A pricing formula for delayed claims: Appreciating the past to value the future
Enrico Biffis, Beniamin Goldys, Cecilia Prosdocimi, Margherita Zanella
aa r X i v : . [ q -f i n . P R ] J a n A Pricing Formula for Delayed Claims:Appreciating the Past to Value the Future
Enrico Biffis ∗ Beniamin Goldys † Cecilia Prosdocimi ‡ Margherita Zanella § January 11, 2019
Abstract
We consider the valuation of contingent claims with delayed dynamics in aBlack & Scholes complete market model. We find a pricing formula that canbe decomposed into terms reflecting the market values of the past and thepresent, showing how the valuation of future cashflows cannot abstract awayfrom the contribution of the past. As a practical application, we provide anexplicit expression for the market value of human capital in a setting with wagerigidity.
Keywords—
Stochastic functional differential equations, delay equations, no-arbitrage pricing, human capital, sticky wages.
AMS Classification—
It is a standard result in asset pricing theory that the absence of arbitrage opportu-nities is essentially equivalent to the existence of an equivalent probability measure ∗ Enrico Biffis ( [email protected] ), Imperial College Business School, Imperial CollegeLondon, South Kensington Campus, London SW7 2AZ, UK. † Beniamin Goldys ( [email protected] ), School of Mathematics and Statistics,The University of Sydney, NSW 2006, Australia. ‡ Cecilia Prosdocimi ( [email protected] ), LUISS ”Guido Carli”, Rome, Italy. § Margherita Zanella ( [email protected] ), LUISS ”Guido Carli”, Rome, Italy. Using our pricingformula, we demonstrate that in the market consistent valuation of future cashflowsthe contribution of the past cannot be neglected.As a practical application of our results, we consider in detail the case in which thecontingent claim represents stochastic wages received by an agent over his/her lifetime(e.g., [17, 6]). 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. In[17] this result is extended to take into account endogenous retirement and borrowingconstraints. It is in general difficult to allow for richer dynamics of labor income,including unspanned sources of risk (e.g., [33]), or state variables capturing wagerigidity (e.g., [17], section 6). The empirical literature on labor income dynamicswidely relies on auto-regressive moving average (ARMA) processes (e.g., [28], [1],[22], [31]): Reiss [35], Lorenz [26], and Dunsmuir et al. [16] show how SFDEs canbe understood as the weak limit of discrete time ARMA processes. We thereforeconsider the introduction of delayed drift and volatility coefficients in a GBM laborincome model to provide a tractable example of wage dynamics that adjusts slowly tofinancial market shocks. We obtain a closed form solution for human capital, whichmakes explicit the contributions of the market value of the past and the present. Ourresults demonstrate that SFDEs are valuable modelling tools that can address thefindings of a large body of empirical literature on wage rigidity (e.g., [30], [25], [13],[3], [27]).Although we discuss the human capital application extensively, the extension The importance of the past in understanding the qualitative feature of a model with delay wasalso emphasized in Fabbri and Gozzi [18], although in a deterministic setting, when solving theendogenous growth model with vintage capital of Boucekkine et al. [7].
2o other applications is immediate. For instance, we provide some references to theliterature on counterparty risk and derivatives valuation, in which analogous dynamicsarise 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 dynamicshas been recently studied by many authors, see for example [2, 29]. Their focushowever 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 inwhich tradable assets have delayed drift and volatility terms.The paper is organized as follows. In the following section, we introduce thesetup, and state our main result. Section 3 presents mathematical tools used to dealwith the non-Markovian nature of a setting with delayed dynamics. In particular,we embed our problem in an infinite dimensional Hilbert space, on which the statevariable process is Markovian. In section 4 we prove our results by following a chainof five lemmas. Section 5 concludes.
