Research Article Credit Derivatives Pricing Model for ...
Transcript of Research Article Credit Derivatives Pricing Model for ...
Research ArticleCredit Derivatives Pricing Model for Fuzzy Financial Market
Liang Wu,1,2 Yaming Zhuang,1 and Xiaojing Lin1
1School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China2Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China
Correspondence should be addressed to Liang Wu; [email protected]
Received 25 April 2015; Accepted 29 September 2015
Academic Editor: Salvatore Alfonzetti
Copyright Β© 2015 Liang Wu et al.This is an open access article distributed under theCreativeCommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
With various categories of fuzziness in the market, the factors that influence credit derivatives pricing include not only thecharacteristic of randomness but also nonrandom fuzziness. Thus, it is necessary to bring fuzziness into the process of creditderivatives pricing. Based on fuzzy process theory, this paper first brings fuzziness into credit derivatives pricing, discusses somepricing formulas of credit derivatives, and puts forward aOne-Factor FuzzyCopula functionwhich builds a foundation for portfoliocredit products pricing. Some numerical calculating samples are presented as well.
1. Introduction
Credit derivatives is a general term for a series of financialengineering technologies which strips, transfers, and hedgescredit risks from basic assets. Its major feature is strippingcredit risks from other financial risks with an approachof transferring. With a history of more than 20 years, thecredit derivatives have made themselves the mainstreamproducts of over-the-counter (OTC) market exchanges withan explosive growth in its market size. Hence, pricing ofcredit derivatives became amajor concern of both the currentfinancial theoretical circle and the practical area.
With a general view of the literatures available aboutcredit derivatives pricing, their common characteristic can besummarized as that all the pricing processes are built on thebase of the theory of random probability. However, financialproduct pricing is a process of mathematical modelingfrom social reality; market environment that its existenceand development rely on is under a lot of fuzziness; andthe human thoughts that describe and judge the pricingmodel are also fuzzy. Just like credit derivatives, the pricingmodeling must be affected by fuzziness produced by thecharacteristics of OTC exchange (such as nonstandardizationof products and the short of strictmanagement system).Thus,with a lot of fuzziness of the market, the factors influencingfinancial products pricing have not only the characteristicof randomness but also nonrandom fuzziness. Therefore, it
is necessary to bring fuzziness into the process of creditderivatives pricing.
Fuzzy theory is a powerful tool to deal with all kindsof fuzziness. Researches on it offer new theoretical foun-dation for financial product pricing. It is a beneficial andnecessary supplement to traditional financial product pricingmethods. Different from randomness, fuzziness is anotherkind of uncertainty in the reality. To depict the nonrandomcharacteristics of fuzziness, Zadeh [1] put forward the fuzzyset theory based on the concept of membership function.After that, the fuzzy set theory is widely used in various fields.In order to measure a fuzzy event, B. Liu and Y.-K. Liu [2]introduced the concept of credibility measure, and, in orderto further deal with the dynamic of fuzzy event over time,Liu [3] founded a fuzzy process, a differential formula, anda fuzzy integral; the related literature can also be seen in You[4], Peng [5], and so forth. Different from random financialmathematics, Liu [3] assumed that the stock price followsgeometric Liu process rather than a geometric Brownianmotion and offered a new European option pricing modelbased on his stock pricemodel. Following that, Qin and Li [6]worked out a European call and put option pricing formulabased on the condition of fuzzy financial markets; Peng [7]also derived an American option pricing formula based onthe condition of fuzzy financial markets; Qin and Gao [8]presented a fractional Liu process in the application of optionpricing. The latest development about fuzzy theory can be
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 879185, 6 pageshttp://dx.doi.org/10.1155/2015/879185
2 Mathematical Problems in Engineering
seen also from Jiao and Yao [9], Ji and Zhou [10], Liu et al.[11], and so forth. Considering the fuzzy uncertainty, Wu andZhuang [12] proposed a reduced-form intensity-basedmodelunder fuzzy environments and presented some applicationsof the methodology for pricing defaultable bonds and creditdefault swap; the model results change into a closed interval.However, the pricing issues about credit derivatives pricingbased on fuzzy theory have not been studied. This paperfirst brings fuzzy process into the model of credit derivativespricing in expectation to match credit derivatives pricingmodel and real financial market better.
