XVA Valuation under Market Illiquidity
NNoname manuscript No. (will be inserted by the editor)
XVA Valuation under Market Illiquidity
Weijie Pang · Stephan Sturm
Received: date / Accepted: date
Abstract
Before the 2008 financial crisis, most research in financial mathe-matics focused on pricing options without considering the effects of counter-parties’ defaults, illiquidity problems, and the role of the sale and repurchaseagreement (Repo) market. Recently, models were proposed to address this bycomputing a total valuation adjustment (XVA) of derivatives; however withoutconsidering a potential crisis in the market. In this article, we include a pos-sible crisis by using an alternating renewal process to describe the switchingbetween a normal financial regime and a financial crisis. We develop a frame-work to price the XVA of a European claim in this state-dependent situation.The price is characterized as a solution to a backward stochastic differentialequation (BSDE), and we prove the existence and uniqueness of this solu-tion. In a numerical study based on a deep learning algorithm for BSDEs, wecompare the effect of different parameters on the valuation of the XVA.
Keywords
Financial crisis · Option pricing · Value adjustments · Arbitragepricing · Backward stochastic differential equations
The 20 th century witnessed the birth and development of financial mathe-matics, but it had been varying dramatically since the 2008 financial crisis.While one decisive cause of this crisis is the unhealthy mortgage market. Thepercentage of sub-prime mortgage, compared with all mortgages, doubled in2004 and continued to increase. During June 2004 and June 2006, the Fed W. PangDepartment of Mathematics and Statistics, McMaster UniversityE-mail: [email protected]. SturmDepartment of Mathematical Science, Worcester Polytechnic InstituteE-mail: [email protected] a r X i v : . [ q -f i n . P R ] N ov Weijie Pang, Stephan Sturm fund rate was increased from 1.00% to 5.25%, and stayed at 5.25% until theFederal Reserve started decrease the rate in September 2007. Considering thehazardous mortgage market at that time, numerous households experiencedgrievous financial pressure, because of shortages of adequately financial capac-ity. Some sold their house and escape from their mortgage luckily, others hadto default on their mortgage loans. In both situations, those houses returnedU.S. real estate market. Because the supply of houses pumped up, prices ofreal estate dropped sharply, which consisted with the Law of Supply and De-mand. Meanwhile many investment companies lost dramatically in their loansin the unhealthy mortgage market and their investments in the real estate.As a result, companies experienced serious financial suffer as well. Several fi-nancial companies even went bankrupt, such as Lehman Brothers. The 2008sub-prime financial crisis resulted out from this cause.According to this situation, many research attempt to take the effect ofdefaults from counterparty and debt into account in the replication frame-work. In order to consider defaults from a institution and its counterparty ina pricing model of an option, the bebt valuation adjustment (DVA) and thecredit valuation adjustment (CVA) become popular. For the spread betweeninterest rates from and to a bank, researcher add a funding valuation adjust-ment (FVA) to their original framework. Many distinct valuation adjustmentscome out after the 2008 financial crisis. Since Basel regulations have requiredto include default risk and cost of collateralization strategies in risk measure-ment ([3]), increasing people work on the total valuation adjustment (XVA),as a combination of CVA, DVA, FVA and several other valuation adjustments.Research on FVA had been starting before the 2008 financial crisis. Beforethe crisis, [23] introduce the setting of different interest rates for borrowingand lending in a stochastic control problem for hedging. To price European calland put options, [30] assume that the borrowing interest ratea is higher thanthe lending one. Based on a similar assumptions of asymmetric interest rates,[24] study super-hedging by nonlinear backward stochastic differential equa-tions. [33] emphasize the necessity of a funding valuation adjustment (FVA)in an incomplete market. [38] discuss the hedging, collateral optimization andreverse stress testing with funding cost and funding benefit adjustment. Inaddition to a funding account, [37] include a sale and repurchase agreement(Repo) market to calculates the adjustment for non-collateralised derivatives.Besides the FVA, there are numerious meaning results in the topics ofCVA and DVA. [9] and [21] study the valuation and hedging of credit defaultswap (CDS) including counterparty credit risk. [20] extend a total valuationadjustment model to several derivitives, using a Markovian pre-default back-ward stochastic differential equation. Besides counterpary credit risk, [13] takefunding cost and collateral service cost into account, and derive a risk-neutralpricing formula. Initially, the pricing model only included a default from acounterparty (unilaterally model), which lead to the asymmetric pricing re-sult. To solve this problem, it is necessary to consider defaults from an com-pany and its counterparty together. From then on, people introduced bilateralmodels and build several framework. [15,16] generalize Piterbarg’s model to
VA Valuation under Market Illiquidity 3 include bilateral credit risk. In theoretical field, [36] prove the existence ofa fair bilateral price. [10] introduce a general semimartingale market frame-work for an arbitrage-free valuation. [4,5,6] include financing from a sale andrepurchase agreement (Repo) market, and construct a backward stochasticdifferential equation representation of European call and put option priceswith bilateral credit risk, asymmetric funding, and collateral rates. Recently,[7] extend the valuation of the XVA with considering the uncertainty bondrates. [8] study the nonlinear arbitrage-free pricing of derivatives, consideringdifferential funding cost, collaterlization, counterparty credit risk, and capitalrequirements.The 2008 financial crisis histronically change pricing models, the study ofthe sale and repurchase agreement (Repo) markets had been altering as well. ASale and Repurchase Agreement (Repo) is the sale of a security combined withan agreement to repurchase the same security at a specied price at the end ofthe contract. [29] point out that the Repo market froze during the financialcrisis and compare distinct performances among U.S., Euro and U.K. Repomarket and deduce the reasons of the differences. Focus on the U.S. Repomarket, [25] conclude that the bilateral Repo market became one importantchannel of the spreading for the sub-prime financial crisis. Contrarily, [31]belive that the funding of securitized assets has enough source, although Repoactivities involving private-label securitized assets sizing down. Furthermore,[26,39] analyze regulations to normalize the Repo market. [19] construct aninfinite-horizon equilibrium model to investigate the fragility of Repo market,according to different microstructures and financial corporates. However, fewresearch include the effect of Repo markets in a financial crisis in their studyof the XVA.In order to seperate a financial crisis from a normal financial crisis, manydifferent indicators have been proposed. [28] introduce a Composite Indicatorof Systemic Stress (CISS), which puts more weight on the stress shared by sev-eral markets at the same time. [42] argues that the CBOE’s Market VolatilityIndex (VIX), introduced in [41], is a good investor fear gauge of the expectedreturn volatility of the S&P 500 index over the next 30 days. [2] introduce thevalue at risk of financial institutions conditional on other institutions being indistress (abbreviated CoVaR) as a new measure for systemic risk. The differ-ence between the three month London Interbank Offered Rate (LIBOR) andthe government’s interest rate for a three-month period (called the Ted spread)changed alot during the financial crisis. As a result, [27] and [1] argue that theTed spread is a good indicator for the liquidity and counterparty risk in theinterbank system. [34,18,25,26] use the Ted spread to measure the capitalconstraints in a secured lending system. In addition, [11] confirm the existenceof a two-regime Ted spread over the period between 2006 and 2011, describingstable and unstable situation by the analysis of historical data. However, thesedifference regimes have not been considered in pricing derivatives so far.Overall, few research includes all of the bilateral credit risk, asymmet-ric interest rates, andfunding and market illiquidity, causing by the frozen ofRepo markets during a financial crisis. Therefore, we are interested in the pric-
Weijie Pang, Stephan Sturm ing of options while considering defaults of an investor and its counterparty,funding illiquidity problem and switching between different financial regimes.In other words, this paper focuses on the total valuation adjustment for oneEuropean option with investor’s and its counterparty’s defaults, different fi-nancial regimes, and funding liquidity problems. Its organization is as follows.In Section 2, we review several topics about the Repo market. In Section 3,we apply an alternating renewal process to describe the switching betweenfinancial regimes. After the discussion of several financial accounts in Section4, we create a hedging portfolio for European options, construct a BSDE toevaluate the arbitrage-free price of a European option and prove the existenceand uniqueness of the solution in Section 5. In Section 6, we construct a BSDEof the XVA and derive a corresponding reduced BSDEs to a smaller filtration.In Section 7, we estimate the parameters of the alternating renewal processby the Ted spread historical data and analyze the sensitivity of the XVA withrespect to the financial states, volatilities and funding rates.
In this section, we review several necessary topics about the Sale and Re-purchase Agreement (Repo) market at first. Then, we analyze its differentperformances during a calm financial time and a financial crisis period as wellas its influence to a stock market.2.1 Structure of Repo marketsA Sale and Repurchase Agreement (Repo) is a sale of a security combined withan agreement to repurchase the same security at a specified price at the endof the contract. Over the last 40 years, the size of the Repo market increaseddramatically. From 2002 to 2007, its capital size even doubled. More details in[26,25].A contract in the Repo market specifies two transactions. At the beginning,one party sells a specific security to the counterparty at a given price. Atthe end of the contract, the party repurchases the same security from itscounterparty at the agreed price, which was decided by two parties during thecontract’s negotiation. Here, the specific security can be seen as a collateral ina collateralized borrowing transaction. The collateral provider is also a cashreceiver, and the collateral receiver is also a cash lender. In this terminology,the above transactions can also be explained in another way. At the initialtime, the cash provider (collateral receiver) lends m dollar to its counterparty(cash receiver, collateral provider). At the same time, the collateral provider(cash receiver) gives a security as collateral to the cash provider (collateralreceiver). At the maturity time, the cash receiver (collateral provider) returnsthe m + r dollars to the cash provider (collateral receiver). At the same time,the collateral receiver (cash provider) returns the collateral to the collateralprovider (cash receiver). VA Valuation under Market Illiquidity 5
Fig. 1: Transactions in Repo Market.In this transaction, a relative difference between the two cash flows ( m + r − m ) m = rm is called a Repo rate. Usually, the market value of the collateral is largerthan the cash transaction. For example, when borrowing m dollars, one needsto provide a collateral with a market price as m + h dollars. The relative differ-ence between the market price of the collateral and the cash lent is called thehaircut, m + h − mm = hm . Based on different confidences played in the collateral,the haircut varies from 0.5% to over 8% in a calm financial period. In theU.S. Repo market, a group of safe collaterals is called the general collateral,it includes, i.e. 10 years U.S. treasury bonds.Based on the numbers of participants and the party to holding the collat-erals, Repo markets have three classical types – Bilateral Repo, Triparty Repoand Hold-in-custody Repo. The Bilateral Repo has been already introducedin the beginning of this section. For the Triparty Repo, there is an agent be-tween two parties in this Repo transactions. The agent helps to prepare thecontract and also hold the collateral in its balance sheet during the lifetime ofthe contract. A Hold-in-custody Repo can be a Bilateral Repo or a TripartyRepo, but the collateral is held on the balance sheet of the collateral providerduring the lifetime of the contract.This market is a significant source of collateralized borrowing of cash andborrowing of any specific securities. When a Repo contract is used to borrowcash, it is called a cash driven Repo activity. On the other hand, many com-panies use the Repo market as a source to borrow a specific security to meettheir liquidity requirements, which is called a security driven Repo activity.To attract collateral providers of some special securities, the cash lender mayprovide very low Repo rate, even a negative value. This unusual Repo rate iscall a special Repo rate. Weijie Pang, Stephan Sturm th , and then extended the banto more 190 companies. The purpose of this ban is to maintain or restore fairand orderly securities markets. Thus, most short stock trades ceased during thefinancial crisis. More information about funding liquidity and market liquiditycan be found at [14]. As we mentioned in Section 2, both the Repo market and stock market havedistinct performance and rules between a calm financial period and a miserableperiod. In order to describe the switching between a normal financial regimeand a financial crisis for these two markets, we need to study a jump stochasticprocess, called as alternating renewal process.3.1 DefinitionAn alternating renewal process is a non-homogeneous Poisson process withoutan independent increments property. This process is well known in engineeringfield, where it is used to describe a service period and a shutdown period ofone machine or one system. Thus, it is also known as On-Off process.An alternating renewal process, denoted as β , is a stochastic process switch-ing between 0 and 1, as shown in Figure 2. The mathematical defition is asfollow. VA Valuation under Market Illiquidity 7
Fig. 2: One path of the process β . Definition 1
An alternating renewal process at time t > is defined as β t = (cid:80) ∞ i =1 ( − i +1 { T i ≤ t } , where odd inter-arrival times T n +1 − T n followan exponential distribution with a constant parameter λ U > , and even inter-arrival times T n − T n − follow an exponential distribution with a constantparameter λ V > . Remark 2
Corresponding to Definition 1, an alternating renewal process β isa stochastic process without independent increments property when λ U (cid:54) = λ V . In this paper, we apply an alternating renewal process to describe thoseswitchings between a normal financial status and a financial crisis status. When β t = 0, financial markets are in a calm period. When β t = 1, financial mar-kets are in a financial crisis period. Based on the definition of the alternatingrenewal processes, a holding time of a calm status follows an exponential dis-tribution with the parameter λ U and a holding time of a financial crisis followsan exponential distribution with the parameter λ V . Remark 3
In this paper, we apply two-regime setting for the financial market.This can be extend to a more general multi-regimes. For example, a three statusof financial market: a calm regime, a stress regime and a crisis period.