Consider a Black-Scholes complete market model defined on our filtered probabilityspace (Ω , F , F , P ). Available for trade are a money market account, S , and n riskyassets with price vector process S . Prices have dynamics described by d S ( t ) = S ( t ) r d t, d S ( t ) = diag( S ( t )) { µ d t + σ d Z ( t ) } ,S (0) = 1 , S (0) ∈ R n> , (1)where Z is an n -dimensional Brownian motion, µ ∈ R n , and σ ∈ R n ⊗ R n , suchthat σσ ⊤ >
0. Here and in what follows, we use the notation R n> for the set { ( x i ) ∈ R n : x i > , i = 1 , . . . , n } . We assume that F := ( F t ) t ≥ is the filtrationgenerated by the Brownian Motion Z , and enlarged with the P -null sets. Definingthe market price of risk as κ := ( σ ⊤ ) − ( µ − r ) , (2)3he stochastic discount factor ξ can be shown to evolve as follows in our setting (see[15]): (cid:26) d ξ ( t ) = − ξ ( t ) r d t − ξ ( t ) κ ⊤ d Z ( t ) ξ (0) = 1 . (3)We consider the valuation of a payment stream represented by the F -adaptedprocess X . Our aim is to give an explicit expression to the following expectation: HC ( t ) := ξ ( t ) − E (cid:18)Z + ∞ t ξ ( t ) X ( t )d t (cid:12)(cid:12)(cid:12) F t (cid:19) . (4)The payment stream can be thought of as capturing the mark-to-market process of atrading account, the flow of profits and losses from a trading strategy, the collateralflows arising from an over-the-counter derivative transaction, or the labor incomereceived by an agent over time. In the latter case, expression (4) represents themarket value of the agent’s human capital (e.g., [17]), which could be extended to abounded horizon to model permanent exit from the labor market (e.g., death, irre-versible unemployment or retirement) along the lines indicated in Remark 2.3 below.We assume that the payment stream X obeys the following stochastic functionaldifferential equation (SFDE): d X ( t ) = h X ( t ) µ + R − d X ( t + s ) φ (d s ) i d t + X ( t ) σ ⊤ + R − d X ( t + s ) ϕ (d s )... R − d X ( t + s ) ϕ n (d s ) ⊤ d Z ( t ) ,X (0) = x ,X ( s ) = x ( s ) for s ∈ [ − d, µ ∈ R , σ ∈ R n , φ, ϕ i are signed measures of bounded variation on [ − d, i = 1 , . . . , n , and x ∈ R > , x ∈ L (cid:0) [ − d, R > (cid:1) . Note that when the paymentstream is understood as labor income, then the SFDE introduces slow adjustmentof wages to market shocks via delay terms in the drift and volatility coefficients of aGBM model. This provides a tractable model to capture the empirical evidence onwage rigidity discussed in the introduction.Equation (5) admits a unique strong solution, as ensured by Theorem I.1 andRemark 4 Section I.3 in [32], and provides a simple, tractable example of income4ynamics adjusting slowly to financial market shocks. Under dynamics (5), the val-uation of (4) can be carried out within a complete market model characterised bya unique stochastic discount process ξ . The results of [2] and [29] suggest that thesame applies to the more general setting in which the risky assets dynamics featuredrift and volatility terms with memory.To provide an explicit expression for (4), and formulate the main result of thispaper, we define the functions K ( λ ) := λ − ( µ − σ ⊤ κ ) − Z − d e λτ Φ(d τ ) , λ ∈ C , (6)˜ K ( λ ) := λ − ( µ − σ ⊤ κ ) − Z − d e λτ | Φ | (d τ ) , λ ∈ C , (7)where the measure Φ on [ − d,
0] is given byΦ( · ) := φ ( · ) − ϕ ( · )... ϕ n ( · ) ⊤ κ , (8)and by | Φ | we mean the total variation measure of Φ.We also define the constants K := K ( r ) = r − µ + σ ⊤ κ − Z − d e rτ Φ(d τ ) , (9)˜ K := ˜ K ( r ) = r − µ + σ ⊤ κ − Z − d e rτ | Φ | (d τ ) . (10)and assume the following conditions to hold throughout the paper. Hypothesis 1. (i) Φ is a signed measure of bounded variation on [ − d, ,(ii) ˜ K is strictly positive, i.e. ˜ K > . (11)We are now ready to state our main result, which provides an explicit decompo-sition of the market value of contingent payment stream X in our setting.5 heorem 2.1. Let ξ be defined as in (3) , and X evolve as in (5) . Then, underHypothesis 1, for any t ≥ we can write HC ( t ) = 1 K (cid:18) X ( t ) + Z − d G ( s ) X ( t + s ) d s (cid:19) , P − a.s., (12) where X ( t ) denotes the solution at time t of equation (5) , K is defined in (9) , and G is given by G ( s ) := Z s − d e − r ( s − τ ) Φ( d τ ) . (13)In expression (12), we recognize an annuity factor, K − , multiplying a term repre-senting current value of X , and a term representing the current market value of thepast trajectory of X over the time window ( t − d, t ). 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 (8). When the delay terms in the drift and volatil-ity coefficients vanish, the valuation of the payment stream reduces to K − X ( t ).Whereas Hypothesis 1 is all we need to provide the explicit valuation result of The-orem 2.1, the particular application to human capital requires labor income to bepositive almost surely. A sufficient condition for this to be the case is provided in thenext remark. Remark 2.2.