This paper will review some preliminary knowledge offuzzy process in Section 2. Some credit derivatives pricingmodels are derived in Section 3 and aOne-Factor Fuzzy Cop-ula function is proposed in Section 4, respectively. Finally, abrief summary is given in Section 5.
2. Preliminary
A fuzzy process is a sequence of fuzzy variables indexed bytime or space, which was defined by Liu [3]. In this section,we will recall some useful definitions and properties aboutfuzzy process.
Definition 1 (Liu [3]). Given an index set π and a credibilityspace (Ξ, π,Cr), then a fuzzy process is a function from π Γ
(Ξ, π,Cr) to the set of real numbers. In other words, a fuzzyprocess π(π‘, π) is a two-variable function. For convenience,we use the symbolπ
π‘instead of the longer notationπ(π‘, π).
Definition 2 (Liu [3]). A fuzzy process ππ‘is said to have
independent increments if
ππ‘1
β ππ‘0
, ππ‘2
β ππ‘1
, . . . , ππ‘π
β ππ‘πβ1
(1)
are independent fuzzy variables for any times π‘0< π‘1< β β β <
π‘π. A fuzzy process π
π‘is said to have stationary increments
if, for any given time π‘ > 0, the increments ππ +π‘
β ππ are
identically distributed fuzzy variables for all π > 0.
Definition 3 (Liu [3]). A fuzzy process πΆπ‘is said to be a Liu
process (or, namely, πΆ process) if
(i) πΆ0= 0,
(ii) πΆπ‘has stationary and independent increments,
(iii) every increment πΆπ +π‘
β πΆπ is a normally distributed
fuzzy variable with expected value ππ‘ and varianceπ2π‘2, whose membership function is
π (π₯) = 2 (1 + exp(π |π₯ β ππ‘|β6ππ‘
))
β1
, π₯ β π . (2)
The parameters π and π are called the drift and diffusioncoefficients, respectively. The Liu process is said to be stan-dard if π = 0 and π = 1.
Definition 4 (Liu [3]). Let πΆπ‘be a standard Liu process; then
the fuzzy process
πΊπ‘= exp (ππ‘ + ππΆ
π‘) (3)
is called a geometric Liu process, or an exponential Liuprocess sometimes. The geometric Liu process is expected tomodel asset values in a fuzzy environment; Li and Qin [13]have derived that πΊ
π‘is of a lognormal membership function:
π (π₯) = 2 (1 + exp(π |lnπ₯ β ππ‘|β6ππ‘
))
β1
, π₯ β₯ 0. (4)
Definition 5 (Liu [14]). The credibility distribution Ξ¦ : π β
[0, 1] of a fuzzy variable π is defined by Ξ¦(π₯) = Cr{π β Ξ |
π(π) β€ π₯}.That is, Ξ¦(π₯) is the credibility in which fuzzy variable π
takes a value less than or equal to π₯. If the fuzzy variableπ is given by a membership function π, then its credibilitydistribution is determined by
Ξ¦ (π₯) =1
2(supπ¦β€π₯
π (π¦) + 1 β supπ¦>π₯
π (π¦)) , βπ₯ β π . (5)
Definition 6 (Liu [2]). Let π be a fuzzy variable. If at least oneof the two integrals is finite in the following formula, then theexpected value of π can be calculated as
πΈ (π) = β«
β
0
Cr {π β₯ π} ππ β β«0
ββ
Cr {π β€ π} ππ. (6)
Let π be a fuzzy variable whose credibility density func-tion π exists. Liu [14] proved that πΈ(π) = β«
+β
ββπ₯π(π₯)ππ₯, pro-
vided that the Lebesgue integral is finite. If limπ₯βββ
Ξ¦(π₯) = 0
and limπ₯β+β
Ξ¦(π₯) = 1, then πΈ(π) = β«+β
ββπ₯πΞ¦(π₯), provided
that the Lebesgue-Stieltjes integral is finite.
Definition 7 (Liu [3]). Let ππ‘be a fuzzy process and let πΆ
π‘
be a standard Liu process. For any partition of closed interval[π, π] with π = π‘
1< π‘2< β β β < π‘
π= π, the interval length can
be expressed asΞ = max1β€πβ€π
|π‘π+1βπ‘π|.Then the fuzzy integral
ofππ‘with respect to πΆ
π‘is
β«
π
π
ππ‘ππΆπ‘= limΞβ0
π
β
π=1
ππ‘π
(πΆπ‘π+1
β πΆπ‘π
) , (7)
provided that the limit exists almost surely and is a fuzzyvariable.
