measuringinterestrateriskwithembeddedoptionusing hpl...

Research ArticleMeasuring Interest Rate Risk with Embedded Option UsingHPL-MC Method in Fuzzy and Stochastic Environment

Enlin Tang 1 and Wei Du 2

1School of Finance and Mathematics Huainan Normal University Huainan 232038 China2School of Mathematics and Statistics Hefei Normal University Hefei 230601 China

Correspondence should be addressed to Wei Du duweihfnueducn

Received 22 August 2020 Revised 21 September 2020 Accepted 25 September 2020 Published 15 October 2020

Academic Editor Ahmed Mostafa Khalil

Copyright copy 2020 Enlin Tang and Wei Du )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Under the condition of continuous innovation of financial derivatives and marketization of interest rate interest rates fluctuatemore frequently and fiercely and the measurement of interest rate risk also attracts more attention Under the premise that thefluctuation of interest rate follows fuzzy stochastic process based on the option characteristics of financial instruments withembedded option this paper takes effective duration and effective convexity as tools to measure interest rate risk when embeddedoptions exist tries to choose CIR extended model as term structure model and uses the Monte Carlo method for hybrid lowdeviation sequences (HPL-MC) to analyze the prepayment characteristics of MBS a representative financial instrument withembedded options when interest rates fluctuate on this basis the effectiveness of effective duration management of interest raterisk is demonstrated with asset liability management cases of commercial banks

1 Introduction

With the further acceleration of the process of global eco-nomic integration the global financial liberalization andfacilitation reform with the financial innovation of devel-oped countries and the financial deepening of developingcountries as the main content is becoming more and moreintense a countryrsquos financial market can no longer developin isolation and it is bound to be integrated into the entireinternational financial market On December 13 2001China officially became a member of the World TradeOrganization (WTO) According to the Financial ServicesAuthority (FSA) of the world trade organization eachmember country must make a commitment to open up thefinancial market and gradually bring 95 of the global fi-nancial service trade into the process of liberalization Asone of the member countries Chinarsquos financial system willalso be integrated into the world economic globalizationChinarsquos monetary policy will be more and more affected bythe international monetary policy and the convergence ofChinarsquos interest rate changes and international market

interest rate changes will become stronger and stronger [1]the interest rate risk caused by this will become one of themost serious risks faced by commercial banks At the sametime the frequent fluctuation of interest rates also gives birthto various derivative financial instruments among themfinancial instruments with the characteristics of embeddedoptions such as deposits that can be withdrawn in advanceloans that can be paid in advance mortgage-backed secu-rities collateralized mortgage obligation and callable bondsare more and more embedded in the balance sheets ofcommercial banks which greatly increases the difficulty ofinterest rate risk management of commercial banks )e gapmodel based on traditional duration and traditional con-vexity is less effective in interest rate risk managementtherefore it is of great significance to measure and managethe interest rate risk of embedded option financial instru-ments [2]

Scholars at home and abroad have made some im-portant achievements in the study of interest rate risk ofembedded options Internationally Brent and Richardproposed the Monte Carlo (MC) simulation pricing

HindawiJournal of MathematicsVolume 2020 Article ID 7410909 13 pageshttpsdoiorg10115520207410909

method for the embedded options belonging to the Asianoption type after estimating the parameters in the modelbased on this method the price of embedded optionbonds was obtained [3] Brooks made a sensitivity analysison the value of embedded options and interest rate level inCDs based on cross-sectional data [4] Ben-Ameurpointed out that the inconsistency of value changes causedby the mismatch of asset and liability convexity is also thecause of embedded option risk and studied the charac-teristics of embedded options in commercial bank de-posits and the characteristics of depositorsrsquo exercise time[5] Domestic scholars have also made importantachievements in the research of embedded optionsZhenlong and Hai decomposed the option characteristicsof assets and liabilities in the balance sheet of commercialbanks and used the basic principles of financial engi-neering to price implicit options without arbitrage andalso made numerical pricing [6] Xia et al built thebenchmark interest rate jump model of deposits andconducted the numerical pricing of embedded optiondeposit and loan based on the Monte Carlo simulationmethod and studied the impact of embedded options oninterest rate risk of Chinarsquos commercial banks by usingcomparative analysis method [7] Li et al proposed thatwith the development of interest rate marketization andthe continuous innovation of financial instruments ofChinarsquos commercial banks embedded options are moreand more appearing in the balance sheet of commercialbanks and at the same time it brings huge interest raterisk to the operation of commercial banks )ereforecommercial banks urgently need to establish a completeinterest rate risk management system in time Howeverthe traditional risk management methods such as interestrate sensitivity gap and duration gap have been unable tomeet the interest rate risk management requirements ofassets and liabilities with embedded options )e review ofthe research results at home and abroad shows that theresearch on embedded options started earlier in foreigncountries although the domestic research started late ithas also made some achievements Domestic researchstudies focus on pricing the value of embedded optionsbut few focus on the measurement and management ofinterest rate risk of embedded options In addition thedifferential equation method and nonarbitrage pricingmethod adopted by predecessors are based on a series ofstrict preconditions )e traditional Monte Carlo simu-lation method is usually used to calculate the price ofembedded option and the effective duration of securitieswith embedded option however the random numbersequence generated by traditional MC in the simulationspace is unevenly distributed which wastes a largenumber of observations and reduces the operation effi-ciency therefore this paper uses a more advanced sim-ulation method namely HPL-MC based on theoptimized interest rate risk management tools namelyeffective duration and effective convexity makes anempirical analysis on the change of characteristics of fi-nancial instruments caused by the existence of embeddedoptions and demonstrates how commercial banks can

effectively measure and manage interest rate risk whenimplicit options exist

2 Analysis on the Option Characteristics ofFinancial Instruments withEmbedded Options

With the increasing types of derivative financial instru-ments financial instruments with embedded options aremore and more embedded in the assets and liabilities ofcommercial banks such as deposits that can be drawn inadvance prepayable loans collateralized mortgage obliga-tion callable bonds mortgage-backed securities and so onHere we will take the prepayment of housing mortgageloans as an example to illustrate

21 Influencing Factors of Prepayment of Housing MortgageLoans From the perspective of borrowers the followingfactors will affect their prepayment behaviors

211 Interest Rate Factor )e interest rate factor is theprimary factor to decide the prepayment behaviors )einterest rate factor here refers to the balance between themarket interest rate and the contract interest rate thecontract interest rate determines the interest expense of theloans and the market interest rate determines the cost ofrefinancing Assuming that the borrower prepays at time tthe refinancing cost is expressed by V(t) the penalty is αand the loan balance at time t is L(t) when W(t) V(t) minus

[L(t) + α] is greater than zero prepayment will be executed

212 Economic Situation With the improvement of peo-plersquos economic situation and the improvement of livingrequirements the prepayment rate will increase and at thesame time it will also cause the increase of house purchaserate and turnover rate In addition when the nationalmacroeconomic operation is good the whole real estatemarket is hot and when the rise of house price exceeds thebalance of mortgage loan prepayment will appear in largenumber

213 Institutional Factors )e minimum amount of pre-payment and the limit of the times of prepayment will re-strain the behavior of prepayment to a certain extentHowever the penalty for prepayment of housing mortgageloan is not high in our country that is to say the cost ofprepayment is small so prepayment often occurs