A sufficient condition for almost sure positivity of X is that φ ≥ and ϕ i = 0 for all i , so that the delay term in the volatility coefficient of (5) vanishes,and hence Φ coincides with φ and is nonnegative. Defining E ( t ) := e ( µ − σ ⊤ σ ) t + σ Z ( t ) , I ( t ) := Z t E − ( u ) Z − d X ( s + u ) φ ( d s ) d u, the variation of constants formula yields X ( t ) = E ( t ) (cid:0) x + I ( t ) (cid:1) , (14) which shows the positivity of labor income X in this special case, as we are consid-ering strictly positive initial conditions x ∈ R > and x ∈ L (cid:0) [ − d, R > (cid:1) . Remark 2.3.
The setup can be extended to the case of payments over a boundedhorizon in some interesting situations. When the payment stream is received until anexogenous Poisson stopping time τ (representing death or irreversible unemployment,for example, in the case of labor income), expression (12) still applies, provided dis-counting is carried out at rate r + δ instead of r , where δ > represents the Poissonparameter. xample 2.4. As a simple example of application of our setup to the context of over-the-counter derivatives, in equation (5) consider the case of n = 1 , µ = 0 , φ = 0 , σ = 0 , and ϕ ( s ) = δ − d ( s ) , where δ a ( s ) indicates the delta-Dirac measure at a , sothat equation (5) reads d X ( t ) = X ( t − d ) d Z ( t ) . (15) Then, for t ∈ [0 , d ) we have X ( t ) = x + Z t X ( s − d ) d Z ( s ) = x + Z t − d − d x ( τ ) d Z ( τ + d ) . (16) In this case X ( t ) is Gaussian, and dynamics (15) could be used to model, for example,the variation margin of an over-the-counter swap, when the collateralization proce-dure relies on a delayed mark-to-market value of the instrument (see [9], page 316,or [10], for example). It will be convenient to embed the labor income X in the infinite dimensional Hilbertspace H H := R × L (cid:0) [ − d, R (cid:1) , endowed with an inner product for x = ( x , x ) , y = ( y , y ) ∈ H defined as h x, y i H := x y + h x , y i L , where h x , y i L := Z − d x ( s ) y ( s ) d s. In what follows we omit the subscript L in the inner product notation.Let us define two operators, A and C , that act on the domain D ( A ) as follows: D ( A ) = D ( C ) := { ( x , x ) ∈ H : x ∈ W , (cid:0) [ − d, R (cid:1) , x = x (0) } , The Sobolev space W , (cid:0) [ − d, R (cid:1) is defined as W , (cid:0) [ − d, R (cid:1) := (cid:8) u ∈ L ([ − d, ∃ g ∈ L ([ − d, u ( θ ) = c + Z θ − d g ( s ) d s (cid:9) . A : D ( A ) ⊂ H → H ,A ( x , x ) := (cid:0) µ x + Z − d x ( s ) φ (d s ) , d x d s (cid:1) , with µ and φ as in (5), and C : D ( A ) ⊂ H → R n × L ([ − d, R n ) ,C ( x , x ) := σ x + R − d x ( s ) ϕ (d s )... R − d x ( s ) ϕ n (d s ) , , with σ and ϕ i as in (5). The following, well known fact (see [12]) is crucial for therest of the paper. Lemma 3.1.