For the alternating renewal process, there are many interesting topics aboutits properties and distribution. Besides the non-independent increment prop-erty, we only introduce one important theorem about its intensity, which proofcan be find in Appendix A. Define a filtration ( F βt ) t ≥ , where F βt = σ ( β s : s ≤ t ). Theorem 4
For an alternating renewal process β , there exists a finite vari-ation stochastic process Λ β such that ˜ β t := β t − Λ βt = β t − (cid:82) t λ βs ds , ˜ β is amartingale with respect to the filtration ( F βt ) t ≥ . Based on the definition and property of the alternating renewal process,we also define a non decresing jump counting process, denoted as J t . Definition 5
For the alternating renewal process β with parameters λ U and λ V , we define a jump counting process J at time t > as J t = (cid:88) s ≤ t { β s − β s − (cid:54) =0 } ( s ) . (1)Similar to the process β , we have the following remark and proposition. Weijie Pang, Stephan Sturm
Remark 6
The jump counting process J is a stochastic process without inde-pendent increments property when λ U (cid:54) = λ V . Proposition 7
The jump counting process J is a square integrable semi-martingale with respect to the filtration ( F βt ) t ≥ .Proof Proof Proof in Appendix A
Proposition 8
For the jump counting process J , there exists a finite variationstochastic process Λ J such that ˜ J t := J t − Λ Jt = J t − (cid:82) t λ Js ds, ˜ J is a squareintegrable martingale with respect to the filtration ( F βt ) t ≥ .Proof Proof Proof in Appendix A
In order to price an European option, we need to construct a replicating port-folio. Besides a classical stock account, we include accounts of two risky bondsto consider credit risks from the debt and the counterparty. In order to con-tain liquidity problems about the funding and the market, we take the fundingaccount and Repo account into our portfolio. In this section, we review theseassets in a filtered probability space ( Ω, F , ( F t ) t ≥ , P ), which is rich enough toprovide all necessary information, containing information about stock prices,default of risky bonds and switching between a normal financial regime anda financial crisis. Here, the probability P is the physical probability measure.Moreover, all interest rates are asymmetric, which means the rates of borrow-ing and lending are distinct.4.1 Repo AccountWe already analyze the structure of the Repo market and its performance dur-ing the 2008 financial crisis. In this paper, an investor apply the Repo marketas a source to borrow or lend cash and stocks. Considering a setting of theasymmetric interest rates, we assume that Repo rates are different constantsfor cash lenders and cash borrowers. For cash lenders, they receive constant in-terest rate r + r in Repo markets. For cash borrowers, they pay constant interestrate r − r , and implement long positions.Let ψ rt be a number of shares of a Repo account, then the Repo rate is r r ( ψ r ) = r − r { ψ r < } + r + r { ψ r > } . In a calm financial regime, we represent the Repo account as B r − r and B r + r for a borrowing and a lending, respectively. Overall, the dynamics of one Repoaccount is dB r ± r t = r ± r B r ± r t dt . So, the value of one Repo account at time t is ψ rt B r r t = ψ rt B r r t ( ψ r ) = ψ rt exp (cid:16) (cid:90) t r r ( ψ rs ) ds (cid:17) . VA Valuation under Market Illiquidity 9
However, the U.S. bilateral Repo market froze during the 2008 financialcrisis, as we mentioned in Section 2. In order to distinguish the values of aRepo account during a calm financial regime and a financial crisis, we containthe alternating renewal process β in Section 3 in the description of the valueof a Repo account. In this paper, we assume all Repo contracts are overnightcontracts. With a frozen bilateral Repo market during a financial crisis, thevalue of the Repo account is zero during a crisis ( β = 1). Thus, we representthe Repo account as(1 − β t ) ψ rt B r r t = (1 − β t ) ψ rt exp (cid:18)(cid:90) t r r ( ψ rs ) ds (cid:19) , The dynamics of the Repo account is(1 − β t ) ψ rt dB r ± r t = (1 − β t ) r ± r ψ rt B r ± r t dt. r + f from the treasury desk. For cash borrowers,they pay a constant interest rate r − f to funding desks. Let ξ ft be a numberof shares in the funding account at time t. So the funding interest rate isrepresented as r f := r f ( ξ ) = r − f { ξ< } + r + f { ξ> } . Similarly with the Repo account, the funding account is ξ ft B r f t := ξ ft B r f t ( ξ f ) = ξ ft exp (cid:18)(cid:90) t r f ( ξ fs ) ds (cid:19) , with the dynamics dB r ± f t = r ± f B r ± f t dt. W P is a standard Brownianmotion under the physical measure P and the inital value is S , then thedynamic of the price is dS t = µS t dt + σS t dW P t , where µ is a constant drift rate and σ is a constant volatility. When denotingthe number of shares of the stock as ξ t , the value of a stock account is ξ t S t ina normal finanical period.Contrarily to a calm regime, the short selling of stocks are banned, as aresult of the combination of a frozen Repo market and short-selling ban duringa financial suffering regime. In other words, the value of a stock account cannotbe a negative number during a financial crisis ( β = 1). Summarizing thesedifferent performances of a stock accounts, we describe the value of one stockaccount as (1 − β t { ξ t < } ) ξ t S t . Here the term (1 − β t { ξ t < } ) is the adjustment, corresponding to the frozenshort trades in a financial crisis. The dynamic of the stock account is(1 − β t { ξ t < } ) ξ t dS t = (1 − β t { ξ t < } ) ξ t ( µS t dt + σS t dW P t ) . τ I and τ C , respectively. We assume that default times follow exponential distributionswith constant intensities h P i , i ∈ { I, C } . Then the default indicator processesof these two risky bonds are described as H it = { τ i ≤ t } , ∀ t ≥ , i ∈ { I, C } . The default stopping times follow an exponential distribution with param-eters r i + h i , i ∈ { I, C } , respectively. So, the two risky bond prices P It and P Ct , underwritten by the investor and the counterparty, have the followingdynamics dP it = ( r i + h P i ) P it dt − P it − dH it , P i = exp( − ( r i + h P i ) T ) , where r i + h P i , i ∈ { I, C } are constant return rates, respectively.Therefore, for these two default indicator processes H tt , i ∈ { I, C } withconstant parameters h P i , i ∈ { I, C } , we have (cid:36) i, P t = H it − (cid:82) t (1 − H iu ) h P i du is a( F t ) t ≥ martingale under the measure P .4.5 Collateral AccountThe collateral account is to protect one party from the counterparty’s default.Similarly with the funding account and the Repo account, the value of thecollateral process C := ( C t : t ≥
0) also has a pair of asymmetric constantcollateral rates. For collateral receivers ( C t < VA Valuation under Market Illiquidity 11 rate r − c . For collateral providers ( C t > r + c . We represent the collateral cash account as B r ± c by representing collateralinterest rates as r c ( c ) = r − c { c< } + r + c { 0) should purchase ψ ct > V be a third partyvaluation of the European claim, i.e. the Black-Scholes price, the collateralaccount is assigned different collateralization levels based on the safe levels ofthe counterparty. This is modeled as C t := α ˆ V , where the collateralization level α is a constant between 0 and 1. When thecounterparty is very reliable, for example the U.S. government, the collat-eralization level α = 0. When the counterparty has low credit rating, thecollateralization level could be α = 1. More detials in [6].4.6 Wealth ProcessAfter reviewing all financial assets in previous section, we build a replicat-ing portfolio when taking an investor and its counterparty’s defaults, liquidityproblems in a Repo market and a stock market, and asymmetric interest ratesinto account. In general, a classical replicating portfolio only two accounts,a underlying stock account and a funding account. However, our replicatingportfolio has four other accounts, the Repo account, two risky bonds accountsand the collateral account. The representations and dynamics of these six ac-counts are already reviewed in previous section. Our investing strategy is a sixdimensional vector, denoted as ϕ := ( ξ t , ξ ft , ξ It , ξ Ct , ψ rt , ψ ct ; t ≥ ξ t rep-resents shares of stocks, ξ ft denotes shares of the funding account, ξ it , i ∈ { I, C } are shares of risky bonds, underwritten by the investor an its counterparty, re-spectively, ψ rt describes shares of the Repo account, and ψ ct symbolizes sharesof the collateral account. By descriptions of all assets in Section 4, we describeour replicating portfolio as V t ( ϕ ) = (1 − β t { ξ t < } ) ξ t S t + ξ It P It + ξ Ct P Ct + ξ ft B r f t + (1 − β t ) ψ rt B r r t − ψ ct B r c t . (3)The dynamics of this wealth process V t is dV t ( ϕ ) = (1 − β t { ξ t < } ) ξ t dS t + ξ It dP It + ξ Ct dP Ct + ξ ft dB r f t +(1 − β t ) ψ rt dB r r t − ψ ct dB r c t . Definition 9 An investment strategy is self-financing if for any t ∈ [0 , T ] , thefollowing identity V t ( ϕ ) = V ( ϕ )+ (cid:90) t (1 − β t { ξ t < } ) ξ t dS t + (cid:90) t ξ It dP It + (cid:90) t ξ Ct dP Ct + (cid:90) t ξ ft dB r f t + (cid:90) t (1 − β t ) ψ rt dB r r t − (cid:90) t ψ ct dB r c t , where V ( ϕ ) is the initial capital. τ I and τ C . Let τ = τ I ∧ τ C ∧ T be the maturity time of our portfolio.Assume 0 ≤ L I , L C ≤ V be a third party valuation of the hedging portfolio, i.e. aclassical Black-Scholes valuation of one option. At