aa r X i v : . [ q -f i n . M F ] A ug Linear Stochastic Dividend Model
Sander Willems ∗ August 2019
Abstract
In this paper we propose a new model for pricing stock and dividend derivatives. Wejointly specify dynamics for the stock price and the dividend rate such that the stock priceis positive and the dividend rate non-negative. In its simplest form, the model features adividend rate that is mean-reverting around a constant fraction of the stock price. Theadvantage of directly specifying dynamics for the dividend rate, as opposed to the morecommon approach of modeling the dividend yield, is that it is easier to keep the distributionof cumulative dividends tractable. The model is non-affine but does belong to the moregeneral class of polynomial processes, which allows us to compute all conditional moments ofthe stock price and the cumulative dividends explicitly. In particular, we have closed-formexpressions for the prices of stock and dividend futures. Prices of stock and dividend optionsare accurately approximated using a moment matching technique based on the principle ofmaximal entropy.
In recent years there has been an increased interest in trading dividend derivatives, in particulardividend futures. Since there also exists an active market for derivatives referencing the price ofthe stock paying the dividends, there is a need for derivative pricing models that can jointly pricederivatives on the stock and on the dividends. Since dividend derivatives typically reference thenominal amount of dividends paid over a window of time, it seems natural to directly specifytractable dynamics for the dividend payments, or the dividend rate if dividends are paid outcontinuously, under a risk-neutral measure. The challenging part of this approach is to keep thestock price positive. Indeed, in absence of arbitrage and frictions such as taxes, the stock pricemust decrease by exactly the amount paid out as dividend, which can push the stock price innegative territory if no connection is made between the dividend and stock price dynamics. Aneasy solution to this problem is to model dividend yields, i.e., the fraction of the stock price thatis paid out as a dividend, instead of dividends themselves. However, such a choice complicatesthe valuation of dividend derivatives, since their payoff now involves the product between thestock price and the dividend yield.In this paper, we consider a stock that pays out dividends continuously at a rate that is stochas-tically varying over time. The dividend rate is defined as a linear function of a multivariate fac- ∗ EPFL and Swiss Finance Institute. E-mail: sander.willems at epfl.ch In reality, dividends are paid out discretely instead of continuously. Most liquid dividend derivatives, however,reference dividends paid by a stock index, in which case continuously paid dividends are considered an acceptableapproximation. − Let X t denote the stock price process and D t the instantaneous dividend rate. Suppose forsimplicity that interest rates are constant. Consider the following dynamics for ( X t , D t ) undera risk-neutral measure Q D t = ⊤ Y t , (1)d X t = ( rX t − D t ) d t + σ ( X t − D t a ) d W t , (2)d Y t = ( bX t + βY t ) d t + r X t − D t a h ν p Y ,t d B ,t , . . . , ν d p Y d,t d B d,t i ⊤ , (3)3here r ∈ R is the short-rate, σ, a > Y t is a d -dimensional factor process, d ≥ b ∈ R d , β ∈ R d × d , ν , . . . , ν d ≥
0, and ( W t , B ,t , . . . , B d,t ) is a (1 + d )-dimensional standard Brownianmotion. The following proposition provides parameter conditions such that (2)-(3) admits aunique solution taking values in E = { ( x, y ) ∈ R d : x > , y ≥ , ⊤ y ≤ ax } . Proposition 2.1.
Denote by x − = min( x, . Suppose that b k + a min l =1 ,...,dl = k β − k,l ≥ , for all k ∈ { , . . . , d } , (4) r − a − max k =1 ,...,d ( ⊤ β ) k − ⊤ ba ≥ . (5) Then for every initial value ( X , Y ) ∈ E there exists a unique in law E -valued solution ( X t , Y t ) to (2) - (3) . The solution satisfies • Y k,t > for all t ≥ if Y k, > and b k + min l =1 ,...,dl = k (cid:18) aβ k,l + ν k (cid:19) − > ν k • aX t > ⊤ Y t for all t ≥ if aX > ⊤ Y and r − a − max k =1 ,...,d ( ν k a + ( ⊤ β ) k ) − ⊤ ba > . (7)We henceforth assume that the inequalities in (4) and (5) are satisfied and ( X , Y ) ∈ E . Theabove proposition shows in particular that we have X t > D t ≥ t ≥
0. Thecondition in (6) can be used to bound D t strictly away from zero, although this is not requiredfrom an economic point of view. The diffusive term of X t is specified such that it vanishes atthe boundary D t = aX t , which is necessary to keep the process inside E . The condition in (7)can be used to bound the stock price volatility strictly away from zero. Remark 2.2.
