A pricing formula for delayed claims: appreciating the past to value the future

We consider the valuation of contingent claims with delayed dynamics in a Samuelson complete market model. We find a pricing formula that can be decomposed into terms reflecting the current market values of the past and the future, showing how the valuation of prospective cashflows cannot abstract away from the contribution of the past. As a practical application, we provide an explicit expression for the market value of human capital in a setting with wage rigidity. The formula we derive has successfully been used to explicitly solve the infinite dimensional stochastic control problems addressed in Biffis et al. (SIAM J Control Optim 58(4):1906–1938, 2020), Djeiche et al. (Stoch Process Appl 145:48–85, 2022) and Biagini et al. (SIAM J Financial Math 13(3):1004–1039, 2022).


Introduction
It is a standard result in asset pricing theory that the absence of arbitrage opportunities is essentially equivalent to the existence of an equivalent probability measure under which the price of any contingent claim is a local martingale after deflation by the money market account; see [14,20,21].In this paper we preserve the standard formulation of arbitrage pricing in a complete market model with security prices evolving as geometric Brownian motions (GBM).The main novelty of our work is that we consider contingent claims that have dynamics described by a stochastic delay differential equation (SDDE).It is perhaps surprising that using the no-arbitrage pricing machinery we are able to derive an explicit valuation formula in the case of dynamics with memory, which is notoriously difficult to study.In particular, we show that the price can be decomposed into a term related to the 'current market value of the past' (in a sense to be made precise below), and a term reflecting the 'market value of the present'.In our setting the contribution of the past is represented by the portion of a contingent claim's past trajectory that shapes its dynamics going forward. 1 Using our pricing formula, we demonstrate that in the market consistent valuation of future cashflows the contribution of the past cannot be neglected.As a practical application of our results, we consider in detail the case in which the contingent claim represents stochastic wages received by an agent over her lifetime (e.g., [17,5]).It is well known that when labor income is spanned by tradable assets, the market value of human capital can be easily derived via risk-neutral valuation.The empirical literature on labor income dynamics widely relies on auto-regressive moving average (ARMA) processes (e.g., [26], [1], [22], [28]); however [32], [24], and [16] show how SDDEs can be understood as the weak limit of discrete time ARMA processes.Therefore specific classes of SDDEs can be chosen as models for labor income, since they are continuous time counterparts of the empirically validated ARMA models.We thus consider the introduction of delayed drift and volatility coefficients in a GBM labor income model to provide a tractable example of wage dynamics that adjusts slowly to financial market shocks.We obtain a closed form solution for human capital, which makes explicit the contributions of the market value of the past and the present.Our results demonstrate that SDDEs are valuable modeling tools that can address the findings of a large body of empirical literature on wage rigidity (e.g., [23], [12], [3], [25]).Moreover, our results become an essential ingredient in finding explicit solutions of an interesting class of agent's lifecycle portfolio choice problems with no-borrowing-without repayment constraint, where the agent receive labor income whose dynamics is path dependent (see Section 4).Although we discuss the human capital application extensively, the extension to other applications is immediate.For instance, we provide some references to the literature on counterparty risk and derivatives valuation, in which analogous dynamics arise in the context of collateralization procedures entailing a delay in the marking-to-market procedure of over-the-counter derivatives (e.g., [9,10]).
It should be noted that no-arbitrage pricing in the case of delayed price dynamics has been recently studied by many authors, see for example [2,27].Their focus however is on proving completeness of the market, hence very different from ours.On the other hand, their work suggests that our results are of broader applicability, in particular to settings where market completeness is preserved, such as the case in which tradable assets have delayed drift and volatility terms (we refer to Section 5 for more comments).The paper is organized as follows.In Section 2 we introduce the setup and state our main result.In Section 3 we prove the result.In Section 4 we provide some applications.Section 5 concludes.In the Appendix we present the proof of some more technical results.