default times, the value ofthe claim is represent as Y := ˆ V τ − C τ = (1 − α ) ˆ V τ . Therefore, the closeoutvalue is ˆ V − L I Y + when the investor defaults, and it is ˆ V − L C Y − whenthe counterparty defaults, where ( · ) + = max(0 , ( · )) and ( · ) − = max(0 , − ( · )).Overall, we have a closeout valuation of our replicating portfolio at time τ as θ ( τ, ˆ V ) := { τ I <τ C } θ I ( ˆ V τ ) + { τ C <τ I } θ C ( ˆ V τ )= ˆ V τ + { τ C <τ I } L C Y − − { τ I <τ C } L I Y + , where θ I ( v ) := v − L I ((1 − α ) v ) + , θ C ( v ) := v + L C ((1 − α ) v ) − . Here, theterm { τ C <τ I } L C Y − is the credit valuation adjustment term after collateralmitigation and the term { τ I <τ C } L I Y + is the debit valuation adjustmentterm. In this section, we evaluate the claim by the principle of No-Arbitrage. Firstly,we convert the dynamics of all financial assets in the real measure P to avaluation measure. Then we discuss necessary and sufficient conditions of theNo-Arbitrage and construct backward stochastic differential equations (BS-DEs) of the wealth process to evaluate the European options on this valuationmeasure. At end of this section, we prove the existence and uniqueness of thesolutions to our BSDEs of the wealth process.5.1 Valuation MeasureIn Section 4, we already give the descriptions of all financial assets in a realprobability P . But in order to apply the principle of No-Arbitrage, we need to VA Valuation under Market Illiquidity 13 convert the stochastic processes of stock and risky bonds account to processeswith a consistent drift rate. Because all interest rates are asymmetric in thispaper, we cannot use the normal risk-free interest rate as the discount rate.Here we assume the investor can choose a discount rate for this valuationmodel, which is denoted as r D . By the Radon–Nikodym derivative, we definethe valuation measure Q with respect to a discount rate r D as d Q d P (cid:12)(cid:12)(cid:12) F t = exp (cid:18) r D − µσ W P t − ( r D − µ ) σ t (cid:19)(cid:18) r I − r D h P I (cid:19) H It exp(( r D − r I ) t ) (cid:18) r C − r D h P C (cid:19) H Ct exp(( r D − r C ) t ) . We denote µ I = r I + h P I and µ C = r C + h P C , which are return rates of riskybonds, underwritten by the investor and counterparty, respectively. The aboveequation becomes d Q d P (cid:12)(cid:12)(cid:12) F t = exp (cid:18) r D − µσ W P t − ( r D − µ ) σ t (cid:19) (cid:18) µ I − r D h P I (cid:19) H It exp(( r D − µ I + h P I ) t ) (cid:18) µ C − r D h P C (cid:19) H Ct exp(( r D − µ C + h P C ) t ) . Under this valuation measure Q , the dynamics of three risky assets are dS t = r D S t dt + σS t dW Q t ,dP It = r D P It dt − P It − d(cid:36) I, Q t ,dP Ct = r D P Ct dt − P Ct − d(cid:36) C, Q t , (4)where W Q t = W P t − r D − µσ t is a Brownian Motion under Q and (cid:36) i, Q t = (cid:36) i, P t + (cid:90) t (1 − H iu )( h P i − h Q i ) du, i ∈ { I, C } are (( F t ) t ≥ , Q ) martingales. Here h Q i = µ i − r D ≥ Q .5.2 Arbitrage-free Assumptions Definition 10 (Arbitrage) The market admits an investor’s arbitrage, ifthere exists a investment strategy ϕ = ( ξ t , ξ It , ξ Ct , ξ ft , ψ rt , ψ ct ; t ≥ such that P [ V t ( ϕ, x ) ≥ exp( r + f t ) x ] = 1 , P [ V t ( ϕ, x ) > exp( r + f t ) x ] > , for a given initial capital x ≥ and a corresponding wealth process V ( ϕ, x ) . Definition 11 (Arbitrage-free Financial Markets) If a financial marketdoes not admit an investor’s arbitrage for any x ≥ , the market is arbitragefree from the investor’s perspective. In Section 4, we already review all financial assets. In order to ban theopportunity of an arbitrage in the financial market, we need the followingassumptions. Assumption 12 Necessary Assumptions of Arbitrage-free Financial Markets1. r + f ≤ r − f , r + f ∨ r D < µ I ∧ µ C , (1 − β t ) r + r ≤ (1 − β t ) r − f (i.e. r + r ≤ r − f in a normal financial status). Remark 13 These three assumptions are necessary to exclude an arbitragepotentiality.1. If r + f > r − f , one can borrow cash from the funding desk at a funding rate r − f , and then lend it to the funding desk at the funding rate r + f . There is apositive arbitrage profit of r + f − r − f > multiplies the amount of cash.2. If r + f > µ I (or r + f > µ C ), one can short sell investor’s (or a counterparty’s)risky bonds with an expected return rate µ I (or µ C ), and then lend themoney to the funding desk, earning an arbitrage profit r + f − µ I > (or r + f − µ C > ) multiplies the shares of the bonds.If r D > µ I (or r D > µ C ) and the investor can trade by the interest rate r D ,the discussion is similar to the cases r + f > µ I (or r + f > µ C ). If the investorcannot trade by r D , then the investor’s default intensity is h Q I = µ I − r D < (or h Q C = µ C − r D < ) under the discount measure Q , which is not realistic.3. In a normal financial status ( β t = 0 ), if r + r > r − f , one can borrow cash fromthe funding desk at the funding rate r − f and lend it to the Repo market atthe Repo rate r + r , earning a positive arbitrage profit r + r − r − f > multipliesthe amount of cash. In a financial crisis status ( β t = 1 ), the inequalityholds trivially. Proposition 14 Suppose that Assumption 12 holds. A financial market isarbitrage-free if (1 − β t ) r + r ≤ (1 − β t ) r + f ≤ (1 − β t ) r − r , which means r + r ≤ r + f ≤ r − r in the normal financial status.Proof Proof In a calm financial market, this arbitrage-free condition is alreadyproved in [6]. During a financial crisis, the result is a direct result from amultiplier 1 − β t and the proof in [6]. Remark 15 If an investor knows the information of the financial status ( β ),this financial market is still arbitrage-free, based on the previous proposition. VA Valuation under Market Illiquidity 15 ψ t B r r t = − ξ t S t , t > , when ξ t < 0. Moreover, when investors need to borrow cash to purchase stocks,they prefer Repo market than the funding desk in general, considering theefficiency and resilience of the Repo market. So the above equation holdswhenever ξ t < ξ t ≥ Assumption 16 In a normal financial status, the stock account is financedby the Repo market, which is represented as (1 − β t ) ψ rt B r r t = − (1 − β t ) ξ t S t . Because borrowing in the Repo market is a collateralized borrowing trade,the borrowing Repo rate r − r is less than the uncolleralized funding rate r − f ingeneral.Contrary to a calm period, we already discuss the performance of the bi-lateral Repo market and stock market during a financial crisis in Section 2and Section 4. Since the bilateral Repo market freeze during a financial crisis,an investor has to borrow money from the funding desk to buy stocks. Con-sidering both of the normal financial status and the financial crisis status, wesummarize the relationship of all asset accounts as(1 − β t ) ψ rt B r r t + β t ξ ft B r f t = β t V t + β t ψ ct B r c t − (1 − β t { ξ t < } ) ξ t S t − β t ξ It P It − β t ξ Ct P Ct . (5)By Equations (3), (4), and the self-financing property of the replicatingportfolio, the dynamics of V t under the valuation measure Q with r D is dV t =(1 − β t { ξ t < } ) ξ t dS t + ξ It dP It + ξ Ct dP Ct + ξ ft dB r f t + (1 − β t ) ψ r r t dB r r t − ψ ct dB r c t = (cid:16) (1 − β t { ξ t < } ) r D ξ t S t + r D ξ It P It + r D ξ Ct P Ct + r f ξ ft B r f t + (1 − β t ) r r ψ r r t B r r t − r c ψ ct B r c t (cid:17) dt + (1 − β t { ξ t < } ) σξ t S t dW Q t − ξ It P It − d(cid:36) I, Q t − ξ Ct P Ct − d(cid:36) C, Q t . Then applying Assumption 16, the above equation is rewrote as dV t = (cid:16) (1 − β t { ξ t < } ) r D ξ t S t + r D ξ It P It + r D ξ Ct P Ct + r f ξ ft B r f t − (1 − β t ) r r ξ t S t − r c ψ ct B r c t (cid:17) dt + (1 − β t { ξ t < } ) σξ t S t dW Q t − ξ It P It − d(cid:36) I, Q t − ξ Ct P Ct − d(cid:36) C, Q t = (cid:16) ( r D − r D β t { ξ t < } − r r + r r β t ) ξ t S t + r D ξ It P It + r D ξ Ct P Ct + r f ξ ft B r f t − r c ψ ct B r c t (cid:17) dt + (1 − β t { ξ t < } ) σξ t S t dW Q t − ξ It P It − d(cid:36) I, Q t − ξ Ct P Ct − d(cid:36) C, Q t . Similarly, for the wealth process itself, we have that ξ ft B r f t = V t − β t (1 − { ξ t < } ) ξ t S t − ξ It P It − ξ Ct P Ct − C t , by Assumption 16 and Equation (2).Plugging the above equation into the dynamics of V t , we get dV t = (cid:16) ( r D − r D β t { ξ t < } − r r + r r β t − r f β t + r f β t { ξ t < } ) ξ t S t + ( r D − r f ) ξ It P It + ( r D − r r ) ξ Ct P Ct + r f V t + ( r c − r f ) C t (cid:17) dt + (1 − β t { ξ t < } ) σξ t S t dW Q t − ξ It P It − d(cid:36) I, Q t − ξ Ct P Ct − d(cid:36) C, Q t . In order to simplify the above dynamics, we define three symbols as following. Z t = (1 − β t { ξ t < } ) σξ t S t ,Z It = − ξ It P It − ,Z Ct = − ξ Ct P Ct − , (6)With these three symbols, second and third term of the drift equation inthe dynamic of V t are represented as( r D − r f ) ξ It P It = − ( r D − r f ) Z It , ( r D − r f ) ξ Ct P Ct = − ( r D − r f ) Z Ct . (7)For the first term of the drift equation in the dynamic of V t , we have( r D − r D β t { ξ t < } − r r + r r β t − r f β t + r f β t { ξ t < } ) ξ t S t = r D σ Z t + ( r r β t − r f β t + r f β t { ξ t < } − r r − r f + r f ) ξ t S t = r D σ Z t − r f σ Z t + ( r r β t − r f β t − r r + r f ) ξ t S t = r D σ Z t − r f σ Z t + ( r f − r r )(1 − β t ) ξ t S t . (8)Next, we want to rewrite (1 − β t ) ξ t S t in a representation by Z t . By somecomparison, we conclude that(1 − β t ) ξ t S t = 1 σ (1 − { Z t > ,β t =1 } ) Z t . Based on this equation, we the