Remark that the more general specification D t = γ ⊤ Y t , for some γ ∈ (0 , ∞ ) d , isequivalent to the one in (1) . Indeed, if we define ˆ Y t = CY t with C = diag( γ , . . . , γ d ) , then wecan write D t = γ ⊤ Y t = ⊤ ˆ Y t . The dynamics of ˆ Y t are of the same form as the dynamics of Y t d ˆ Y t = (ˆ bX t + ˆ β ˆ Y t ) d t + r X t − D t a (cid:20) ˆ ν q ˆ Y ,t d B ,t , . . . , ˆ ν d q ˆ Y d,t d B d,t (cid:21) ⊤ , with ˆ b = Cb , ˆ β = CβC − , and ˆ ν k = √ γ k ν k , k = 1 , . . . , d . If we define the dividend yield δ t = D t X t , then we have0 ≤ δ t ≤ a. The dividend yield is therefore bounded from above by a parameter a > x t = log( X t ) is given byd x t = (cid:18) r − δ t − σ − δ t a ) ) (cid:19) d t + σ (1 − δ t a ) d W t . a largeenough.The following proposition shows that our model does not contain a bubble in the stock pricedynamics. Proposition 2.3.
The discounted gains process G t = e − rt X t + R t e − rs D s d s is a martingale. If ⊤ b > , then we have for all t ≥ E t (cid:20)Z ∞ t e − r ( s − t ) D s d s (cid:21) = X t . (8)Equation (8) shows that the stock price is equal to the present value of all future dividends in ourmodel. It is important to note that this is not a trivial relationship. Indeed, from no-arbitrageprinciples, it only follows that the present value of future dividends must be lower than or equalto the stock price, see e.g. Filipovi´c and Willems (2018). In general, if the discounted gainsprocess is a martingale, then X t = E t [e − r ( T − t ) X T ] + E t (cid:20)Z Tt e − r ( s − t ) D s d s (cid:21) , T ≥ t. A positive difference between the stock price and the present value of future dividends can beinterpreted as the present value of a terminal payment at an infinite time horizon, which isdifficult to reconcile with standard economic theory. Proposition 2.3 shows that, in our model,we have lim T →∞ E t [e − r ( T − t ) X T ] = 0 if ⊤ b >
0. The derivation of this result relies on the lineardrift structure of ( X t , Y t ) and the geometry of E , which are a key ingredients of our model. InExample 2.5 in the next section, we illustrate a parameterization where the assumption ⊤ b > Remark 2.4.
The processes X t and Y t have zero quadratic covariation. Note that this doesnot mean that dividends are independent of the stock price, since X t still enters in the drift anddiffusion function of Y t . The dynamics of X t can be generalized to allow for non-zero quadraticcovariation with Y t as follows d X t = ( rX t − D t ) d t + σ ( X t − D t a ) d W t + r X t − D t a d X k =1 η k p Y k,t d B k,t , for some parameters η k ∈ R , k = 1 , . . . , d . All the results in the paper are easily adjusted toaccommodate this generalization. For d = 1, we obtain the following model dynamicsd X t = ( rX t − D t ) d t + σ ( X t − D t a ) d W t , (9)d D t = ( bX t + βD t ) d t + ν r D t ( X t − D t a ) d B t . (10)5f β <
0, then D t is mean-reverting around − bβ X t , with an upper bound of aX t . The inwardpointing drift conditions (4) and (5) become0 ≤ b ≤ a ( r − a − β ) . (11)Boundary non-attainment is satisfied if 0 < D < aX and ν < b < a ( r − a − β ) − ν . The dividend yield becomes an autonomous diffusion with the following dynamicsd δ t = (cid:18) b + ( β − r ) δ t + δ t + σ δ t (1 − δ t a ) (cid:19) d t − σδ t (1 − δ t a )d W t + ν r δ t (1 − δ t a )d B t . Remark that the dividend yield process is not a polynomial diffusion, due to the terms δ and δ in the drift, and δ and δ in the diffusion function. However, since δ t is typically in the order ofpercentage points, higher powers of δ contribute relatively little to the dynamics. In particular,the dividend yield δ t has approximately a linear drift b + ( β − r + σ ) δ t .We end this section with an example where the assumption in Proposition 2.3 is violated. Example 2.5. If b = 0 , then we have E t [ D T ] = e β ( T − t ) D t for all T ≥ t , and (11) becomes r − β ≥ a > . The present value of future dividends is E t (cid:20)Z ∞ t e − r ( s − t ) D s d s (cid:21) = Z ∞ t e ( β − r )( s − t ) d s D t = D t r − β . Using aX t ≥ D t and r − β ≥ a we obtain E t (cid:20)Z ∞ t e − r ( s − t ) D s d s (cid:21) ≤ ar − β X t ≤ X t . The present value of future dividends is therefore lower than or equal to the stock price, asrequired by absence of arbitrage. If aX t > D t or r − β > a , then we have an example where thepresent value of future dividends is strictly below the stock price, i.e., E t [e − r ( T − t ) X T ] does notgo to zero as T → ∞ . The above example shows that the presence of X t in the drift of D t is crucial for the stock priceto be equal to the present value of future dividends. In this section we show how to compute prices of derivatives referencing the stock price and/orthe dividends paid over some time interval.