214 Other Factors Factors such as relocation mortgagecharacteristics and seasons

22-e Influence of Prepayment ofHousingMortgage LoanonCommercial Banks

221 Positive Impact Firstly the prepayment of housingmortgage loan reduces the possibility of nonperforming loan

2 Journal of Mathematics

and reduces the credit risk Secondly the prepayment ofhousing mortgage loan increases the liquidity of commercialbank assets and improves the response ability of commercialbanks to long-term and short-term capital demand and theefficiency of capital utilization

222 Negative Effect Firstly prepayment reduces the in-terest margin of deposit and loan greatly which is the mainpart of profit for commercial banks so prepayment willreduce the profit of commercial banks Secondly the risk ofreinvestment of commercial banks increases when themarket interest rate decreases commercial banks can onlylook for investment projects with low return rate and thereinvestment income is greatly reduced )irdly prepay-ment destroys the original capital structure and the assetliability structure based on duration and convexity is nolonger matched resulting in risk exposure Fourthly whenthe market interest rate drops the prepayment of mortgageloan makes the market value of bank assets have an upperlimit As shown in Figure 1 two curves represent the value-interest curve of mortgage the solid line indicates non-prepayment and the dotted line indicates prepaymentWhen the contract stipulates that the mortgage loan cannotbe paid in advance the decrease of interest rate will cause theasset value of the bank to rise however when the interestrate falls to Rlowast (Rlowast refers to the interest rate level that causesthe borrower to prepay that is to exercise the embeddedoption) the value of assets will encounter a bottleneck whenthe mortgage loan can be repaid in advance and it willdecrease gradually after reaching Pprime which will lead to thedecrease of net assets of commercial banks

23 Analysis on the Option Characteristics of HousingMortgage Loan For the option characteristics of prepay-ment of mortgage loans the following is an example of Case1

Case 1 A borrower and a commercial bank sign a loancontract with a term of T at zero time in which the loanamount is unit 1 the penalty is α and the lending rate isexpressed by continuous compound interest rate r0T Whenprepayment is not considered the total borrowing cost iser0TT assuming that the benchmark interest rate changes tor1T at time t prepayment occurs and the refinancing cost is(er0Tt + α)er1T(Tminus t) if er0TT ge (er0Tt + α)er1T(Tminus t) the rationalborrower will prepay otherwise the prepayment will not becarried out )is method can be used for similar analysiswhenever interest rate adjustment occurs

If expressed by mathematical equation the borrowingcost of the borrower at time t can be expressed as follows

minus min er0Tt

+ α1113872 1113873er1T(Tminus t)

1113872 1113873 er0TT

1113960 1113961

minus er0TT

minus min er0Tt

+ α1113872 1113873er1T(Tminus t)

minus er0TT

01113960 1113961

minus er0TT

+ max er0TT

minus er0Tt

+ α1113872 1113873er1T(Tminus t)

01113960 1113961

(1)

where minus er0TT represents a loan contract with default risk andmax[er0TT minus (er0Tt + α)er1T(Tminus t) 0]prime represents the interest

rate call option with the strike price of K er0TT implied inthe loan contract

)is kind of housingmortgage loan can be understood asthat the borrower issues bonds that can be paid in advance tothe bank )is bond contains options and prepayment canoccur at any time before the end of the loan contracttherefore such a loan contract can be broken down into twoparts (as shown in Figure 2)

A the borrower issues callable bonds to the bankB the bank issues an American callable option to theborrower

However banks often ignore B and do not charge optionfees to borrowers

Specifically with the decline of interest rate when thecost of refinancing is less than the interest expense ofmaintaining the original loan prepayment means ldquore-demptionrdquo behavior will occur such ldquoredemptionrdquo right isinterest rate put option

3 InterestRateRiskMeasurementofEmbeddedOption Financial Instruments EffectiveDuration and Effective Convexity

31 Duration-Convexity Gap Model By matching the du-ration and convexity of assets and liabilities commercialbanks can realize the immunity of net assets to interest ratechanges and avoid interest rate risk exposure [8] Let A

denote assets L denote liabilities and E denote net assetsthen dE dA minus dL Next we take the second derivative of A

and L to get the following equation

dA dA

dRA

dRA +12d2AdR

2A

dRA( 11138572 (2)

dL dL

dRL

dRL +12d2LdR

2L

dRL( 11138572 (3)

Let the initial interest rates of A and L be equal that isRA RL R and dRA dRL dR We can getDA minus dAAdR DL minus dLLdR CA minus d2AAdR2 andCL minus d2LLdR2 then the expression of dE is as follows

dE minus DA minus KDL( 1113857AdR +12

CA minus KCL( 1113857A(dR)2 (4)

where DA and DL are the duration of assets and duration ofliabilities CA and CL are the convexity of assets and con-vexity of liabilities respectively and K LA is the leveragecoefficient that is the asset liability ratio DGAP DA minus KDL

is duration gap which is used to measure the matchingdegree of asset duration and liability duration CGAP CA minus

KCL is convexity gap which is used tomeasure thematchingdegree of asset convexity and liability convexity

It can be seen from equation (4) that ADGAP DA minus KDL CGAP CA minus KCL and R are the mostimportant factors affecting the net asset value of commercialbanks in addition it can be learned that when the fluctu-ation range of interest rate is small the influence of

Journal of Mathematics 3

convexity on net assets can be ignored )e specific con-clusions are shown in Table 1

When the interest rate changes greatly the influence ofconvexity on net assets will be highlighted the specificconclusions are shown in Table 2

32-e Influence ofEmbeddedOptiononDuration-ConvexityGap Model Net assets of commercial banks are equal toassets minus liabilities assuming that the initial net assets ofcommercial banks are greater than 0 the duration andconvexity of assets and liabilities are matched )e impact ofthe existence of embedded options on the net assets ofcommercial banks can be illustrated by the curve of rela-tionship between market value of assets (liabilities) andinterest rate as shown in Figure 3

At the beginning when banks manage interest rate riskaccording to the duration-convexity gap model the assetliability structure is matched when the interest rate changesbetween Rlowast and Rlowastlowast the embedded option is not executedthe bankrsquos net assets are immune to interest rate changesand the net assets of the bank remain unchanged at ElowastWhen interest rates fall below Rlowastlowast the curve of relationshipbetween market value of liabilities and interest rate does notchange however the duration of bank assets decreases dueto the borrowerrsquos prepayment and the absolute value of theslope of the curve becomes smaller that is the convexity ofassets becomes negative which makes the rise of asset valueproduce bottleneck when the interest rate drops while theconvexity of liabilities will not be affected therefore the netasset decreases from Elowast to E1 resulting in interest rate riskexposure

When the interest rate rises higher than Rlowast the curveof relationship between market value of assets and interestrate remains unchanged when the interest rate rises thedepositor withdraws the deposit in advance which leadsto the decrease of the absolute value of duration andconvexity of the liability and the curve becomes gentle)e decrease speed of asset value is faster than that ofliability value which leads to the decrease of net assetvalue from Elowast to E2 thus resulting in interest rate riskexposure

Combining Figure 3 with equation (4) when the interestrate drops below Rlowastlowast the embedded option is executed andthe duration gap DGAP DA minus KDL becomes negativewhich means that the bankrsquos net asset value decreases whenthe interest rate drops in addition when the embeddedoption is executed the convexity gap CGAP CA minus KCL

becomes negative and the net asset value of the bank de-creases when the interest rate drops

