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Catastrophe Bond Pricing for the Two-Factor Vasicek Interest Rate Model with Automatized Fuzzy Decision Making

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Catastrophe Bond Pricing for the Two-Factor Vasicek Interest Rate Model with Automatized Fuzzy Decision Making View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Soft Comput DOI 10.1007/s00500-015-1957-1 METHODOLOGIES AND APPLICATION Catastrophe bond pricing for the two-factor Vasicek interest rate model with automatized fuzzy decision making Piotr Nowak1 · Maciej Romaniuk1,2 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract Catastrophe bonds are financial instruments, [e.g., tsunami in Japan (2011)]. This results in high values of which enable to transfer the natural catastrophe risk to finan- damages. One can mention Hurricane Andrew (1992) with cial markets. This paper is a continuation of our earlier losses estimated at USD 30 billion [see, e.g., Muermann research concerning catastrophe bond pricing. Weassume the (2008)]. On the other hand, devastating floods are a signifi- absence of arbitrage and neutral attitude of investors toward cant problem in Europe. catastrophe risk. The interest rate behavior is described by the Natural catastrophes losses have a negative impact on two-factor Vasicek model. To illustrate and analyze obtained financial stability of insurers, e.g., after Hurricane Andrew results, we conduct Monte Carlo simulations, using parame- more than 60 insurance companies became insolvent [see, ters fitted for real data on natural catastrophes. Besides the e.g., Muermann (2008)]. This is caused not only by the huge crisp cat bond pricing formulas, we obtain their fuzzy coun- value of losses, but also the classical insurance mechanisms terparts, taking into account the uncertainty on the market. [see, e.g., Borch (1974)] are not suitable for losses caused by Moreover, we propose an automated approach for decision natural disasters. The classical insurance approach applied making in fuzzy environment with relevant examples pre- to the construction of the insurer’s portfolio can lead directly senting this method. to bankruptcy of this enterprise [see, e.g., Ermoliev et al. (2001); Romaniuk and Ermolieva (2005)]. Keywords Catastrophe bonds · Stochastic processes · One of the methods of solving the mentioned above Monte-Carlo simulations · Vasicek model · Automated problems is application of other financial and insurance decision making · Fuzzy numbers mechanisms. Such instruments are, among others, catastro- phe bonds [also called cat bonds, see, e.g., George (1999); Nowak and Romaniuk (2009)]. They transfer risk connected 1 Introduction with natural catastrophe losses (i.e., the insurance risk) to financial markets. Nowadays, natural catastrophes occur more frequently than The number of papers dedicated to catastrophe bonds before. Additionally, they threaten densely populated areas pricing is relatively low. A significant problem is the incom- pleteness of the financial market on which catastrophe Communicated by V. Loia. instruments are traded. In many papers, simplified models or behavioral finance methods are used. In turn, many stochastic B Piotr Nowak approaches do not take into account the change of probability [email protected] measure in the martingale method or apply the utility func- Maciej Romaniuk tion, which choice can be an additional problem in practice. [email protected] An interesting stochastic approach proposed by Vaugirard in 1 Systems Research Institute Polish Academy of Sciences, ul. Vaugirard (2003) overcame the problem of the incomplete- Newelska 6, 01-447 Warsaw, Poland ness of the financial market. For more detailed references we 2 The John Paul II Catholic University of Lublin, Lublin, refer the reader to Nowak and Romaniuk (2013b). Poland 123 P. Nowak, M. Romaniuk In this paper, we solve the catastrophe bond pricing prob- rithms, and the theory of chaos, in business, finance and lem under the assumption of absence of arbitrage on the economics are discussed in Dostál (2012). An interesting financial market. We derive and prove the cat bond valu- example presented in Dostál (2012) is application of the- ation expressions in crisp case. Then, assuming that some ory of chaos to simulation of market price of a share on the parameters of the model cannot be precisely described, we stock market. In Beynon and Clatworthy (2013), the problem obtain fuzzy counterparts of the crisp pricing formulas and of understanding the relationship between company stock we apply them to an automatized method of decision making. returns and earnings components is discussed. The authors We illustrate our theoretical results by numerical examples. apply the classification and ranking belief simplex (CaRBS) Our approach in this paper can be treated as an essential and a development of this data analysis technique, called extension of Vaugirard’s method. In analogous catastrophe RCaRBS. In turn, the author of Gordini (2014) uses a genetic bond valuation formulas considered earlier [see Nowak et al. algorithms approach for small enterprise default prediction (2012); Nowak and Romaniuk (2010b, 2013b)], one-factor modeling. A widely applicable multi-variate decision sup- models of the spot interest rate rt were used. A new model port model for market trend analysis and prediction, based of rt proposed in this paper is the two-factor Vasicek model on a time series transformation method in combination with (introduced by Hull and White) with a stochastic process Bayesian logic and Bayesian network, is presented in Mrši´c describing deviation of the current view on the long-term (2014). In Besbes et al. (2015), the authors propose a two- level of rt from its average view. Additionally, in comparison phase mathematical approach, involving the application of a with the mentioned approaches, we introduce more detailed stochastic programming model, to effective product-driven formulas describing prices of cat bonds (see Theorem 2)for supply chain design. In Vasant (2013b), a new and interest- arbitrary time moment t, on the basis of Dynkin’s Lemma. ing approach to the industrial production planning problems For this purpose, we apply zero-coupon bond pricing for- is discussed. Finally, various generalized soft methods are mulas from Munk (2011). However, the proof of Theorem 2 applied in many other scientific areas, see, e.g., Kowalski required considering the case of ar = aε (see Lemma 1, case et al. (2008), Kulczycki and Charytanowicz (2005, 2010). b), not considered in Munk (2011). Furthermore, contrary Section 2 contains lists of symbols, expressions and oper- to Nowak et al. (2012), Nowak and Romaniuk (2010b), we ators used in the paper. In Sect. 3, we discuss basic notions conduct Monte Carlo simulations, applying real parameters and mechanisms connected with catastrophe bonds. In Sect. for rt and distributions of catastrophe losses. We also apply a 4, catastrophe bond pricing formulas are proved for the two- non-homogeneous Poisson process for quantity of catastro- factor Vasicek interest rate model and a necessary theoretical phe events. As mentioned previously, apart from the standard base is presented. Section 5 is devoted to cat bond pricing valuation formulas, we obtain their fuzzy counterparts, tak- in fuzzy environment. In Sect. 6, we propose an automated ing into account the uncertainty on the market. We propose an method of decision making with application of the fuzzy automated approach for decision making and present exam- valuation expressions. Section 7 is dedicated to algorithms ples to illustrate it for a given set of market parameters. used in Monte Carlo simulations. These algorithms are then As noted above, applying fuzzy set theory, we consider applied in Sect. 8 during analysis of some numerical exam- different sources of uncertainty, not only the stochastic one. ples of cat bond pricing and automatized decision making. In particular, we take into account that volatility parameters and the correlation coefficient, used for description of rt , are determined by fluctuating financial market and often the uncertainty does not have a stochastic character. Therefore, 2 List of symbols, expressions and operators we apply fuzzy counterparts of some market parameters. As result, price obtained by us has the form of a fuzzy number, 2.1 List of symbols and expressions in order of which can be used for investment decision making. Similar appearance approach was applied in the case of options in Wu (2004) and Nowak and Romaniuk (2010a, 2013c, 2014b), where (,F, P) probability space (see Sect. 4), the Jacod-Grigelionis characteristic triplet [see, e.g., Nowak P probability measure (see Sect. 4), (2002); Shiryaev (1999)] was additionally used. [0, T ] time horizon (see Sect. 4), The literature devoted to soft computing methods is very T date of maturity and payoff of catastrophe bonds (see Def- rich. Books (Vasant et al. 2012; Vasant 2013a, 2014, 2015) inition 1, Definition 2), 1 2 n W = W , W ,...,W standard, n-dimensional provide a wide overview of soft computing algorithms and t t t t t∈[0,T ] their applications. We mention only certain selected exam- Brownian motion (see Sect. 4), ( )∞ ples of the presented approaches, focusing on financial and Ui i=1 sequence of independent random variables (see economic problems. Various optimization methods using soft Sect. 4), computing such as fuzzy logic, neural networks, genetic algo- Ui value of losses during ith catastrophic event (see Sect. 4), 123 Catastrophe bond pricing for the two-factor Vasicek interest... ˜ λ ,λ ,...,λ Nt compound Poisson process describing aggre- 1 2 n constants defining the market price of risk ∈ , t [0 T ] process (see Sect. 4), gated catastrophic losses (see (1)), . the Euclidean norm in Rn (see Sect. 4), ( ) Nt t∈[0,T ] Poisson process describing number of T ¯ ¯ λt dWt Itô integral of λ with respect to W (see (2)), catastrophic events (see Sect. 4), 0 2 2 κ :[0, T ]→R+ intensity function of Poisson process T λ¯ λ¯ 0 t dt Lebesgue integral of (see (2)), (N ) (see Sect. 4 and Sect. 7.1), t t∈[0,T ] dQ Radon–Nikodym derivative of Q with respect to P (see (F ) filtration with respect to which stochastic dP t t∈[0,T ] (2)), processes considered in the paper are adapted (see P-a.s.
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