last term in Equation (8) becomes( r f − r r )(1 − β t ) ξ t S t = r f − r r σ (1 − { Z t > ,β t =1 } ) Z t . (9) VA Valuation under Market Illiquidity 17 Plugging Equations (6), (7), (8), (9) into the dynamics of V t , we get that dV t = (cid:16) r D − r f + ( r f − r r )(1 − { Z t > ,β t =1 } ) σ Z t − ( r D − r f ) Z It − ( r D − r f ) Z Ct + r f V t + ( r c − r f ) C t (cid:17) dt + Z t dW Q t + Z It d(cid:36) I, Q t + Z Ct d(cid:36) C, Q t = (cid:16) r f ( V t − { Z t > ,β t =1 } σ Z t + Z It + Z Ct − C t ) − r D Z I − r D Z C + r c C t + ( r D − r r (1 − { Z t > ,β t =1 } )) Z t σ (cid:17) dt + Z t dW Q t + Z It d(cid:36) I, Q t + Z Ct d(cid:36) C, Q t = (cid:16) r + f ( V t − { Z t > ,β t =1 } σ Z t + Z It + Z Ct − C t ) + − r − f ( V t − { Z t > ,β t =1 } σ Z t + Z It + Z Ct − C t ) − + r D − r − r (1 − { Z t > ,β t =1 } ) σ ( Z t ) + − r D − r + r (1 − { Z t > ,β t =1 } ) σ ( Z t ) − + r + c ( α ˆ V t ) + − r − c ( α ˆ V t ) − − r D Z I − r D Z C (cid:17) dt + Z t dW Q t + Z It d(cid:36) I, Q t + Z Ct d(cid:36) C, Q t . V t , the drift term has a very complicatestructure. In order to simplify the equation, we define a generator function f ( t, v, z, z I , z C ; β, ˆ V ) as followings f + ( t, v, z, z I , z C ; β, ˆ V ) = − (cid:16) r + f ( v − { z> ,β =1 } σ z + z I + z C − α ˆ V ) + − r − f ( v − { z> ,β =1 } σ z + z I + z C − α ˆ V ) − + r D − r − r (1 − { z> ,β =1 } ) σ z + − r D − r + r (1 − { z> ,β =1 } ) σ z − + r + c ( α ˆ V ) + − r − c ( α ˆ V t ) − − r D z I − r D z C (cid:17) ,f − ( t, v, z, z I , z C ; β, ˆ V ) = − f + ( t, − v, − z, − z I , − z C ; β, − ˆ V ) . Moreover, we need the following assumptions. Assumption 17 We assume that(i) r − f < √ T ,(ii) ( r − f + r D ∨| r D − r − r | ) ∨| r D − r + r | σ ∧ < T ,(iii) r − f − r D < √ λ I ∧√ λ C T . Now we construct two BSDEs with generator functions f ± : Ω × [0 , T ] × R × { , } → R , ( ω, t, v, z, z I , z C ; β, ˆ V ) (cid:55)→ f ± ( t, v, z, z I , z C ; β, ˆ V ) as (cid:40) − dV + t = f + ( t, V + t , Z + t , Z I, + t , Z C, + t ; β, ˆ V ) dt − Z + t dW Q t − Z I, + t d(cid:36) I, Q t − Z C, + t d(cid:36) C, Q t ,V + τ = θ I ( ˆ V τ ) { τ I <τ C ∧ T } + θ C ( ˆ V τ ) { τ C <τ I ∧ T } + Θ { τ = T } , (10) and (cid:40) − dV − t = f − ( t, V − t , Z − t , Z I, − t , Z C, − t ; β, ˆ V ) dt − Z − t dW Q t − Z I, − t d(cid:36) I, Q t − Z C, − t d(cid:36) C, Q t ,V − τ = θ I ( ˆ V τ ) { τ I <τ C ∧ T } + θ C ( ˆ V τ ) { τ C <τ I ∧ T } + Θ { τ = T } , (11)where ˆ V is a third party valuation of V with E (cid:2) (cid:82) T ˆ V s ds (cid:3) < ∞ and Θ is theterminal value at time T without defaults.The process V + describes the wealth process to replicate the selling po-sition of a claim with zero initial capital and a terminal payoff Θ . Then, theprocess V − t describes the wealth process to replicate the holding position ofthe claim. But in a financial crisis status, we can only super-hedge this claim,because of the frozen short selling trades of stock.5.5 Existence and Uniqueness of Solutions for the Valuation BSDEsIn this section, we want to prove that there exists an unique solution of thevaluation BSDE (10) and BSDE (11). In order to prove it, we need a martin-gale decomposition theorem including non-independent increment process β .A literal statement of this theorem is listed here, but the mathematical versionof this theorem is in Appendix A. Theorem 18 Let M be a ( F t ) t ≥ martingale with sup t ≤ T E [ M t ] < ∞ . Then,there exists a unique decomposition of M as M t = (cid:90) t X s dW s + (cid:90) t X Is d(cid:36) Is + (cid:90) t X Cs d(cid:36) Cs + (cid:90) t X βs d ˜ J s + Y t , where X, X I , X C , X β are B ([0 , t ]) × F t predictable processes with the squareintegrity, and Y is orthogonal with all other stochastic integrals. We state the existence and uniqueness of solutions as the following theorem. Theorem 19 Given a filtered probability space ( Ω, ( F t ) t ≥ , F , Q ) , the BSDE (10) admits an unique solution ( V + , Z + , Z I, + , Z C, + ) ∈ S × M . The solutionsatisfies the following equation V + t = V + τ + (cid:90) τt f + ( s, V + s , Z + s , Z I, + s , Z C, + t ; β, ˆ V ) ds − (cid:90) τt Z + s dW Q s − (cid:90) τt Z I, + s d(cid:36) I, Q t − (cid:90) τt Z C, + s d(cid:36) C, Q t . To prove Theorem 19, we need to prove the Lipschitz continuity of thegenerator function f + at first. Lemma 20 For any given < t < T, β, ˆ V , the generator functions f ± areLipschitz continuous in v, z, z I , z C .Proof Proof Given (cid:15) > t ≥ ω ∈ Ω , for v , v , z , z , z I , z I , z C , z C with | v − v | < (cid:15), | z − z | < (cid:15) , | z I − z I | < (cid:15) and | z C − z C | < (cid:15) , we can prove thislemma by dividing it into three case. VA Valuation under Market Illiquidity 19 1. When z , z are nonnegative, we have | f + ( t, v , z , z I , z C ; β, ˆ V ) − f + ( t, v , z , z I , z C ; β, ˆ V ) |≤ ( r + f ∨ r − f ) (cid:12)(cid:12)(cid:12) ( v − { z > ,β =1 } σ z + z I + z C − α ˆ V t ) − ( v − { z > ,β =1 } σ z + z I + z C − α ˆ V t ) (cid:12)(cid:12)(cid:12) + r D σ | z +1 − z +2 | + r D | z I − z I | + r D | z C − z C |≤ r − f (cid:12)(cid:12)(cid:12) ( v − { z > ,β =1 } σ z + z I + z C − α ˆ V t ) − ( v − { z > ,β =1 } σ z + z I + z C − α ˆ V t ) (cid:12)(cid:12)(cid:12) + r D σ | z − z | + r D | z I − z I | + r D | z C − z C |≤ r − f | v − v | + r − f + r D σ | z − z | + (cid:16) r − f + r D (cid:17) | z I − z I | + (cid:16) r − f + r D (cid:17) | z C − z C |≤ A (cid:0) | v − v | + | z − z | + | z I − z I | + | z C − z C | (cid:1) , where A = r − f + r D σ ∧ .2. When z , z are negative, we have | f + ( t, v , z , z I , z C ; β, ˆ V ) − f + ( t, v , z , z I , z C ; β, ˆ V ) |≤ ( r + f ∨ r − f ) (cid:12)(cid:12)(cid:12) ( v + z I + z C − α ˆ V t ) − ( v + z I + z C − α ˆ V t ) (cid:12)(cid:12)(cid:12) + | r D − r + r | σ | z − − z − | + r D | z I − z I | + r D | z C − z C |≤ r − f | v − v | + | r D − r + r | σ | z − z | + ( r − f + r D ) | z I − z I | + ( r − f + r D ) | z C − z C |≤ A (cid:0) | v − v | + | z − z | + | z I − z I | + | z C − z C | (cid:1) , where A = ( r − f + r D ) ∨ | r D − r + r | σ ∧ .3. Without loss of generality, we assume that z > z < 0, we have | z + z | ≤ | z + | z || = | z − z | < (cid:15) and | z | ≤ | z + | z || = | z − z | < (cid:15) . | f + ( t, v , z , z I , z C ; β, ˆ V ) − f + ( t, v , z , z I , z C ; β, ˆ V ) |≤ ( r + f ∨ r − f ) (cid:12)(cid:12)(cid:12) ( v + z I + z C − α ˆ V t ) − ( v − { z > ,β =1 } σ z + z I + z C − α ˆ V t ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) r D − r − r (1 − { z > ,β =1 } ) σ z +1 + r D − r + r σ z − (cid:12)(cid:12)(cid:12) + r D | z I − z I | + r D | z C − z C |≤ r − f (cid:12)(cid:12)(cid:12) ( v + z I + z C − α ˆ V t ) − ( v − { z > ,β =1 } σ z + z I + z C − α ˆ V t ) (cid:12)(cid:12)(cid:12) + r D ∨ | r D − r + r | σ | z + | z || + r D | z I − z I | + r D | z C − z C |≤ r − f | v − v | + r − f σ | z | + r D ∨ | r D − r − r | σ | z + | z || + ( r − f + r D ) | z I − z I | + ( r − f + r D ) | z C − z C | ≤ r − f | v − v | + r − f σ | z − z | + r D ∨ | r D − r − r | σ | z − z | + ( r − f + r D ) | z I − z I | + ( r − f + r D ) | z C − z C |≤ r − f | v − v | + r − f + ( r D ∨ | r D − r − r | ) σ | z − z | + ( r − f + r D ) | z I − z I | + ( r − f + r D ) | z C − z C |≤ A (cid:0) | v − v | + | z − z | + | z I − z I | + | z C − z C | (cid:1) , where A = r − f +( r D ∨| r D − r − r | ) σ ∧ ∨ ( r − f + r D ).Overall, the function f + satisfies the Lipschitz condition in v, z, z I , z C suchthat | f + ( t, v , z , z I , z C ; β, ˆ V ) − f + ( t, v , z , z I , z C ; β, ˆ V ) | ≤ K (cid:0) | v − v | + | z − z | + | z I − z I | + | z C − z C | (cid:1) , where K = A ∨ A ∨ A = ( r − f + r D ∨| r D − r − r | ) ∨| r D − r + r | σ ∧ independently for β andˆ V . We already proved the Lipschitz condition of the generator function f + ,we will prove the Theorem 19 here by Lemma 20 and Theorem 18. Proof Proof of Theorem 19To prove this theorem, BSDE (10) has to satisfies the Lipschitz Condition,Terminal Condition and Integrability Condition of Assumption 42 in AppendixA.1. Lipschitz ConditionBy the Lemme 20, we have the generator function f + has the inequality | f + ( t, v , z , z I , z C ; β, ˆ V ) − f + ( t, v , z , z I , z C ; β, ˆ V ) |≤ r − f | v − v | + ( r − f + r D ∨ | r D − r − r | ) ∨ | r D − r + r | σ | z − z | + ( r − f + r D ) | z I − z I | + ( r − f + r D ) | z C − z C | . By Assumptions 17, the above inequality becomes | f + ( t, v , z , z I , z C ; β, ˆ V ) − f + ( t, v , z , z I , z C ; β, ˆ V ) | < √ T | v − v | + √ λ I T | z I − z I | + √ λ C T | z C − z C | + 15 T | z − z | = 15 T (cid:18) √ T | v − v | + | z − z | + √ λ I | z I − z I | + √ λ C | z C − z C | (cid:19) . So our generator function f + satisfies the Lipschitz condition in Assump-tion 42.2. Terminal ConditionBy the definition of V τ , we have V τ ∈ L ( Ω, F τ , Q ), so the closeout valua-tion satisfies the terminal condition in Assumption 42. VA Valuation under Market Illiquidity 21 3. Integrability ConditionBy the definition of f + , we have f + ( t, , , , β, ˆ V ) = − (cid:16) r + f ( − α ˆ V t ) + − r − f ( − α ˆ V t ) − + r + c ( α ˆ V t ) + − r − c ( α ˆ V t ) − (cid:17) . For T < ∞ , E (cid:104) (cid:90) T | f + ( s, , , , β, ˆ V ) | ds (cid:105) < ∞ , so f + ( t, , , , β, ˆ V ) ∈ H satisfies the integrability condition in Assump-tion 42.By Theorem 18 and Theorem 44 in Appendix A, this valuation BSDE(10) with the generator f + admits a unique solution ( V, Z, Z I , Z C , Z β , Y ) ∈ S × M . V + t = V + τ + (cid:90) τt f ( ω, s, V s − , Z s , Z Is , Z Cs , Z βs ) ds + (cid:90) τt Z s dW Q s + (cid:90) τt Z Is d(cid:36) I, Q t + (cid:90) τt Z Cs d(cid:36) C, Q t + (cid:90) τt Z βs d ˜ J s + Y τ − Y t . Based on the specific form of the valuation BSDE with the generator f + ,the solution does not depend on the stochastic integral with respect to ˜ J andthe orthogonal term Y . So the solution ( V, Z, Z I , Z C ) to BSDE (10) is givenby V + t = V + τ + (cid:90) τt f + ( s, V + s , Z + s , Z I, + s , Z C, + s ; β, ˆ V ) ds − (cid:90) τt Z + s dW Q s − (cid:90) τt Z I, + s d(cid:36) I, Q t − (cid:90) τt Z C, + s d(cid:36) C, Q t . Theorem 21 Given ( Ω, ( F t ) t ≥ , F , P ) , BSDE (11) admits a unique solution ( V − , Z − , Z I, − , Z C, − ) ∈ S × M . The solutions satisfies the following equation V − t = V − τ + (cid:90) τt f − ( s, V − s , Z − s , Z I, − s , Z C, − t ; β, ˆ V ) ds − (cid:90) τt Z − s dW Q s − (cid:90) τt Z I, − s d(cid:36) I, Q t − (cid:90) τt Z C, − s d(cid:36) C, Q t . Proof Proof The proof is similar to the proof of Theorem 19. In this section, we develop a BSDEs framework to price the XVA of one Eu-ropean option and prove the existence and uniqueness of its solution. Thetotal valuation adjustment (XVA) is an adjustment made to the fair value ofa derivative contract to take into account funding and credit risk. We want tocompute the total valuation adjustment, which is added to the Black-Scholesprice of a European option, considering an investor’s and its counterparty’sdefaults, funding liquidity and asymmetric interest rates. Definition 22 The seller’s XVA is a value adjustment between the seller’svaluation of the claim, compared with a third party valuation ˆ V , which is astochastic process defined as XVA + t := V + t − ˆ V t . Similarly, the buyers’ XVA is the total valuation adjustment between the buyer’svaluation and a third party valuation ˆ V , which is a stochastic process definedas XVA − t := V − t − ˆ V t . XVA + is a valuation adjustment by the trader to hedge a long position in theoption, and XVA − is a valuation adjustment by the trader to hedge a shortposition in the option.In this paper, ˆ V is a solution of the Black-Scholes model (cid:40) − d ˆ V t = − r D ˆ V t dt − ˆ Z t dW Q t , ˆ V T = Θ, (12)where Θ is the terminal value of the claim from the point of view the thirdparty.In Section 5, we constructed a BSDEs framework to evaluate the price ofone claim, considering the defaults from the investor and its counterparty, theswitching between different financial regimes and asymmetric interest rates.By Definition 22 and BSDEs (10) (11), we have the BSDEs for the XVA ± as (cid:40) − d XVA ± t = ˜ f ± ( t, XVA ± t , ˜ Z ± t , ˜ Z I, ± t , ˜ Z C, ± t ; β, ˆ V , ˆ Z ) dt − ˜ Z ± t dW Q t − ˜ Z I, ± t d(cid:36) I, Q t − ˜ Z C, ± t d(cid:36) C, Q t , XVA ± τ = ˜ θ I ( ˆ V τ ) { τ I <τ C ∧ T } + ˜ θ C ( ˆ V τ ) { τ C <τ I ∧ T } , (13)where ˜ Z ± t := Z ± t − ˆ Z t , ˜ Z I, ± t := Z I, ± t , ˜ Z C, ± t := Z C, ± t , ˜ θ I (ˆ v ) := − L I ((1 − α )ˆ v ) + , ˜ θ C (ˆ v ) := L C ((1 − α )ˆ v ) − , (14) VA Valuation under Market Illiquidity 23 and the generators are˜ f + ( t, xva, ˜ z, ˜ z I , ˜ z C ; β, ˆ V , ˆ Z )= − (cid:16) r + f ( xva − { ˜ z + ˆ Z> ,β =1 } σ (˜ z + ˆ Z ) + ˜ z I + ˜ z C + (1 − α ) ˆ V ) + + r D − r − r (1 − { ˜ z + ˆ Z> ,β =1 } ) σ (˜ z + ˆ Z ) + − r − f ( xva − { ˜ z + ˆ Z> ,β =1 } σ (˜ z + ˆ Z ) + ˜ z I + ˜ z C + (1 − α ) ˆ V ) − − r D − r + r (1 − { ˜ z + ˆ Z> ,β =1 } ) σ (˜ z + ˆ Z ) − + r + c ( α ˆ V t ) + − r − c ( α ˆ V t ) − − r D ˜ z I − r D ˜ z C (cid:17) + r D ˆ V t , ˜ f − ( t, xva, ˜ z, ˜ z I , ˜ z C ; β, ˆ V , ˆ Z ) := − ˜ f + ( t, − xva, − ˜ z, − ˜ z I , − ˜ z C ; β, − ˆ V , − ˆ Z ) . (15)Based on the definition of the generator function of BSDEs (10) and (11), wehave˜ f ± ( t, xva, ˜ z, ˜ z I , ˜ z C ; β, ˆ V , ˆ Z ) = f ± ( t, xva, ˜ z + ˆ Z, ˜ z I , ˜ z C ; β, ˆ V ) ± r D ˆ V . (16) Theorem 23 Given a filtered probability space ( Ω, ( F t ) t ≥ , F , Q ) . The XVABSDEs (13) admit a unique solution ( XVA ± , ˜ Z ± , ˜ Z I, ± , ˜ Z C, ± ) . The solutionsatisfies the following equation XVA ± t = XVA ± τ + (cid:90) τt ˜ f ± ( s, XVA ± s , ˜ Z ± s , ˜ Z I, ± s , ˜ Z C, ± t ; β, ˆ V , ˆ Z ) ds − (cid:90) τt ˜ Z + s dW Q s − (cid:90) τt ˜ Z I, ± s d(cid:36) I, Q t − (cid:90) τt ˜ Z C, ± s d(cid:36) C, Q t . Proof Proof The result is a direct consequence of Theorem 19 and the Black-Scholes formula.6.2 Reduced XVA BSDEsThe XVA BSDEs (13) has a BSDEs witha jump terminal condition in thefiltration ( F t ) t ≥ . In this paper, we want to rewrite the XVA BSDEs (13) toa reduced XVA BSDEs with a smaller filtration ( F W,β ) t ≥ and a continuousterminal condition. The reduced XVA BSDEs are (cid:40) − d ˘ U ± t = ˘ g ± ( t, ˘ U ± t , ˘ Z ± t ; β, ˆ V , ˆ Z ) dt − ˘ Z ± t dW Q t , ˘ U ± T = 0 , (17)in the filtration ( F W,βt ) t ≥ without default events with˘ g + ( t, ˘ u, ˘ z ; β, ˆ V , ˆ Z ) := h Q I (˜ θ I ( ˆ V t ) − ˘ u ) + h Q C (˜ θ C ( ˆ V t ) − ˘ u ) + ˜ f + ( t, ˘ u, ˘ z, ˜ θ I ( ˆ V t ) − ˘ u, ˜ θ C ( ˆ V t ) − ˘ u ; β, ˆ V , ˆ Z ) , ˘ g − ( t, ˘ u, ˘ z ; β, ˆ V , ˆ Z ) := − ˘ g + ( t, − ˘ u, − ˘ z ; β, − ˆ V , − ˆ Z ) . (18) Theorem 24 The reduced XVA BSDE (17) admits a unique solution ( ˘ U ± , ˘ Z ± ) .When ( XVA ± , ˜ Z ± , ˜ Z I, ± , ˜ Z C, ± ) is a unique solution of BSDEs (13) , then ( ˘ U ± , ˘ Z ± ) defined as ˘ U ± t := XVA ± t ∧ τ − , ˘ Z ± t := ˜ Z ± t { t<τ } , are solutions to the reduced XVA BSDE (17) . When ( ˘ U ± , ˘ Z ± ) are unique solu-tions to the reduced XVA BSDEs (17) , then ( XVA ± , ˜ Z ± , ˜ Z I, ± , ˜ Z C, ± ) , definedasXVA ± t := ˘ U ± t { t<τ } + (cid:16) ˜ θ I ( ˆ V τ I ) { τ I <τ C ∧ T } + ˜ θ C ( ˆ V τ C ) { τ C <τ I ∧ T } (cid:17) { t ≥ τ } , ˜ Z ± t := ˘ Z ± t { t<τ } , ˜ Z I, ± t := (cid:16) ˜ θ I ( ˆ V t ) − ˘ U ± t (cid:17) { t ≤ τ } , ˜ Z Ct := (cid:16) ˜ θ C ( ˆ V t ) − ˘ U ± t (cid:17) { t ≤ τ } , are unique solutions of the XVA BSDEs (13) .Proof Proof Similary to the proof of Theorem 19, we need the Lipschitz con-dition of the generator functions ˘ g ± . By Equation (18), it’s obvious. Based onthe definition of ˘ g ± , the integrability and terminal conditions are trivial. ByTheorem 44, the reduced XVA BSDEs (17) admit a unique solution.The equivalence between the XVA BSDEs (13) and the reduced XVA BS-DEs (17) follows from Theorem 4.3 in [22] and Theorem 37. In this section, we illustrate some simulation results of the alternating renewalprocess β and the XVA valuation of a European call option. We want to es-timate the parameters of the alternating renewal process by using historicaldata and compare the XVA with and without considering the different per-formances of the Repo account and stock account during different financialstatuses.7.1 Estimations of Alternating Renewal ProcessesIn this section, we want to estimate the parameters λ U and λ V of the alternat-ing renewal process β , which are expected lengths of a normal financial regimeand a financial crisis. So we need to select historical data from some financialstress index to estimate these parameters. As we reviewed in Section 1, thereare several indicators of financial distress, such as VIX, CoVaR and the Tedspread. For Ted spread, [12] confirm the existence of a two-regime Ted spreadsfrom January 2006 to December 2011. Their estimation of the threshold forthe regime switching is 0.48 basis points. Here we also use the historical dataof the Ted spread to estimate the parameters λ U and λ V .Setting the threshold at .48 basis points, we claim that the financial marketenters a crisis status when the Ted spread is larger than the 48 basis points.When the Ted spread is smaller or equal to the .48 basis points, we claimthat the financial market is in a normal status. Assuming all financial statusare independent, we use the sample mean to estimate the expectation of thelengths of the financial statuses. The results are given in Table 1. In thisresult, the estimates are relatively small. In Figure ?? , we can see that theTed spread crossed the threshold (green line) 10 times, which means both VA Valuation under Market Illiquidity 25(a) Threshold 0.48. (b) Thresholds 0.48 and 0.8.l 2 Fig. 3: Ted spread from Jan 2006 to Dec 2011.statuses appeared five times in the dataset. But this number contradicts tothe real financial condition from 2006 to 2011. The cause of this contradictionis the noise information during a financial stress period. The Ted spread datarepresents those noise information, so there were more several oscillation atthe beginning of the financial crisis. Normal Financial Statuses Financial Crisis StatusesNumber of Statuses 5 5Average Length (days) 179 172Estimates for λ U and λ V Table 1: Estimates of λ U and λ V when threshold is .48 basis points.In order to remove the effect of noise information, [11] mention that the0.8 basis points is also a meaningful threshold, which is the threshold for thecentral bank responds to a financial crisis. We combine these two thresholds0.48 and 0.8 basis points to eliminate the effect of the Ted spread’s noisemovements. When the Ted spread up-crosses the .80 basis points (red line inFigure ?? ), we claim that the financial market enters a financial crisis. Whenthe Ted spread down-crosses the .48 basis points (green line in Figure ?? ), weclaim that the financial market enters a normal financial regime. Based on thesetting of .48 and .80 basis points, we estimate the parameters λ U and λ V ofthe alternating renewal process β , given in Table 2.Besides using the Ted spread, we can also use other financial stress indica-tors. It would be interesting to analyze these financial indicators statisticallyand find reasonable thresholds of different financial regimes in future. λ U and λ V Table 2: Estimations of λ U and λ V when threshold is .48 and .8 basis points.7.2 Simulation Results of XVAsIn this section, we want to simulate the XVA + Assume that an initial stockprice S = $1, the strike price K = $1 and the terminal time of the optionis three months ( T = 0 . 