Define the cumulative dividend process as C t = Z t D s d s, t ≥ , , t ]. In contrast to the instan-taneous dividend rate D t , the cumulative dividend C t is observable in practice. The process( C t , X t , Y t ) is jointly a polynomial diffusion, so we are able to compute all conditional momentsin closed form, see e.g. Filipovi´c and Larsson (2016) for details. Let Pol n denote the set ofpolynomials p : R d → R with 1 ≤ deg( p ) ≤ n . Applying the infinitesimal generator G of( C t , X t , Y t ) to a twice differentiable function f ( c, x, y ) gives G f = ( ⊤ y, rx − ⊤ y, ( bx + βy ) ⊤ ) ∇ f + 12 σ ( x − ⊤ ya ) f xx + 12 d X k =1 ν k y k ( x − ⊤ ya ) f y k y k , where the subscripts of f denote partial derivatives, ∇ f the gradient of f , and we have omittedthe function arguments for brevity. It is easily verified that G Pol n ⊆ Pol n for any n ∈ N .Therefore, if we fix a vector of polynomial basis functions H n = ( h , . . . , h N n ) ⊤ for Pol n , with N n = dim(Pol n ), then we can find a unique matrix G n such that for all ( c, x, y ⊤ ) ⊤ ∈ R d G H n ( c, x, y ) = G n H n ( c, x, y ) . By definition of the infinitesimal generator, we obtain the following moment formula E t [ H n ( C T , X T , Y T )] = e G n ( T − t ) H n ( C t , X t , Y t ) , ∀ T ≥ t. (12)In particular, we can compute all the F t -conditional mixed moments of ( C T , X T ) in closed formfor all T ≥ t . Remark 3.1.
If one is only interested in the moments of X T , then there is no need to augmentthe state with C t , since ( X t , Y t ) is already a polynomial diffusion on its own. For n = 1, we can without loss of generality choose the basis H ( c, x, y ) = ( c, x, y ⊤ ) ⊤ . Thematrix G then becomes G = ⊤ r − ⊤ b β . The most actively traded linear derivatives are stock futures and dividend futures. Stock futuressettle on the stock price at some terminal date T and dividend futures settle on the dividendspaid in a time interval [ T , T ]. The moment formula (12) can be used to compute prices of stockfutures and dividend futures. Indeed, futures prices are given by the risk-neutral expectation ofthe terminal settlement price because of continuous marking-to-market, so we get E t [ X T ] = e ⊤ e G ( T − t ) H ( C t , X t , Y t ) , (13) E t [ C T − C T ] = e ⊤ (cid:16) e G ( T − t ) − e G ( T − t ) (cid:17) H ( C t , X t , Y t ) , (14)where e k denotes the k -th canonical basis vector in R d , T ≥ t , and T ≥ T ≥ t . In case thereference period of the dividend futures has already started, i.e., 0 ≤ T ≤ t ≤ T , we get E t [ C T − C T ] = e ⊤ e G ( T − t ) H ( C t , X t , Y t ) − C T . (15)Without loss of generality we can assume that T = 0, in which case C T = 0 and C t is theamount of dividends already paid. 7emark that the volatility parameters σ and ν , . . . , ν d do not enter into the prices of dividendfutures, which is a consequence of the linear drift structure of Y t . This allows us, for example,to calibrate b and β to dividend futures first, and subsequently use σ and ν , . . . , ν d to calibratenon-linear derivatives such as stock and dividend options. The parameter a also does not appearin the prices of dividend futures, however it should be noted that the value of a affects the valuesthat b and β are allowed to take, because of the inequalities (4) and (5) that we assume to betrue. Consider a derivative on the stock price with discounted payoff at time T given by F ( X T ), forsome function F . In absence of arbitrage, its price at time t ≤ T is given by π t = E t [ F ( X T )] . The probability density function of X T is not known explicitly, so we cannot compute π t bydirect integration in general. We do