Evaluation of equity-based debt obligations
aa r X i v : . [ q -f i n . P R ] J a n Evaluation of equity-based debt obligations
Alexander Fromm ∗† Institute for Mathematics, University of Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany
January 9, 2019
Abstract
We consider a class of participation rights, i.e. obligations issued by a company toinvestors who are interested in performance-based compensation. Albeit having desirableeconomic properties equity-based debt obligations (EbDO) pose challenges in accountingand contract pricing. We formulate and solve the associated mathematical problem ina discrete time, as well as a continuous time setting. In the latter case the problem isreduced to a forward-backward stochastic differential equation (FBSDE) and solved usingthe method of decoupling fields.
Keywords. participation rights, mezzanine capital, forward-backward stochastic differentialequation, decoupling field.
Introduction
Equity-based debt obligations (EbDOs) are a form of participation rights: They constitutea legal arrangement between an investor and a company according to which the investor ispromised a share of the company’s profits. Contrary to common shares an EbDO does notconstitute co-ownership of a company. It is legally a form of debt. At the same time thevolume of the debt is not fixed but dependent on the future performance of the company.EbDOs are unique among other types of participation rights in that they offer the mostdirect and reliable access to the company’s future equity: An EbDO is defined as any obligationaccording to which the money owed is an increasing function of the company’s equity at somemoment in the future (maturity). Here the equity of a company is defined as the sum of allits assets minus the total volume of all its outstanding debt at a given moment in time.We discuss the motivation behind EbDOs in Section 1 and compare them to other formsof participation. A key advantage of EbDOs is that the company and the respective investorboth win or loose at the same time. The possibility of one of the sides taking advantage ofthe other is, thus, greatly reduced. ∗ A. Fromm acknowledges support from the
German Research Foundation through the project AN 1024/4-1. † [email protected] gross equity of a company, i.e. the value of its assets minus the value of all non-performance-based liabilities, and given a list of pay-off functions for the outstanding EbDOs calculateexplicitly how the gross equity is to be divided between the company and the investors, suchthat the share remaining with the company, which is the net equity , yields the shares of theinvestors when applying the pay-off functions to it. A more fundamental problem is to showthat there is a unique solution to this problem in the first place.Being able to resolve this is essential to the applicability of EbDOs in practice: Firstly, com-panies are usually legally obliged to know and to report their equities. More importantly, acompany must know its equity at the time an EbDO matures, as otherwise the volume of thepay-off cannot be determined. Finally, it is highly useful to know how a change in the grossequity impacts the actual equity: This would for instance allow the management of the com-pany to determine to what price a new EbDO can be sold. The price must be such that theincrease in the company’s assets due to raised funds outweighs the expansion in outstandingEbDO debt such that the actual equity increases as a consequence of the transaction.We solve the problem of evaluating EbDO debt and calculating the net equity in twodifferent settings: under the discrete time model and the continuous time model. While thetwo models have their advantages and disadvantages both methods can be used in practice.The calculations in the discrete time case, however, are mathematically less challenging andmore straightforward to implement. The continuous time model, however, is more flexible andallows extensions to more complex problems.In the course of studying the continuous time model we reduce the EbDO debt evalua-tion problem to a forward-backward stochastic differential equation (FBSDE). Unfortunately,the resulting system is coupled, i.e. neither the forward nor the backward equation can besimulated independently of the other. Furthermore, such coupled systems are not necessarilywell-posed. It is a longstanding challenge to find conditions guaranteeing that a given fullycoupled FBSDE possesses a solution. Sufficient conditions are provided e.g. in [7], [11], [10],[12], [3], [8] (see also references therein). The method of decoupling fields, developed in [4] (seealso the precursor articles [9] and [8]), is practically useful for determining whether a solutionexists. A decoupling field describes the functional dependence of the backward part Y on theforward component X . If the coefficients of a fully coupled FBSDE satisfy a Lipschitz condi-tion, then there exists a maximal non-vanishing interval possessing a solution triplet ( X, Y, Z ) and a decoupling field with nice regularity properties. The method of decoupling fields consistsin analyzing the dynamics of the decoupling field’s gradient in order to determine whetherthe FBSDE has a solution on the whole time interval [0 , T ] . The method can be successfullyapplied to various problems involving coupled FBSDE: In [6] solutions to a quadratic stronglycoupled FBSDE with a two-dimensional forward equation are constructed to obtain solutionsto the Skorokhod embedding problem for Gaussian processes with non-linear drift. In [5] theproblem of utility maximization in incomplete markets is treated for a general class of utilityfunctions via construction of solutions to the associated coupled FBSDE. In the more recentwork [2], the method is used to obtain solutions for the problem of optimal control of diffusioncoefficients. In this paper we follow a similar methodology in showing that our FBSDE is in2act well-posed.This paper is structured as follows: In Section 1 we discuss EbDOs from a purely economicpoint of view and consider their key features. In Section 2 we mathematically formulate theEbDO evaluation problem in discrete time. In Section 3 we solve the discrete time problemin the sense that its well-posedness is shown and a simple numerical scheme to calculatesolutions is deduced. In Section 4 the continuous time problem is formulated in the formof a coupled FBSDE. Since we rely on the method of decoupling fields to study this systemwe briefly introduce this method and its underlying theory in Section 5. Finally, in Section6 existence and uniqueness of solutions to the FBSDE from Section 4 is shown. Moreover,the solution is obtained explicitly for a simple illustrating example. Note that we do notprovide a numerical scheme for approximating solutions to the FBSDE introduced in Section4: numerical treatment of FBSDE is a separate topic and is usually considered in a moregeneral context. As touched upon in the introduction the motivation behind studying equity-based debt obli-gations is what one may refer to as the investor’s participation problem : An investor ownsassets that could help a company to meet its production goals. Assume that the investor doesnot need or use these assets himself or herself at a given moment and could provide them tothe company assuming there is sufficient return on his or her investment. Clearly, this returnshould come from claims to the company’s assets based on a legal arrangement between theinvestor and the company. The problem consists of designing the contract such that bothsides benefit or may expect to benefit on average.The first precondition for any successful arrangement is of course that the company hasa sound business strategy such that assets inside the company may be expected to grow dueto profits. Note that such profits are often rooted not only in the company’s existing know-how and expertise, but also in the fact that through investors’ participation the company isable to amass various resources needed to implement a non-trivial production scheme in thefirst place. In other words, the pooling of resources in itself, e.g. through the issuance ofparticipation rights, may significantly contribute to the business’s capability and profitability.Mathematically, profits may be defined as changes of the company’s equity in time. Onemight further distinguish between gross and net profits depending on what is meant by equity(e.g. gross equity or net equity). Apart from assumptions on profitability the second precon-dition for a successful arrangement is the existence of a clear definition of what it means thatthe investor receives a share of the company’s future profits. It is natural to agree to a payoffwhich is a function of the overall profit over the period of time beginning with the momentthe arrangement is made and ending at a well-defined maturity. Since the current equity isknown and only the future equity is a random variable, this is equivalent to assuming thatthe payoff is a function of the future equity. Unlike other participation rights an EbDO is afunction of the net equity rather than of any other notion of equity. Note here, that the netequity is the true equity of the company as this is what remains after all existing obligationsare considered