The operator A generates a strongly continuous semigroup in H .Proof. The operator A can be written in the form A ( x , x ) = (cid:18)Z − d x ( θ ) a (d θ ) , d x d s (cid:19) , (17)where a (d θ ) = µ δ (d θ ) + φ (d θ ) , and δ is the delta-Dirac measure at zero. The measure a defines a finite measure on[ − d,
0] and the lemma follows immediately from Proposition A.25 in [12].The labor income in (5) can be equivalently defined as the first component of thesolution in H of the following equation (see [11]) d X ( t ) = AX ( t )d t + ( CX ( t )) ⊤ d Z ( t ) ,X (0) = x ,X (0 , s ) = x ( s ) for s ∈ [ − d, , (18)with A and C defined above, and x , x as in (5).8 Proof of the Main Result
The proof of Theorem 2.1 will follow by a chain of five lemmas stated below. To provethe theorem we will consider the conditional mean of the labor income X under anequivalent probability measure. We will show that this quantity obeys a deterministicdifferential equation described in terms of the operator A defined below. Let D ( A ) := (cid:8) ( x , x ) ∈ H : x ( · ) ∈ W , (cid:0) [ − d, R (cid:1) , x = x (0) (cid:9) , and A : D ( A ) ⊂ H −→ H A ( x , x ) := (cid:16) ( µ − σ ⊤ κ ) x + Z − d x ( s )Φ(d s ) , d x d s (cid:17) , (19)with ( µ − σ ⊤ κ ) ∈ R and Φ defined in (8). Replacing µ with µ − σ ⊤ κ and φ with Φ we infer from Lemma 3.1 and Hypothesis 1 (ii) that A generates a stronglycontinuous semigroup ( S ( t )) in H . Let (cid:0) M ( t ; 0 , m , m ) , M ( t, s ; 0 , m , m ) (cid:1) be thesolution at time t of the following differential equation d M ( t )d t = A M ( t ) ,M (0) = m ,M (0 , s ) = m ( s ) , s ∈ [ − d, , (20)with m ∈ R > and m ∈ L (cid:0) [ − d, R > (cid:1) . Then by definition S ( t ) (cid:18) m m (cid:19) = (cid:18) M ( t ; 0 , m , m ) M ( t, s ; 0 , m , m ) (cid:19) . (21)Denote by ρ ( A ) and R ( λ, A ) = ( λ − A ) − , the resolvent set and the resolventof A respectively and by σ ( A ) the spectrum of A . It is known (see for exampleProposition 2.13 on p. 126 of [4] or Proposition A.25 in [12]) that the spectrum of A is given by σ ( A ) = { λ ∈ C : K ( λ ) = 0 } , where K ( · ) is defined in (6). Moreover it is known that σ ( A ) is a countable set andevery λ ∈ σ ( A ) is an isolated eigenvalue of finite multiplicity. Let λ = sup { Re λ : K ( λ ) = 0 } (22)9e the spectral bound of A .At this point, in order to prove the chain of lemmas (that we employ to proveTheorem 2.1) we need to introduce a new operator ˜ A . Let D ( ˜ A ) = D ( A ) , and ˜ A : D ( A ) ⊂ H → H , ˜ A ( x , x ) := (cid:0) ( µ − σ T κ ) x + Z − d x ( s ) | Φ | (d s ) , d x d s (cid:1) , Appealing to Lemma 3.1 we infer that ˜ A generates a strongly continuous semigroupin H . Denote by ρ ( ˜ A ) and ˜ R ( λ, ˜ A ) = ( λ − ˜ A ) − , the resolvent set and the resolventof ˜ A respectively and by σ (cid:16) ˜ A (cid:17) the spectrum of ˜ A . Arguing as for A we havethat the spectrum of ˜ A is given by σ (cid:16) ˜ A (cid:17) = n λ ∈ C : ˜ K ( λ ) = 0 o , where ˜ K ( · ) is defined in (7). σ (cid:16) ˜ A (cid:17) is a countable set and every λ ∈ σ (cid:16) ˜ A (cid:17) is anisolated eigenvalue of finite multiplicity. Let λ = sup n Re λ : ˜ K ( λ ) = 0 o (23)be the spectral bound of ˜ A . Lemma 4.1.