however know all the moments of X T through the momentformula (12). In particular, if F is a polynomial, then we can compute π t explicitly. If F is nota polynomial, we approximate π t using the available stock price moments and the principle ofmaximum entropy, similarly as in Filipovi´c and Willems (2018). Specifically, denote by M n = E t [ X nT ], n = 1 , . . . , N , the first N ≥ f which has the same first N moments as X T and has maximal entropy:max f − Z ∞ f ( x ) ln f ( x ) d x s . t . Z ∞ x n f ( x ) d x = M n , n = 0 , . . . , N, (16)where we set M = 1 so that the density integrates to one. Jaynes (1982) motivates such achoice by noting that maximizing entropy incorporates the least amount of prior informationin the distribution, other than the imposed moment constraints. In this sense it is maximallynoncommittal with respect to unknown information about the distribution. Straightforwardfunctional variation with respect to f gives the following unique solution f ( N ) to the optimizationproblem in (16) f ( N ) ( x ) = exp − N X n =0 λ n x n ! , where the Lagrange multipliers λ , . . . , λ N have to be solved numerically from the momentconstraints. Finally, we approximate π t by numerically computing the integral π ( N ) = Z ∞ F ( x ) f ( N ) ( x ) d x. We can use exactly the same approach to price dividend derivatives with discounted payoff attime T given by P ( C T − C T ), for some function P . All we need are the moments of C T − C T ,which can be computed explicitly using the law of iterated expectations and the moment formula812) as follows E t [( C T − C T ) n ] = n X k =0 (cid:18) nk (cid:19) E t [( − C T ) n − k E T [ C kT ]]= n X k =0 (cid:18) nk (cid:19) E t [( − C T ) n − k w ⊤ k e G k ( T − T ) ˜ H k ( C T , X T , Y T )]= n X k =0 (cid:18) nk (cid:19) v ⊤ k e G n ( T − t ) H n ( C t , X t , Y t ) , where v k and w k are the unique vectors satisfying w ⊤ k H k ( c, x, y ) = c k and v ⊤ k H n ( c, x, y ) =( − c ) n − k w ⊤ k e G k ( T − T ) H k ( c, x, y ). As an example, we calibrate the single factor model (9)–(10) using a snapshot of real marketdata on 21/12/2015. The stock in the calibration exercise is the Euro Stoxx 50 index, theleading blue-chip stock index in the Eurozone. The Euro Stoxx 50 index is well suited forcalibrating our model since it has a liquid dividend derivatives market associated with it. TheEuro Stoxx 50 dividend futures contracts are exchange traded on Eurex and reference the sumof the declared ordinary gross cash dividends (or cash-equivalent) on index constituents thatgo ex-dividend during a given calendar year, divided by the index divisor on the ex-dividendday. There are always ten adjacent annual contracts available for trading, with maturities everythird Friday of December. We use all ten contracts in the calibration. Euro Stoxx 50 dividendoptions are also exchange traded on Eurex. They are options on the futures contracts, where thematurity of the option coincides with the maturity of the futures contract, which makes themeffectively options on the dividends realized in a calendar year. In the calibration, we use theBlack implied volatility of the option on the first dividend futures contract with at-the-moneystrike (i.e., strike equal to the dividend futures price). We also use the Black-Scholes impliedvolatility of the option on the Euro Stoxx 50 index level with maturity in three months andat-the-money strike. The prices of the dividend futures and the implied volatility of the indexand dividend option are shown in the second column of Table 1. Remark that the impliedvolatility of the dividend option is substantially lower ( ≈ ≈ r = 0 .