and subtracted, including outstanding EbDOs themselves. We further assumethat this payoff function is increasing: the larger the future net equity the more the investorwill get at maturity. It is also natural to assume that there is no payoff in case of zero net3quity. A typical EbDO payoff function is h ( y ) := α · ( y − y ) + , where y ≥ , α > are fixed constants and where · + refers to the positive part of a realnumber. In other words, the payoff is equal to α ( y − y ) if the future equity y is larger than y , the latter being typically set to the equity at the time the contract is entered. If y ≤ y , onthe other hand, there is no payoff at all. Now assume that for an EbDO with payoff function h maturity is reached. Then the gross equity x , i.e. the sum of all assets minus all fixed, i.e.non-performance-based, liabilities is known or can be straightforwardly calculated. The value y is still unknown and is implicitly given by the condition y + h ( y ) = x , assuming there isonly one EbDO and no other performance based debt. If y = 0 and x > then the uniquesolution is y = x α . Accordingly, h ( y ) = x α α . Note that y is positive regardless of howlarge α > was chosen. Thus, there is no limit to the volume of EbDOs a company can issue.Apart from EbDOs there are other participation schemes used for essentially the sameeconomic purpose. As examples let us name common shares, preferred stock and participationrights where the payoff is a function of the gross equity or changes of the gross equity overperiods of time. As a specific example one might consider a payoff h ( x ) := α ( x − x ) + , where x ≥ refers to the gross equity of the company at a fixed future moment in time. In this casethe net equity would be y = x − h ( x ) , assuming there is no other performance based debt.Observe that α > should not exceed as otherwise bankruptcy occurs for sufficiently large x ! There are various reasons for EbDOs being a superior solution compared to the aforemen-tioned alternatives, in the sense that EbDOs as a class of arrangements provide a better dealfor both the investor and the company. We now briefly discuss the key advantages: Predictability and measurability:
As we shall see in the following sections the investorcan calculate, based on his or her expectations about the company’s (future) profitability,explicitly how much he or she will receive as EbDO holder on average. Other moments ofthis random variable can be calculated as well. Apart from that, the investor knows when thepayoff occurs as the maturity is agreed on in the contract. This is a significant advantage overe.g. common shares: In the latter case the investor might not know when dividend paymentswill occur and whether there will be any dividend payments at all. Also, such paymentsare not necessarily tied to the company’s performance: The company making profits doesnot immediately imply that they will be paid out in the form of dividends. This is likely todepend on decisions made by the management and/or other investors (shareholders) and isoften merely insignificantly influenced by a given investor. This might be of major concernfor minority shareholders in particular. In case the company is not paying any dividends andthere are no reliable mechanisms to force it to do so, holding shares of such a company mightbe of interest for pure speculators only and not for actual investors.
Incentivisation:
A key advantage of EbDOs is that they prevent conflicts of interest betweenthe company and the respective investors. This is because the payoff is an increasing functionof what remains in the company after the payoff (and all other payoffs and/or expected valuesof future payoffs). As a consequence there is no action a company can take to reduce thepayoff without reducing its own net equity. Conversely, there is no action the investor cantake to increase its payoff without also helping the company to grow. In other words, thecompany and the investor "sit in the same boat". By extension all EbDO holders "sit in thesame boat". The net equity of the company serves as the reference value everyone seeks to4aximize.The above property of EbDOs cannot be underestimated as it makes them highly suitable forcompensations of executives and other persons exercising control over the company or other-wise influencing its economic performance. It pushes these individuals towards cooperation aseveryone is interested in increasing the same value, which is the net equity.
Flexibility:
Unlike common shares an EbDO can be tailor made to suit the needs of indi-vidual investors by choosing the respective payoff function and maturity date appropriatelyto reflect the preferences of a particular investor.
Easy issuance:
A major problem with common shares is dilution: The more shares areissued the more the value of already issued shares is expected to decrease, especially if sharesare sold at a relatively low price. This means that, on the one hand, the equity of the com-pany increases due to the funds raised through public offerings, but on the other that existinginvestors do not necessarily benefit from this increase. There is no straightforward criterionto decide under which circumstances it is or is not appropriate for the management to issuenew shares in light of its commitment to protect the interests of existing investors.With EbDO’s, however, this problem does not arise: Any action that increases the company’snet equity is good both for the company and existing EbDO holders. In other words, thecompany’s management is free to just concentrate on raising the net equity, e.g. through theissuance of EbDOs. It must merely make sure that no EbDO is sold below the so-called risk-neutral price (see Section 3).Another problem along these lines occurs with participation rights, where the gross equityis used as the underlying: There is a natural limit on how many such participation rightscan be issued as all of the gross equity is eventually claimed by and divided between existinginvestors and the remaining net equity becomes insignificant in comparison. At the same timethe company might become unable to raise more funds through the issuance of additionalparticipation rights as existing investors would still get their share of the gross equity and itsincrements, even if the contribution of newcomers is more significant. This may lead to an"investment deadlock", as the company is unable to offer a reasonable deal to new investors,i.e. a deal which would not amount to immediate exploitation of new investors by the existing.This problem, however, does not arise with EbDO’s as there is no natural limit to the overallvolume of outstanding EbDOs. Whatever the existing EbDO - debt structure is it is impossiblefor the gross equity to be completely "consumed" such that the net equity reaches zero:Otherwise the EbDO - debt would be zero as well and, thus, the net equity actually equal tothe gross equity. In other words, there is always room to accommodate new investors. Also,unlike in the case of participation rights based on the gross equity, the company cannot bedriven into bankruptcy through EbDOs alone. On the contrary, EbDOs have an amortizingeffect and help to protect the company from insolvency.
Neutrality in terms of corporate governance:
Unlike common shares EbDOs do not goalong with any controlling interest. In other words, they do not constitute co-ownership ofthe company nor do they entitle to any right to influence the management’s decision makingor the composition of the management. Although at first one might see it as a disadvantage,there is actually none, since the respective investor might still be allowed to exercise controlover the company due to additional arrangements or because of an existing or scheduled roleas a manager.In general, it is neither necessary nor always desirable for an investor to exercise control and5nterfere in the company’s operations. Whatever the internal corporate governance proceduresare, they should be designed to ensure economically sound decision making. In particular, thecompany should be run by whoever is most competent in achieving stated business objectives.Obviously, this is not necessarily the person or the group of people who provide the largestamount of assets. It is worth noting that especially for significantly diversified investors it isneither desirable nor possible to bear responsibility for the management of every company inthe portfolio. On the contrary, an investor might be reluctant to invest or even investigatewhether the company should be invested in, if the company raises funds through shares orparticipation rights which lead to co-ownership: the company might end up under the controlof future investors who’s identities are still unknown such that this investor is being offeredto buy a "cat in the sack".To sum up, a company’s internal constitution, composition and compliance standards arean important, but completely separate topic. Whatever the chosen corporate design is, it isdesirable that investment vehicles used by the company to raise funds do not interfere withinternal mechanisms of decision making or at least that this interference is not "hard-coded".This gives freedom to choose the most effective set of internal rules and practices.
