=Paper=
{{Paper
|id=Vol-1774/MIDAS2016_paper11
|storemode=property
|title=A probabilistic approach for financial IoT data
|pdfUrl=https://ceur-ws.org/Vol-1774/MIDAS2016_paper11.pdf
|volume=Vol-1774
|authors=Salvatore Cuomo,Pasquale De Michele,Vittorio Di Somma,Giovanni Ponti
|dblpUrl=https://dblp.org/rec/conf/pkdd/CuomoMSP16
}}
==A probabilistic approach for financial IoT data==
https://ceur-ws.org/Vol-1774/MIDAS2016_paper11.pdf
A probabilistic approach for financial IoT data
Salvatore Cuomo1 , Pasquale De Michele1 ,
Vittorio Di Somma1 , and Giovanni Ponti2
1
University of Naples Federico II, 80126, Naples, Italy
2
ENEA Portici Research Center, 80055 Portici, Naples, Italy
Abstract. The extraction of information from the Internet of Things (IoT)
plays a fundamental role in many research fields. In this work we focus
our attention on financial data, used to describe self-financing 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 deter-
mine 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.
1 Introduction
The Internet of Things (IoT) [3] denotes a system based on the link between the
real world and Internet and characterized by big data flows, as occurs for sys-
tems managing financial data. The number of internet services, as home banking
and on-line trading, which let people access Internet databases and do financial
transactions, is increasing more and more. This implies the necessity to have
very fast and efficient methods able to solve problems by available informations
in real time. As a case study, we propose the problem of pricing of derivatives. A
derivative is a kind of contract whose value (represented by a function F named
Payoff function) depends on another entity, called underlying. Derivatives are
used especially in risk management and speculation field. Common examples of
derivatives are options. More in detail, an option gives the right to buy (Call
option) or to sell (Put option) its underlying in future dates by paying a strike
price. In the following section we briefly describe the analytical model and apply
a numerical approach to an example.
2 Analytical and Numerical Model
A portfolio (or strategy) is an integrable stochastic process rappresenting the
shares of the assets and identified by its value function. In particular, self-
financing portfolios, are strategies where a change of its value depends only
on a change in the values of assets. The following equation, known as Black-
Scholes equation [1], characterizes the self-financing portfolios f depending both
on present time and the risk asset:
σ 2 s2
∂ss f (t; s) + rs∂s f (t; s) + ∂t f (t; s) = rf (t; s) ∀(t; s) ∈ [0; T [×R+
2
where r ∈ R+ is the free risk interest rate. An arbitrage strategy is a self-
financing 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 differential equation dSt = µSt dt + σSt dWt with µ, σ ∈ 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 first step consists in estimating
σ. We extract a sample of past values of the underlying from a sample database
andcalculate
the corresponding values of the normal process X = (Xt )t∈[0;T ] =
log SS0t . Here, we have V ar(X) = T σ 2 , where V ar(X) can be approximated
by the sample variance. The second
√step consists
in determining
the simulations
σ2
of the underlying S̃k = S0 exp σ T Z̃k + r − 2 T , where Z̃k are normal
standard casual numbers. The last step consists in finding the value of P0 by the
−rT Pn
Law of big numbers: P0 ≈ e n k=1 F (S̃k ). 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 first row and in the second simulations of underlying.
Table 1. Extraction of historical data and Simulations of underlying.
His. Val. 101,8 109,7 119,3 109,2 113,4 108,1 107,9 109,9 105,9 115,0
Sim. Val. 110,4 110,6 110,2 110,3 110,6 110,0 110,1 110,56 110,5 110,3
From the first row we obtain σ ≈ 0, 002. Since the derivative is an option, by
using the values of the second row, the expression of the price becomes:
e−0,1 X
P0 = (S̃n − 100) ≈ 9, 4
10
S̃n >K
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)