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2017 | OriginalPaper | Buchkapitel

2. The Signal-Based Framework

verfasst von : Nadi Serhan Aydın

Erschienen in: Financial Modelling with Forward-looking Information

Verlag: Springer International Publishing

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Abstract

This chapter, assembles some fundamental properties of random bridge processes and justifies their use in modelling forwardlooking financial information. Although this chapter is essentially based on Brody et al. (Int J Theor Appl Financ 11(1):107–142, 2008), Brody et al. (Beyond hazard rates: a new framework for credit-risk modelling. Birkhäuser, Boston, 2007) and Brody et al. (Proc Math Phys Eng Sci 464(2095):1801–1822, 2008), it contributes to the existing literature by recovering the necessary properties of the signal-based framework in a much greater detail, and presenting a useful information-theoretic analysis to quantify the information component.

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Vorheriges Kapitel Introduction

Nächstes Kapitel A Signal-Based Heterogeneous Agent Network

Zero-noise at initial date is still intuitive since single point will have no prediction power.

See [2] for bridges on a random intervals [0, τ].

The part \(\mathbb{E}[\beta _{t}] = 0\) is indeed trivial, whereas \(\mathbb{V}\left [\beta _{t}\right ] = \mathbb{E}\left [B_{t}^{2} - \frac{t} {T}B_{t}B_{T} + \frac{t^{2}} {T^{2}} B_{T}^{2}\right ] = t - 2\frac{t^{2}} {T} + \frac{t^{2}} {T} = t\kappa _{t}^{-1}\).

See [18] for a definition of Lévy random bridge instead.

Note that \(\Lambda _{t}\) is also forward-looking.

\(\mathbb{E}\left [\mathbb{E}\left [X\vert \mathcal{F}_{u}\right ]\vert \mathcal{F}_{t}\right ] = \mathbb{E}\left [X\vert \mathcal{F}_{t}\right ]\) for t ≤ u and increasing set of σ-algebras \(\left (\mathcal{F}_{t}\right )_{t\geq 0}\).

Indeed; \(\mathbb{V}\left [\beta _{u}\kappa _{u} -\beta _{t}\kappa _{t},\beta _{t}\right ] = \mathbb{E}\left [(\beta _{u}\kappa _{u} -\beta _{t}\kappa _{t})\beta _{t}\right ] =\kappa _{u}\mathbb{E}\left [\beta _{u}\beta _{t}\right ] -\kappa _{t}\mathbb{E}\left [\beta _{t}^{2}\right ] = t\kappa _{u}\kappa _{u}^{-1} - t\kappa _{t}\kappa _{t}^{-1} = 0.\)

These are [27]:

\(\int _{\mathbb{X}}\exp (-x^{2}/2)\exp (ax - bx^{2})\text{d}x = \sqrt{2\pi }(2b + 1)^{-1/2}\exp (a^{2}/(2(2b + 1))\), and

\(\int _{\mathbb{X}}x\exp (-x^{2}/2)\exp (ax - bx^{2})\text{d}x = \sqrt{2\pi }a(2b + 1)^{-3/2}\exp (a^{2}/(2(2b + 1)))\),

where \(\mathbb{X} = (-\infty,\infty )\).

It reads \(S_{t} = \mathbf{1}_{\left \{t<T\right \}}S_{0}\exp \left (-\mu (T - t) + \left (\mu -\frac{\nu ^{2}} {2} \right )T + \frac{\nu ^{2}T} {2(1+\sigma ^{2}\kappa _{t}t)} + \frac{\nu \sqrt{T}\kappa _{t}\sigma } {1+\sigma ^{2}\kappa _{t}t}\xi _{t}\right )\).

\(S_{t} = \mathbf{1}_{\left \{t<T\right \}}e^{-r(T-t)}\int _{0}^{\infty }x\pi _{t}(x)\text{d}x\).

It can rewritten as \(\text{d}\xi _{t} = \text{d}W_{t} -\kappa _{t}\left (\xi _{t}T^{-1} -\sigma \mathbb{E}_{t}\left [\phi (X_{T})\right ]\right )\text{d}t\) where, again, W _t is a martingale under the pricing measure.

It reads \(\text{d}\xi _{t} = \left (\sigma \phi (x) - T^{-1}\int _{0}^{t}\kappa _{s}\text{d}W_{s}^{{\prime}}\right )\text{d}t + \text{d}W_{t}^{{\prime}} = \left (\sigma \phi (x) - T^{-1}\kappa _{t}\xi _{t}\right )\text{d}t + \text{d}W_{t}^{{\prime}}\).

It reads \(\phi (X_{T}) = S_{0}\exp \left [\left (\mu -\frac{1} {2}\nu ^{2}\right )T +\nu \sqrt{T}X_{T}\right ]\), where \(X_{T} \sim \mathcal{N}(0, 1)\).

We complete the definition (2.87) by adding the conditions \(0\ln \left (0/0\right ) = 0\), \(0\ln \left (0/p(\cdot )\right ) = 0\), and \(p(\cdot )\ln \left (\,p(\cdot )/0\right ) = \infty\).