Setup and statement of the main result
We work in the framework of a Black and Scholes complete market model.On the filtered probability space (Ω, F , F, P) we consider the F-adapted vector valued process (S 0 , S), representing the price evolution of a money market account, S 0 , and n risky assets, S = (S 1 , ..., S n ) ⊤ , whose dynamics is the following ( Here Z is an n-dimensional Brownian motion and we assume that F := {F t } t≥0 is the filtration generated by Z, and enlarged with the P-null sets.µ ∈ R n , and the matrix σ ∈ R n×n is invertible.We are interested in the valuation of a payment stream represented by the F-adapted process y.The payment stream can be thought of as capturing the mark-to-market process of a trading account, the flow of profits and losses from a trading strategy, the collateral flows arising from an over-the-counter derivative transaction, or the labor income received by an agent over time.We assume the payment stream y to obey the following stochastic differential equation with delay in both the drift and the diffusion terms: Here µ y ∈ R, σ y ∈ R n , φ, ϕ i , for i = 1, . . ., n, are signed measures of bounded variation on [−d, 0] and The income stream y provides a simple, tractable example of income dynamics adjusting slowly to financial market shocks; the measures φ, ϕ 1 , ...ϕ n can be thought as representing the impact that the past has on the present value of the payment stream y. Existence of a unique solution of ( 2) is ensured by the following result.
Proof of Proposition (2.1) is given in Appendix A.1.Since the market is complete and the stream process y is spanned by the stock S, a certainty equivalent value for the stream of future (stochastic) payments can be found.It is in fact well known that in a complete market, the value of future uncertain cash flows is obtained by discounting their expected value under the unique risk neutral measure.This value is where the process ξ represents the unique stochastic discount factor whose time evolution is described (see e.g.[15]) by the following stochastic differential equation: The constant κ that appears in the above equation, represent the market price of risk.It is known that in a Black and Scholes market model where by 1 = (1, . . ., 1) ⊤ we mean the unitary vector in R n .Our aim is to provide an explicit formula for the quantity H(t 0 ) given in (3).The challenging aspect of the problem lies in the fact that we consider a payment stream whose dynamics is path dependent.This makes the problem considerably different from the ones studied in the literature and much harder to prove.Now we have all the ingredients to state the main result of the paper.Throughout the paper we will let the following assumption be in force Assumption 2.2.(i) The measure Φ on [−d, 0] defined as is a signed measure of bounded variation. (ii) Assumption 2.2 will be explained in details in Section 3.5.
Theorem 2.3.Under Assumption 2.2, for any t 0 ≥ 0, the quantity H(t 0 ) defined in (3) has the following explicit form where y(t 0 ) denotes the solution at time t 0 of equation (2), and G is given by In expression (8), we can recognize an annuity factor, K −1 , multiplying a term representing current value of y, and a term representing the current market value of the past trajectory of y over the time window (t 0 − d, t 0 ).The 'market value of the past' trades off the returns on the payment stream against its exposure to financial risk, as can be seen from expression (6).When the delay terms in the drift and volatility coefficients vanish, the valuation of the payment stream reduces to K −1 y(t 0 ).
Remark 2.4.The setup described above can be extended to the case of payments over a bounded horizon in some interesting situations.When the payment stream is received until an exogenous Poisson stopping time τ δ (representing for example death or irreversible unemployment, in the case payment stream y represents the labor income), expression (8) still applies, provided discounting is carried out at rate r + δ instead of r, where δ > 0 represents the Poisson parameter.This extension has been already considered in [5], [6] and [13].
The case in which payments are received until a finite (deterministic) time (representing for example permanent retirement from the labor market, in the case payment stream y represents the labor income) has been addressed in [7] (see Proposition 3.4).

Proof of the result
Within this Section we assume that (2) has a unique continuous F-adapted solution y, as ensured by Proposition 2.1.The proof of Theorem 2.3 can be divided in the following steps: • incorporate the discount factor ξ in the equivalent risk-neural probability measure P and rewrite the dynamics of y under P. Derive the deterministic delayed equation satisfies by the quantity Ẽ[y(t)|F t0 ]. (Subsection 3.1).
• Rewrite the delayed equation for Ẽ[y(t)|F t0 ] as a deterministic evolution equation taking values in a suitable Hilbert space that incorporate in its structure the past and the present.We will appeal to the so-called product-space framework for path-dependent equations (Subsection 3.2).
• Exploit suitable spectral properties of the operator that appears in the above mentioned infinite-dimensional formulation of the problem, to obtain expression (8) for H(t 0 ) (Subsections 3.3 and 3.4).
• Clarify the relation between the used spectral properties and our Assumption 2.2 (Subsection 3.5).
The above first three steps will lead to Proposition 3.8, whereas the last step will be formalized in Lemmas 3.10 and 3.11.Theorem 2.3 will then follow as an immediate consequence of these results.For readability, the proof of some technical Lemmas in the following Sections will be postponed to the Appendix.