25 year). We set the following benchmark coefficients: r + r = r − r = 0 . r + c = r − c = 0 . r + f = 0 . , r D = 0 . , µ I = 0 . , µ C =0 . , σ = 0 . , h Q I = 0 . , h Q C = 0 . 15, and L I = L C = 0 . 5, as in [6]. Firstly, wewant to analyze the effect of the financial regimes ( β ), the different collateral-ization levels α and the borrowing funding rates ( r − f ) on the XVA. Then wewant to analyze the change of XVA, corresponding of different estimation of λ U . (a) β t = 0. (b) β t = 1. Fig. 4: XVA + when σ = 0 . β t = 0),we compute the XVA + for different collateralization levels α between 0 and1 and different funding rates r − f = 0 . , . , and 0 . 1, the result is plottedin Figure ?? . The XVA + increases corresponding to the increasement in thecollateralization level α . The increments of the XVA + for the different fundingrates r − f increase slightly for collateralization levels. This result means thatthe hedger invest their money from selling stocks in the Repo market, whichis consistent with the no-arbitrage assumption r + r > r + f . When α increases, VA Valuation under Market Illiquidity 27 the investor needs to hold more shares in the collateral account C t , whichleads to borrowing more cash from the funding account. In this situation, theXVA also increases due to the higher funding cost incurred the hedging of theinvestor’s and its counterparty’s default risks. However, in a financial crisis,the XVA + increase sharply corresponding with the increase of r − f . Because theRepo market freezes during a financial crisis, all borrowing of cash has to beceased through the funding market. Therefore, the increasement in the fundingrate r − f effect the XV A + dramatically. The relationship between XVA + andthe collateralization level α is simplified to a linear relationship. The resultsare plotted in Figure ?? . Compared XVA + for different financial regimes,theXVA + in a crisis is nearly double the size of the XVA + in a normal finan-cial regime. Therefore, it is important to differentiate the different financialstatuses when pricing an option.Fig. 5: XV A + for β t = 0 and a dynamic beta process.We also compare the XVA + with different estimations of λ U and collater-alization levels α , given in Figure 5. Since λ U is the expectation length of thenormal financial regimes, the larger λ U means a longer stable financial regime.This result shows that investor need larger XVA + to hedge the incoming fi-nancial crisis, especially when a financial crisis is close (smaller λ U ). However,when the collateral account is a fully collateralized account, the XVA + doesn’tchange corresponding to an incoming financial crisis. Since we assume the col-lateralization in collateral account is cash, we ignore the fire sales in this paper.In future, it’s interest to include the market illiquidity problem into the collat-eral account. In that case, XVA maybe change corresponding with the changeof the estimation of λ U when α = 1. In this paper, we discussed pricing of options emerging in the wake of the 2008financial crisis. We provided the arbitrage free pricing of one claim, consideringcredit risk, asymmetric interest rates and the different performances of severalfinancial accounts during different financial statuses. To model the frozen bilateral Repo market and short-selling of stocks dur-ing the financial crisis, we used an alternating renewal process to describethe switching between different financial regimes. With a replicating portfolioincluding risky bonds from the investor and its counterparty, we constructeda BSDE with respect to a martingale without independent increments prop-erty to price the claim and the corresponding XVA. We proved the existenceand uniqueness of the solution to these BSDEs. In the empirical application,we estimated the length of different financial periods by Ted spread historicaldata. We also analyzed the sensitivity of the XVA to collateralization levels,expected length of a normal financial regime ( λ U ) and the borrowing fundingrates ( r − f ). In a simulation study, the XVA in the financial crisis increased100%, compared with the XVA in a calm financial market. A Alternating Renewal Process β By the definition of the alternating renewal process, the stochastic process β is a rightcontinuous Markov process with left limits ( c ` adl ` ag ), switching between status “0” and status“1”. At each alternating time T n , the jump direction of the process β depends on the statusof β T n − . When β T n − = 0, then β T n = 1 as a result of an upward jump. When β T n − = 1,then β T n = 0 as a result of a downward jump. Therefore, the alternating renewal process β does not have an independent increments property. Because β has only countable manyjumps almost surely, it has one c ` agl ` ad modification. We decompose β as the sum of twoprocesses β + and β − : β + t = (cid:88) s ≤ t { β s − <β s } , β − t = − (cid:88) s ≤ t { β s − >β s } . (19)The process β + is a jump counting process of the upward jumps and the process β − isa jump counting process of a downward jumps.Since β − is a non-increasing c ` adl ` ag process, it is a supermartingale. By the definition,the inter-arrival times of β − are exponential distributed random variables with parameter λ , where λ = λ U λ V λ U + λ V . In the same way, we know that β + is a submartingale. Contrary to β − , it does not have the independent increments property. It is a sum of one exponentialprocess with parameter λ U and a Gamma process Gamma ( n, λ ), where λ = λ U λ V λ U + λ V . Weget the probability distribution function of β + t as follows: P ( β + t = 0) = e − λ U t . (20) P ( β + t = 1) = λ U λ − λ U (cid:0) exp( − λ U t ) − exp( − λt ) (cid:1) . (21)For n ≥ P ( β + t = n +1) = λ U λ n exp( − λ U t )( λ − λ U ) n +1 − n − (cid:88) k =0 λ U λ n exp( − λt )( λ − λ U ) n − k +1 k (cid:88) j =0 t j j ! λ k − j +1 − λ U exp( − λt ) λ − λ U n (cid:88) k =0 t k λ k k ! . (22) Proposition 25 For the stochastic process β + , there exist a finite variation stochasticprocess Λ + such that ˜ β + t := β + t − Λ + t = β + t − (cid:82) t λ + s ds, ˜ β + is a martingale with respect tothe natural filtration ( F βt ) t ≥ , F βt = σ ( β s : s ≤ t ) . VA Valuation under Market Illiquidity 29 Proof Proof By the definition of the process β + , it is square integrable. Since β + t is anondecreasing process, it is a submartingale. By the Doob–Meyer Decomposition Theorem,there exist a finite variation process Λ + t such that ˜ β + t = β + t − Λ + t , ˜ β + is a square integrablemartingale with respect to the filtration ( F βt ) t ≥ .Then, by the intensity of Poisson processes λ + t = lim h → P ( β + t + h − β + t =1) h and Λ + t := (cid:82) t λ + s ds , we havelim h → P ( β + t + h − β + t = 1) h = lim h → P ( β + h = 1 | t < T ) h { t For an alternating renewal process β , there exists a finite variation stochasticprocess Λ β such that ˜ β t := β t − Λ βt = β t − (cid:82) t λ βs ds , ˜ β is a martingale with respect to thefiltration ( F βt ) t ≥ .Proof Proof By the definition of process β , the integrability is trivial. Since − β − is a Poissonprocess with parameter λ , we have that process ˜ β − t := β − t + λt, ˜ β − is a martingale withrespect to the filtration ( F βt ) t ≥ . By Proposition 25, ˜ β + is a martingale with respect to thefiltration ( F βt ) t ≥ . Since β t = β + t + β − t , we define Λ t = Λ + t + λt and λ βt = λ + t + λ , thenthe expectation is E (cid:104) β t − (cid:90) t λ βu du | F βs (cid:105) = E (cid:104) β + t − (cid:90) t λ + u du | F βs (cid:105) + E (cid:104) β − t + (cid:90) t λdu | F βs (cid:105) = β + s − (cid:90) s λ + u ds + β − s + λs = β s − (cid:90) s λ βu du. So ˜ β t := β t − Λ βt = β t − (cid:82) t λ βs ds, ˜ β is a martingale with respect to the filtration ( F βt ) t ≥ . A.1 Jump Counting Processes In Section 3, we already define the jump counting processes and discuss several importantproperties of it. In this section, we will give the proof these propositions. Proposition 27 For the jump counting process J , there exists a finite variation stochasticprocess Λ J such that ˜ J t := J t − Λ Jt = J t − (cid:82) t λ Js ds, ˜ J is a square integrable martingale withrespect to the filtration ( F βt ) t ≥ .Proof Proof Since J = β + − β − , we define Λ Jt := Λ + t + λt, λ J = λ + + λ − . Similar to theproof of Theorem 26, ˜ J t := J t − Λ Jt = J t − (cid:82) t λ Js ds is a martingale with respect to thefiltration ( F βt ) t ≥ by Proposition 25.By Proposition 7, we have E [( ˜ J t ) ] = E [( J t − (cid:82) t λ Js ds ) ] ≤ E [( J t ) ] + 2 E [( (cid:82) t λ Js ds ) ] < ∞ . So ˜ J t is a square integrable martingale with respect to the filtration ( F βt ) t ≥ .We call the stochastic process ˜ J as a compensated jump counting process of J .0 Weijie Pang, Stephan Sturm A.2 Stochastic Calculus With Respect To Compensated Jump CountingProcesses Since the compensated jump counting process ˜ J is a martingale with respect to the