Invariance w.r.t. jurisdiction:
Unlike common shares the economic meaning of an EbDO isalways the same regardless of the jurisdiction. The rights of a holder of common shares and theobligations of the company towards a shareholder, but also the relationship between minorityand majority shareholders might vary greatly from jurisdiction to jurisdiction. In addition,there is the threat of new laws being enacted changing the nature of existing investments. Aparticipation right in the form of a well-defined financial obligation with a fixed maturity andpay-off function greatly reduces such uncertainties.
Tax efficiency:
Note that a company usually pays corporate taxes on positive changes of itsequity but not necessarily on increments of the gross equity. If an increase of the gross equityoccurs an increase of the value of outstanding EbDO-debt occurs at the same time. Thisreduces the company’s net profits and impacts the corporate taxes it has to pay. Dependingon the jurisdiction this might make EbDOs (and other participation rights) significantly moreattractive in terms of corporate taxation compared to common shares.To sum up, EbDOs are, from a purely economic view, a highly desirable arrangement asthey couple the profit of the investor to the overall economic success, over a given period oftime, of the respective company in the most effective and reliable way possible. At the sametime the nature of this arrangement creates a non-trivial evaluation and accounting problem:Since the payoff is a function of the net equity and the net equity depends on the payoffneither can be determined independently of the other. In other words, it is not a priori clearhow a given gross equity is to be divided between the investors and the company such that aninvestor’s payoff is a prescribed function of what remains in the company. Although for specialcases, e.g. if there is only one investor and the EbDO matures immediately, the problem has asimple solution, some level of complexity is needed to treat the general case. In the followingsections we formulate, study and solve this problem in a rigorous mathematical setting.
For n ∈ N let ≤ T < T < . . . < T n be deterministic future times. In addition, assumethat for each i ∈ { , . . . , n } there is a pay-off function h i : [0 , ∞ ) → [0 , ∞ ) , which determines6he total pay-off of all EbDOs which mature at time T i . This pay-off is a function of the netequity Y i of the company at time T i . Each h i is monotonically increasing. Furthermore, weassume that h i (0) = 0 for all i . Otherwise, we could split up the pay-off into a constant partand a monotonically increasing part which starts at . The constant part is then added tothe fixed debt (i.e. debt which does not depend on the performance of the company) and isconsidered when calculating the gross equity X ≥ at time . This value is deterministicand can be calculated by summing up the value of all assets of the company and subtractingall the non EbDO-debt from it.Our primary aim is to calculate the equity Y of the company at time . In case T = 0 the company would pay out the amount h ( Y ) to the holders of the EbDOs which mature attime T . More generally, the current equity of the company must be known at all times, notjust for reporting purposes, but also in order to be able to tell how much must be payed out tothe EbDO holders at a given moment. Our secondary aim is to calculate the expected pay-off E [ h i ( Y i )] ≥ of the EbDOs which mature at a future time T i for arbitrary i = 1 , . . . , n .We denote by X i the gross equity at time T i and by X ′ i := X i − h i ( Y i ) the gross equityimmediately after the payoffs occurring at time T i , i = 1 , . . . , n . We must have X ′ n = Y n asthe gross equity coincides with the net equity after all EbDO debt was served. Like the grossequity X i and the net equity Y i the random variable X ′ i is required to be non-negative.The basis of our calculation is the current equity before EbDO-debt X = X ′ ∈ [0 , ∞ ) ,which unlike the values X i , Y i , X ′ i , i = 1 , . . . , n , is a priori known, and which we refer to as thegross equity at time T . In addition, we must postulate the dynamics according to whichthe gross equity evolves between payoffs. The most simple model is the model of a geometricBrownian motion: We assume that X i is equal to X ′ i − · Z i , where Z i ∼ LN (cid:18)(cid:18) µ − σ (cid:19) · ( T i − T i − ) , σ · ( T i − T i − ) (cid:19) has a log-normal distribution with the parameters specified above. µ ∈ R determines the trendof the gross equity and σ ∈ [0 , ∞ ) its volatility in time. Thus, the gross equity does not havea drift (neither to the upside nor to the downside) if and only if µ = 0 . Z i is deterministic ifand only if σ ( T i − T i − ) = 0 , otherwise it is stochastic. σ reflects the uncertainty about thefuture evolution of the gross equity due to the company’s intrinsic performance. We assumethat the Z i , i = 1 , . . . , n , are independent random variables.For accounting purposes it is necessary to assume that µ = 0 since expectations aboutfuture profits cannot be included in the calculation of the current equity. An investor couldset µ to a positive value to calculate the expected pay-off of his or her EbDO under theassumption that the company grows with the rate µ per unit of time on average. However,this would be a subjective view, different from a neutral stance to be taken for accountingpurposes. In the same context, we must require Y i , i ∈ { , . . . , n } , to be a martingale . In otherwords the current net equity must be equal to the expectation of a future net equity given theinformation already available. The martingale property must hold w.r.t. the filtration givenby F i := σ ( Z j , j = 1 , . . . , i ) , i ∈ { , . . . , n } , where F is trivial.Now, given σ , X and the pay-off functions h i the problem of calculating Y as well as E [ h i ( Y i )] under the assumption that µ = 0 can be solved which is done in the next section.The solution entails that suitable adapted processes X i , Y i , X ′ i , i = 0 , . . . , n , such that all ofthe above is satisfied, exist and are unique in the first place. We refer to such X i , Y i , X ′ i , i = 0 , . . . , n , as the solution to the EbDO evaluation problem . To sum up, such a triplet mustsatisfy: 7. ( X i ) , ( X ′ i ) , ( Y i ) are non-negative and adapted w.r.t. ( F i ) i ∈{ ,...,n } ,2. X i = X ′ i − · Z i a.s. for all i ∈ { , . . . , n } ,3. X ′ i = X i − h i ( Y i ) a.s. for all i ∈ { , . . . , n } ,4. ( Y i ) i ∈{ ,...,n } is a martingale w.r.t. ( F i ) i ∈{ ,...,n } ,5. Y n = X ′ n a.s. and X ′ = X , where X ∈ [0 , ∞ ) is given. We denote by Id the identity mapping on [0 , ∞ ) . Let us define a function f n − : [0 , ∞ ) → [0 , ∞ ) via f n − ( x ′ ) := E [(Id + h n ) − ( x ′ Z n )] , x ′ ≥ . Note that the function
Id + h n is equal zero at zero, is strictly increasing and, therefore,is invertible. Its inverse is also strictly increasing and is zero at zero. As a consequence f n − : [0 , ∞ ) → [0 , ∞ ) is strictly increasing such that f n − (0) = 0 . Next, we define f n − ( x ′ ) := E [( f − n − + h n − ) − ( x ′ Z n − )] , for arbitrary x ′ ≥ . Again, ( f − n − + h n − ) − is well-defined, strictly increasing and vanishesat zero. Therefore, f n − has the same properties. Similarly, we define recursively f i − ( x ′ ) := E [( f − i + h i ) − ( x ′ Z i )] , x ′ ≥ , for every i = 1 , . . . , n − . This holds true for i ∈ { n − , n } as well, after setting f n := Id . Atthe end of this backwards recursion we obtain f : [0 , ∞ ) → [0 , ∞ ) . All f i : [0 , ∞ ) → [0 , ∞ ) are strictly increasing and vanish at .We now claim that Y can be calculated simply as f ( X ) : Theorem 3.1.
