ]) to evaluate the price formula. The rst step consists in estimating . We extract a sample of past values of the underlying from a sample database and calculate the corresponding values of the normal process X = (Xt)t2[0;T ] = log SS0t . Here, we have V ar(X) = T 2, where V ar(X) can be approximated by the sample variance. The second step consists in determining the simulations of the underlying S~k = S0exp pT Z~k + r 22 T , where Z~k are normal standard casual numbers. The last step consists in nding the value of P0 by the Law of big numbers : P0 e nrT Pkn=1 F (S~k). We apply the previous numerical results to determine an approximation of a Call with a generic underlying S and S0 = 100; K = 100; T = 1; r = 0; 1. Table 1 contains some historical values of S in the rst row and in the second simulations of underlying. 109,7 110,6 From the rst row we obtain 0; 002. Since the derivative is an option, by using the values of the second row, the expression of the price becomes: P0 = 10

X (S~n S~n>K 100)

References

1. Black , F. , Scholes , M.: The pricing of options and corporate liabilities . Journal of Political Economy 81 ( 3 ), 637 { 654 ( 1973 )

2. Glasserman , P. : Monte Carlo Methods in Financial Engineering. Applications of mathematics : stochastic modelling and applied probability , Springer ( 2004 )

3. Kevin , A. : That \internet of things" thing, in the real world things matter more than ideas . RFID Journal 22 ( 2009 )