Equivalent probability measure
We find more convenient to incorporate the discount factor ξ in the equivalent probability measure P. It is in fact well known that, in a complete market model, pricing formulas can be obtained, in a equivalent way, working under the (unique) risk-neutral measure P or working under the objective measure P but discounting by the state price process ξ.
We start by considering the equivalent probability measure Ps on F s such that 2 d Ps By [?, Lemma 3.5.3]we can write and thus where Φ is defined in (6).Integrating between t 0 and t we obtain and therefore, by taking the conditional expected value on both sides, we get The following Lemma guarantees that the stochastic integral with respect to Z is a martingale, and has zero mean.The proof is provided in Appendix A.2 3 Recall (see (3)) that our aim is to evaluate We will prove that ds is equal to the r.h.s. of ( 8) and then justify the equality We thus obtain that and, by definition of conditional mean and by Fubini's Theorem, the expression in ( 13) reduces to Therefore, defining we have that M t0 satisfies for t ≥ t 0 the equation (with random initial conditions) Existence of a unique solution of the above system is guaranteed by the following generalization of [4, Part II, Chapter 4, Theorem 3.2] to random initial conditions.

Reformulation of the problem in an infinite-dimensional framework
We now reformulate the differential equation with delay ( 16) as an evolution equation with values in the so called Delfour-Mitter Hilbert space, defined as whose elements are denoted as x = (x 0 , x 1 ).H is a Hilbert space when endowed with the inner product (x 0 , x 1 ), (y 0 , y 1 ) H = x 0 y 0 + x 1 , y 1 , the latter being the usual inner product of L 2 (−d, 0; R).We define the operator A : D(A) ⊂ H → H as with Φ defined in (6).We can then reformulate equation ( 16) as an evolution equation in H.
Consider, again for any fixed F t0 -measurable H-valued random variable m = (m 0 , m 1 ), the Hvalued process M(t 0 ; •) that is the solution on [t 0 , +∞) of We collect in the following Proposition some useful results about the above equation (for more details see e.g.[11, Appendix A]).

Proposition 3.3. (i)
The operator A generates a strongly continuous semigroup {S(t)} t≥0 in H.
(ii) S(t) is a compact operator for every t ≥ d.
(iii) For every F t0 -measurable H-valued random variable m the process is the unique weak (in distributional sense) solution of (19); in particular (iv) The Cauchy problem ( 19) is equivalent to (17).

Proof. (i)
The operator A can be written in the form (iii) Existence and uniqueness of a weak solution given by (20) for deterministic m is a classical result (see [11,Proposition A.5]).One can then easily generalize the result to random m.Property (21) follows from uniqueness of the solution.
As an immediate consequence of the above result we obtain the desired equivalence between equations ( 19) and ( 16).
Corollary 3.4.Let y be a solution of (2) on [0, t 0 ]; when choosing m as (m 0 , m 1 ) = (y(t 0 ), y(t 0 + •)), ( 19) is equivalent to (16) and in this case we have From now on we thus will work with formulation (19).The spectral properties of the operator A, that appears in this infinite-dimensional formulation, will be crucial to prove our result.We devote the next Section to the analysis of these properties.

Spectral properties of A
In the present Section we collect some technical results concerning the spectral properties of the operator A. Proof of Theorem 2.3 is based on the Lemmas presented here.The technical proofs are postponed to the Appendix.
Lemma 3.5.The spectrum of the operator A is given by where The spectrum σ (A) is a countable set and every λ ∈ σ (A) is an isolated eigenvalue of finite multiplicity.
The spectral bound of A is Proof.See [19, Chapter 7, Lemma 2.1 and Theorem 4.2] We ca explicitly compute the resolvent operator of A.
Lemma 3.6.Let ρ(A) denote the resolvent set of A and let λ ∈ R ∩ ρ (A).The resolvent operator of A at λ, denoted by R(λ, A) is given by with Proof.See Appendix A.3.
Lemma 3.7.For every real λ such that λ > λ 0 and every m = (m 0 , m 1 ) ∈ H we have Proof.Identity ( 27) is well known to hold for all real λ larger than the type of S(t).Since S(t) is compact for every t ≥ d, its type is actually equal to its spectral radius λ 0 .For a reference see e.

