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4.1Changing the exposureFirst, we explore the impact of changing the risk exposure by letting Y log G and introducing the random variable 12Hτ exp η(X0 ) · Wτ η(X0 ) τ .2(8)This random variable has conditional mean one conditioned on X0 . Note that 12Gt Hτ exp log Gt η(X0 ) · Wτ η(X0 ) τ .2 We thus use Hτ to increase the exposure of the stochastic growth process G to the nextperiod shock Wτ in the direction η (X0 ). The direction vector is normalized to satisfy E η (X0 ) 2 1.Our interest lies in comparing expected cash flows for cash flow processes with differentexposures to risk. Motivated by the construction of elasticities, we focus on the proportionalimpact of changing the exposure expressed in terms of the ratioE [Gt Hτ X0 x]E [Gt X0 x]or expressed as the difference in logarithmslog E [Gt Hτ X0 x] log E [Gt X0 x] .For a marginal counterpart of this expression, we localize the change in exposure byconsidering a family of random variables parameterized by a ‘perturbation’ parameter r: 1 22Hτ (r) exp rη(X0 ) · Wτ r η(X0) τ .2(9)Following Borovička and Hansen (2013) we construct the derivativedlog E [Gt Hτ (r) X0 x]drr 0 η(x) ·E [Gt Wτ X0 x]E [Gt X0 x]that represents the proportional change in the expected cash flow to a marginal increase inthe exposure to the shock Wτ in the direction η (x). This leads us to label this derivativea shock-exposure elasticity for the cash flow process G. This elasticity depends both on thematurity of the cash flow t as well as on the current state X0 x.5

4.1.1A change of measure interpretationThe construction of the shock-exposure elasticity has a natural interpretation as a change ofmeasure that provides a close link to the impulse response functions that we delineated inSection 3. Multiplication of the stochastic growth process G by the positive random variableHτ constructed in (8) prior to taking expectations is equivalent to changing the distributionof Wτ from a normal Q with mean zero and covariance τ I to a normal Qη with mean η(x)and covariance τ I. As a consequence,E [Gt Hτ X0 x] ZΦgh (t, x, w)Q(dw) (10) E [E [Gt Wτ ; X0 x] Hτ X0 x] ZΦg (t, x, w)Qη (dw x)where the function Φg is defined as in (4) with φ(Yt ) exp(Yt ) Gt . The first row inexpression (10) uses the baseline Q distribution for Wτ . On the other hand, in the secondrow of (10) we use the perturbed distribution Qη distribution as a computational device toalter the risk exposure of the process G.Expression (10) relates the shock-exposure elasticity to the nonlinear impulse responsefunctions. We compute shock elasticities by altering the exposure of the stochastic growthprocess G to the shock Wτ , or, equivalently, by changing the distribution of this shock.Since we are interested in computing the effects of a marginal change in exposure, weuse the family Hτ (r) from equation (9) to define a family of measures Qηr as in (7). Theshock-exposure elasticity can then be computed asdlogdr4.1.2ZΦg (t, x, w)Qηr (dw x)A special caseRΦg (t, x, w)wQ(dw). η(x) · RΦg (t, x, w)Q(dw)(11)Linear vector-autoregressions (VARs) are models (3) with parameters that satisfyµ (x) µ̄xσ (x) σ̄β (x) β̄ · xα (x) ᾱX is a linear vector-autoregression with autoregression coefficient µ̄τ I and shock exposurematrix σ̄. Let η(x) η̄ where η̄ is a vector with norm one. The (the scalar product is nota vector) impulse response function of Yt where t jτ for the linear combination of shocks6