There exists a unique solution X i , Y i , X ′ i , i = 0 , . . . , n to the EbDO evaluationproblem. Furthermore, these processes can be obtained explicitly using the following forwardrecursion: For X ∈ [0 , ∞ ) , set Y = f ( X ) and X ′ = X . Then, for i ∈ { , . . . , n } , set X i := X ′ i − · Z i ,Y i := ( f − i + h i ) − ( X i ) , (1) X ′ i := X i − h i ( Y i ) . Proof.
Firstly, note that a random variable X ′ i is non-negative if the last two equations of (1)are satisfied with X i , Y i ≥ : We have ( f − i + h i )( Y i ) = X i , so X ′ i = X i − h i ( Y i ) = f − i ( Y i ) ≥ , i = 1 , . . . , n. This shows in particular that recursion (1) is well defined and the resulting processes arenon-negative. Now let us verify that such processes are in fact a solution to the problem:Clearly, X i , Y i , X ′ i are adapted (inductive argument). Note that X ′ n = f − n ( Y n ) = Y n . Sincethe properties X i = X ′ i − · Z i and X ′ i = X i − h i ( Y i ) are given we must merely show that Y i ,8 ∈ { , . . . , n } , is a martingale, i.e. Y i − = E [ Y i |F i − ] : Using the definition of f i − we have infact E [ Y i |F i − ] = E [( f − i + h i ) − ( X ′ i − · Z i ) |F i − ] = f i − ( X ′ i − ) , since σ ( Z i ) and F i − are independent and X ′ i − is measurable w.r.t. F i − . Now if i = 1 wehave f i − ( X ′ i − ) = f ( X ′ ) = Y . Otherwise, X ′ i − = f − i − ( Y i − ) yields f i − ( X ′ i − ) = Y i − ,which verifies the martingale property.On the other hand, we can show that for any solution to the EbDO evaluation problem,i.e. for any three non-negative and adapted processes X i , Y i , X ′ i , i = 0 , . . . , n , such that theproperties X i = X ′ i − · Z i , X ′ i = X i − h i ( Y i ) , Y n = X ′ n , X = X ′ and the martingale propertyfor ( Y i ) are satisfied, recursion (1) must already hold:Without even using the martingale property we first obtain Y n = ( f − n + h n ) − ( X n ) = (Id + h n ) − ( X n ) from Y n = X ′ n = X n − h n ( Y n ) . Next consider the property E [ Y n |F n − ] = Y n − : Using Y n = (Id + h n ) − ( X n ) and X n = X ′ n − · Z n we obtain Y n − = E [(Id + h n ) − ( X ′ n − Z n ) |F n − ] = f n − ( X ′ n − ) . This implies f − n − ( Y n − ) = X ′ n − = X n − − h n − ( Y n − ) , which then yields (1) for i = n − usinga straightforward transformation. Similarly, consider the martingale property E [ Y j +1 |F j ] = Y j for a j ∈ { , . . . , n − } while assuming that (1) is already verified for all i ∈ { j + 1 , . . . , n } .Using X j +1 = X ′ j · Z j +1 we have Y j = E [( f − j +1 + h j +1 ) − ( X ′ j Z j +1 ) |F j ] = f j ( X ′ j ) . This implies f − j ( Y j ) = X ′ j = X j − h j ( Y j ) , which then yields (1) for i = j using a straightfor-ward transformation. This completes an inductive argument showing that (1) holds for any i ∈ { , . . . , n } . Finally, considering E [ Y |F ] = Y we obtain Y = E [( f − + h ) − ( X ′ Z ) |F ] = f ( X ′ ) = f ( X ) . Remark 3.2.
It is worth noting that once the functions f i have been calculated the recursion(1) can be used to simulate the gross and net equities forward into the future. Using MonteCarlo simulation it is straightforward to obtain estimates for E [ h i ( Y i )] . This expected payoffrepresents the total estimated value of EbDOs which mature at time T i . By performing thesame simulation, but with a trend µ different from an investor can calculate or estimate theexpected payoff E [ h i ( Y i )] under the assumption that the company performs with some rate µ > due to profits expected on average. This allows to assign to each EbDO two prices:The risk neutral price , i.e. the expected payoff under the assumption that the assets underthe control of the company neither grow nor shrink, and the market price , which is the pricean investor is willing to pay based on his or her expectations regarding the company’s futureperformance. The risk neutral price is the one to be reported in the company’s official balancesheet, while the market price is a subjective value used, for instance, by a profit orientedanalyst. These two values can considerably differ, as an EbDO might be essentially worthlessif no growth occurs, but might become very valuable if the company grows at a steady paceover a prolonged period of time until the EbDO matures.9his difference between the risk neutral price and the subjective market price is the imme-diate reason why a deal between the company and the investor takes place and the EbDO issold for a given price: This sum is, from the point of view of the company, more valuable thenthe EbDO, since the company must assume a risk neutral view in its books. The investor onthe other hand may consider the EbDO more valuable than the fixed amount of money paidto purchase it, because the investor is working with a positive µ and thereby with a higherprice.Conversely, it is possible that investors predominantly work with a negative µ , due to poorexpectations regarding the company’s future performance. In this case both the company andthe investor might be interested in the opposite transaction, i.e. in a buy-back of outstandingEbDOs. This might allow the company to improve its balance sheet by removing outstandingobligations from it, while the investor protects himself or herself from future losses by meansof selling currently held positions. Remark 3.3.