01 and a ∈ { . , . , . } . By fixing a , the parameter constraint in (11) becomesa linear inequality in the free parameters b and β , which most optimization routines can easilydeal with. In our model, a determines the upper bound for the dividend yield process δ t . InFigure 2 we plot a proxy of the (unobservable) dividend yield δ t over time, which we calculate bydividing the price of the first to expire dividend futures contract (which has a time to maturityvarying between 1 day and 1 year) by the index level. We observe that between 2010 and 2016,the dividend yield proxy moves roughly between 3% and 6%, well below the three values thatwe consider for a .We use N = 6 moments to compute the dividend and stock option prices using the maximalentropy method described in Section 3.3. We use the gradient-free Nelder-Mead simplex opti-9bsolute errorsData a = 0 . a = 0 . a = 0 . k represents the dividendfutures contract with expiry on the third Friday of December (2015 + k ). IV stock is the Black-Scholes implied volatility of the stock option. IV dividend is the Black implied volatility of thedividend option. All data comes from Bloomberg. The last three columns show the absoluteerrors of the calibrated models. Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan160.0250.030.0350.040.0450.050.0550.060.065 D i v i dend y i e l d p r o xy Figure 1: This figure plots the historical dividend yield, which we proxy by the price of the firstto expire dividend futures contract divided by the index level.10 b β σ ν D a .mization algorithm to find the optimal parameters b, β, σ, ν , and D . The calibrated parametersare shown in Table 2 and the absolute pricing errors are shown in the last three columns of Table1. The calibrated values of b , β , and D are almost identical for different values of a . This isnot surprising, since these parameters mainly control the term structure of dividend futuresprices, and a does not enter in the pricing formula (14) for the dividend futures. However,from (9) and (10) it is clear that a has an impact on the volatility of the stock price and thedividend rate. Indeed, if a increases, all else being equal, then the volatility of the stock priceand the dividend rate increases. To offset this effect, the calibrated parameters of σ and ν aresmaller for larger a . From the absolute errors in Table 1, we can see that the choice of a doesnot really matter for the quality of the calibration, since the absolute pricing errors are almostidentical. The maximal relative error in the dividend futures contracts is less than 2%, which isa remarkably good fit for a single factor model. Figure 2 visualizes the good fit of the calibratedmodel with the dividend futures term structure. The option prices are matched perfectly. Thisis a consequence of the fact that the dividend futures prices do not depend on the martingalepart of X t and D t . The parameters σ and ν therefore remain free to calibrate to the dividendand stock option.Figure 3 plots a simulation of the dividend yield process δ t over a ten years horizon with dailydiscretization. We use the model parameters from Table 2 with a = 0 .
2, however the plot looksidentical when using the calibrated parameters with a = 0 . a = 0 .
3. The process is roughlymean-reverting around b/ ( r − β − σ ) = 3 . δ t whenignoring the higher order terms in the drift. Remark that the range of values that δ t takes inthe simulation is very similar to the range of values in Figure 1.Figure 4 shows the option price approximation as a function of the number of moments N . Asa benchmark, we run a Monte-Carlo simulation with daily time steps and 10 sample paths. Inorder to reduce the variance of the Monte-Carlo estimator, we use a degree one polynomial in theunderlying as a control variate, where we determine the coefficients of the polynomial througha least squares regression of the simulated payoff paths on the simulated