Deriving the explicit formula for H
In this Section we exploit the results derived in the above Sections to prove the following.
Proposition 3.8.Assume r > λ 0 , then for any t 0 ≥ 0, the quantity H(t 0 ) defined in (3) has the following explicit form where y(t 0 ) denotes the solution at time t 0 of equation (2), and G is given by Remark 3.9.Notice that the statement of the above result is the same of Theorem 2.3, but the assumptions here are different: we assume r > λ 0 instead of Assumption 2.2.An explanation of why we do actually consider Assumption 2.2 in Theorem 2.3, will be provided in the next Section.
Proof.Let m = (m 0 , m 1 ) = (y(t 0 ), y(t 0 + •)).We denote here by Π the projection on the first Starting from (10), we have ( From the above equalities we infer, in particular, the P-integrability of In fact, by the characteristic property of the conditional mean, and Fubini's Theorem we have that, for any Defining now K := K(r) and recalling (3), ( 28) and (29), by (30) and (31), the result immediately follows.

Motivations for Assumption 2.2
In Proposition 3.8 we proved our main result under the Assumption r > λ 0 .This requirement is difficult to check in practice, since it requires an explicit computation of the spectral bound λ 0 .In the present Section we therefore look for some sufficient conditions easier to check.Set for λ ∈ C where by |Φ| we denote the total variation measure of Φ. Set We note that λ 0 is the spectral radius of the operator A : D( A) ⊂ H → H defined as follows: Lemma 3.10.The function K, restricted to the real numbers, is strictly increasing and the spectral bound λ 0 is the only real root of the equation K(ξ) = 0.In particular, Proof.See Appendix A.4.
Recall the definition of K given in (23) and the definition of the spectral bound of A, λ 0 , given (24).
Thanks to the above two Lemmas it becomes now clear why we work under Assumption 2.2 in Theorem 2.3.It provides a sufficient condition for the condition r > λ 0 , imposed in Proposition 3.8, to hold.In fact, assume K > 0 as in Assumption 2.2, then by Lemmas 3.10 and 3.11 we immediately get r > λ 0 .