When the interest rate is higher than Rlowast the embeddedoption is executed and the duration gap DGAP DA minus KDL

becomes positive and the bankrsquos net asset value decreaseswhen the interest rate rises meanwhile the convexity gapCGAP CA minus KCL also becomes positive and the bankrsquos netasset value decreases when the interest rate rises

From the above analysis we can see that whether themarket interest rate rises or falls the existence of embeddedoptions will lead to the decrease of net asset value of

commercial banks and such embedded options are valuableto customers Commercial banks often ignore the existenceof embedded options and give them to customers free ofcharge thus reducing their own profits In addition em-bedded options will also cause duration mismatch andconvexity mismatch between assets and liabilities and thetraditional duration and traditional convexity are not ac-curate in measuring interest rate risk so we need to in-troduce more effective measurement tools namely effectiveduration and effective convexity

33 Calculation of Effective Duration and Effective ConvexityBased on Embedded Options When assets and liabilitiescontain embedded options the traditional duration andtraditional convexity will lose accuracy in measuring interestrate risk it needs to be replaced by effective duration (Deff )and effective convexity (Ceff ) and their expressions are asfollows

Deff P+ minus Pminus

2ΔrP0 (5)

Ceff P+ minus 2P0 + Pminus

Δr2P0 (6)

where Δr is the interest rate spread expressed by base pointP0 is the initial bond value and Pminus and P+ are the bondprices when the interest rate moves downward and upwardby a certain basis point respectively

Pminus 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i minus Δr + OAS1113872 1113873 (7)

P+ 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rji + Δr + OAS1113872 1113873

(8)

where m is the total number of interest rate change paths t

is the time when the cash flow is generated j is the j-thpath of interest rate change n is the total contract termCFj

t is the t-th cash flow under the j-th interest rate pathr

ji is the i-th market interest rate under the j-th interestrate path and OAS is the option-adjustment spread [9]which is the basis for the calculation of effective durationand effective convexity Specifically OAS is the bond riskcompensation when the bond market value is equal to thetheoretical value which mainly refers to the credit riskliquidity risk and other risk compensation after excludingthe embedded option risk it can be calculated by thefollowing equation

P 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i + OAS1113872 1113873 (9)

where P represents the bond market price 1m is the meanvalue under the risk neutral probability measure and CFj

t

and rji have the same meanings as above


)e calculation flow of OAS in equation (9) is shown inFigure 4

)e calculation of OAS depends on the simulation ofinterest rate path [10] when simulating the interest ratepath the selected interest rate model should not only meetthe initial interest rate term structure but also conform to theassumed stochastic process [11] Based on the calculatedcash flow and discount factor Pminus and P+ are obtained andthen the effective duration and effective convexity arecalculated

34 Selection of HPL-MC in the Calculation of Effective Du-ration and Effective Convexity )ere are many differentmethods to select and simulate the sample interest rate pathmentioned above such as the trigeminal tree method exceltable method and Monte Carlo simulation method In thispaper the Monte Carlo simulation method is used tosimulate the sample interest rate path and calculate the valueof OAS

Using MC to calculate the price effective duration andeffective convexity of securities with embedded option isactually to calculate the multidimensional numerical inte-gration on the path space of a sample interest rate )erandom number sequence generated by traditional MC isunevenly distributed in the simulation space which leads tothe clustering effect of random numbers wastes a large

number of observations and reduces the computationalefficiency therefore some scholars have proposed the quasi-MC (QMC) method which uses the predetermined deter-ministic method to generate some low deviation points inspace to replace the independent random points in MonteCarlo simulation Due to the uniform distribution of thesesimulation points in space it can produce better conver-gence effect with less simulation times )e research showsthat QMC simulation can produce obvious improvementeffect only when the maintenance simulation is low andsometimes the simulation effect is not as good as MCsimulation when the simulation dimension is high [12]therefore a hybrid method for generating interest rate pathof QMC and MC HPL-MC is proposed

)e method is based on the principal componentmethod It is assumed that the simulation interval [0 T] isdivided into dimension d ie 0 t0 lt t1 lt middot middot middot lt td T )esimulated interest rate path is a Wiener vectorWt (wt

1 wt2 wt

d)T where A is the transformationmatrix and Z (z1 z2 zd)T Suppose that the covari-ance matrix of the standard normal vector Zi is Σ thenΣ AAT Principal component analysis is used to constructa matrix B so that BBT AAT Σ and each standardnormal variable produced by wt BZ can reveal the in-formation contained in all the original d standard normalvariables to the maximum extent According to the idea ofprincipal component analysis matrix B should be in thefollowing formB VΛ among them the matrix Λ is adiagonal matrix composed of the square roots of all ei-genvalues of Σ the order of the square roots of eigenvalues isfrom large to small along the diagonal all column vectors ofmatrix Λ are the unionized eigenvectors of matrix Σ and thei-th column vector corresponds to the i-th eigenvalue on thediagonal of matrix V

After getting the matrix HPL-MC generates the first k-term quasirandom numbers zlowasti (i 1 2 k) with QMCand then generates the remaining d minus k dimensionalpseudorandom numbers zi(i k + 1 k + 2 d) by MCsimulation In this way the resulting hybrid analog sequenceis

Wi 1113944k

i1bizlowasti + 1113944

d

ik+1bizi (10)

where bi is the column vector of matrix B)e hybrid sequence not only ensures that the principal

component terms which play a leading role in the fluctuationproperties of the sequence can be evenly distributed in spacebut also avoids the disadvantage that QMC has no obviouseffect in simulating high-dimensional sequences so as toimprove the simulation accuracy and speed up the con-vergence speed

4 An Empirical Analysis of Interest Rate RiskMeasurement of Financial Instruments withEmbedded Option

)is part takes the financial instruments with embeddedoption in American capital market as an example which is

rRlowast

Plowast

P

No prepayment

Prepayment

Figure 1 )e curve of relationship between market value ofmortgage and interest rate

Callable bonds issued to borrowers

Regular bonds-right of redemption

Com

mer

cial

ban

k

Borr

owerA issue of American callable options to borrowers

B issuance of bonds to banks

Figure 2 Decomposition of housing mortgage business


mortgage-backed security (MBS) MBS is a kind of fixedincome securities which has obvious option characteristicsits future cash flow is affected by many factors such asmarket interest rate loan type macroeconomic situationand season which will bring interest rate risk of embeddedoption to commercial banks and other financial institutionsBefore the empirical analysis of MBS we need to determineprepayment model of MBS cash flow model of MBS andterm structure models of interest rates

41-e PrepaymentModel ofMBS OTSrsquos prepayment modelof MBS [13] Richard and Rollrsquos prepayment model and im-proved Goldman Sachs model are used for reference and it isconcluded that the prepayment rate is related to four mainfactors namely refinancing incentives seasoning or age of themortgage factor seasonality or the month factor and burnoutfactor If CPR (conditional annual prepayment rate) is used torepresent the annualmortgage prepayment rate then CPR is theproduct of the above four factors the expression is as follows

CPR (refinancing incentives)lowast (seasoning factor)

lowast (seasonality factor)lowast (burnout factor)(11)

)e expression of each factor (equation (11)) is asfollows

411 Refinancing Incentives It is measured by dividing theannual weighted average coupon rate of mortgage-backed bythe refinancing rate