polynomial. The solidline shows the Monte-Carlo estimator and the dashed lines are the corresponding 95% confidenceintervals. For the stock option, the maximal entropy approximation with four moments is alreadywithin the confidence bands and the one with six moments is exactly equal to the Monte-Carloestimator. For the dividend option, using only two moments already provides a very accurateoption price approximation. The dividend option price is easier to approximate because thevolatility of the dividend rate is much lower than the volatility of the stock price. Indirectly, the dividend futures prices are to some extent affected by a through the inequality (11) that hasto be satisfied. ec16 Dec17 Dec18 Dec19 Dec20 Dec21 Dec22 Dec23 Dec24 Dec25 Expiry D i v i dend f u t u r e s p r i c e MarketModel
Figure 2: Market prices and model implied prices of dividend futures. The model implied pricesare calculated using the parameter in Table 2 with a = 0 . t Figure 3: Simulation the dividend yield process δ t over ten years with daily discretization. Themodel parameters are those in Table 2 with a = 0 . Number of moments matched B l a ck - S c ho l e s i m p li ed v o l a t ili t y Max EntMCMC CI (a) Stock option
Number of moments matched B l a ck i m p li ed v o l a t ili t y Max EntMCMC CI (b) Dividend option
Figure 4: Option price approximations for varying number of moments N . The solid linesrepresents the Monte-Carlo estimates and the dashed lines represent the corresponding 95%confidence intervals. We can enrich the model dynamics by adding jumps to X t as followsd X t = ( rX t − D t ) d t + ( X t − − D t − a ) ( σ d W t + d J t ) , (17)where D t is the same as before and J t is a compensated compound Poisson process with arrivalintensity λ ≥ F (d z ) that is assumed to have moments in closed-form of all orders and a support S ⊆ ( − , ∞ ). Remark that D t is still a continuous process, sothat D t − = D t . Let τ denote a jump time of J t and suppose that ( X τ − , Y τ − ) ∈ E . From theassumption on the support of F , we have X τ = X τ − + ∆ X τ = X τ − + ( X τ − − D τ − a )∆ J τ ≥ D τ − a , where equality only holds if aX τ − = D τ − , in which case ∆ X τ = 0. Therefore, the resultsin Proposition 2.1 remain valid since X t behaves as in (2) in between jump times, we have( X τ , Y τ ) ∈ E so the process cannot jump outside of E , and aX τ > D τ if aX τ − > D τ − so jumpsto the boundary are not possible.If we denote by G J the the infinitesimal generator of ( C t , X t , Y t ) in the case with jumps, thenwe get for a twice differential function f G J f = G f + λ Z S f (cid:18) c, x + xz − ⊤ ya z, y (cid:19) − f − (cid:18) x − ⊤ ya (cid:19) zf x F (d z ) , where we assume that f is such that the integral is finite and we have again omitted the functionarguments for brevity, except in the first term of the integrand. Since the amplitudes of thejumps in X t depend linearly on X t and Y t , it follows immediately that G J Pol n ⊆ Pol n . Therefore,( C t , X t , Y t ) belongs to the class of polynomial jump-diffusions and we can compute all conditionalmoments in closed form. 13ince X t enters in the dynamics of Y t , the jumps also indirectly impact the dynamics of D t . Themagnitude of the effect of a jump in X t on the drift of D t is determined by b . A stylized fact ofindex options and index dividend options is that both have a negative skew in implied volatilities.Choosing a distribution F with a sufficiently negative mean produces a negative skew in impliedvolatilities for both stock and dividend options. We leave a calibration to option skews for futurework. Remark 5.1.