Optimal portfolio problems with path dependent labor income
As a first practical application of our results, we consider the case in which the contingent claim y, given in (2), represents stochastic wages received by an agent over her lifetime.As mentioned in the Introduction, stochastic delay differential equations allow for a realistic description of the labor income evolution in continuous time models.The path-dependency of the dynamics of the labor income (2) is in line with the empirical literature showing that wages adjust slowly to financial markets shocks.In fact, as outlined by Keynes in the article The General Theory of Employment, Interest and Money (1936), wages and prices do not adjust immediately to shocks in the economy.
There is a vast literature on the topic and we refer to [5] for a comprehensive list of relevant papers.In this context, expression (3) represents the present value of future discounted labor income and it has to be interpreted as the market value of the agent's human capital (see e.g., [17]).In Theorem 2.3 we therefore obtain a closed form solution for human capital, which makes explicit the contributions of the market value of the past and the present.Our valuation formula results in an explicit expression of human capital demonstrating the importance of appreciating the past to quantify the current market value of future labor income.Whereas Assumption 2.2 is all we need to provide the explicit valuation result of Theorem 2.3, the particular application to human capital requires labor income to be positive almost surely.A sufficient condition for this to be the case is provided in the following remark (for the proof see [5, Proposition 2.7]).
Remark 4.1.When ϕ i = 0 for all i = 1...n, that is when the delay term in the volatility coefficient of (2) vanishes, the variation of constants formula yields an explicit representation for y: where In this case one can see that, if x 0 > 0, x 1 ≥ 0 a.s., φ ≥ 0 a.s., then y(t) > 0 P-a.s.
The results of Theorem 2.3 and the approach followed in this paper show how tools from infinitedimensional analysis can be successfully used to address valuation problems that are non-Markovian, and hence beyond the reach of conventional approaches.For instance, Theorem 2.3 becomes an essential ingredient for solving some optimal control problems as the ones addressed in [5], [13] and [6].In the following Examples we briefly recall the results of these paper, that show how the findings of Theorem 2.3 (or suitable generalization of it) can be successfully used to address an interesting class of optimal control problems.
Example 4.2.In [5] the authors consider, and completely solve, an infinite horizon portfolio problem with borrowing constraints, in which an agent receives labor income which adjusts to financial market shocks in a path dependent way.We briefly describe the framework of [5], emphasizing how our Theorem 2.3 is crucial in the derivation of their results.
In the framework of a Black&Scholes complete market model described by (1), a representative agent is endowed with initial wealth w ≥ 0 and receives wages till her death.The time of death τ δ is modeled as an exponential random variable of parameter δ > 0. The wealth of the agent at time t ≥ 0 is denoted by W (t) and the wage rate is y(t).She can invest in the riskless and risky assets, and can consume at a rate c(t) ≥ 0. The wealth allocated to the risky assets is θ(t) ∈ R n at each time t ≥ 0. The agent has a bequest target B(τ δ ) at death, where the bequest process B(•) ≥ 0 is also chosen by the agent.To cover the gap between bequest and wealth at death: B(τ δ ) − W (τ δ ), the agent pays an instantaneous life insurance premium of δ(B(t) − W (t)) for t < τ δ (for more details see [17]).The agent's wealth (before death) is assumed to obey to the standard dynamic budget constraint of the Merton portfolio model, but with the labor income and insurance premium terms added in the drift, exactly as in [17]; thus the agent's wealth W satisfies the SDE In line with the empirical findings recalled above, to reflect a realistic economic setting, the dynamics of labor incomes adjust slowly to financial market shocks and it is modeled as a path-dependent delayed diffusion process of the form where µ y ∈ R, σ y ∈ R n , the functions φ(•), x 1 (•) belong to L 2 (−d, 0; R) (and thus (37) is a particular case of (2)).
Denoted by k > 0 the intensity of preference for leaving a bequest, γ ∈ (0, 1) ∪ (1, +∞) the riskaversion coefficient and ρ > 0 the discount rate, the aim is to maximize the expected utility from lifetime consumption and bequest, which is a no-borrowing-without-repayment constraint as the second term in (39) represents the agent's market value of human capital at time t.Human capital can be pledged as collateral, and represents the agent's maximum borrowing capacity.Notice that here ξ satisfies equation (4), with a drift of the form −ξ(t)(r + δ), as explained in Remark 2.4.Under such budget constraint, the authors find an explicit solution to expected power utility maximization from consumption and bequest.The proof of the result relies on the resolution of an infinite-dimensional Hamilton-Jacobi-Bellman (HJB) equation, that can be considered an infinitedimensional version of the classical Merton problem.From a technical point of view, the key idea is to extend the state space so to include the past path of y.In this way the problem becomes infinite dimensional and Markovian in the current wealth and in the path of y over the time window [−d, 0].In this infinite-dimensional reformulation of the problem it becomes essential to rewrite the constraint (39) by decomposing it in its present and past components.This kind of decomposition, provided by our Theorem 2.3, is the essential ingredient that allows to find an explicit solution to the HJB equation, which in turns allows to completely solve the problem and to find explicitly the optimal controls in feedback form (see [5, Theorem 5.1, Section 5]).
Example 4.3.In [6] is studied a robust version of the life-cycle optimal portfolio choice problem presented in Example 4.2.Theorem 2.3 is needed in order to face the infinite-dimensional robust Merton problem the authors derive and to obtain an explicit solution of it.
Example 4.4.Another generalization of the problem addressed in [5] is considered in [13].Similarly to [5], the authors consider a life-cycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky assets and a riskless bond subject to a borrowing constraint.However here the dynamics of the labor income has two main features.First, labor income adjust slowly to financial market shocks, as in [5].Second, the labor income y i of an agent i is benchmarked against the labor incomes of a population y n := (y 1 , y 2 , . . ., y n ) of n agents with comparable tasks and/or ranks.This last feature is faced taking the limit when n → +∞ so that the problem falls into the family of optimal control of infinite dimensional McKean-Vlasov Dynamics.The problem in studied in a simplified case where, adding a suitable new variable, the authors are able to find explicitly the solution of the associated infinite-dimensional HJB equation and find the optimal feedback controls.A necessary step to solve the problem is to provide a suitable reformulation of the no-borrowing without repayment constraint (39), where now the labor income y follows a stochastic delay differential equation where the drift contains not only a path-dependent term but also a mean reverting term.This issue is analyzed in [13,Section 3] where the authors provide a suitable generalization of formula (8) carefully readapting the technique we use in Section 3.

Counterparty risk and derivatives valuation
Example 4.5.As a simple example of application of our setup to the context of over-the-counter derivatives, in equation (2) consider the case of n = 1, µ 0 = 0, φ = 0, σ 0 = 0, and ϕ(s where δ a (s) indicates the delta-Dirac measure at a, so that equation (2) reads dX 0 (t) = X 0 (t − d)dZ(t). (40) Then, for t ∈ [0, d) we have In this case X 0 (t) is Gaussian, and dynamics (40) could be used to model, for example, the variation margin of an over-the-counter swap, when the collateralization procedure relies on a delayed markto-market value of the instrument (see [9], page 316, or [10], for example).