RI(t) 031234 minus 02025

times arctan 8157 times minusWAC + SE

P(t) + F+ 1207611113888 11138891113890 1113891

(12)

where WAC represents the average coupon rate of MBS t

represents the number of months and SE represents theservice fee rate provided by the service organization usuallya fixed ratio is drawn based on the balance of the loan poolthis ratio is generally 50ndash100 basis points here 50 basispoints are taken Let Prin(t) represent the book balance ofthe outstanding principal of the mortgage-backed bond loanagreement at the beginning of month t and S(t) representthe service fee paid in month t then

S(t) 0005 times Prin(t) times112

(13)

P(t) represents the refinancing rate in which the refi-nancing rate is represented by the ten-year zero coupon rate

P(t) minusln(p(t12 tt12 + 10))

10 (14)

P(t T) represents the bond price with the maturity dateT at time t then

p(t T) A(t T)eminus B(tT)r

(15)

B(t T) 1 minus e

minus a(Tminus t)

a (16)

lnA(t T) lnp(0 T)

p(0 t)minus B(t T)

z ln p(0 t)

ztminus

14a

3σ2

middot eminus aT

minus eminus at

1113872 11138732

e2at

minus 11113872 1113873

(17)

Table 1 Interest rate risk faced by duration gap

Duration gap )e direction of interest rate change Changes in net assets

Positive number Rise DeclineDecline Rise

Negative Rise DeclineDecline Rise

Table 2 Interest rate risk faced by convexity gap

Convexity gap )e direction of interest rate change Changes in net assets

Positive number Rise RiseDecline Rise

Negative Rise DeclineDecline Decline

P

R1 RlowastRlowastlowast R2

E2

Elowast

E1

r

Assets without embedded options

Assets with embedded options

Liabilities without embedded options

Liabilities with embedded options

Figure 3 )e curve of relationship between market value of assets(liabilities) and interest rate


In equation (12) F represents the refinancing expensesrelated to refinancing mortgage loans

412 Seasoning or Age of the Mortgage Factor )e defi-nition of this factor by the Office of)rift Supervision (OTS)is as follows

seasoning(t) min(t times 00333 1) (18)

413 Seasonality Factor )e definition of this factor by theOffice of )rift Supervision (OTS) is as follows

seasonality 1 + 02 times sin 1571 timesmonth + t minus 3

31113888 1113889 minus 11113896 1113897

(19)

where month is the number of months (from 1 to 12)

414 Burnout Factor At present the functional expressionof this factor is not uniform and even some prepayment

models do not involve this factor Here the burnout factor isexpressed as follows

burnout(t) eminus 0115(WACP(t))

(20)

)e meaning of each parameter in equation (20) is thesame as before and this factor is only calculated when theloan age factor reaches 1 then the monthly conditionalprepayment rate (MCPR) is expressed as follows

MCPR(t) 1 minus (1 minus CPR(t))112

(21)

42 Cash FlowModel of MBS According to the OTS modelthe sum of the discounted value of the cash flow during thevalidity period of MBS is calculated as follows [14]

PV 1113944WAM

i1dis(t) middot (TPP(t) + IP(t) + S(t)) (22)

WAM in equation (22) means weighted averagematurity

Term structure model of

interest rate

Initialyield curve

Simulated sample

interest rate

Option characteristics of securities

Cash flow on each path

Sum of discounted

value of cash

Find the value of

OAS

Market price of

securities

Comparison between

market price and

theoretical price of

securities

UnequalEqual

Figure 4 Calculation flow chart of option-adjusted spread


dis(t) represents the discount factor of MBS in month tand its expression is as follows

dis(t) 1

1113945t

i1(1 + r(i))112

(23)

IP(t) in equation (22) represents the interest paid ofMBS in month t

IP(t) Prin(t) middot(WAC minus SE)

12 (24)

In equation (24) SE is the service rate which is 50 bpsIn equation (22) TPP(t) represents the total principal

paid of MBS in month t

TPP(t) SPP(t) + PA(t) (25)

In equation (25) SPP(t) represents the scheduledprincipal paid of MBS in month t

SPP(t) Pmt(t) minus IP(t) minus S(t) (26)

In equation (26) Pmt(t) represents the monthly pay-ment of MBS in month t when prepayment is considered

Pmt(t) Prin(t) middotWAC12

1 minus 1(1 + WAC12)WAMminus t+1

1113872 1113873 (27)

When prepayment is not considered the followingequation is used


1 minus 1(1 + WAC12)WAM

1113872 1113873 (28)

In equation (25) PA(T) represents the prepaymentamount of MBS in the month t

PA(t) MCPR(t) middot Prin(t) minus SPP(t) (29)

In equation (29) MCPR(t) and Prin(t) have the samemeanings as the variables in equations (21) and (13) Inaddition the book balance of the outstanding loan principalof MBS at the beginning of month t + 1 is

Prin(t + 1) Prin(t) minus SPP(t) (30)

In equation (30) SPP(t) has the same meaning as inequation (25)

43 Selection of Interest Rate Term Structure

431 Introduction of Fuzzy Stochastic Process It should benoted that many scholars use stochastic models to describethe uncertainty of financial markets andmodernmethods todeal with these stochastic models mainly include theprobability theory method (mainly considering from ran-dom process) and fuzzy set method In this paper wecombine randomness and fuzziness and consider that in-terest rate and volatility are fuzzy random numbers of n-thparabola distribution Next the theory of the coexistence of

random and fuzzy namely the mixed process theory isproposed here [15ndash17]

Definition 1 (Θ P Cr) is credibility space (Ω A Pr) isprobability space and opportunity space is defined as(Θ P Cr) times (Ω A Pr) )en the mixed variable is a mea-surable function from the chance space(Θ P Cr) times (Ω A Pr) to the set of real numbers If T is anindex set then a measurable function from T times (Θ P Cr) times

(Ω A Pr) to the set of real numbers is called a mixedprocess

Definition 2 If the mixing process Xt has independentincrements if for any time t0 lt t1 lt middot middot middot lt tk Xt1

minus Xt0 Xt2

minus

Xt1 Xtk

minus Xtkminus 1are independent mixed variables If the

fuzzy process Xt has a stable increment if tgt 0 at any timeand for any sgt 0 Xs+t minus Xs is a mixed variable with the samedistribution

Definition 3 Wt is a Wiener process Ct is a Liu process (itwas put forward by Liu Baoding) [18 19] and Dt (Wt Ct)

is called a D process When Wt and Ct are both standardprocesses the mixing process Xt et + βWt + βlowastCt is astandard D process where e Wt and Ct are independent ofeach other )e parameter e is called the drift coefficient β iscalled the random diffusion coefficient and βlowast is called thefuzzy diffusion coefficient

432 -e Choice of Interest Rate Model In choosing theinterest rate model we compare and analyze the advantagesof the HullndashWhite model (generalized Vasicek model) andHullndashWhite extended CIR model and finally chooseHullndashWhite extended CIR model the next part is thespecific implementation process

Considering that the general initial income curve isslightly upward sloping curve the initial term structureselected in the simulation is the most typical initial termstructure

f(t) 006 + 05 timesln(t)

100 (31)

In equation (31) f(t) represents the initial termstructure function

Many mathematicians have chosen the HullndashWhitemodel (generalized Vasicek model) as the term structuremodel of interest rate its expression is as follows

dr(t) [θ(t) minus ar(t)]dt + σdWt (32)

In equation (32) a is a constant which represents themean regression rate of interest rate or the expected value ofdrift rate of interest rate σ is also a constant representing thestandard deviation of interest rate and Wt is a Brownianmotion which follows the Markov process the variation ofWt on a small time interval Δt is represented by ΔWt thenΔWt and Δt satisfy the following relationship