It reads \(\pi _{t}(x) = p(x)e^{\kappa _{t}\left (\sigma x\xi _{t}^{\alpha }-\frac{1} {2} \sigma ^{2}x^{2}t\right )}/\int _{ \mathbb{X}}p(x)e^{\kappa _{t}\left (\sigma x\xi _{t}-\frac{1} {2} \sigma ^{2}x^{2}t\right )}\text{d}x.\)

It reads \(\text{d}\phi (X_{T}) =\sigma \kappa _{t}\mathbb{C}\text{ov}_{t}\left (\phi (X_{T}),X_{T}\right )\left [\kappa _{t}\left (T^{-1}\xi _{t} -\sigma \mathbb{E}_{t}\left [\phi (X_{T})\right ]\right )\text{d}t + \text{d}\xi _{t}\right ]\).

Back K, Pedersen H (1998) Long-lived information and intra-day patterns. J Financ Mark 1:385–402CrossRef

Bedini M, Buckdahn R, Engelbert H-J (2016) Brownian bridges on random intervals. arXiv:1601.01811v1

Belen S, Kropat E, Weber G-W (2009) On the classical Maki-Thompson rumour model in continuous time. CEJOR 19(1):1–17CrossRef

Bhar R (2010) Stochastic filtering with applications in finance. World Scientific, SingaporeCrossRef

Bommel J (2003) Rumors. J Financ LVIII(4):1499–1519

Brody D, Hughston L (2013) Lévy information and the aggregation of risk aversion. Proc R Soc Lond A Math Phys Sci 469(2154). http://rspa.royalsocietypublishing.org/content/469/2154/20130024

Brody D, Law Y (2015) Pricing of defaultable bonds with random information flow. Appl Math Financ 22(5):399–420CrossRef

Brody D, Hughston LP, Macrina A (2007) Beyond hazard rates: a new framework for credit-risk modelling. In: Advances in mathematical finance. Applied and numerical harmonic analysis, chapter III. Birkhäuser, Boston, pp 231–257

Brody D, Hughston LP, Macrina A (2008) Dam rain and cumulative gain. Proc Math Phys Eng Sci 464(2095):1801–1822CrossRef

10.

Brody D, Hughston L, Macrina A (2008) Information-based asset pricing. Int J Theor Appl Financ 11(1):107–142CrossRef

11.

Brody D, Davis M, Friedman R, Hughston L (2009) Informed traders. Proc R Soc A 465:1103–1122CrossRef

12.

Brody D, Hughston L, Macrina A (2011) Modelling information flows in financial markets. Advanced mathematical methods for finance. Springer, Berlin, pp 133–153

13.

Brody D, Hughston L, Yang X (2013) Signal processing with Lévy information. Proc R Soc Lond A469:20120433

14.

Çaglar M, Sezer AD (2014) Analysis of push-type epidemic data dissemination in fully connected networks. Perform Eval 77:21–36CrossRef

15.

ChunXia Y, Sen H, BingYing X (2012) The endogenous dynamics of financial markets: interaction and information dissemination. Physica A 391(12):3513–3525CrossRef

16.

Cover TM, Thomas JA (2006) Elements of information theory, 2nd edn. Wiley, Hoboken

17.

Girsanov I (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Probab Appl 5(3):285–301CrossRef

18.

Hoyle A (2010) Information-based models for finance and insurance, Ph.D. Thesis, Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

19.

Hoyle A, Hughston L, Macrina A (2011) Lévy random bridges and the modelling of financial information. Stoch Process Appl 121:856–884CrossRef

20.

Hughston L, Macrina A (2008) Information, inflation, and interest. In: Advances in mathematics of finance, vol 83. Banach Center Publications, Warszawa

21.

Kalman R (1960) A new approach to linear filtering and prediction problems. J Basic Eng (Ser D) 82:35–45CrossRef

22.

Kyle A (1985) Continuous auctions and insider trading. Econometrica 53(6):1315–1335CrossRef

23.

Liang X (2013) The Liang-Kleeman information flow: theory and applications. J Entropy 15(1):327–360CrossRef

24.

Macrina A (2006) An information-based framework for asset pricing: X-factor theory and its applications, Ph.D. Thesis, University of London

25.

Macrina A, Parbhoo P (2010) Security pricing with information-sensitive discounting, Cornell University Library ArXiv e-prints: 1001.3570

26.

Mengütürk L (2012) Information-based jumps, asymmetry and dependence in financial modelling, Ph.D. Thesis, Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

27.

Moll VH (2014–2015) Special integrals of Gradshteyn and Ryzhik: the proofs. Monographs and research notes in mathematics, vol 1–2. Chapman and Hall/CRC Press, London/Boca Raton

28.

Øksendal B (1998) Stochastic differential equations: an introduction with applications, 5th edn. Springer, BerlinCrossRef

29.

Renyi A (1984) A diary on information theory. Wiley series in probability and mathematical statistics. Wiley, New York

30.

Rutkowski M, Yu N (2007) An extension of the Brody–Hughston–Macrina approach to modeling of defaultable bonds. Int J Theor Appl Financ 10(3):557–589CrossRef

31.

Shannon C (1948) A mathematical theory of communication. Bell Syst Techn J 27:379–423CrossRef

32.

Stein C (1981) Estimation of the mean of a multivariate normal distribution. Ann Stat 9(6):1135–1151CrossRef

33.

Yang X (2013) Information-based commodity pricing and theory of signal processing with Lévy information, Ph.D. Thesis, Department of Mathematics, Imperial College London and Shell International, London SW7 2AZ, United Kingdom

Titel: The Signal-Based Framework
verfasst von: Nadi Serhan Aydın
Verlag: Springer International Publishing
Buch: Financial Modelling with Forward-looking Information
Print ISBN: 978-3-319-57146-1

Electronic ISBN: 978-3-319-57147-8

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-57147-8_2

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