The function R ∋ ξ −→ ˜ K ( ξ ) ∈ R , is strictly increasing and the spectral bound λ is the only real root of the equation ˜ K ( ξ ) = 0 . In particular, ˜ K defined by (10) is positive if and only if r > λ .Proof. The function ˜ K ( · ) : R → R is differentiable and˜ K ′ ( ξ ) = 1 + Z − d e ξτ | τ | | Φ | (d τ ) > , ξ ∈ R . It is easy to see that lim ξ →±∞ ˜ K ( ξ ) = ±∞ , K ( ξ ) = 0 has exactly one real solution ξ . Clearly, wehave ξ ≤ λ . To show that ξ = λ consider an arbitrary λ = x + iy such that˜ K ( λ ) = 0. Then 0 = x − µ + σ ⊤ κ − Z − d e xτ cos( yτ ) | Φ | (d τ ) ≥ x − µ + σ ⊤ κ − Z − d e xτ | Φ | (d τ )= ˜ K ( x ) . Therefore, ˜ K ( x ) ≤ x = Re λ ≤ ξ , hence λ ≤ ξ . Finally, exploitingthe fact that ˜ K is an increasing function, we immediately get λ < r if and only if˜ K ( r ) > Lemma 4.2.
Let K and ˜ K be defined as in (6) and (7) and let λ and λ be thespectral bounds of the operators A and ˜ A (respectively). It holds λ ≥ λ . Proof.
Exploiting the fact that ˜ K is an increasing function (see Lemma 4.1), in orderto prove that λ ≤ λ , it is sufficient to prove ˜ K ( λ ) ≤ ˜ K ( λ ). Recall that fromLemma 4.1 we have that λ ∈ R and actually λ coincides with the only real root ofthe equation ˜ K ( λ ) = 0. Therefore, we just have to prove that ˜ K ( λ ) ≤ λ = x + iy be a complex root of K ( λ ) = 0. In particular this means that its realpart satisfies the following equation x − ( µ + σ T κ ) − Z − d e xτ cos( yτ ) Φ(d τ ) = 0 . Let us show that ˜ K (Re( λ )) = ˜ K ( x ) ≤
0. Keeping in mind the previous equality, wehave that˜ K ( x ) = x − ( µ + σ T κ ) − Z − d e xτ | Φ | (d τ )= x − ( µ + σ T κ ) − Z − d e xτ cos( yτ ) Φ(d τ ) − Z − d e xτ | Φ | (d τ ) + Z − d e xτ cos( yτ ) Φ(d τ ) ≤ − Z − d e xτ (1 − cos( yτ )) Φ(d τ ) .
11t this point, writing Φ = Φ + − Φ − , with Φ + and Φ − the positive and negative partof Φ, respectively, we have˜ K ( x ) ≤ − Z − d e xτ (1 − cos( yτ )) Φ(d τ )= − Z − d e xτ (1 − cos( yτ )) Φ + (d τ ) + Z − d e xτ (1 − cos( yτ )) Φ − (d τ ) ≤ . Since λ was a generic element of the spectrum of A we have that ˜ K ( λ ) ≤
0. Thisconcludes the proof.
Lemma 4.3.
Let λ ∈ R ∩ ρ ( A ) . Then the resolvent R ( λ, A ) is given by R ( λ, A ) (cid:18) m m (cid:19) = (cid:18) u u (cid:19) (24) with u = 1 K ( λ ) (cid:2) m + Z − d Z s − d e λ ( τ − s ) Φ( d τ ) m ( s ) d s (cid:3) ,u ( s ) = e λs K ( λ ) (cid:18) m + Z − d Z s − d e λ ( τ − s ) Φ( d τ ) m ( s ) d s (cid:19) + Z s e − λ ( s − s ) m ( s ) d s . (25) Proof.