Theorem 3.1 confirms that the key to solving the evaluation problem is obtain-ing the functions f i : [0 , ∞ ) → [0 , ∞ ) and ( f − i +1 + h i +1 ) − : [0 , ∞ ) → [0 , ∞ ) , i = 0 , . . . , n − .Note that the former is obtained from the latter by calculating an expectation. More precisely, f i ( x ′ ) = E [( f − i +1 + h i +1 ) − ( x ′ Z i +1 )] for any x ′ ≥ , where Z i +1 has a log-normal distribution.This motivates a simple numerical scheme to calculate or approximate the functions f i :Assume that we have two piecewise linear approximations of f i +1 and h i +1 , such thatthese approximations are increasing and equal zero at zero. We also assume that the piecewiselinear approximation of f i +1 is strictly increasing. Then the corresponding approximation of ( f − i +1 + h i +1 ) − is also piecewise linear, as inverse functions of piecewise linear functions arethemselves piecewise linear. It is also strictly increasing and equal zero at zero. By a slightabuse of notation, let us now imagine that ( f − i +1 + h i +1 ) − is piecewise linear. This means thatit is a linear combination of functions which are either equal or equal the identity on someinterval and everywhere else. So, for a given x ′ > the value E [( f − i +1 + h i +1 ) − ( x ′ Z i +1 )] canbe calculated as a linear combination of integrals of the form Z ba ρ ( v ) d v and/or Z ba e v ρ ( v ) d v, where −∞ < a < b < ∞ are constants that are calculated explicitly and where ρ is the densityof a normal distribution. This statement is true because ( f − i +1 + h i +1 ) − is piecewise linear and Z i +1 is equal to the exponential function applied to a normally distributed random variable.Observe further that expressions of the form R ba ρ ( v ) d v and R ba e v ρ ( v ) d v can be calculatedexplicitly using the cumulative function of the standard normal distribution.Since according to the above the function f i ( x ′ ) can be calculated explicitly for any given x ′ it is straightforward to obtain a strictly increasing piecewise linear approximation of f i andthen repeat the whole process to obtain an approximation of f i − and so on until a piecewiselinear approximation of f is obtained.While the above numerical scheme is straightforward to implement, the calculations mightbe somewhat slow if there is a large number of maturities T i : A new function f i : [0 , ∞ ) → [0 , ∞ ) needs to be calculated for each i . At the same time, if there is such a high densityof maturities the payoff process j P ji =1 h i ( Y i ) might increasingly resemble a continuousprocess with a payoff rate which is a function of the respective net equity. This motivateswhat we refer to as the continuous time model introduced and studied in the following sections.10 Problem formulation in continuous time
Under the continuous time model we assume that the EbDOs do not mature at finitely manypoints ≤ T < T < . . . < T n in time, but instead postulate that the pay-off due to EbDOstakes place continuously in time according to a rate function h : [0 , T ] × [0 , ∞ ) → [0 , ∞ ) overa given time period [0 , T ] . In other words the total pay-off over the time [0 , t ] , where t ∈ [0 , T ] ,is given by Z t h ( s, Y s ) d s, where Y s is the (a priori unknown) net equity of the company at time s . Here T > is largerthan the maturity of every EbDO, such that all outstanding EbDOs mature during the timeinterval [0 , T ] . Note that the value Y s is to be determined or modeled stochastically. As amatter of fact, our primary goal is to calculate Y . We assume that h ( t, · ) is monotonicallyincreasing and satisfies h ( t,
0) = 0 , where t ∈ [0 , T ] is arbitrary. Remark 4.1.
Observe that if the company has an EbDO with a fixed maturity T i andpayoff function h i on its books then the associated payoff h i ( Y T i ) can be approximated by thevalue ε R T i T i − ε h i ( Y s ) d s with some small ε > . This approximation fits mathematically thecontinuous time setting with the payoff rate function being ( s, y ) ε h i ( y ) [ T i − ε,T i ] ( s ) .In addition to the payoff rate h we have the gross equity X ≥ at time as the startingpoint of our calculation. Finally, we need to postulate the dynamics of the gross equity X s in time. Clearly, the gross equity is continuously diminished due to the pay-off rate h ( s, Y s ) .Apart from that we assume that there are random fluctuations characteristic for a geometricBrownian motion without a drift. Thus, we obtain that X has the dynamics X s = X − Z s h ( r, Y r ) d r + Z s X r · σ d W r , s ∈ [0 , T ] , where W is a Brownian motion and where σ ∈ [0 , ∞ ) is a fixed parameter determining theuncertainty about the future evolution of the gross equity due to the company’s intrinsicperformance. Note that we work under the assumption that on average the company neithershrinks nor grows in time as expectations about future profits cannot be included in thecalculations and known future expenditures were already incorporated in the calculation of X . This means that the total wealth t X t + Z t h ( s, Y s ) d s is a martingale. For accounting purposes we must also postulate that the net equity Y t , t ∈ [0 , T ] , is a martingale as well. Note that X T = Y T holds since we assume that there are nomore outstanding EbDOs beyond time T . Now, the martingale representation theorem yields Y s = X T − Z Ts Z r d W r , s ∈ [0 , T ] , with some square-integrable process Z . To sum up, we have to solve the following coupledforward-backward system: X s = X − R s h ( r, Y r ) d r + R s σX r d W r ,Y s = X T − R Ts Z r d W r , a.s. for all s ∈ [0 , T ] . (2)11n order to show existence and uniqueness of solutions X, Y we use the so-called the methodof decoupling fields, which was designed for the purpose of analyzing coupled systems. Webriefly introduce the theory of decoupling fields in the following section. A cornerstone of thismethod is the construction of a time-dependent random field u which connects X and Y via u ( s, X s ) = Y s . Remark 4.2.
Before introducing the theoretical backbone of our analysis and then actuallysolving the above problem in Section 6 let us point out that strongly coupled FBSDEs ingeneral are a powerful and flexible tool allowing to formulate and study more general andmore complicated problems than the one given by (2). For instance, one might be interestedin a problem where the pay-off rate h also depends on ω ∈ Ω or on the gross equity X . It isalso possible to study multi-dimensional problems where net equities of different companies areto be determined simultaneously, for instance due to two or more companies holding EbDOsof each other. This may occur if there is a set of affiliated companies forming a group. Wereserve such considerations and generalizations to future research and concentrate on the basictime-continuous problem provided by (2). As a key result of this paper we prove in Section 6 the solvability of (2). Even under Lipschitzassumptions for h , it is not trivial to show well-posedness of (2) due to its coupled nature. Bythis we mean that the forward equation, which describes the dynamics of X , depends on Y via h , while the backward equation, describing the dynamics of Y , depends on X via the condition Y T = X T . This means that neither of the two processes can be simulated or calculatedindependently of the other. Furthermore, coupled systems are not always solvable, even underLipschitz conditions. It is, thus, necessary to take more subtle structural properties intoaccount to conduct the proof. Our argumentation will be based on the method of decouplingfields which we briefly sum up in this section.For a fixed finite time horizon T > , we consider a complete filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) , where F consists of all null sets, ( W t ) t ∈ [0 ,T ] is a -dimensional Brownianmotion and F t := σ ( F , ( W s ) s ∈ [0 ,t ] ) with F := F T . The dynamics of an FBSDE is given by X s = X + Z s µ ( r, X r , Y r , Z r ) dr + Z s σ ( r, X r , Y r , Z r ) dW r ,Y t = ξ ( X T ) − Z Tt f ( r, X r , Y r , Z r ) dr − Z Tt Z r dW r , for s, t ∈ [0 , T ] and X ∈ R , where ( ξ, ( µ, σ, f )) are measurable functions such that ξ : Ω × R → R , µ : [0 , T ] × Ω × R × R × R → R ,σ : [0 , T ] × Ω × R × R × R → R , f : [0 , T ] × Ω × R × R × R → R , Throughout the whole section µ , σ and f are assumed to be progressively measurable withrespect to ( F t ) t ∈ [0 ,T ] .A decoupling field comes with an even richer structure than just a classical solution ( X, Y, Z ) . 12 efinition 5.1. Let t ∈ [0 , T ] . A function u : [ t, T ] × Ω × R → R with u ( T, · ) = ξ a.e. iscalled decoupling field for ( ξ, ( µ, σ, f )) on [ t, T ] if for all t , t ∈ [ t, T ] with t ≤ t and any F t -measurable X t : Ω → R there exist progressively measurable processes ( X, Y, Z ) on [ t , t ] such that X s = X t + Z st µ ( r, X r , Y r , Z r ) dr + Z st σ ( r, X r , Y r , Z r ) dW r ,Y s = Y t − Z t s f ( r, X r , Y r , Z r ) dr − Z t s Z r dW r ,Y s = u ( s, X s ) , (3)a.s. for all s ∈ [ t , t ] . In particular, we want all integrals to be well-defined.Some remarks about this definition are in place. • The first equation in (3) is called the forward equation , the second the backward equation and the third will be referred to as the decoupling condition . • Note that, if t = T , we get Y T = ξ ( X T ) a.s. as a consequence of the decoupling conditiontogether with u ( T, · ) = ξ . • If t = T we can say that a triplet ( X, Y, Z ) solves the FBSDE, meaning that it satisfiesthe forward and the backward equation, together with Y T = ξ ( X T ) . This relationship Y T = ξ ( X T ) is referred to as the terminal condition .For the following we need to introduce further notation.Let I ⊆ [0 , T ] be an interval and u : I × Ω × R → R a map such that u ( s, · ) is measurablefor every s ∈ I . We define L u,x := sup s ∈ I inf { L ≥ | for a.a. ω ∈ Ω : | u ( s, ω, x ) − u ( s, ω, x ′ ) | ≤ L | x − x ′ | for all x, x ′ ∈ R } , where inf ∅ := ∞ . We also set L u,x := ∞ if u ( s, · ) is not measurable for every s ∈ I . Onecan show that L u,x < ∞ is equivalent to u having a modification which is truly Lipschitzcontinuous in x ∈ R .We denote by L σ,z the Lipschitz constant of σ w.r.t. the dependence on the last component z . We set L σ,z = ∞ if σ is not Lipschitz continuous in z .By L − σ,z = L σ,z we mean L σ,z if L σ,z > and ∞ otherwise.For an integrable real valued random variable F the expression E t [ F ] refers to E [ F |F t ] ,while E t, ∞ [ F ] refers to ess sup E [ F |F t ] , which might be ∞ , but is always well defined as theinfimum of all constants c ∈ [ −∞ , ∞ ] such that E [ F |F t ] ≤ c a.s. Additionally, we write k F k ∞ for the essential supremum of | F | .In practice it is important to have explicit knowledge about the regularity of ( X, Y, Z ) .For instance, it is important to know in which spaces the processes live, and how they reactto changes in the initial value. Definition 5.2.