Definition 8 (Liu [3]). Suppose πΆπ‘is a standard Liu process,
and π and π are two given functions. Then
πππ‘= π (π‘, π
π‘) ππ‘ + π (π‘, π
π‘) ππΆπ‘
(8)
is called a fuzzy differential equation. A solution is a fuzzyprocessπ
π‘that satisfies (8) identically in π‘.
Let ππ‘be the bond price and π
π‘the stock price. Assume
that stock price ππ‘follows a geometric Liu process (Liu [15]).
Then Liuβs stock model is written as below:
πππ‘= πππ‘ππ‘,
πππ‘= πππ‘ππ‘ + ππ
π‘ππΆπ‘,
(9)
Mathematical Problems in Engineering 3
where π is the riskless interest rate, π is stock drift, π is stockdiffusion, and πΆ
π‘is a standard Liu process. This is the fuzzy
stockmodel for fuzzy financialmarket and is used to simulatefuture bond stocks or financial derivatives.
3. Defaultable Bond and CDS Pricing
3.1. Defaultable Bond Pricing. The defaultable bond is acontract that pays full face value at maturity date, as long asthe bondβs issuer is not default. If a default occurs during thevalidity period of the contract, then the recovery is paid to thebondβs holder, and the contract is ended. Let the face valueof zero-coupon defaultable bond is 1 unit and the maturitydate is π. The interest rate and recovery rate are π and πΏ.Take advantage of Liuβs stock model; we have the followingresults.
In order to get this defaultable bond price of Liuβs stockmodel, we need to derive the bondβs default distributionfunction firstly. If π
π‘is the price of the underlying defaultable
bond, then it is easy to solve (9) in which ππ‘= π0exp(ππ‘ +
ππΆπ‘). Without loss of generality, we assume that the default
thresholdπΎ is a positive real numberπΎ β π +.Then, based on
Mertonβs structuralmodel [16], the bondβs default distributionfunction is derived as follows:
(i) If 0 < πΎ < π0, we have
Ξ¦π‘(πΎ) = π (π β€ π‘) = Cr {π
π‘β€ πΎ}
= Cr {π0exp (ππ‘ + ππΆ
π‘) β€ πΎ}
= Cr{exp (ππ‘ + ππΆπ‘) β€
πΎ
π0
}
=1
2( supπ¦β€πΎ/π
0
π (π¦) + 1 β supπ¦>πΎ/π
0
π (π¦))
= (1 + exp(π (ππ‘ β ln (πΎ/π
0))
β6ππ‘))
β1
,
(10)
where π represents the bondβs default time.
(ii) It is obvious that Ξ¦π‘(πΎ) = 1, when πΎ β₯ π
0(the initial
bond price is lower than the default threshold).
Theorem 9. Under an equivalent martingale measure π, thepresent fuzzy value of a defaultable bond π
π‘is given as
ππ‘= exp (βπ (π β π‘))
β (1 + (πΏ β 1) (1 + exp( ππ
β6π))
β1
) .
(11)
Table 1: Fuzzy defaultable bond.
π 0.5 1 3 5 10π0
0.8483 0.8273 0.7486 0.6773 0.5275
Proof. By the definition of expected value of fuzzy variable,we have
ππ‘= πΈπ[exp (βπ (π β π‘)) 1
{π>π}+ exp (βπ (π β π‘))
β πΏ1{πβ€π}
] = πΈπ[exp (βπ (π β π‘)) (1 β 1
{πβ€π})
+ exp (βπ (π β π‘)) πΏ1{πβ€π}
] = πΈπ[exp (βπ (π β π‘))
+ exp (βπ (π β π‘)) (πΏ β 1) 1{πβ€π}
] = exp (βπ (π β π‘))
+ β«
+β
ββ
exp (βπ (π β π‘)) π ((πΏ β 1)Ξ¦π(πΎ))
= exp (βπ (π β π‘)) + (πΏ β 1)
β β«
π0
0
exp (βπ (π β π‘)) πΞ¦π(πΎ)
= exp (βπ (π β π‘)) (1 + (πΏ β 1) (1
+ exp( ππ
β6π))
β1
) ,
(12)
where 1{πβ€π}
is an indicator function; that is, if the bond isdefault then the function value is 1; otherwise the functionvalue is 0.