To compute R ( λ, A ), we will consider for a fixed (cid:18) m m (cid:19) ∈ H the equation( λ − A ) (cid:18) u u (cid:19) = (cid:18) m m (cid:19) , (26)that by definition of A is equivalent to ( λ − ( µ − σ ⊤ κ )) u − Z − d u ( τ )Φ(d τ ) = m λu − d u d s = m . Then u ( s ) = e λs u + Z s e − λ ( s − s ) m ( s ) d s , s ∈ [ − d, , u is determined by the equation (cid:0) λ − ( µ − σ ⊤ κ ) (cid:1) u = h m + Z − d (cid:18) e λτ u + Z τ e − λ ( s − τ ) m ( s ) d s (cid:19) Φ(d τ ) i or equivalently, u is given by the equation K ( λ ) u = m + Z − d Z s − d e λ ( τ − s ) Φ(d τ ) m ( s )d s , with K ( λ ) defined in (6). Thus for K ( λ ) = 0 the equation (26) is invertible and theresult follows.Recall that by S ( t ) we denote the strongly continuous semigroup generated by A .The following fact is well known. Lemma 4.4.
For any λ with Re( λ ) > λ we have Z ∞ e − λt S ( t ) (cid:18) m m (cid:19) d t = R ( λ, A ) (cid:18) m m (cid:19) . (27) Proof.
Formula (27) is standard for any strongly continuous semigroup provided λ isbig enough. To show that we can take λ > λ we invoke the fact that the semigroup S ( t ) is eventually compact, hence for the generators of the delay semigroups thegrowth bound and the spectral bound λ coincide, see Corollary 2.5 on p. 121 of[4]. For λ ∈ R such that K ( λ ) = 0, let (cid:0) f ( λ ) , g ( λ ) (cid:1) be defined as f ( λ ) := 1 K ( λ ) ,g ( λ, s ) := 1 K ( λ ) Z s − d e − λ ( s − τ ) Φ(d τ ) . (28) Lemma 4.5.
Fix t ≥ . Let M = ( M , M ) ∈ H be a solution to the followingdifferential equation d M ( t ) d t = A M ( t ) ,M ( t ) = m ,M ( t , s ) = m ( s ) , s ∈ [ − d, . (29)13 ith ( m , m ) ∈ R × L ([ − d, R ) . Then for any λ ∈ R , λ > λ we have Z + ∞ t e − λt M ( t ) d t = e − λt h ( f ( λ ) , g ( λ, · )) , ( m , m ) i H . Proof.
We first prove the result for t = 0. Recalling Lemma 4.3 and Lemma 4.4, wehave Z ∞ e − λt M ( t )d t = Z ∞ e − λt S ( t ) m d t = R ( λ, A ) m = 1 K ( λ ) (cid:20) m + Z − d Z s − d e λ ( τ − s ) Φ(d τ ) m ( s )d s (cid:21) = h ( f ( λ ) , g ( λ, · )) , ( m , m ) i H . (30)Now, consider t ≥
0, and let (cid:0) M ( t ; t , m , m ) , M ( t ; t , m , m ) (cid:1) be a solution toequation (29) starting at time t from ( m , m ). Then we have M ( t ; t , m , m ) = M ( t − t ; 0 , m , m ) . By (30), it holds Z + ∞ t e − λt M ( t ; t , m , m )d t = Z + ∞ e − λ ( s + t ) M ( s ; 0 , m , m )d s = e − λt Z + ∞ e − λs M ( s ; 0 , m , m )d s = e − λt h ( f ( λ ) , g ( λ )) , ( m , m ) i H . In order to prove Theorem 2.1 we also need the following technical lemma.
Lemma 4.6.
It holds that E (cid:16) Z tt (cid:13)(cid:13)(cid:13) X ( s ) σ + R − d X ( s + τ ) ϕ ( d τ ) ... R − d X ( s + τ ) ϕ n ( d τ ) (cid:13)(cid:13)(cid:13) R n d s (cid:17) < + ∞ . Proof.
Let us denote with σ i the i -th component of σ , and let us show that E (cid:16) Z tt (cid:2) X ( s ) σ i + Z − d X ( s + τ ) ϕ i (d τ ) (cid:3) d s (cid:17) < + ∞ .