It is possible to introduce jumps in Y t as well, although one should be careful withsimultaneous jumps where D t jumps up and X t jumps down in order to avoid jumping out of E .We do not consider this extension in this paper. We have introduced a model for jointly pricing stock and dividend derivatives. The noveltyof our approach lies in the fact that we directly model the dividend rate while guaranteeing apositive stock price. This is accomplished by upper bounding the dividend rate by a constantfraction of the stock price, so that the dividend rate goes to zero as the stock price approacheszero. The model belongs to the class of polynomial diffusions, which leads to closed-form pricesfor stock and dividend futures, and efficient approximations for stock and dividend options. Wehave calibrated a single factor model to data on dividend futures and at-the-money stock anddividend options. Future research includes calibrating the model to stock and dividend optionswith a range of strikes using the extension with jumps outlined in Section 5, as well as extendingour framework to discrete dividend payments. 14
Proofs
A.1 Proof of Proposition 2.1
Existence of an R d -valued solution follows from (Ikeda and Watanabe, 1981, Theorem IV.2.4),since the drift and dispersion coefficient of ( X t , Y t ) satisfy a linear growth condition. It remainsto show that a solution starting in E also stays in E .Denote by µ : E → R and Σ : E → R d × d respectively the drift and dispersion function of Y t ,i.e. d Y t = µ ( X t , Y t ) d t + Σ( X t , Y t ) d B t . We need to verify that µ k ( x, y ) ≥ x, y ) ∈ E with y k = 0, so that the drift pushes Y k,t awayfrom the zero boundary again. Using the fact that 0 ≤ y k ≤ ax for all ( x, y ) ∈ E , we have forall ( x, y ) ∈ E with y k = 0 that µ k ( x, y ) = b k x + X l = k β k,l y l ≥ b k x + min l = k β − k,l X l = k y l ≥ ( b k + a min l = k β − k,l ) x ≥ k,l ( x, y ) = 0, l = 1 , . . . , d , for ( x, y ) ∈ E with y k = 0,shows that Y k,t ≥ t ≥ k = 1 , . . . , d . Indeed, Y k,t starts in E and has aninward pointing drift and vanishing diffusion at the boundary. Using the same argument for aX t − ⊤ Y t instead of Y k,t , it follows that aX t ≥ ⊤ Y t for all t ≥
0. As a consequence we alsohave X t ≥ X t , we use a stochasticcomparison argument. Define the process Z t = − log X t if X t > Z t = ∞ if X t = 0. Definethe process ˜ Z t through the following SDEd ˜ Z t = ( a − r + 12 σ ) d t − σ d W t , ˜ Z = Z . From Theorem 1.3 in Hajek (1985) we get for all c ∈ R and t > P ( Z t ≥ c ) ≤ P ( ˜ Z t ≥ c ) . Since lim c →∞ P ( ˜ Z t ≥ c ) = 0, we have lim c →∞ P ( Z t ≥ c ) = 0 and therefore X t > Y t . Theorem A.1 (Theorem 5.7(i) Filipovi´c and Larsson (2016)) . Denote by G the infinitesimalgenerator and by m ( x, y ) the diffusion function of ( X t , Y t ) . Let p ( x, y ) be a polynomial and let h ( x, y ) be a vector of polynomials such that m ( x, y ) ∇ p ( x, y ) = h ( x, y ) p ( x, y ) for all ( x, y ) ∈ R d .If there exists a neighborhood U of E ∩ { p = 0 } such that for all ( x, y ) ∈ E ∩ U G p ( x, y ) − h ( x, y ) ⊤ ∇ p ( x, y ) ≥ , (18) then p ( X t , Y t ) > for all t > . Y k,t >
0. For p ( x, y ) = y k , we have h ( x, y ) = (0 , . . . , ν k ( x − ⊤ y/a ) , . . . , ⊤ , with the non-zero element in the ( k + 1)-th component. For some ǫ >
0, consider the followingneighborhood of E ∩ { p = 0 } U = { ( x, y ) ∈ R d : | y k | ≤ ǫ } . For ( x, y ) ∈ E ∩ U = { ( x, y ) ∈ R d : x > , y k ≤ ǫ, ax ≥ ⊤ y, y ≥ } we have2 G p ( x, y ) − h ( x, y ) ⊤ ∇ p ( x, y ) = 2( b k x + d X l =1 β k,l y l ) − ( x − ⊤ y/a ) ν k ≥ b k x + X l = k β k,l y l ) − ( x − X l = k y l /a ) ν k + min(2 β k,k + ν k /a, ǫ = (2 b k − ν k ) x + X l = k (2 β k,l + ν k /a ) y l + min(2 β k,k + ν k /a, ǫ ≥ (2 b k − ν k + min l = d (2 aβ k,l + ν k ) − ) x + min(2 β k,k + ν k /a, ǫ. (19)If 2 β k,k + ν k /a ≥
0, then (19) is non-negative if 2 b k − ν k + min l = k (2 aβ k,l + ν k ) − ≥
0. If2 β k,k + ν k /a <
0, then we can always find an ǫ > b k − ν k +min l = k (2 aβ k,l + ν k ) − > aX t > ⊤ Y t . For p ( x, y ) = ax − ⊤ y we have h ( x, y ) = ( σ ( x − ⊤ y/a ) , − y ν /a, . . . , − y d ν d /a ) ⊤ . For some ǫ >