Conclusion
In recent years mathematical finance literature has seen the development of dynamics models that take into consideration the influence of past events on the current and future state of the system.Systems with delay find their well-deserved place in finance since delay in the dynamics can represent memory or inertia in the financial system.On the other hand, this kind of models are not new and they appear in many applications, think for instance to population dynamics models in biology where delays occur naturally for biological reasons, or to epidemic models where time delays are introduced to model constant sojourn times in a state, for example, the infective or immune state.The applications studied in the literature generally make rather clear that, when introduced in an explicit way, time delays may change the qualitative behavior of a model.From a mathematical perspective, delay systems has, in general, an infinite-dimensional nature.Problems connected to equations with delays are thus a challenging research area since the natural general approach to solve them is infinite-dimensional.The findings of this paper perfectly fit into this research area.On the one hand, our result highlights, once again, how the knowledge of the past affects the knowledge of the future.On the other hand, it points out how tools from infinite-dimensional analysis can be effectively used to address problems involving delays.

Concluding remarks
In this paper we have considered a standard Black&Scholes complete market model where securities evolve as geometric Brownian motions.Despite the classical setup we started from, we have been able to derive an explicit pricing formula for stream of payments with a delayed dynamics, by means of techniques from infinite-dimensional analysis.Our valuation formula results in an explicit expression demonstrating the importance of appreciating the past to quantify the current market value of the future.The approach followed in this paper highlights how tools from infinite-dimensional analysis can be successfully used to address valuation problems that are non-Markovian, and hence beyond the reach of conventional approaches.Moreover, as highlighted in Section 4, our results and the techniques we developed here, have already been successfully used to explicitly solve an interesting class of infinite dimensional stochastic optimal control problems.

Future work
From a financial perspective it may be interesting to address the problem of what happens if a path dependent object like ( 2) is made available for trade.A problem of this kind has actually already been studied in [2].There, the authors assume that the stock prices satisfy stochastic functional differential equations and derive an explicit formula for the valuation of an European call option on a given stock.The problem addressed in [2] could be consider a sort of counterpart of our setup.In fact, there the setup is non standard since the authors consider securities with path dependent dynamics.However, once the market has proved to be complete with a path-dependent market price of risk, the pricing of an European call becomes reasonably standard.The valuable contribution of [2] lies in showing that, if the securities of the market have pathdependent dynamics, then the market price of risk will have a path-dependent structure.As a consequence, pricing formulas can be obtained working under the objective measure P and discounting by a path-dependent state price process ξ, suitable related to the market-price of risk.
Bearing in mind this finding, we could argue as in [2] to see that, if an object of the form ( 2) is made available for trade, then the market price of risk would change becoming path-dependent.
The formal proof of this fact is outside the scope of this paper and it would be nothing innovative being just an extension of the results contained in [2].Anyway, the setup considered in [2] raises up some interesting questions since one may now ask how the valuation formula we derived in Theorem 2.3 becomes if the stock prices have themselves a delayed dynamics.That is, suppose to consider, on the filtered probability space (Ω, F , F, P), the F-adapted vector valued process (S 0 , S), representing the price evolution of a money market account, S 0 with rate of return r ≥ 0, and a single (for simplicity, as in [2]) asset whose price S(s) at time s satisfies the following stochastic differential equation with delay: where µ and σ are positive constants, φ S and φ S 1 are measure of bounded variation on [−d, 0], δ 0 is the delta Dirac function and Z is a one-dimensional Brownian motion (assume that F := {F t } t≥0 is the filtration generated by Z, and enlarged with the P-null sets).Arguing as in [2] one should expect the market price of risk to be of the following form: Thus the question one may ask is how the valuation formula we derived in Theorem 2.3 changes if the market price of risk has a stochastic and path-depentent form like in (43) (or like the one derived in [2] if one wants to work in their setup). 5This problem somehow would combine the novelties presented in [2] and the ones presented in this paper since both the stock prices and the stream of payments would have a stochastic delayed dynamics.So the problem would be to derive an explicit pricing formula for (3), with y as in (2), when the securities have themselves a path dependent dynamics of the form (42) or the form considered in [2].This is not a trivial extension of our result Theorem 2.3 because of the stochasticity and path dependency of the market price of risk.We leave it for future work. 4We emphasize that the formal proof passes through an application of the Girsanov Theorem.In order for the hypothesis of the Girsanov Theorem to be satisfied, it is reasonable to impose suitable restrictions on µ, σ, φ S and φ S 1 .For instance, a sufficient condition for (43) to be well defined is to assume φ S 1 ≡ 0, s 0 > 0, s 1 ≥ 0 a.s., φ S ≥ 0 a.s.which ensures (see Remark 4.1) that S(t) > 0 P-a.s..Here we just proceed heuristically. 5Let us point out that a different issue is to understand if y itself is tradable (for instance in the case it is not regarded as a stream of cashflows).In this case it is not difficult to verify that, an object y of the form (2) would be tradable under a delayed Sharpe ratio condition of the following form: Notice that, when the delay part is set to zero we recover the classical Sharpe ratio condition µ−r σ = µ 0 −r σ 0 . Moreover, mutatis mutandis, (44) has the same structure of the Sharpe ratio condition derived in [2].