Example 10. Suppose that the riskless interest rate π is 5% perannum, the drift π is 0.25, the diffusion π is 0.25, and therecovery is 0.4. Then, we can calculate the defaultable bondprice that expires in different years (see Table 1).
From Table 1, it can be known that these results are theportrayal of reality with the time increase; the more thefuzziness of the financial markets becomes intense, the morethe peopleβs ability to predict the future becomes weak andfuzzy, so the present value of defaultable bond becomes lowerwith time increasing.
3.2. CDS Pricing. A credit default swap (CDS) is a financialswap agreement that the seller of the CDS will compensatethe buyer in the event of a loan default. The buyer of the CDSmakes a series of payments (the CDS βfeeβ or βspreadβ) to theseller and, in exchange, receives a payoff if the loan defaults.Letπ be the notional amount, πΏ the recovery rate, π the creditspread, and π
1< π2< β β β < π
πthe sequence payment dates,
with ππ= π. The CDS running spread is computed such that
the fair price of the CDS equals zero at initiation. That is, thepresent value of the default leg equals the present value of thefixed leg: PV (default leg) = PV (fixed leg). In this case, wederive the following theorem.
4 Mathematical Problems in Engineering
Theorem 11. The present fuzzy credit spread π is given as
π =(1 β πΏ) exp (βπ (π β π‘))
Ξππexp (ππ/β6π)βπ
π=1exp (βπ (π
πβ π‘))
. (13)
Proof. By the definition of expected value of fuzzy variable,we have
ππ (default leg) = πΈπ[(1 β πΏ)π exp (βπ (π β π‘))
β 1{πβ€π‘}
] = (1 β πΏ)πβ«
π0
0
exp (βπ (π β π‘)) πΞ¦π‘(πΎ)
= (1 β πΏ)π exp (βπ (π β π‘)) (1 + exp( ππ
β6π))
β1
,
ππ (fixed leg) = πΈπ[π π
π
β
π=1
1{π>ππ}Ξππ
β exp (βπ (ππβ π‘))] = πΈ
π[π πΞπ
π
π
β
π=1
(1 β 1{πβ€ππ})
β exp (βπ (ππβ π‘))] = π πΞπ
π
π
β
π=1
[exp (βπ (ππβ π‘))
β β«
π0
0
exp (βπ (ππβ π‘)) πΞ¦
ππ(πΎ)]
= π πΞππ
π
β
π=1
[exp (βπ (ππβ π‘)) β exp (βπ (π
πβ π‘))
β (1 + exp( ππ
β6π))
β1
]
= π πΞππ
π
β
π=1
[exp (βπ (ππβ π‘))
β (
exp (ππ/β6π)
1 + exp (ππ/β6π))] .
(14)
Then, based on no arbitrage pricing principles, we can provethis theorem.
In the following, we will simulate Theorem 11 and givea numerical example and then employ the model results tocompare with the market data for analysis. The parametersinvolved in the simulation process are assumed as follows:suppose that the riskless interest rate π is 4.6% per annum,the drift π is 0.01, the diffusion π is 0.02, and the recovery rateπΏ is 0.5. Then, through the use of MATLAB R2010a, we cancalculate the present running spread that expires in differentyears and get the following contrast results (see Figure 1).
In Figure 1, the point line (data item 1) represents thenumerical results of the model derived fromTheorem 11 andthe solid line (data item 2) represents the actual market data(the data come from the report of the international clearingbank (BIS)βOTC derivatives statistics at end-December2014). Through the comparison and analysis we found that
Data item 1Data item 2
0.5 1.5 21100
200
300
400
500
600
700
800
Cred
it sp
read
s
T (years)
Figure 1: The contrast analysis of model results.
the model results and market data are all gradually decreasedwith time and they are all falling credit curves; this is inaccordance with the current market environment of financialcrisis. Meanwhile, in the fuzzy random environment, themodel results fell faster than the market data; this shows thatthe fuzziness of the market environment has increased thepessimism of investorsβ expectations in the future.