ΔWt εΔt

radic (33)


In equation (33) ε is a random value taken from thestandard normal distribution For any two different timeintervals of Δt the values of ΔWt are independent of eachother

In equation (32) θ(t) is the time function to ensure thatthe model conforms to the initial term structure its ex-pression is as follows

θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)

)e HullndashWhite model (generalized Vasicek model)has a significant drawback that is the interest ratesimulated by the model may be negative )erefore inorder to make up for the defect that the interest rate maybe negative and make the model conform to the initialyield curve accurately the HullndashWhite extended CIRmodel is selected as follows

dr(t) [θ(t) minus af(t)]d(t) + σr(t)

1113968dWt (35)

In equation (35) θ(t) is the time function to ensure thatthe model conforms to the initial term structure the interestrate volatility in this model is different from that in theprevious model therefore the following expression isobtained

dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)

1113968in equation (35) represents the volatility of in-

terest rates that is the standard deviation of interest rates[20ndash22]

Although the HullndashWhite extended CIR model is morecomplicated than the HullndashWhite model (generalizedVasicek model) the HullndashWhite extended CIR model cansolve the problem of negative r

In order to more clearly illustrate the advanced natureof the HullndashWhite extended CIR model we compare thesimulation results shown in Figure 5 where the asteriskline is the interest rate value simulated under the gen-eralized Vasicek model and the grid line is the interest ratevalue simulated under the HullndashWhite extended CIRmodel

It can be clearly seen from Figure 5 that when theHullndashWhite extended CIR model is adopted the possiblenegative interest rate can be effectively avoided

When simulating the path of interest rate r equation(33) needs to be discretized as follows

r(t + 1) minus r(t) [θ(t) minus ar(t)] + σr(t)

1113968ΔWt (37)

)e selected initial term structure is brought intoequation (34) to get the following equation

θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)

Incorporating equation (38) into equation (37) gives thefollowing equation

r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)

2aminus ar(t)1113891Δt + σ

r(t)

1113968ε

Δt

radic

(39)

Among themΔti ti+1 minus ti is fuzzy random variablewhich can be realized by the value of fuzzy random numberfunction

Let us estimate the value of σ and write σr12 in the modelas follows

σr σr12

(40)

According to historical data the average volatility σr ofinterest rates in the US market is 001 and the average yieldof short-term Treasury bills is 006 so the value of σ inequation (39) can be determined based on historical data

Transform equation (40) as follows

ln σr( 1113857 ln σ + 12 ln r (41)

It can be seen that there is a linear relationship betweenln σr and ln σ and the estimated value of σ can be deter-mined by the method of linear regression through thecalculation of historical data the estimated value of σ is0041

In this way by combining the prepayment model andcash flow model mentioned above effective duration andeffective convexity can be calculated through simulation

44 Empirical Simulation and Result Analysis )is articletakes mortgage-backed bonds in the US securities market asan example which is issued on the basis of a loan pool calledFNCL702342 the weighted average coupon rate of this loanpool is 005888 the weighted average maturity is 360months and the face price is $100

441 Graphic Analysis of Simulation Results )rough thesimulation calculation of FNCL702342 and the data ob-tained by MATLAB the final results are as follows

When prepayment is not considered the cash flow chartof MBS is shown in Figure 6

It can be seen that when the prepayment rate is 0 that isthere is no embedded option (as shown in Figure 6) themonthly repayment of principal and interest is a constantand the monthly interest repayment amount and themonthly service fee payable decrease month by monththerefore the monthly principal repayment amount in-creases month by month

When the prepayment factor is considered the cash flowchart of MBS is shown in Figure 7

Due to the existence of prepayment of principal themonthly cash flow will change In this case the monthlyplanned repayment of principal and interest is no longer aconstant but slowly decreases month by month and theamount of prepayment of principal also decreases Due tothe existence of prepayment of principal the total monthlyprincipal repayment has increased


442 Calculation of OAS Effective Duration and EffectiveConvexity of MBS Based on HPL-MC In the actual simu-lation QMC is used to simulate the random terms corre-sponding to the eigenvalues of the first k terms (k 8 12)and MC is used to simulate the remaining random terms Inthe empirical simulation of FNCL702342 we take the the-oretical price obtained by MC simulation for 40000 times asthe real value of the security price to observe the simulationconvergence taking k 8 and k 12 we use HPL-MC tosimulate the interest rate path and the results are shown inTable 3

It can be found from Table 3 that no matter what thevalue of k is the results of HPL-MC simulation are betterthan those of MC simulation and the effect of MC simu-lation for 5000 times is only equivalent to that of HPL-MCsimulation of 1000 times In addition the higher the value ofk the better the simulation effect and the faster the

convergence speed When k 12 the OAS of MBS is 00264that is 264 basis points After adding and subtracting 5 basispoints to the adjusted discount rate of options the corre-sponding securities prices are calculated to be 857796 and863163 respectively and the effective duration is 63148 andthe effective convexity is minus 7176643 Without consideringthe prepayment factor the traditional Macaulay duration ofMBS is 96022 it can be found that the effective duration ofMBS with embedded option is much smaller than the tra-ditional duration that is the value of securities is moresensitive to the change of interest rate

Because of this difference when there are implicit op-tions in the assets or liabilities of a bank if the traditionalMacaulay duration is still used to measure and manage theinterest rate risk and the duration gap is zero on the surfaceit seems that the bank has realized the risk immunity but infact the real duration gap is not zero and the interest raterisk is not really eliminated

45 Asset Liability Management Cases of Commercial Bankswith Embedded Options )e empirical study here is basedon the MBS with embedded option characteristics howeverthere are also embedded options in the assets and liabilitiesof commercial banks such as time deposits that can bedrawn in advance and loans that can be repaid in advance Atthis time if commercial banks use the traditional durationgap method to manage interest rate risk the apparent du-ration gap is zero the actual duration gap is not zero andthere is still interest rate risk exposure In other words ifthere are embedded options in the balance sheet of com-mercial banks the traditional Macaulay duration will lose itseffectiveness in gap management Here we will choose a caseto demonstrate the effectiveness of effective duration inmanaging interest rate risk when there are embeddedoptions

0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r

Figure 5 Comparison of interest rates simulated by generalizedVasicek model and extended CIR model

0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353

PrincipalService fee

InterestPrincipal and interest

Month

Dol

lar

Figure 6 Cash flow chart without prepayment

0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar

Cash flowService feeTotal principal repaymentPlanned principalrepayment

Principal and interestPrepayment of principalInterest payment

Figure 7 Cash flow chart with prepayment


Suppose that the initial assets of commercial bank A arecash 30-year FNCL and 10-year loan and the liabilities are10-year deposit and 3-year deposit )e duration value ofeach asset and liability is shown in Table 4 Finally accordingto the calculation equation of duration gap that isDGAP DA minus KDL the duration gap of bank A is minus 001879(the calculation process is omitted)

According to the equationΔE minus (DA middot PA minus DL middot PL) middot (ΔR1 + R) since the durationgap is less than 0 when the interest rate moves downwardthe net asset value of the bank will decrease In order to makethe net asset value of the bank unaffected by the change ofinterest rate it is necessary to adjust the structure of thebankrsquos assets or liabilities so that its duration gap is about 0We adopt the method of adjusting the debt structure to getthe following (Table 5)