0, consider the following neighborhood of E ∩ { p = 0 } U = { ( x, y ) ∈ R d : | ax − ⊤ y | ≤ ǫ } . For ( x, y ) ∈ E ∩ U = { ( x, y ) ∈ R d : x > , ≤ ax − ⊤ y ≤ ǫ, y ≥ } we have2 G p ( x, y ) − h ( x, y ) ⊤ ∇ p ( x, y )= 2 a ( rx − ⊤ y ) − ⊤ ( bx + βy ) − σ a ( x − ⊤ y/a ) − y ν /a − · · · − y d ν d /a = (2 ar − aσ − ⊤ b ) x − ((2 a − σ ) ⊤ + ( ν , . . . , ν d ) /a + 2 ⊤ β ) y ≥ (2 ar − aσ − ⊤ b ) x − max k =1 ,...,d (2 a − σ + ν k /a + 2 ⊤ β k ) ⊤ y ≥ (2 a ( r − a ) − ⊤ b − a max k =1 ,...,d ( ν k /a + 2 ⊤ β k )) x + ǫ min(0 , a − σ + max k =1 ,...,d ( ν k /a + 2 ⊤ β k )) , where the last line follows from ax − ǫ ≤ ⊤ y ≤ ax . If 2 a − σ + max k =1 ,...,d ( ν k /a + 2 ⊤ β k ) ≥ G p − h ⊤ ∇ p ≥ E ∩ U if2 a ( r − a ) − ⊤ b − a max k =1 ,...,d ( ν k /a + 2 ⊤ β k ) ≥ . If 2 a − σ + max k =1 ,...,d ( ν k /a + 2 ⊤ β k ) <
0, then we can always find an ǫ > G p − h ⊤ ∇ p ≥ E ∩ U if2 a ( r − a ) − ⊤ b − a max k =1 ,...,d ( ν k /a + 2 ⊤ β k ) > . X t , Y t ), note that Y t X t is an autonomous diffusion with0 ≤ Y t X t ≤ a for all t ≥
0. A straightforward application of Itˆo’s lemma shows that the process(log( X t ) , Y t X t ) has a uniformly bounded drift and diffusion function, so that uniqueness in lawfor (log( X t ) , Y t X t ), and therefore for ( X t , Y t ), follows from (Ikeda and Watanabe, 1981, TheoremIV.3.3). A.2 Proof of Proposition 2.3
To proof that G t is a martingale, we can use Novikov’s condition. An application of Itˆo’s lemmagives d G t = σ e − rt ( X t − D t a ) d W t , which shows that G t is a local martingale. Since the volatility of log( G t ) is uniformly bounded, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ e − rt ( X t − D t a )e − rt X t + R t e − rs D s d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ σ, Novikov’s condition is trivially satisfied and we conclude that G t is a martingale. Remark A.2.
The process G t represents the discounted value of a trading strategy of a longposition in the stock and investing all the dividends in the risk-free account. Alternatively, wecould also re-invest all the dividends in the stock itself. This strategy has a discounted valueprocess G ∗ t = e − rt + R t δ s d s X t , which is again a martingale by Novikov’s condition. Next, we show that the present value of all future dividends is equal to the stock price. Define˜ X t = e − rt X t and ˜ Y t = e − rt Y t . The dynamics of ˜ X t and ˜ Y t becomesd ˜ X s = − ⊤ ˜ Y s d s + · · · d W s , d ˜ Y s = ( b ˜ X s + ( β − r Id) ˜ Y s ) d s + · · · d B s . Taking conditional expectations and denoting f ( s ) = E t [ ˜ X s ] and g ( s ) = E t [ ˜ Y s ], s ≥ t , gives thefollowing linear first order ODE (cid:18) f ′ g ′ (cid:19) = (cid:18) − ⊤ b β − r Id (cid:19) (cid:18) fg (cid:19) . Using the properties of E , we have that f ( s ) > g ( s ) ≥
0, and ⊤ g ( s ) ≤ af ( s ) for all s ≥ t .In particular, we have that f is a non-increasing function and hence f and g are uniformlybounded 0 < f ( s ) ≤ f ( t ) , ≤ g ( s ) ≤ af ( t ) , ∀ s ≥ t. Moreover, all derivatives of f and g are uniformly bounded as well since (cid:18) f ( n ) g ( n ) (cid:19) = (cid:18) − ⊤ b β − r Id (cid:19) n (cid:18) fg (cid:19) , for all n ∈ N . Since f is a non-increasing positive function, we have lim s →∞ f ( s ) = ξ ∈ [0 , f ( t )].Since f ′′ is bounded, f ′ must be uniformly continuous. By Barbalat’s lemma (see e.g., Lemma17.2 in Khalil (2002)) we therefore have that lim s →∞ f ′ ( s ) = 0. Since f ′ ( s ) = − ⊤ g ( s ) and g ( s ) ≥ s →∞ g ( s ) = 0 componentwise. Taking the limit of f ′′ giveslim s →∞ f ′′ ( s ) = lim s →∞ − ⊤ g ′ ( s ) = − ⊤ b lim s →∞ f ( s ) − ⊤ ( β − r Id) lim s →∞ g ( s ) = − ⊤ bξ. Since f ′′′ is bounded, f ′′ must be uniformly continuous, and by Barbalat’s lemma we havelim s →∞ f ′′ ( s ) = 0. Since ⊤ b > ξ = 0. This concludes the proofsince 0 = lim s →∞ f ( s ) = lim s →∞ E t [ ˜ X s ] = ˜ X t − E t (cid:20)Z ∞ t e − ru D u d u (cid:21) . References
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