Let u : [ t, T ] × Ω × R → R be a decoupling field to ( ξ, ( µ, σ, f )) .1. We say u to be weakly regular if L u,x < L − σ,z and sup s ∈ [ t,T ] k u ( s, · , k ∞ < ∞ .13. A weakly regular decoupling field u is called strongly regular if for all fixed t , t ∈ [ t, T ] , t ≤ t , the processes ( X, Y, Z ) arising in (3) are a.e. unique and satisfy sup s ∈ [ t ,t ] E t , ∞ [ | X s | ] + sup s ∈ [ t ,t ] E t , ∞ [ | Y s | ] + E t , ∞ (cid:20)Z t t | Z s | ds (cid:21) < ∞ , (4)for each constant initial value X t = x ∈ R . In addition they are required to bemeasurable as functions of ( x, s, ω ) and even weakly differentiable w.r.t. x ∈ R n suchthat for every s ∈ [ t , t ] the mappings X s and Y s are measurable functions of ( x, ω ) andeven weakly differentiable w.r.t. x such that ess sup x ∈ R sup s ∈ [ t ,t ] E t , ∞ h | ∂ x X s | i < ∞ , ess sup x ∈ R sup s ∈ [ t ,t ] E t , ∞ h | ∂ x Y s | i < ∞ , ess sup x ∈ R E t , ∞ (cid:20)Z t t | ∂ x Z s | d s (cid:21) < ∞ . (5)3. We say that a decoupling field on [ t, T ] is strongly regular on a subinterval [ t , t ] ⊆ [ t, T ] if u restricted to [ t , t ] is a strongly regular decoupling field for ( u ( t , · ) , ( µ, σ, f )) .Under suitable conditions a rich existence, uniqueness and regularity theory for decouplingfields can be developed. Assumption (SLC): ( ξ, ( µ, σ, f )) satisfies standard Lipschitz conditions (SLC) if1. ( µ, σ, f ) are Lipschitz continuous in ( x, y, z ) with Lipschitz constant L ,2. k ( | µ | + | f | + | σ | ) ( · , · , , , k ∞ < ∞ ,3. ξ : Ω × R → R is measurable such that k ξ ( · , k ∞ < ∞ and L ξ,x < L − σ,z .In order to have a notion of global existence we need the following definition: Definition 5.3.
We define the maximal interval I max ⊆ [0 , T ] of the problem given by ( ξ, ( µ, σ, f )) as the union of all intervals [ t, T ] ⊆ [0 , T ] , such that there exists a weakly regulardecoupling field u on [ t, T ] .Note that the maximal interval might be open to the left. Also, let us remark that wedefine a decoupling field on such an interval as a mapping which is a decoupling field onevery compact subinterval containing T . Similarly we can define weakly and strongly regulardecoupling fields as mappings which restricted to an arbitrary compact subinterval containing T are weakly (or strongly) regular decoupling fields in the sense of the definitions given above.Finally, we have global existence and uniqueness on the maximal interval: Theorem 5.4 ([4], Theorem 5.1.11, Lemma 5.1.12 and Corollary 2.5.5) . Let ( ξ, ( µ, σ, f )) sat-isfy SLC. Then there exists a unique strongly regular decoupling field u on I max . Furthermore,either I max = [0 , T ] or I max = ( t min , T ] , where ≤ t min < T . In the latter case we have lim t ↓ t min L u ( t, · ) ,x = L − σ,z . (6)14 oreover, for any t ∈ I max and any initial condition X t = x ∈ R there is a unique solution ( X, Y, Z ) of the FBSDE on [ t, T ] satisfying sup s ∈ [ t,T ] E [ | X s | ] + sup s ∈ [ t,T ] E [ | Y s | ] + E (cid:20)Z Tt | Z s | ds (cid:21) < ∞ . Equality (6) allows to verify global existence, i.e. I max = [0 , T ] , via contradiction. Werefer to this approach as the method of decoupling fields. In order to apply the results of the previous section we need to allow arbitrary real initialvalues X = x ∈ R in (2). In addition, we extend the domain of h by setting it to zerowhenever x ≤ . Finally, we assume that h is uniformly Lipschitz continuous in x . Under thisassumption the problem (2) satisfies (SLC).We use the method of decoupling fields for proving that there exists a solution of (2) on [0 , T ] . Since the parameters of (2) satisfy the (SLC), there exists a maximal interval I max witha weakly regular decoupling field u (see Theorem 5.4).In the following fix t ∈ I max . Let ( X, Y, Z ) = ( X t ,x , Y t ,x , Z t ,x ) be the solution of (2)on [ t , T ] with initial value x ∈ R such that Y t = u ( t, X t ) a.s. for all ( t, x ) ∈ [ t , T ] × R .According to strong regularity u is weakly differentiable w.r.t. the initial value x ∈ R .In the following we denote by u x a version of the weak derivative of u w.r.t. x such that itcoincides with the classical derivative at all points for which it exists and with everywhereelse. Moreover, the processes ( X, Y, Z ) are weakly differentiable w.r.t. x . We can formallydifferentiate the forward and the backward equation in (2). One can verify that one caninterchange differentiation and integration and that a chain rule for weak derivatives applies(see Sections A.2 and A.3 in [4]). We thus obtain that for every version ( ∂ x X, ∂ x Y, ∂ x Z ) =( ∂ x X t ,x , ∂ x Y t ,x , ∂ x Z t ,x ) of the weak derivative, such that for every s ∈ [ t , T ] ( ∂ x X s , ∂ x Y s ) is a weak derivative of ( X s , Y s ) , we have for every t ∈ [ t , T ] : ∂ x X t =1 − Z tt h y ( s, Y s ) ∂ x Y s d s + Z tt σ · ∂ x X s d W s (7)and ∂ x Y t = ∂ x X T − Z Tt ∂ x Z s d W s , (8)for P ⊗ λ - almost all ( ω, x ) ∈ Ω × R .By redefining ( ∂ x X, ∂ x Y ) as the right-hand-sides of (7) and (8) respectively, we