After adjustment the 10-year deposit is 3 million yuanand the 3-year deposit is 9 million yuan According to the

calculation equation of duration gap which isDGAP DA minus KDL the duration gap of bank A is minus 000004Only from the perspective of traditional duration the du-ration gap of bank A can be approximately zero that is theimmunity of interest rate risk has been basically realizedHowever considering the prepayment nature of FNCL theeffective duration gap is used tomeasure the interest rate riskof FNCL under the existing asset liability structure andTable 6 is obtained

As the 30-year FNCL has the characteristics of pre-payment its effective duration becomes 63148 and theeffective duration of other assets and liabilities is the same asthe traditional duration Similarly according to the durationgap calculation equation the duration gap of bank A isminus 06582

It can be seen that the duration gap of bank A is not zeroand the bank is still exposed to interest rate risk As theduration gap is negative when the interest rate drops the net

Table 3 Simulation results of MC and HPC

Simulation parameters f(t) 005888 a 03 and σ 0041Simulation times 100 500 1000 5000 10000 40000

Simulation methodMC 868452 868546 865955 863424 861682 860492

HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487

Table 4 Balance sheet of a commercial bank

Assets Marketvalue

Interestrate

Macaulayduration

Liabilities and shareholdersrsquoequity

Marketvalue

Interestrate

Macaulayduration

Cash 200 0 0 10-year deposit 200 006 7801630-yearFNMA 300 005888 96022 3-year deposit 1000 005 81077

10-year loan 1000 01 67588 Total liabilities 1200 80566

Total 1500 64264 Shareholdersrsquo equity 300Total 1500

Table 5 Adjusted balance sheet of a commercial bank

Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration




Table 6 Balance sheet and effective duration of a commercial bank

Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration



Total 1500 Shareholdersrsquo equity 300Total 1500


asset value of the bank will decrease )erefore once thereare embedded options the banks which are immune tointerest rate risk will have risk exposure )erefore forcommercial banks with embedded options in assets andliabilities the effective duration is more accurate and ef-fective in measuring and managing interest rate risk

5 Conclusions

)is paper takes the financial instruments with embeddedoptions as the research object empirically analyzes therelevant characteristics of prepayable mortgage-backedbonds under the background of frequent interest ratefluctuations and on this basis demonstrates the effective-ness of effective duration in managing interest rate risk ofcommercial bank )e conclusions are as follows

On the premise that the change of interest rate followsthe fuzzy stochastic process this paper selects the Hull-ndashWhite extended CIR model through comparative analysisFrom the result chart of empirical simulation the Hull-ndashWhite extended CIR model can effectively solve theproblem that interest rate may be negative in Monte Carlosimulation and the effect of the HullndashWhite extended CIRmodel is better

In Monte Carlo simulation the random number se-quence generated by traditional MC is unevenly dis-tributed in the simulation space and QMC simulation canproduce obvious improvement effect only in lower di-mensional simulation )erefore the HPL-MC methodwhich is a hybrid of QMC and MC is adopted From thesimulation results no matter what the value of k theresults of HPL-MC simulation are better than those of MCsimulation

When the prepayment of MBS is considered theembedded option will lead to the decrease of bond du-ration and the value of MBS is more sensitive to thechange of interest rate

Under the condition of continuous innovation of fi-nancial derivatives and marketization of interest rate in-terest rates fluctuate more frequently and fiercely In theprocess of asset and liability management effective durationcan reduce interest rate risk exposure position and completethe task of interest rate risk management

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is study was supported by the Humanities and SocialSciences Project of Anhui Education Department(SK2018A0520) and Anhui Province Scientific Officersquos SoftScientific Research Project (1706a02020042)

References

[1] K Drakos ldquoAssessing the success of reform in transitionbanking 10 years later an interest margins analysisrdquo Journalof Policy Modeling vol 25 no 3 p 309 2003

[2] A G Cornyn and R A Klein Controlling and ManagingInterest-Rate Risk NYIF New York NY USA 1997

[3] W Brent and J Richard ldquoEmbedded options in the mortgagecontractrdquo Pennsylvania University Philadelphia PA USAWorking Paper 305 1998

[4] R Brooks ldquoComputing yields on enhanced CDsrdquo FinancialServices Review vol 5 no 1 pp 31ndash42 1996

[5] H Ben-Ameur M Breton and L Karoui ldquoA dynamicprogramming approach for pricing options embedded inbondsrdquo Journal of Economic Dynamics ampControl vol 31no 7 pp 2212ndash2233 2007

[6] Z Zheng and L Hai ldquoDecomposition and pricing of em-bedded options in assets and liabilities of bankrdquo FinancialResearch no 7 pp 24ndash32 2004 in Chinese

[7] H Xia M Zhou and X Wang ldquo)e influence of embeddedoptions in deposit and loan on interest rate risk of commercialbanks in Chinardquo Financial Research no 9 pp 138ndash150 2007in Chinese

[8] C Wang and W Zhang ldquoInterest rate risk measurement andmanagement of commercial banks with embedded optionsconvexity gap modelrdquo Journal of Management Science vol 10pp 21ndash29 2001 in Chinese

[9] Y Fan M Gu and X Zheng ldquoApplication of option adjustedspread model (OAS) in interest rate risk management ofcommercial banksrdquo Technology and Management no 2pp 72ndash75 2003 in Chinese

[10] R W Kopparasch ldquoOption-adjusted spread analysisgoingdown the wrong pathrdquo Financial Analysis Journal vol 50no 3 pp 42ndash47 1994

[11] R Chen and X Guo ldquoResearch on option adjusted spread(OAS) and its applicationrdquo Financial Analysis Journal vol 8pp 24ndash33 1992 in Chinese

[12] C Wang Financial Market Risk Management Tianjin Uni-versity Press Tianjing China 2001 in Chinese

[13] Office of )rift Supervision -e OTS Net Portfolio ValueModel OTS Washington DC USA 1994

[14] J Daniel ldquoA fixed-rate loan prepayment model for Australianmortgagesrdquo Australian Journal of Management vol 35 no 1pp 99ndash112 2010

[15] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[16] H Liu S Li H Wang and G Li ldquoAdaptive fuzzy syn-chronization for a class of fractional-order neural networksrdquoChinese Physics B vol 26 no 3 Article ID 030504 2017

[17] B Liu ldquoFuzzy process hybrid process and uncertain processrdquoJournal of Uncertain Systems vol 2 no 1 pp 3ndash16 2008

[18] Z Zmeskal ldquoApplication of the fuzzy-stochastic methodologyto appraising the firm value as ardquo European Journal of Op-erational Research vol 135 no 2 pp 303ndash310 2001

[19] Y Yoshida M Yasuda J-i Nakagami and M Kurano ldquoA newevaluation ofmean value for fuzzy numbers and its application toAmerican put option under uncertaintyrdquo Fuzzy Sets and Systemsvol 157 no 19 pp 2614ndash2626 2006

[20] H-C Wu ldquoEuropean option pricing under fuzzy environ-mentsrdquo International Journal of Intelligent Systems vol 20no 1 pp 89ndash102 2005


[21] C Lee G Tzeng and S Wang ldquoA new application of fuzzy settheory to the Black-Scholes option pricing modelrdquo ExpertSystems with Applications vol 29 no 2 pp 330ndash342 2005