14y the trivial inequality ( a + b ) ≤ a + b ), it is sufficient to show that E (cid:16) Z tt X ( s )( σ i ) d s (cid:17) < + ∞ , (31)and E (cid:16) Z tt (cid:20)Z − d X ( s + τ ) ϕ i (d τ ) (cid:21) d s (cid:17) < + ∞ . (32)To show (31), by Theorem 7.4 in [12] we can write E (cid:16) Z tt X ( s )( σ i ) d s (cid:17) ≤ ( σ i ) ( t − t ) E (cid:0) sup s ∈ [ t ,t ] X ( s ) (cid:17) < + ∞ . To show (32), by the H¨older inequality (cid:18)Z − d X ( s + τ ) ϕ i (d τ ) (cid:19) ≤ (cid:18)Z − d | X ( s + τ ) | ϕ i (d τ ) (cid:19) (cid:18)Z − d ϕ i (d τ ) (cid:19) = ϕ i ([ − d, (cid:18)Z − d | X ( s + τ ) | ϕ i (d τ ) (cid:19) . Thus E (cid:16) Z tt (cid:20)Z − d X ( s + τ ) ϕ i (d τ ) (cid:1) d s (cid:21) ≤ ϕ i ([ − d, Z tt Z − d E (cid:16) | X ( s + τ ) | (cid:17) ϕ i (d τ )d s ≤ (cid:16) ϕ i ([ − d, (cid:17) ( t − t ) sup τ ∈ [ − d, sup s ∈ [ t ,t ] E (cid:16) | X ( s + τ ) | (cid:17) . By Theorem 7.4 in [12], the expression above is finite.We can now provide the proof of Theorem 2.1.
Proof.
We have E (cid:16) Z + ∞ t ξ ( s ) X ( s )d s | F t (cid:17) = Z + ∞ t E (cid:0) ξ ( s ) X ( s ) | F t (cid:1) d s P -a.s. (33)15n fact, using the characteristic property of the conditional mean, and Fubini’s The-orem together with Theorem 7.4 in [12], for any F ∈ F t we have Z F E (cid:16) Z + ∞ t ξ ( s ) X ( s )d s | F t (cid:17) d P = Z F Z + ∞ t ξ ( s ) X ( s )d s d P = Z + ∞ t Z F ξ ( s ) X ( s )d P d s = Z + ∞ t Z F E (cid:0) ξ ( s ) X ( s ) | F t (cid:1) d P d s = Z F Z + ∞ t E (cid:0) ξ ( s ) X ( s ) | F t (cid:1) d s d P . To compute E (cid:0) ξ ( s ) X ( s ) | F t (cid:1) , let us consider the equivalent probability measured ˜ P ( s ) := e − | κ | s − κ ⊤ Z s d P , defined on F s . Note that d ˜ P ( s )d P = e − | κ | s − κ ⊤ Z s = e rs ξ ( s ) , and hence by Lemma 3.5.3 in [23] we can write E (cid:16) ξ ( s ) X ( s ) | F t (cid:17) = ξ ( t ) e − r ( s − t ) ˜ E (cid:16) X ( s ) | F t (cid:17) , where ˜ E denotes the mean under the measure ˜ P ( s ). Our aim is to evaluate Z + ∞ t E (cid:16) ξ ( s ) X ( s ) | F t (cid:17) d s = ξ ( t ) e rt Z + ∞ t e − rs ˜ E (cid:16) X ( s ) | F t (cid:17) d s. (34)Let ˜ P denote the measure, such that ˜ P (cid:12)(cid:12)(cid:12) F s = ˜ P ( s ) for all s ≥
0. By the GirsanovTheorem, the process ˜ Z ( t ) = Z ( t ) + κt (35)is an n -dimensional Brownian motion under the measure ˜ P , and the dynamics of X under ˜ P is dX ( s ) = (cid:2) ( µ − σ ⊤ κ ) X ( s ) + R − d X ( s + τ ) Φ(d τ ) (cid:3) d s + X ( t ) σ ⊤ + R − d X ( s + τ ) ϕ (d τ )... R − d X ( s + τ ) ϕ n (d τ ) ⊤ d ˜ Z ( s ) , t and t we obtain X ( t ) = X ( t ) + Z tt ( µ − σ ⊤ κ ) X ( s )d s + Z tt Z − d X ( s + τ )Φ(d τ )d s + Z tt X ( s ) σ ⊤ + R − d X ( s + τ ) ϕ (d τ )... R − d X ( s + τ ) ϕ n (d τ ) ⊤ d ˜ Z ( s ) , (36)and therefore˜ E (cid:16) X ( t ) | F t (cid:17) = X ( t ) + ( µ − σ ⊤ κ ) ˜ E (cid:16) Z tt X ( s )d s | F t (cid:17) + ˜ E (cid:16) Z tt Z − d X ( s + τ )Φ(d τ )d s | F t (cid:17) + ˜ E Z tt X ( s ) σ ⊤ + R − d X ( s + τ ) ϕ (d τ )... R − d X ( s + τ ) ϕ n (d τ ) ⊤ d ˜ Z ( s ) | F t . (37)By Lemma 4.6, which still applies after the change of measure, the stochastic integralwith respect to ˜ Z is a martingale, and has zero mean. By definition of conditionalmean and by Fubini’s Theorem, the expression in (37) gives˜ E (cid:16) X ( t ) | F t (cid:17) = X ( t ) + ( µ − σ ⊤ κ ) Z tt ˜ E (cid:0) X ( s ) | F t (cid:1) d s + Z tt Z − d ˜ E (cid:0) X ( s + τ ) | F t (cid:1) Φ(d τ )d s. (38)Deriving (38) with respect to t , we obtain the following, for t > t :d ˜ E (cid:16) X ( t ) | F t (cid:17) d t = ( µ − σ ⊤ κ ) ˜ E (cid:16) X ( t ) | F t (cid:17) + Z − d ˜ E (cid:16) X ( t + τ ) | F t (cid:17) Φ(d τ ) . (39)17e then see that ˜ E (cid:0) X ( t ) | F t (cid:1) must be a solution of d M d t ( t ) = ( µ − σ ⊤ κ ) M ( t ) + R − d M ( t + s ) Φ(d s ) , t > ,M ( t ) = m ,M ( t , s ) = m ( s ) , s ∈ [ − d, . (40)By Hypothesis 1 and Lemmas 4.1 and 4.2 we have r > λ , hence invoking Lemma4.5 we obtain Z + ∞ t e − rt ˜ E (cid:0) X ( t ) | F t (cid:1) d t = e − rt h (cid:0) f ( r ) , g ( r, · ) (cid:1) , ( m , m ) i H . Recalling (33) and (34), we can write E (cid:16) Z + ∞ t ξ ( s ) X ( s )d s | F t (cid:17) = ξ ( t ) e rt R + ∞ t e − rs ˜ E (cid:16) X ( s ) | F t (cid:17) d s = ξ ( t ) h (cid:0) f ( r ) , g ( r, · ) (cid:1) , ( m , m ) i H . Note that (cid:0) f ( r ) , g ( r, · ) (cid:1) = ( K , K G ( · )), with ( f, g ) defined in (28), K in (9), and G in(13). The proof is thus complete. In this paper, we have provided a valuation formula for streams of payments withdelayed dynamics in an otherwise standard, complete market model with risky assetsdriven by a GBM. As a practical example, we have discussed the application of ouranalysis to the valuation of human capital in a setting with sticky wages, where wagerigidity is obtained by introducing delay terms in the drift and volatility coefficients ofan otherwise standard GBM labor income dynamics. Our valuation formula results inan explicit expression of human capital demonstrating the importance of appreciatingthe past to quantify the current market value of future labor income. More generally,the approach followed in this paper shows how tools from infinite-dimensional analysiscan be successfully used to address valuation problems that are non-Markovian, andhence beyond the reach of coventional approaches.18 eferences [1] Abowd, J. M., and D. Card, On the Covariance Structure of Earnings and HoursChanges, (1989), Econometrica, 57(2), 411-445.[2]
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