chosen by the vector η̄ is given byE [Yt X0 x, Wτ η̄] E [Yt X0 x] κ̄j · η̄.(12)whereκ̄j 1 κ̄j τ β̄ ζ̄jζ̄j 1 (I τ µ̄) ζ̄jwith initial conditions ζ̄1 σ̄ and κ̄1 ᾱ. Thusζ̄j (I τ µ̄)j 1 σ̄ hi ′κ̄j ᾱ µ̄ 1 (I τ µ̄)j 1 I σ̄ β̄The first term, ᾱ · η̄, represents the “instantaneous” impact arising from the current shock,while the second term captures the subsequent propagation of the shock through the dynamics of the model.Now consider the shock elasticity for the multiplicative functional G exp(Y ). Usingthe formula for the expectation of a lognormal random variable, we have 1Φg (t, x, w)2R exp κ̄j κ̄j · w .2Φg (t, x, v)Q(dv)Using formula (11), we obtain the shock-exposure elasticity for G:η̄ ·Z w11 ′2 exp κ̄j κ̄j · w exp w w η̄ · κ̄j222πnThus for a linear model, our shock-exposure elasticity for G coincides with the linear impulse response function for Y log G, with the direction vector η̄ selecting a particularcombination of shocks in Wτ .4.2Pricing the exposureGiven our interest in pricing we are led to the study of the sensitivity of expected returnsto shocks. We will utilize the family of cash flows Gt Hτ (r) indexed by the perturbationparameter r and construct a local measure of this sensitivity as the pricing counterpart ofthe shock-exposure elasticity (11).A stochastic discount factor S is a stochastic process that represents the valuation of7

payoffs across states and time. Therefore, the value at time zero of a cash flow Gt Hτ (r) maturing at time t (or cost of purchasing an asset with such a payoff) is E [St Gt Hτ (r) X0 x].Since we assume that S and G are multiplicative functionals, so is their product SG.The logarithm of the expected return for the cash flow Gt Hτ (r) maturing at time t islog E [Gt Hτ (r) X0 x] log E [St Gt Hτ (r) X0 x] .log expected payofflog costSince both components of the expected return are distorted by the same random variableHτ (r), we can write the (log) expected return on cash flow Gt Hτ (r) aslogZΦg (t, x, w)Qηr (dw x) logZΦsg (t, x, w)Qηr (dw x).We localize the change in exposure by computing the derivative of this expression withrespect to r and evaluating this derivative at r 0. This calculation yields the discrete-timeshock-price elasticityRRΦg (t, x, w)wQ(dw)Φsg (t, x, w)wQ(dw)η(x) · R η(x) · R.Φg (t, x, w)Q(dw)Φsg (t, x, w)Q(dw)(13)The shock-price elasticity is the difference between a shock-exposure elasticity and ashock-cost elasticity. Locally, the impulse response for both components of the shock-priceelasticity is given by the covariance of the impulse response Φ with the shock Wτ . Inwhat follows we will show how a continuous-time formulation gives us an alternative way tolocalize shock exposures in a convenient way.The shock-price elasticity for the one-period horizon has particularly simple representation and is given by:η(x) · αg (x) η(x) · [αg (x) αs (x)] η(x) · [ αs (x)]In this formula the entries of αs (x) give the vector of “prices” assigned to exposures ofeach of the entries of Wτ . These entries are often referred to as the (local) price of risk.Our shock-price elasticity function captures the term structure of the price of risk, in thesame way as an impulse response function captures the dynamic adjustment of an economyover time in response to a shock.In the special case of a log normal model discussed in Section 4.1.2, we can construct amodel of the stochastic discount factor S exp(Y ). In this case, the linearity of the resultsimplies that the shock-price elasticity (13) will correspond to the impulse response function8

for log S.5Returning to continuous timeBy taking continuous-time limits, we achieve some simplicity given the Brownian motioninformation structure. In this section, we proceed informally to provide economic intuition.A more formal treatment follows in Section 6.Given a Markov diffusion X such as (1), the Malliavin derivative D0 Xt allows us toexamine the contribution of a shock dW0 to the value of that diffusion at time t 0. Wecalculate the Malliavin derivative recursively by computing what is called the first variationprocess associated to the diffusion.4 The first variation process for X is an n n-dimensionalprocess Z x that measures the impact of the change in initial condition X0 x on futurevalues of the process X: Xt. x′This process solves a linear stochastic differential equation obtained by differentiating theZtx coefficients:dZtx k X x µ(Xt ) Zt dt [σ (Xt )]·i Ztx dWi,t′′ x xi 1 (14)where [σ (x)]·i is the i-th column of σ (x) and Wi is the i-th component of the Brownianmotion. The initial condition for the first variation process is Z0x In . The Malliavinderivative satisfiesD0 Xt Ztx σ (x) ,since σ (x) gives the impact of the the shock dW0 on dX0 .The construction of the Malliavin derivative can be extended to the additive functionalY . Since (X, Y ) form a Markov diffusion, and the coefficients in this Markov diffusion areindependent of Y , we obtain a 1 n process Z y that satisfies:dZty k X x β(Xt ) Zt dt α (Xt ) Ztx dWi,t′ i x′ xi 1(15)with initial condition Z0y 01 n . Consequently, the Malliavin derivative of Y is an 1 kvector given byD0 Yt Zty σ (x) α (x)′4See e.g. Property P2 on page 395 of Fournié et al. (1999). Fournié et al. (1999) assume that the diffusioncoefficients have bounded derivatives, which is not verified in this example. A precise justification wouldrequire extending their theorem to our setup.9