4. One-Factor Fuzzy Copula Function
The key point of single credit derivative pricing is theabovementioned default timemodeling with just one variableneeded. As for portfolio credit products, its earnings and thedefault correlation in the investment portfolio are of strongsensibility. Thus, to confirm the default correlation amongthe assets, which is the correlation of default risks, is thecore part of portfolio credit derivatives pricing. Moreover,One-Factor Gaussian Copula function in portfolio creditasset pricing model has particular advantages in establishingunited default probability distribution functionβthe modelis simple, convenient to calculate, and easy to be extended.Motivated by the definition of Liβs One-Factor GaussianCopula model [17], One-Factor Fuzzy Copula function canbe put forward as follows.
Definition 12. Suppose that the factors influencing companyasset value can be divided into two parts, system factors andidiosyncrasy factors; and both of the two kinds of factors arefuzzy processes and obey the standard Liu processπΆ
π‘, and the
asset value of company ππ(latent variable) can be expressed
as the following One-Factor Fuzzy Copula model:
ππ= βπππ + β1 β π
πππ, π = 1, 2, . . . , π. (15)
In the above model, system factor π represents macroeco-nomic situation, and idiosyncrasy factor π
πrepresents the
factor that is only effective for company asset π, and theyare mutually independent. The marginal distribution of the
Mathematical Problems in Engineering 5
company asset value ππis independent of the condition that
factor π is known. βππ is the correlation coefficient betweenthe company asset and system factors. And on the conditionthat system factorsπ and idiosyncrasy factorsπ
πβs all obey the
standard Liu process πΆπ‘, the company asset π
πwill also obey
the standard Liu process.
According to the structural model by Merton [16], whenthe company assetπ
πis under a certain default threshold value
πΎπβ π +, a company default event will take place. Through
Copula function, the condition default probability of singlecompany asset can be expressed as follows:
Ξ¦π‘(πΎπ) = ππ|π
π‘= π (π
πβ€ πΎπ| π)
= π (βπππ + β1 β ππππβ€ πΎπ| π)
= π(ππβ€πΎπβ βπππ
β1 β ππ
| π)
= (1 + exp( π
β6π‘
(πΎπβ βπππ)
β1 β ππ
))
β1
.
(16)
Theorem 13. On the assumed condition of One-Factor FuzzyCopulamodel, the united default probability for every companyis
π = π (π1β€ πΎ1, π2β€ πΎ2, . . . , π
πβ€ πΎπ)
= β«
+β
ββ
π
β
π=1
ππ|π
π‘πΞ¦ (π¦) ,
(17)
where ππ|ππ‘is the expression of formula (16).
Proof. On the condition that the system factorπ is known, wecan put forward, independently of the expectation-orientedquality and company default,
π = π (π1β€ πΎ1, π2β€ πΎ2, . . . , π
πβ€ πΎπ)
= πΈ (1{π1β€πΎ1,π2β€πΎ2,...,ππβ€πΎπ})
= πΈ (πΈ (1{π1β€πΎ1,π2β€πΎ2,...,ππβ€πΎπ}) | π = π¦)
= πΈ (πΈ (1{π1β€πΎ1}| π = π¦) β β β πΈ (1
{ππβ€πΎπ}| π = π¦))
= πΈ(
π
β
π=1
ππ|π
π‘) = β«
+β
ββ
π
β
π=1
ππ|π
π‘πΞ¦ (π¦) .
(18)
According to the conclusion of Theorem 13, portfoliocredit products can be priced, for example, ITRAXX, CDXseries, and collateralized debt obligations (CDO). As for theidea that the detailed pricing process is similar to CDSpricing, it will not be discussed in this paper.
5. Conclusion
In this paper, we have investigated the credit derivativespricing problems for fuzzy financial market. Defaultablebond and CDS price formulas have been calculated for Liuβsstock model, and a One-factor Fuzzy Copula function hasbeen proposed. Some numerical examples of these formulashave been studied. Through the comparison and analysiswe can find that the model results and market data are allgradually decreased with time, and they are all falling creditcurves; this is in accordance with the current market envi-ronment of financial crisis. Meanwhile, in the fuzzy randomenvironment, the model results fell faster than the marketdata; this shows that the fuzziness of the market environmenthas increased the pessimism of investorsβ expectations inthe future. Therefore, in the management of credit risk, weshould taking into account both the market fuzziness andrandomness.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (no. 71171051) and the FundamentalResearch Funds for the Central Universities and the Fundingof Jiangsu Innovation Program for Graduate Education (no.KYLX 0213).
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