obtainprocesses ( ∂ x X, ∂ x Y ) that are continuous in time for all ( ω, x ) but remain weak derivativesof X, Y w.r.t. x . From now on, we always assume that ∂ x X and ∂ x Y are continuous in time.We also assume that for fixed t ∈ [ t , T ] the mappings ∂ x X t and ∂ x Y t are weak derivatives of X t and Y t w.r.t. x ∈ R . In particular ∂ x X t = 1 a.s. for almost all x ∈ R .In order to obtain bounds on the weak derivative u x , we study the process V t := u x ( t, X t ) , t ∈ [ t , T ] .Recall that Y t = u ( t, X t ) a.s. for all ( t, x ) ∈ [ t , T ] × R . Therefore, for fixed t ∈ [ t , T ] , theweak derivatives of the two sides of the equation w.r.t. x ∈ R must coincide up to a P ⊗ λ -15ull set. The chain rule for weak derivatives (see Corollary 3.2 in [1] or Lemma A.3.1. in [4])implies, for any fixed t ∈ [ t , T ] , that we have for P ⊗ λ - almost all ( ω, x ) ∂ x Y t { ∂ x X t > } = u x ( t, X t ) ∂ x X t { ∂ x X t > } = V t ∂ x X t { ∂ x X t > } . (9)Now, choose a fixed x ∈ R such that ∂ x X t = 1 a.s., (9), (7), (8) are satisfied for almostall ( ω, t ) ∈ [ t , T ] × Ω and, in addition, (9) is satisfied for t = t , P - almost surely. Note that,since ∂ x X , ∂ x Y are continuous in time, (7) and (8) in fact hold for all t ∈ [ t , T ] , P - almostsurely.Observe that V t is bounded since u x is bounded. We now turn to the dynamics of V . Lemma 6.1.
The process ( V t ) t ∈ [ t ,T ] has a time-continuous version which is an Itô process.Moreover, there exists a square-integrable progressive process b Z such that ( V, b Z ) is the uniquesolution of the BSDE V t = 1 − Z Tt b Z s dW s − Z Tt (cid:16) V s h y ( s, Y s ) − σ b Z s (cid:17) d s, t ∈ [ t , T ] . Proof.
Let τ n = T ∧ inf { t ≥ t : ∂ x X t ≤ n } . On [ t , τ n ] we have V t = ∂ x Y t ∂ x X t , a.e. Hence V has a version which is an Itô process on [ t , τ n ] . We denote the Itô process decomposition by V t = u x ( t , x ) + Z tt b Z s d W s + Z tt κ s d t, t ∈ [ t , τ n ] . The product formula yields, on [ t , τ n ] , d( V t ∂ x X t ) = V t ( − h y ( t, Y t ) ∂ x Y t d t + σ · ∂ x X t d W t )+ ∂ x X t (cid:16) κ t d t + b Z t d W t (cid:17) + σ · ∂ x X t b Z t d t. Observe that V t ∂ x X t = ∂ x Y t . The drift and diffusion coefficients coincide with the coefficientsin (8). This implies: ∂ x Z t = V t σ∂ x X t + ∂ x X t b Z t and − V t h y ( t, Y t ) ∂ x Y t + ∂ x X t κ t + σ · ∂ x X t b Z t . Using straightforward transformations we obtain b Z t = ∂ x Z t ∂ x X t − σV t and κ t = V t h y ( t, Y t ) V t − σ b Z t , again on the stochastic interval [ t , τ n ] . It remains to show that τ := lim n →∞ τ n = T a.s. Tothis end note that, according to (7), ∂ x X t satisfies, on [ t , τ ) , the linear SDE d ∂ x X t = α t ∂ x X t d t + σ∂ x X t d W t , where α t = − h y ( t, Y t ) V t is uniformly bounded. Consequently, ∂ x X t ∧ τ n = exp (cid:18)Z t ∧ τ n t (cid:18) α s − σ (cid:19) d s + Z t ∧ τ n t σ d W s (cid:19) . (lim n →∞ ∂ x X τ n ) ( ω ) = 0 for some ω , then lim n →∞ | W t ∧ τ n ( ω ) | = ∞ would hold for thesame ω . This, however, is false for almost all ω . In other words, the continuous process ∂ x X does not reach with probability and, therefore, lim n →∞ τ n = T a.s.In particular V t = ∂ x Y t ∂ x X t a.e. and V has a time-continuous version.In the following we assume that V refers to the time-continuous version of Lemma 6.1.Note that there exists a probability measure Q ∼ P such that d Q d P = exp (cid:18)Z Tt σ d W t − Z Tt σ d t (cid:19) . By Girsanov’s theorem W Qt := W t − R tt σ d s , t ∈ [ t , T ] , is a Brownian motion w.r.t. Q .Observe that V satisfies V t = 1 − Z Tt b Z s dW Qs − Z Tt V s h y ( s, Y s ) d s, t ∈ [ t , T ] . Working under the new probability measure we now prove:
Lemma 6.2.
For all t ∈ [ t , T ] we have a.s. q ≤ V t ≤ , where q := exp ( − T k h y k ∞ ) ∈ (0 , . Proof.
Define the cut-off function c ( v ) := (( v ∨ ∧ and consider the solution ( ˇ V , ˇ Z ) to theLipschitz BSDE ˇ V t = 1 − Z Tt ˇ Z s dW Qs − Z Tt c ( ˇ V s ) h y ( s, Y s ) d s, t ∈ [ t , T ] . It is not a priori clear that ˇ V and V are the same. The comparison theorem, applied to ( ˇ V , ˇ Z ) and the BSDE with parameters (1 , , implies ˇ V t ≤ . A comparison with the BSDEwith parameters (1 , −k h y k ∞ c ) , where −k h y k ∞ c ≤ − c h y ( s, Y s ) refers to the generator, yields ˇ V t ≥ q . Having established that ˇ V assumes values in [0 , we have c ( ˇ V s ) = ˇ V s such that thebounded processes ˇ V and V satisfy the same local-Lipschitz BSDE and are, therefore, thesame. As a consequence, we obtain q ≤ V t ≤ .Note that we have chosen a version of V such that V t = ∂ x Y t ∂ x X t for all t ∈ [ t , T ] . Moreover,for t = t , we have V t = ∂ x Y t = u x ( t , x ) , a.s.Since x was chosen arbitrarily outside a λ - null set, we have that the u x ( t , · ) is essentiallybounded by . Since the bound does not depend on t , by Theorem 5.4 it must hold that I max = [0 , T ] , which concludes the proof of well-posedness of the FBSDE (2). Moreover, thefollowing holds true: Proposition 6.3.