A.1 Proof of Proposition 2.1
The existence and uniqueness result for (2) is not covered by the extant literature.When the initial datum x = (x 0 , x 1 ), seen as a function on [−d, 0], is continuous, existence and uniqueness of the strong solution to the SDDE for y is proved by [29,Theorem I.2].When the initial datum x ∈ R × L 2 ([−d, 0), dt; R), with the additional requirement that φ and (ϕ 1 , ...ϕ n ) are absolutely continuous w.r.t the Lebesgue measure, that is dφ = ϕdt, d(ϕ 1 , ...ϕ n ) = (φ 1 , ...φ n ) dt and the Radon-Nikodym densities ϕ, φ 1 , ...φ n ∈ L 2 ([−d, 0), dt), the existence and uniqueness result follows by [29,Remark I.3(iv)].We need to extend this latter result to the case in which φ and (ϕ 1 , ...ϕ n ) are signed measures of bounded variation on [−d, 0].We will prove the result by means of the same procedure employed for the proof of [6, Proposition B.2].There the authors prove the existence and uniqueness of the solution for an equation similar to (2), under more general assumptions on the measure φ, but with no delay in the diffusion term.Let us start by introducing the standard notation for the past path at t of a (deterministic) function The past path of y at t for the realization ω is thus y t (s, ω) := y(t + s, ω) s ∈ [−d, 0].As usual, we hide the dependence of the process on ω and write the delay terms in the drift and in the diffusion as follows: The delay parts in (2), given by ( 45) and (46) can be then expressed in terms of (an extension of) the following linear operators ok kernel type: which, in general, is not a continuous function, but only square integrable.Therefore, the operators L and G introduced in (47) and (48) cannot be applied to y t since the integrals in ( 45) and (46) may not be well defined.The first issue is thus to show that L and G admit continuous extensions to the square integrable functions on [−d, 0], as made precise in the following lemma.but for the sake of completeness, we prove the result for the operator G.For the operator L one follows the same reasoning. 1.
We estimate the i-th component (i = 1...n) of the above expression exploiting the Fubini Theorem and the Hölder inequality.We are now ready to prove Proposition 2.1.
Proof of Proposition 2.1.The proof of the result relies on a contraction type argument.The same argument has been used in the proof of [6,Proposition B.2].There the authors consider a SDDE of type (2) with no delay in the diffusion term.On the other hand they work in a more general setting considering a measure valued process φ in the delay integral of the drift term.
We provide here a sketch of the proof referring to [6] for more details.We will give just the details of the estimates concerning the delay integral in the diffusion term that is missing in [6].
Let us fix the initial condition (x 0 , x 1 ) ∈ R We consider the space L p (Ω; S T ), p ≥ 2, endowed with the norm .
In the sequel we will denote by p ′ := p p−1 the conjugate exponent to p and by p * := p p−2 the conjugate exponent to p 2 .Given y ∈ L p (Ω; S T ), let Here L and G are the continuous linear operators introduced in Lemma A.1 and ȳx1 ∈ L p (Ω; L 2 ([−d, T ]; R)) is defined as follows: We aim at proving that F maps L p (Ω; S T ) into itself for any p ≥ 2 and that it is a contraction on the same space when p > 4.
Let us start by proving that F maps L p (Ω, S T ), p ≥ 2, into itself.We write The boundedness of the terms that appears in the r.h.s. of (53), except the last one, can be proved following the lines of [6,Proposition B.2].We estimate the last term in the r.h.s. of (53) by means of the Burkholder-Davies-Gundy inequality where in the last inequality we exploited (50) of Lemma A.1.
Let us now prove that, for p > 4, F defines a contraction in L p (Ω, S T ).We endow this space by the equivalent norm where α > 0 will be chosen later on.Once we proved that F defines a contraction, by the Banach fixed point Theorem, we can infer the existence of a unique y ∈ L p (Ω; S T ) such that y = F (y), i.e. (56) 6 For more details on the estimates, the interested reader can consult that paper.
For the second term we get (57) For the third term, by means of the so called factorization method (see e.g.[?, Section 5.3]), for a given δ ∈ 1 p , 1 Let us now come to the estimate of the fourth term in (55).Exploiting the factorization method, for η ∈ 1 p , p−2 2p8 we rewrite that stochastic integral as follows Thanks to the Hölder inequality we estimate .
Therefore we obtain Now, recalling the definition of G and that, when r < d, zx1 r (s) − ȳx1 r (s) = 0 for s ∈ [−d, −r) (see (52)), by means of the Burkholder-Davis-Gundy (BDG) and the Hölder (H) inequalities, we obtain for all u ∈ [0, T ], where C i (α) → 0 as α → ∞, for i = 1, ..., 4. So, by taking α > 0 sufficiently large, this proves that F is a contraction, thus there exists a unique fixed point of it.In this way we prove the existence and uniqueness of the solution in the space L p (Ω, S T ) for p > 4. Since, for such p, L p (Ω, S T ) ⊂ L 2 (Ω, S T ), such solution also belongs to L 2 (Ω, S T ).To get uniqueness in the space L 2 (Ω, C([0, T ]; R)) one can take two solutions y and ỹ in this space and take their difference.Using the fact that both are fixed points of F , by means of the Gronwall Lemma one gets sup t∈[0,T ] E |y(t) − ỹ(t)| 2 = 0 and this concludes the proof.
A.2 Proof of Lemma 3.1 Proof.Let us denote with σ i y the i-th component of σ y , and let us show that Then the result immediately follows.
A.4 Proof of Lemma 3.10 Since K is an increasing function, we can infer Reλ ≤ ξ and taking the supremum in the definition of λ 0 we obtain λ 0 ≤ ξ.By the same argument, the relation K(r) > 0 ⇐⇒ r > λ 0 immediately follows.
A.5 Proof of Lemma 3.11 Proof.Since by Lemma 3.10 we know that K is an increasing function and K( λ 0 ) = 0, we just need to prove that K(λ 0 ) ≤ 0. Recalling the definition of λ 0 it is enough to show that, for every λ = a+ib ∈ C such that K(λ) = 0, it holds K(a) ≤ 0. We have that This concludes the proof.