[22] S Muzzioli and C Torricelli ldquoA multiperiod binomial model forpricing options in a vague worldrdquo Journal of Computational andApplied Mathematics vol 223 no 2 pp 552ndash566 2009


method for the embedded options belonging to the Asianoption type after estimating the parameters in the modelbased on this method the price of embedded optionbonds was obtained [3] Brooks made a sensitivity analysison the value of embedded options and interest rate level inCDs based on cross-sectional data [4] Ben-Ameurpointed out that the inconsistency of value changes causedby the mismatch of asset and liability convexity is also thecause of embedded option risk and studied the charac-teristics of embedded options in commercial bank de-posits and the characteristics of depositorsrsquo exercise time[5] Domestic scholars have also made importantachievements in the research of embedded optionsZhenlong and Hai decomposed the option characteristicsof assets and liabilities in the balance sheet of commercialbanks and used the basic principles of financial engi-neering to price implicit options without arbitrage andalso made numerical pricing [6] Xia et al built thebenchmark interest rate jump model of deposits andconducted the numerical pricing of embedded optiondeposit and loan based on the Monte Carlo simulationmethod and studied the impact of embedded options oninterest rate risk of Chinarsquos commercial banks by usingcomparative analysis method [7] Li et al proposed thatwith the development of interest rate marketization andthe continuous innovation of financial instruments ofChinarsquos commercial banks embedded options are moreand more appearing in the balance sheet of commercialbanks and at the same time it brings huge interest raterisk to the operation of commercial banks )ereforecommercial banks urgently need to establish a completeinterest rate risk management system in time Howeverthe traditional risk management methods such as interestrate sensitivity gap and duration gap have been unable tomeet the interest rate risk management requirements ofassets and liabilities with embedded options )e review ofthe research results at home and abroad shows that theresearch on embedded options started earlier in foreigncountries although the domestic research started late ithas also made some achievements Domestic researchstudies focus on pricing the value of embedded optionsbut few focus on the measurement and management ofinterest rate risk of embedded options In addition thedifferential equation method and nonarbitrage pricingmethod adopted by predecessors are based on a series ofstrict preconditions )e traditional Monte Carlo simu-lation method is usually used to calculate the price ofembedded option and the effective duration of securitieswith embedded option however the random numbersequence generated by traditional MC in the simulationspace is unevenly distributed which wastes a largenumber of observations and reduces the operation effi-ciency therefore this paper uses a more advanced sim-ulation method namely HPL-MC based on theoptimized interest rate risk management tools namelyeffective duration and effective convexity makes anempirical analysis on the change of characteristics of fi-nancial instruments caused by the existence of embeddedoptions and demonstrates how commercial banks can

effectively measure and manage interest rate risk whenimplicit options exist

2 Analysis on the Option Characteristics ofFinancial Instruments withEmbedded Options

With the increasing types of derivative financial instru-ments financial instruments with embedded options aremore and more embedded in the assets and liabilities ofcommercial banks such as deposits that can be drawn inadvance prepayable loans collateralized mortgage obliga-tion callable bonds mortgage-backed securities and so onHere we will take the prepayment of housing mortgageloans as an example to illustrate

21 Influencing Factors of Prepayment of Housing MortgageLoans From the perspective of borrowers the followingfactors will affect their prepayment behaviors

211 Interest Rate Factor )e interest rate factor is theprimary factor to decide the prepayment behaviors )einterest rate factor here refers to the balance between themarket interest rate and the contract interest rate thecontract interest rate determines the interest expense of theloans and the market interest rate determines the cost ofrefinancing Assuming that the borrower prepays at time tthe refinancing cost is expressed by V(t) the penalty is αand the loan balance at time t is L(t) when W(t) V(t) minus

[L(t) + α] is greater than zero prepayment will be executed

212 Economic Situation With the improvement of peo-plersquos economic situation and the improvement of livingrequirements the prepayment rate will increase and at thesame time it will also cause the increase of house purchaserate and turnover rate In addition when the nationalmacroeconomic operation is good the whole real estatemarket is hot and when the rise of house price exceeds thebalance of mortgage loan prepayment will appear in largenumber

213 Institutional Factors )e minimum amount of pre-payment and the limit of the times of prepayment will re-strain the behavior of prepayment to a certain extentHowever the penalty for prepayment of housing mortgageloan is not high in our country that is to say the cost ofprepayment is small so prepayment often occurs

214 Other Factors Factors such as relocation mortgagecharacteristics and seasons

22-e Influence of Prepayment ofHousingMortgage LoanonCommercial Banks

221 Positive Impact Firstly the prepayment of housingmortgage loan reduces the possibility of nonperforming loan







minus min er0Tt

+ α1113872 1113873er1T(Tminus t)

1113872 1113873 er0TT

1113960 1113961

minus er0TT

minus min er0Tt

+ α1113872 1113873er1T(Tminus t)

minus er0TT

01113960 1113961

minus er0TT

+ max er0TT

minus er0Tt

+ α1113872 1113873er1T(Tminus t)

01113960 1113961

(1)











dA dA

dRA

dRA +12d2AdR

2A

dRA( 11138572 (2)

dL dL

dRL

dRL +12d2LdR

2L

dRL( 11138572 (3)






















2ΔrP0 (5)


Δr2P0 (6)


Pminus 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i minus Δr + OAS1113872 1113873 (7)

P+ 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rji + Δr + OAS1113872 1113873

(8)





P 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i + OAS1113872 1113873 (9)


t









1 wt2 wt



Wi 1113944k


d

ik+1bizi (10)





rRlowast

Plowast

P

No prepayment

Prepayment




Com

mer

cial

ban

k

Borr











RI(t) 031234 minus 02025


P(t) + F+ 1207611113888 11138891113890 1113891

(12)




(13)



10 (14)



(15)

B(t T) 1 minus e

minus a(Tminus t)

a (16)

lnA(t T) lnp(0 T)

p(0 t)minus B(t T)

z ln p(0 t)

ztminus

14a

3σ2

middot eminus aT

minus eminus at

1113872 11138732

e2at

minus 11113872 1113873

(17)









P


E2

Elowast

E1

r












31113888 1113889 minus 11113896 1113897

(19)





(20)



(21)


PV 1113944WAM




interest rate

Initialyield curve

Simulated sample

interest rate



Sum of discounted

value of cash

Find the value of

OAS

Market price of

securities

Comparison between

market price and


securities

UnequalEqual




dis(t) 1

1113945t

i1(1 + r(i))112

(23)



12 (24)









1113872 1113873 (27)




1113872 1113873 (28)












minus Xt0 Xt2

minus

Xt1 Xtk








100 (31)





ΔWt εΔt

radic (33)




θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References






























minus min er0Tt

+ α1113872 1113873er1T(Tminus t)

1113872 1113873 er0TT

1113960 1113961

minus er0TT

minus min er0Tt

+ α1113872 1113873er1T(Tminus t)

minus er0TT

01113960 1113961

minus er0TT

+ max er0TT

minus er0Tt

+ α1113872 1113873er1T(Tminus t)

01113960 1113961

(1)











dA dA

dRA

dRA +12d2AdR

2A

dRA( 11138572 (2)

dL dL

dRL

dRL +12d2LdR

2L

dRL( 11138572 (3)






















2ΔrP0 (5)


Δr2P0 (6)


Pminus 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i minus Δr + OAS1113872 1113873 (7)

P+ 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rji + Δr + OAS1113872 1113873

(8)