There exists a unique weakly regular decoupling field u on [0 , T ] to theproblem given by (2) . In addition to being strongly regular u has the following properties: • u is deterministic, i.e. it is a function of ( s, x ) only. • u ( s, x ) = x whenever x ≤ . • The weak derivative u x takes values in [ q, only. In particular, u is monotonicallyincreasing in x . inally, for any X = x ∈ [0 , ∞ ) there exists a unique solution ( X, Y, Z ) of the FBSDE (2) on [0 , T ] satisfying sup s ∈ [0 ,T ] E [ | X s | ] + sup s ∈ [0 ,T ] E [ | Y s | ] + E (cid:20)Z T | Z s | ds (cid:21) < ∞ . This unique solution satisfies ≤ Y s = u ( s, X s ) ≤ X s for all s ∈ [0 , T ] .Proof. The existence of ( X, Y, Z ) follows from Theorem 5.4. The fact that u x assumes valuesin [ q, is a direct consequence of Lemma 6.2, which holds for any t ∈ [0 , T ] . Also, u isdeterministic because h has this property (see e.g. Lemma 2.5.13. of [4]).We next show u ( t , x ) = x for x ≤ and t ∈ [0 , T ] : In this case the solution to (2) canbe provided explicitly: X t = x exp (cid:18)Z tt σ d W s − Z tt σ d s (cid:19) ,Y t = E [ X T |F t ] = X t , t ∈ [ t , T ] . It is straightforward to verify that these processes, being non-positive martingales, in factsolve the FBSDE, since h ( t, Y t ) = 0 . As a result u ( t , x ) = u ( t , X t ) = Y t = X t = x .It remains to show ≤ Y s = u ( s, X s ) ≤ X s for x ≥ : Since u ( s,
0) = 0 and since u isincreasing in x with the derivative being at most , we have ≤ u ( s, x ) ≤ x , for all x ≥ . An illustrating example
In general problem (2) cannot be solved in closed form and the use of a numerical scheme forcoupled FBSDE is necessary to calculate or approximate the decoupling field u : [0 , T ] × R → R .Once the decoupling field is obtained, the forward equation, i.e. the first equation in (2), isstraightforward to simulate and Y is obtained from Y s = u ( s, X s ) .However, in special cases the solution can be obtained explicitly: Assume that h is time-homogeneous and linear (for non-negative net equities), i.e. h ( s, y ) = γy + with some constant γ > . Let t ∈ [0 , T ] and x = X t ∈ [0 , ∞ ) . According to Proposition 6.3 (applied to theinterval [ t , T ] ) the process Y is non-negative, so Lemma 6.1 yields V t = 1 − Z Tt b Z s dW Qs − Z Tt V s γ d s, t ∈ [ t , T ] . Since and γ are deterministic it is natural to conjecture that V depends on time only, suchthat b Z vanishes. This would mean that V solves a quadratic ODE the solution for whichis straightforward to obtain. In deed, it is straightforward to check that t γ ( T − t ) is abounded solution to the BSDE satisfied by V in Lemma 6.1, which means that b Z = 0 and V t = 11 + γ ( T − t ) , t ∈ [ t , T ] . In particular, u x ( t , x ) = V t = γ ( T − t ) for all x ∈ [0 , ∞ ) and all t ∈ [0 , T ] . This implies u ( t, x ) = x γ ( T − t ) , ( t, x ) ∈ [0 , T ] × [0 , ∞ ) using Proposition 6.3. In particular Y = u (0 , X ) = X (1 + γT ) − .18ext we observe (2) and conclude that for X ∈ [0 , ∞ ) the corresponding process X on [0 , T ] is the unique solution to the linear SDE X s = X − Z s γ γ ( T − r ) X r d r + Z s σX r d W r . This allows to simulate the gross equity X under the assumption that there is no intrinsicgrowth or shrinkage. The interested reader is encouraged to apply the Itô formula to u toverify that t Y t = u ( t, X t ) is then indeed a martingale.If an investor wishes to work under the assumption of positive growth, given by an exponentialgrowth rate parameter µ > , the SDE to simulate would be X s = X + Z s (cid:18) µ − γ γ ( T − r ) (cid:19) X r d r + Z s σX r d W r . Regardless of µ ∈ R the function s E [ X s ] satisfies a linear ODE. Also, the respective valueof EbDOs maturing over a time interval [ a, b ] ⊆ [0 , T ] is calculated via Z ba E [ h ( s, Y s )] d s = Z ba E [ h ( s, u ( s, X s ))] d s = Z ba γ E [ X s ]1 + γ ( T − s ) d s. Note that in the above example the volatility parameter σ influences the volatility of theprocesses X, Y but not their expected values. Neither is the expected payoff (resp. value ofEbDOs) influenced. This is characteristic for the linear case only.
References [1] L. Ambrosio and G. Dal Maso. A general chain rule for distributional derivatives.
Proc.Am. Math. Soc. , 108(3):691–702, 1990.[2] S. Ankirchner and A. Fromm. Optimal control of diffusion coefficients via decouplingfields.
SIAM Journal on Control and Optimization , 56(4):2959–2976, 2018.[3] F. Delarue. On the existence and uniqueness of solutions to FBSDEs in a non-degeneratecase.
Stochastic Process. Appl. , 99(2):209–286, 2002.[4] A. Fromm.
Theory and applications of decoupling fields for forward-backward stochasticdifferential equations . PhD thesis, Humboldt-Universität zu Berlin, 2015.[5] A. Fromm and P. Imkeller. Utility maximization via decoupling fields.
PreprintarXiv:1711.06033 , 2017.[6] A. Fromm, P. Imkeller, and D. Prömel. An FBSDE approach to the Skorokhod embeddingproblem for Gaussian processes with non-linear drift.
Electron. J. Probab. , 20:38 pp., 2015.[7] J. Ma, P. Protter, and J. Yong. Solving forward-backward stochastic differential equationsexplicitly – a four step scheme.
Probab. Theory Relat. Fields , 98(3):339–359, 1994.[8] J. Ma, Z. Wu, D. Zhang, and J. Zhang. On well-posedness of forward-backward SDEs—aunified approach.
Ann. Appl. Probab. , 25(4):2168–2214, 2015.199] J. Ma, H. Yin, and J. Zhang. On non-Markovian forward-backward SDEs and backwardstochastic PDEs.
Stochastic Process. Appl. , 122(12):3980–4004, 2012.[10] J. Ma and J. Yong.
Forward-backward stochastic differential equations and their applica-tions , volume 1702 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1999.[11] E. Pardoux and S. Tang. Forward-backward stochastic differential equations and quasi-linear parabolic PDEs.
Probab. Theory Relat. Fields , 114(2):123–150, 1999.[12] S. Peng and Z. Wu. Fully coupled forward-backward stochastic differential equations andapplications to optimal control.