Notation 1 .
Given a, b ∈ R n , by a • b we denote the scalar product in R n and by • we denote the corresponding norm.Given a signed measure of bounded variation φ on [−d, 0], by |φ| we denote its total variation.The space of Lebesgue square integrable (deterministic) functions on [−d, 0] is denoted by L 2 (−d, 0; R), that is the measure dt is understood.By W 1,2 (−d, 0; R) we denote the Sobolev space of weakly differentiable square integrable functions.Given φ, ψ ∈ L 2 (−d, 0; R)), by •, • we denote the usual L 2 scalar product.If functions a, b ≥ 0 satisfy the inequality a ≤ C(A)b with a constant C(A) > 0 depending on the expression A, we write a A b.

2
Recall that the state price density ξ characterizes the Radon-Nikodym derivative that defines the change of probability measure from objective P to risk-neutral measure P via the relation: ξ(s) = e −rs ρ(s) = e −rs d P dP (s).
Since the operators L and G are well-defined only on the space of continuous functions C([−d, 0]; R), when the initial datum does not belongs to C([−d, 0]; R) but just to L 2 ([−d, 0); R) problems may arise.In fact, consider the initial datum (x 0 , x 1 ) ∈ R × L 2 (−d, 0; R) and proceed by assuming that the solution to (2) exists.Denote the past path on the window [t − d, t] by y t : [−d, 0] → R, y t (s) := y(t + s) a.e.t ≥ 0, s ∈ [−d, 0].Then, for 0 ≤ t < d, the past path is

2 .
The existence of the bounded linear extension of L and G to L 2 ([−d, t]; R) is a consequence of inequalities (49) and (50) and the fact that C([−d, T ]; R) is dense in L 2 ([−d, T ]; R).