P 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i + OAS1113872 1113873 (9)


t









1 wt2 wt



Wi 1113944k


d

ik+1bizi (10)





rRlowast

Plowast

P

No prepayment

Prepayment




Com

mer

cial

ban

k

Borr











RI(t) 031234 minus 02025


P(t) + F+ 1207611113888 11138891113890 1113891

(12)




(13)



10 (14)



(15)

B(t T) 1 minus e

minus a(Tminus t)

a (16)

lnA(t T) lnp(0 T)

p(0 t)minus B(t T)

z ln p(0 t)

ztminus

14a

3σ2

middot eminus aT

minus eminus at

1113872 11138732

e2at

minus 11113872 1113873

(17)









P


E2

Elowast

E1

r












31113888 1113889 minus 11113896 1113897

(19)





(20)



(21)


PV 1113944WAM




interest rate

Initialyield curve

Simulated sample

interest rate



Sum of discounted

value of cash

Find the value of

OAS

Market price of

securities

Comparison between

market price and


securities

UnequalEqual




dis(t) 1

1113945t

i1(1 + r(i))112

(23)



12 (24)









1113872 1113873 (27)




1113872 1113873 (28)












minus Xt0 Xt2

minus

Xt1 Xtk








100 (31)





ΔWt εΔt

radic (33)




θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References






































2ΔrP0 (5)


Δr2P0 (6)


Pminus 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i minus Δr + OAS1113872 1113873 (7)

P+ 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rji + Δr + OAS1113872 1113873

(8)





P 1m

1113944

m

j11113944

n

t1

CFjt

1113945t

i1 1 + rj

i + OAS1113872 1113873 (9)


t









1 wt2 wt



Wi 1113944k


d

ik+1bizi (10)





rRlowast

Plowast

P

No prepayment

Prepayment




Com

mer

cial

ban

k

Borr











RI(t) 031234 minus 02025


P(t) + F+ 1207611113888 11138891113890 1113891

(12)




(13)



10 (14)



(15)

B(t T) 1 minus e

minus a(Tminus t)

a (16)

lnA(t T) lnp(0 T)

p(0 t)minus B(t T)

z ln p(0 t)

ztminus

14a

3σ2

middot eminus aT

minus eminus at

1113872 11138732

e2at

minus 11113872 1113873

(17)









P


E2

Elowast

E1

r












31113888 1113889 minus 11113896 1113897

(19)





(20)



(21)


PV 1113944WAM




interest rate

Initialyield curve

Simulated sample

interest rate



Sum of discounted

value of cash

Find the value of

OAS

Market price of

securities

Comparison between

market price and


securities

UnequalEqual




dis(t) 1

1113945t

i1(1 + r(i))112

(23)



12 (24)









1113872 1113873 (27)




1113872 1113873 (28)












minus Xt0 Xt2

minus

Xt1 Xtk








100 (31)





ΔWt εΔt

radic (33)




θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References































1 wt2 wt



Wi 1113944k


d

ik+1bizi (10)





rRlowast

Plowast

P

No prepayment

Prepayment




Com

mer

cial

ban

k

Borr











RI(t) 031234 minus 02025


P(t) + F+ 1207611113888 11138891113890 1113891

(12)




(13)



10 (14)



(15)

B(t T) 1 minus e

minus a(Tminus t)

a (16)

lnA(t T) lnp(0 T)

p(0 t)minus B(t T)

z ln p(0 t)

ztminus

14a

3σ2

middot eminus aT

minus eminus at

1113872 11138732

e2at

minus 11113872 1113873

(17)









P


E2

Elowast

E1

r












31113888 1113889 minus 11113896 1113897

(19)





(20)



(21)


PV 1113944WAM




interest rate

Initialyield curve

Simulated sample

interest rate



Sum of discounted

value of cash

Find the value of

OAS

Market price of

securities

Comparison between

market price and


securities

UnequalEqual




dis(t) 1

1113945t

i1(1 + r(i))112

(23)



12 (24)









1113872 1113873 (27)




1113872 1113873 (28)












minus Xt0 Xt2

minus

Xt1 Xtk








100 (31)





ΔWt εΔt

radic (33)




θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References































RI(t) 031234 minus 02025


P(t) + F+ 1207611113888 11138891113890 1113891

(12)




(13)



10 (14)



(15)

B(t T) 1 minus e

minus a(Tminus t)

a (16)

lnA(t T) lnp(0 T)

p(0 t)minus B(t T)

z ln p(0 t)

ztminus

14a

3σ2

middot eminus aT

minus eminus at

1113872 11138732

e2at

minus 11113872 1113873

(17)









P


E2

Elowast

E1

r












31113888 1113889 minus 11113896 1113897

(19)





(20)



(21)


PV 1113944WAM




interest rate

Initialyield curve

Simulated sample

interest rate



Sum of discounted

value of cash

Find the value of

OAS

Market price of

securities

Comparison between

market price and


securities

UnequalEqual




dis(t) 1

1113945t

i1(1 + r(i))112

(23)



12 (24)









1113872 1113873 (27)




1113872 1113873 (28)












minus Xt0 Xt2

minus

Xt1 Xtk








100 (31)





ΔWt εΔt

radic (33)




θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References






























31113888 1113889 minus 11113896 1113897

(19)





(20)



(21)


PV 1113944WAM




interest rate

Initialyield curve

Simulated sample

interest rate



Sum of discounted

value of cash

Find the value of

OAS

Market price of

securities

Comparison between

market price and


securities

UnequalEqual




dis(t) 1

1113945t

i1(1 + r(i))112

(23)



12 (24)









1113872 1113873 (27)




1113872 1113873 (28)












minus Xt0 Xt2

minus

Xt1 Xtk








100 (31)





ΔWt εΔt

radic (33)




θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References


























dis(t) 1

1113945t

i1(1 + r(i))112

(23)



12 (24)









1113872 1113873 (27)




1113872 1113873 (28)












minus Xt0 Xt2

minus

Xt1 Xtk








100 (31)





ΔWt εΔt

radic (33)




θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References



























θ(t) df(t)

dt+ af(t) +

σ2

2a1 minus e

minus 2at1113872 1113873 (34)



1113968dWt (35)


dr(t) df(t)

dt+ af(t) +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873 (36)

σr(t)








1113968ΔWt (37)


θ(t) 12t

+ a 006 + 05 timesln(t)

1001113888 1113889 +

σ2r(t)

2a1 minus e

minus 2at1113872 1113873

(38)


r(t + 1) r(t) +12t

+ a 006 + 05 timesln(t)

1001113888 11138891113890

+σ2r(t)


r(t)

1113968ε

Δt

radic

(39)



σr σr12

(40)



ln σr( 1113857 ln σ + 12 ln r (41)















0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References






























0 50 100 150 200 250 300 350 400ndash01

ndash005

0

005

01

015

02

025

Time

r


0

02

04

06

08

1

12

14

1 23 45 67 89 111

133

155

177

199

221

243

265

287

309

331

353



Month

Dol

lar


0

05

1

15

2

25

1 20 39 58 77 96 115

134

153

172

191

210

229

248

267

286

305

324

343

Month

Dol

lar














HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References


































HPL-MC K 8 862644 862614 861546 860982 860479K 12 862346 862245 861525 860979 860487


Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration





Assets Marketvalue

Interestrate

Macaulayduration


Marketvalue

Interestrate

Macaulayduration






5 Conclusions






Data Availability




Acknowledgments


References


























5 Conclusions






Data Availability




Acknowledgments


References

























measuringinterestrateriskwithembeddedoptionusing hpl...

Documents