filtration( F βt ) t ≥ , we can define the stochastic integral with respect to ˜ J ; see [35, Chapter 4] fordetails. Let X be a predictable process with respect to the filtration ( F βt ) t ≥ , and wedenote the stochastic integral with respect to the compensated jump counting process ˜ J as (cid:82) t X s d ˜ J t . Proposition 28 (Isometry) Let X be a predictable process with respect to the filtration ( F βt ) t ≥ and [ ˜ J ] t be the quadratic variation of the compensated jump counting process ˜ J .We have E (cid:20)(cid:16)(cid:82) t X s d ˜ J s (cid:17) (cid:21) = E (cid:104)(cid:82) t ( X s ) d [ ˜ J ] s (cid:105) = E (cid:104)(cid:82) t ( X s ) λ Js ds (cid:105) . Proof Proof By Proposition 27, we know that ˜ J is a square integrable martingale withrespect to the filtration ( F βt ) t ≥ . By Proposition 18.13 in [35], we have this result. A.2.1 The Space of Square Integrable Martingales To prove that a square integrable martingale has a representation of stochastic integralswith nonindependent increments processes, we introduce several assumptions and notationsat first. Assumption 29 Let W be a Brownian motion, β be an alternating renewal process, (cid:36) I , (cid:36) C be two compensated processes with a single exponential distributed jump, which are inde-pendent and strongly orthogonal. Notation 30 – F t = σ ( W s , β s , (cid:36) Is , (cid:36) Cs : s ≤ t ) . – H β, = { X | X is B ([0 , t ]) ⊗ F βt predictable process with (cid:107) X (cid:107) H t < ∞ , for ∀ t ≤ T } , where (cid:107) · (cid:107) H T = E [ (cid:82) t X s ds ] . – H = { ( X, X I , X C , X β ) | X, X I , X C , X β are B ([0 , t ]) ⊗ F t predictable process with (cid:107) X (cid:107) H T < ∞ , (cid:107) X I (cid:107) H T < ∞ , (cid:107) X C (cid:107) H T < ∞ and (cid:107) X β (cid:107) H T < ∞ , for ∀ t ≤ T } . – M β = { M | M is a ( F βt ) t ≥ martingale with sup t ≤ T E [ M t ] < ∞ , for ∀ t ≤ T } . – M = { M | M is a ( F t ) t ≥ martingale with sup t ≤ T E [ M t ] < ∞ , for ∀ t ≤ T } . – M β, ∗ T = { M T | M T is a F βT measurable random variable with M T := I βT ( X ) = (cid:82) T X s d ˜ J s , for sup t ≤ T E [ M t ] < ∞ , X ∈ H β, } . – M β, ∗ = { M | M ∈ M β and M t := I βt ( X ) = (cid:82) t X s d ˜ J s with X ∈ H β, , for ∀ t ≤ T } . – M ∗ T = { M T | M T is a F T measurable random variable with M T =: I T ( X ) = (cid:82) T X s dW s + (cid:82) T X Is d(cid:36) It + (cid:82) T X Cs d(cid:36) Ct + (cid:82) T X βs d ˜ J s , for sup t ≤ T E [ M t ] < ∞ , ( X, X I , X C , X β ) ∈ H T } . – M ∗ = { M | M ∈ M and M t =: I t ( X ) = (cid:82) t X s dW s + (cid:82) t X Is d(cid:36) Is + (cid:82) t X Cs d(cid:36) Cs + (cid:82) t X βs d ˜ J s with ( X, X I , X C , X β ) ∈ H , for ∀ t ≤ T } . Proposition 31 Given ( Ω, F , ( F βt ) t ≥ , P ) , ( M β, ∗ , (cid:107) (cid:107) ) is a Banach space with the norm (cid:107) · (cid:107) = E [ M t ] for M ∈ M β, ∗ .Proof ProofI. We prove first that ( M β, ∗ , (cid:107) (cid:107) ) is a vector space. Let M (1) t , M (2) t ∈ M β, ∗ , thenwe have two predictable square integrable processes X (1) and X (2) such that M (1) t =VA Valuation under Market Illiquidity 31 (cid:82) t X (1) s d ˜ J s , M (2) t = (cid:82) t X (2) s d ˜ J s . Let µ , µ ∈ R and X = µ X (1) + µ X (2) , then X isstill predictable and square integrable. So µ M (1) t + µ M (2) t = (cid:82) t X s d ˜ J s ∈ M β, ∗ .II. Let M ( n ) t ∈ M β, ∗ be a Cauchy sequence. Since this vector space is a topologicalnormed space, the limit of M ( n ) t exists, denoted by M t = lim n →∞ M ( n ) t . Then we needto prove that M t ∈ M β, ∗ . Since M ( n ) t ∈ M β, ∗ , there exists a sequence of adapted squareintegrable processes X ( n ) such that M ( n ) t = I βt ( X ( n ) ) = (cid:82) t X ( n ) s d ˜ J s .i) Let X = lim n →∞ X ( n ) , we need to prove it exist and X is predictable and squareintegrable. Since M ( n ) t is a Cauchy sequence, for any given (cid:15) > 0, there exists N ∈ N such that (cid:107) M ( n ) t − M ( m ) t (cid:107) < (cid:15) for any n, m > N . Since M ( n ) t , M ( m ) t ∈ M β, ∗ , there exist X ( n ) , X ( m ) such that M ( n ) t = (cid:82) t X ( n ) s d ˜ J s , M ( m ) t = (cid:82) t X ( m ) s d ˜ J s . Based on the definition of H β, and the isometry property, we have (cid:107) X ( n ) − X ( m ) (cid:107) H t ≤ λ E (cid:20)(cid:90) t | X ( n ) s − X ( m ) s | λ Js ds (cid:21) = 1 λ E (cid:20)(cid:90) t | X ( n ) s − X ( m ) s | d [ ˜ J ] s (cid:21) = 1 λ E (cid:104)(cid:16) (cid:90) t | X ( n ) s − X ( m ) s | d ˜ J s (cid:17) (cid:105) = 1 λ (cid:107) M ( n ) − M ( m ) (cid:107) t <(cid:15). So the sequence X ( n ) is a Cauchy sequence. Since H β, is complete, the limit X =lim n →∞ X ( n ) exists. By the definition of X , the predictability and square integrabilityproperties are trivial.ii) For any (cid:15) > 0, there exists N ∈ N such that (cid:107) M ( n ) t − M t (cid:107) < (cid:15) for any n > N . Weneed to prove the square integrability. For any n > N be given, we have (cid:107) M (cid:107) ≤ ( (cid:107) M t − M ( n ) t (cid:107) + (cid:107) M ( n ) t (cid:107) ) ≤ ( (cid:15) + (cid:107) M ( n ) t (cid:107) ) < ∞ , which proved the square integrability. By thesquare integrability, M t = lim n →∞ M ( n ) t = lim n →∞ (cid:82) t X ( n ) s d ˜ J s = (cid:82) t lim n →∞ X ( n ) s d ˜ J s = (cid:82) t X s d ˜ J s , which proved M t ∈ M β, ∗ . So any Cauchy sequence M ( n ) t ∈ M β, ∗ is convergent in the space ( M β, ∗ , (cid:107) (cid:107) ), whichmeans ( M β, ∗ , (cid:107) (cid:107) ) is close.Overall, ( M β, ∗ , (cid:107) (cid:107) ) is a Banach space. Remark 32 By the square integrability in this Banach space, it is also a Hilbert space fora inner produce < M t , M t > = E [ M t M t ] . Proposition 33 M β, ∗ t ∈ M β , which means for any M t := I βt ( X ) ∈ M β, ∗ , I β ( X ) is asquare integrable martingale with respect to the filtration ( F βt ) t ≥ .Proof Proof I. By Proposition 31, the square integrability of I βt ( X ) follows the squareintegrability of X .II. For a given t ≥ 0, we need to prove the martingale property of I βt ( X ). By thedefinition of process ˜ J , We have E [ I βT ( X ) | F βt ] = E (cid:104) (cid:90) t X s d ˜ J s | F βt (cid:105) + E (cid:104) (cid:90) Tt X s d ˜ J s | F βt (cid:105) = I βt ( X ) + E (cid:104) (cid:90) Tt X s d ˜ J s | F βt (cid:105) (23)2 Weijie Pang, Stephan SturmNext, We show that the second term in the above equation is equal 0 in two steps.i) Assume that X is an elementary process, i.e. X = (cid:80) Ni =1 a t i { t i ≤ s 0= 0 . So we proved the martingale property of I β ( X ).Assume that the stochastic processes W, β, (cid:36) I , (cid:36) C are independent and strongly or-thogonal, Proposition 31 and Proposition 33 can be extended to the martingale M withrespect to the natural filtration ( F t ) t ≥ . Proposition 34 Given ( Ω, F , ( F t ) t ≥ , P ) , ( M ∗ , (cid:107) (cid:107) ) is a Banach space with the norm (cid:107) · (cid:107) = E [ M t ] for M t ∈ M ∗ .Proof Proof I. First we prove that ( M ∗ , (cid:107) (cid:107) ) is a vector space. Let M t , M t ∈ M ∗ and µ , µ ∈ R . By the linearity of stochastic integrals, we have µ M t + µ M t = (cid:90) t ( µ X s + µX s ) dW s + (cid:90) t ( µ X ,Is + µ X ,Is ) d(cid:36) Is + (cid:90) t ( µ X ,Cs + µ X ,Cs ) d(cid:36) Cs + (cid:90) t ( µ X ,βs + µ X ,βs ) d ˜ J s ∈ M ∗ t . II. Since ( F t ) t ≥ , P ) is a normed space, we want to prove the completeness of this space.By the definition of (cid:107) · (cid:107) , Proposition 31 and the strongly orthogonality of the processes W, β, (cid:36) I , (cid:36) C , we have the completeness of ( M ∗ , (cid:107) (cid:107) ).Overall, we proved that ( M ∗ , (cid:107) (cid:107) ) is a Banach space. Proposition 35 M ∗ ∈ M , which means for any M t = I t ( X ) ∈ M ∗ , I ( X ) is a squareintegrable martingale with respect to the filtration ( F t ) t ≥ .Proof Proof I. For the square integrability, since ( X, X I , X C , X β ) ∈ H , we have thisproperty.II. Then we want to prove the martingale property of I ( X ). By the linearity of theexpectation, independence and Proposition 33, we have E (cid:104) I T ( X ) | F t (cid:105) = E (cid:104) (cid:90) T X s dW s | F Wt (cid:105) + E (cid:104) (cid:90) T X Is d(cid:36) Is | F It (cid:105) + E (cid:104) (cid:90) T X Cs d(cid:36) Cs | F Ct (cid:105) + E (cid:104) (cid:90) T X βs d ˜ J s | F βt (cid:105) = (cid:90) t X s dW s + (cid:90) t X Is d(cid:36) Is + (cid:90) t X Cs d(cid:36) Cs + (cid:90) t X βs d ˜ J s = I t ( X ) . Thus, M ∗ is a subspace of M .4 Weijie Pang, Stephan Sturm A.2.2 Martingale Decomposition Theorem In this section, we will prove the existence and uniqueness of a decomposition of a square in-tegrable martingale. Any square integrable martingale with respect to the filtration ( F t ) t ≥ can be write in a form as the sum of stochastic integrals with respect to W, ˜ J, (cid:36) I , (cid:36) C anda orthogonal term. Similar to Proposition 4.1 in [32], we will prove a decomposition for arandom variable M T and then extend the decomposition ot any martingale M ∈ M t . Proposition 36 Let M T be an F βT measurable random variable with sup t ≤ T E [ M t ] < ∞ , ( M ∗ T ) ⊥ be the orthogonal space with respect to M ∗ T . Then, there exists a unique pair I T ( X ) ∈ M ∗ T and Y ∈ ( M ∗ T ) ⊥ such that M T = I T ( X ) + Y T .Proof Proof I. First we want to prove the existence of the decomposition M T = I T ( X )+ Y T .Because M ∗ T is complete subspace of M T , there exists an orthogonal space of M ∗ T in M T ,denoted as ( M ∗ T ) ⊥ .Let I T ( X ) = proj M ∗ T M T , Y = M T − I T ( X ). We have M T = I T ( X ) + Y with (cid:107) I T ( X ) Y (cid:107) = (cid:107) I T ( X )( M T − I T ( X )) (cid:107) = (cid:107) ( proj M ∗ T M T ) − ( proj M ∗ T M T ) (cid:107) = 0 . So we have Y ∈ ( M ∗ T ) ⊥ . We proved the existence of the decomposition.II. Then, we want to prove the uniqueness of the decomposition. Assume that thereexists two decompositions, I T ( X ) , I T ( X ) ∈ M ∗ T and Y , Y ∈ ( M ∗ T ) ⊥ such that M T = I T ( X )+ Y = I T ( X )+ Y . Thus, I T ( X ) − I T ( X ) = Y − Y . Since M ∗ T and ( M ∗ T ) ⊥ are linearspaces and I T ( X ) − I T ( X ) ∈ M ∗ T and Y − Y ∈ ( M ∗ T ) ⊥ , we have I T ( X ) − I T ( X ) = Y − Y ∈ M ∗ T ∩ ( M ∗ T ) ⊥ . By the property of an orthogonal space, I T ( X ) − I T ( X ) = Y − Y = 0.Therefore, we get I T ( X ) = I T ( X ) ∈ M ∗ T and Y = Y ∈ ( M ∗ T ) ⊥ . So, we proved theuniqueness.By the unique decomposition of M T ∈ M T , we have the following decomposition of amartingale M ∈ M . Theorem 37 Let M ∈ M t . Then, there exists a unique decomposition M = I ( X ) + Y within the space ( M , (cid:107) (cid:107) ) , where I ( X ) ∈ M ∗ and Y ∈ ( M ∗ ) ⊥ .Proof ProofI. We want to prove the existence of a decomposition of M ∈ M . By Proposition 36,for any M T ∈ M T , we have M T = Y T + I T ( X ), where I T ( X ) = (cid:82) T X s dW s + (cid:82) T X Is d(cid:36) It + (cid:82) T X Cs d(cid:36) Ct + (cid:82) T X βs d ˜ J s ∈ M ∗ T , and Y T ∈ ( M ∗ T ) ⊥ .Since M is a martingale, by Proposition 35, we have M t = E [ M T | F t ] = E [ Y T + I T ( X ) | F t ] = E [ Y T | F t ] + I t ( X ) . Define a stochastic process Y by Y t := E [ Y T | F t ]. So wehave a decomposition of M t = Y t + I t ( X ). Then, we want to prove that Y ∈ ( M ∗ ) ⊥ . Since I ( X ) ∈ M ∗ , we denote I t ( X ) = (cid:82) t X s dW s + (cid:82) t X Is d(cid:36) It + (cid:82) t X Cs d(cid:36) Ct + (cid:82) t X βs d ˜ J s ∈ M ∗ .Then E [ Y t I t ( X )] = E (cid:104) E [ Y T | F t ] (cid:16) (cid:90) t X s dW s + (cid:90) t X Is d(cid:36) Is + (cid:90) t X Cs d(cid:36) Cs + (cid:90) t X βs d ˜ J s (cid:17)(cid:105) = E (cid:104) E (cid:104) Y T (cid:16) (cid:90) t X s dW s + (cid:90) t X Is d(cid:36) Is + (cid:90) t X Cs d(cid:36) Cs + (cid:90) t X βs d ˜ J s (cid:17) | F t (cid:105)(cid:105) = E (cid:104) Y T (cid:16) (cid:90) t X s dW s + (cid:90) t X Is d(cid:36) Is + (cid:90) t X Cs d(cid:36) Cs + (cid:90) t X βs d ˜ J s (cid:17)(cid:105) . Define X (cid:48) s = { s ≤ t } X s , X (cid:48) Is = { s ≤ t } X Is , X (cid:48) Cs = { s ≤ t } X Cs , X (cid:48) βs = { s ≤ t } X βs , ∀ s ∈ [0 , T ] , ( X (cid:48) s , X (cid:48) Is , X (cid:48) Cs , X (cid:48) βs ) ∈ H , we have I t ( X ) = I T ( X (cid:48) ) ∈ M ∗ T . Hence E (cid:104) Y t I t ( X ) (cid:105) = E (cid:104) Y T (cid:16) (cid:90) T X (cid:48) s dW s + (cid:90) T X (cid:48) Is d(cid:36) Is + (cid:90) T X (cid:48) Cs d(cid:36) Cs + (cid:90) T X (cid:48) βs d ˜ J s (cid:17)(cid:105) = 0 . VA Valuation under Market Illiquidity 35Since Y ⊥ I ( X ) for any I ( X ) ∈ M ∗ , we proved Y ∈ ( M ∗ ) ⊥ .II. Next, we want to prove the uniqueness of this decomposition. For any given M ∈ M ,we assume that I ( X ) , I ( X ) ∈ M ∗ and Y , Y ∈ ( M ∗ ) ⊥ such that M t = Y t + I t ( X ) = Y t + I t ( X ), for 0 ≤ t ≤ T < ∞ . Then 0 = M t − M t becomes0 = Y t − Y t + (cid:90) t ( X s − X s ) dW s + (cid:90) t ( X I, s − X I, s ) d(cid:36) Is + (cid:90) t ( X C, s − X C, s ) d(cid:36) Cs + (cid:90) t ( X β, s − X β, s ) d ˜ J s , which means that Y t − Y t = (cid:90) t ( X s − X s ) dW s + (cid:90) t ( X I, s − X I, s ) d(cid:36) Is + (cid:90) t ( X C, s − X C, s ) d(cid:36) Cs + (cid:90) t ( X β, s − X β, s ) d ˜ J s ∈ M ∗ . Since Y − Y ∈ ( M ∗ ) ⊥ ∩ M ∗ , we have Y − Y = 0 within the space ( M , (cid:107) (cid:107) ). And I ( X ) − I ( X ) = 0. So the decomposition is unique in the space ( M , (cid:107) (cid:107) ). B Backward Stochastic Differential Equations In this appendix, we will define a general backward stochastic differential equation (BSDE),including a stochastic integral with respect to the process ˜ J . Because ˜ J does not havethe independent increments property, we need to apply Theorem 37 and some results of afixed point problem to prove the existence and uniqueness of the solutions. Then we willrewrite the original BSDEs with the filtration ( F t ) t ≥ to a smaller filtration and show theirequivalence.For convenience, we define several notations here. Notation 38 – H = { X | X : Ω × [0 , T ] → R is a predictable process with E (cid:2) (cid:82) T | X s | ds (cid:3) < ∞} . – S = { X | X : Ω × [0 , T ] → R is a c ` adl ` ag adapted processes with E (cid:2) sup s ∈ [0 ,T ] | X s | (cid:3) < ∞} . – M = { M | M is a martingale with respect to ( F t ) t ≥ in S } . B.1 Construction of the General BSDEs Let N I , N C be two Poisson processes with parameters λ I , λ C . Define τ I = inf { t : N It =1 } , τ C = inf { t : N Ct = 1 } and τ = τ I ∧ τ C . Let ˜ J a corresponding compensated jumpcounting process with parameter λ J . Assume that W, ˜ J, (cid:36) I and (cid:36) C are independent andstrongly orthogonal. In this section, we study a general BSDE on the filtered probabilityspace ( Ω, F , ( F t ) t ≥ , P ), where F t = σ ( W s , β s , N Is , N Cs : s ≤ t ) as following form. (cid:40) dV t = f ( ω, t, V t − , Z t , Z It , Z Ct , Z βt ) dt − Z t dW t − Z It d(cid:36) It − Z Ct d(cid:36) Ct − Z βt d ˜ J t ,V T = θ, (25)where 0 ≤ t ≤ T < ∞ , ( Z, Z I , Z C , Z β ) ∈ H .To simplify notations, we define a martingale M of a specific form and a generator F t ( V, M ) as follows. Definition 39 Define M t := (cid:82) t Z s dW s + (cid:82) t Z Is d(cid:36) It + (cid:82) t Z Cs d(cid:36) Ct + (cid:82) t Z βs d ˜ J s , for ( Z, Z I , Z C , Z β ) ∈ H and ≤ t ≤ T < ∞ . Proposition 40 M is a martingale with respect to ( F t ) t ≥ .Proof Proof Since ( Z, Z I , Z C , Z β ) ∈ H , we have that Z t , Z It , Z Ct , Z βt are B ([0 , t ]) ⊗ F t predictable and square integrable. By Proposition 33, we have that M is a martingale.6 Weijie Pang, Stephan Sturm Definition 41 Define a generator as a function F t ( V, M ) : H × M ∗ → S with F t ( V, M ) = (cid:82) t f ( ω, s, V, Z, Z I , Z C , Z β ) ds . For general BSDE (25), we have the following assumptions: Assumption 42 (i) (Lipschitz Condition) The generator f : Ω × [0 , T ] × R → R , ( ω, t, v, z, z I , z C , z β ) (cid:55)→ ( ω, t ) is predictable and Lipschitz continuous in v, z, z I , z C , z β , i.e. for ( v , z , z I , z C , z β ) , ( v , z , z I , z C , z β ) ∈ R , we have | f ( ω, t, v , z , z I , z C , z β ) − f ( ω, t, v , z , z I , z C , z β ) | < T (cid:18) √ T | v − v | + | z − z | + √ λ I | z I − z I | + √ λ C | z C − z C | + √ λ J | z β − z β | (cid:19) . (ii) (Terminal Condition) The terminal value satisfies θ ∈ L ( Ω, F T , P ) . (iii) (Integrability Condition) f ( ω, t, , , , , ∈ H . Proposition 43 The generator F t ( V, M ) satisfies the following inequality (cid:107) F t ( V , M ) − F t ( V , M ) (cid:107) S < (cid:16) (cid:107) V − V (cid:107) S + (cid:107) Z − Z (cid:107) S + (cid:107) Z I − Z I (cid:107) S + (cid:107) Z C − Z C (cid:107) S + (cid:107) Z β − Z β (cid:107) S (cid:17) . Proof Proof Since the function f satisfies the Lipschitz condition in Assumptions 42 and W, ˜ J, (cid:36) I , (cid:36) C are orthogonal, we have (cid:107) F t ( V , M ) − F t ( V , M ) (cid:107) S < E (cid:104) sup ≤ t ≤ T (cid:90) t (cid:16) T | V − V | + | Z − Z | + λ I | Z I − Z I | + λ C | Z C − Z C | + λ J | Z β − Z β | (cid:17) ds (cid:105) ≤ T E (cid:104) sup ≤ t ≤ T (cid:90) t | V − V | ds (cid:105) + 15 E (cid:104) sup ≤ t ≤ T (cid:16) (cid:90) t | Z − Z | ds + (cid:90) t λ I | Z I − Z I | ds + (cid:90) t λ C | Z C − Z C | ds + (cid:90) t λ J | Z β − Z β | ds (cid:17)(cid:105) . By the properties of isometry and orthogonality, the second term is15 E (cid:104) sup ≤ t ≤ T (cid:16) (cid:90) t | Z − Z | ds + (cid:90) t λ I | Z I − Z I | ds + (cid:90) t λ C | Z C − Z C | ds + (cid:90) t λ J | Z β − Z β | ds (cid:17)(cid:105) ≤ (cid:107) M − M (cid:107) S Therefore, we have (cid:107) F t ( V , M ) − F t ( V , M ) (cid:107) S ≤ T E (cid:104) T sup ≤ t ≤ T | V − V | (cid:105) + 15 (cid:107) M − M (cid:107) S = 15 (cid:16) (cid:107) V − V (cid:107) S + (cid:107) M − M (cid:107) S (cid:17) . B.2 Existence and Uniqueness of Solution In this section, we will prove the existence and uniqueness of the solutions of the generalBSDE. For BSDEs, we can prove the existences of solutions of a BSDE by proving theexistence of the solutions for a corresponding fixed point problem, more details in [17]. Theorem 44 If the BSDE (25) satisfies Assumption 42, then the BSDE (25) admits aunique solutions ( V, Z, Z I , Z C , Z β ) ∈ S × M . VA Valuation under Market Illiquidity 37 Proof Proof By Proposition 43, the generator F t ( V, M ) satisfies (cid:107) F t ( V , M ) − F t ( V , M ) (cid:107) S < (cid:16) (cid:107) V − V (cid:107) S + (cid:107) M − M (cid:107) S (cid:17) . Since the generator satisfies the terminal condition and integrability condition, by the the-orem 3.1 in [17], the BSDE (25) admits a unique solution ( V, M ) ∈ S × M . ( Comparedwith the notation F t ( k )( V, M ) in [17], we only need F t ( k )( V, M ) : ≡ F t ( V, M ) in our case.)Since M ∈ M is a martingale with respect to the filtration ( F t ) t ≥ , by Theorem 37, wecan rewrite M ∈ M as M T − M t = (cid:90) Tt Z s dW s + (cid:90) Tt Z Is d(cid:36) It + (cid:90) Tt Z Cs d(cid:36) Ct + (cid:90) Tt Z βs d ˜ J s + Y T − Y t . 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