where α (x) represents the initial contribution of the shock η(x) · dW0 to Yt and Zty σ (x) η(x)captures the propagation of the shock through the dynamics of the state vector X.When conditional mean coefficients (µ, β) are linear in state vector, and the exposurecoefficients (σ, α) on the Brownian increment are constant and the η vector is constant, theMalliavin derivatives only depend on the date zero state and not on the Brownian incrementsthat follow. In this case the Malliavin derivative calculations will yield the continuous timecounterpart to the calculations in Section 4.1.2. More generally, the Malliavin derivativesdepend on the Brownian increments. We could compute the “average” responses usingΦy (t, x) η(x) · E [D0 Yt X0 x]for t 0 which still depends on the initial state x but not on the Brownian increments.We have featured η as a device for selecting which (conditional) linear combination ofincrements to target for the response function Φy . In fact, Malliavin derivatives are typicallycomputed by introducing drift distortions for the Brownian increment vector dW0 . Thus anequivalent interpretation of the role of η in computation of Φy is that of a date zero driftdistortion, the counterpart to the mean shift for a normally distributed random vector thatwe used in our discrete-time constructions.Since we are interested in the process M exp(Y ), averaging the random responses oflog M will not be of direct interest in our analysis of elasticities. This leads us to modify ourcalculation of average responses.The Malliavin derivative of the function of a process Ut φ(Yt ) is well defined providedthe function φ is sufficiently regular. In this case we may use a “chain-rule”,Du Ut φ′ (Yt )Du Yt .The function φ exp is of particular interest to us since M exp(Y ). For the shockη(x) · dW0 ,D0 Mt η(x) exp(Yt )D0 Yt η(x)gives the date t distributional response of Mt to the date zero shock η(x) · dW0 .Next we construct a nonlinear counterpart to a moving-average representation, which for acontinuous-time diffusion is the Haussmann–Clark–Ocone representation. The Haussmann–Clark–Ocone representation uses Malliavin derivatives to produce a “moving-average” representation of M with state-dependent coefficients and typically expressed as:Mt Z0tE [Du Mt Fu ] · dWu E[Mt ].10(16)

Many of the random variables we consider depend on X0 x along with the Brownianmotion W , including random variables on both sides of (16). We hold X0 x fixed for thecalculation of the Malliavin derivatives, and rewrite equation (16) as:Mt Zt0E [Du Mt Fu , X0 x] · dWu E[Mt X0 x].(17)The notation E[ · X0 x] should remind readers that the computation of the expectationover the function of the Brownian motion depends on the choice of initial conditions.This convenient result represents Mt as a response to shocks with “random coefficients”E [Du Mt Fu , X0 x] that are adapted to Fu whereas in linear time series analysis thesecoefficients are constant. With representation (17), a continuous-time analogue to the impulse response functions computed in Section 3 measures the impact on φ(Yt ) of a “shock”dW0 :Φm (t, x) η(x) · E [exp(Yt )D0 Yt X0 x] ,(18)for t 0. The term D0 Yt can be computed using the recursive calculations outlined above.The weighting by the nonstationary process M exp(Y ) may be important, because Mgrows or decays stochastically over time.Next we consider shock elasticities in continuous time. In this paper we build theseelasticities in a way that is consistent with those given in Borovička et al. (2011), but wederive them in a more direct way.5 In Section 6 we will show that the elasticities of interestcan be expressed asε(t, x) η(x) ·E [exp(Yt )D0 Yt X0 x]E [exp(Yt ) X0 x)(19)where Y log G in the case of an shock exposure elasticity and Y log S log G in thecase of a cost elasticity. The numerator is the same as the impulse response for exp(Yt )given in (18). Consistent with our interest in elasticities, we divide by the conditionalexpectation of exp(Yt ). In accordance with this representation, the elasticities we justify areweighted averages of the impulse responses for Y weighted by exp(Y ). Asymptotic resultsfor t can be obtained using a martingale decomposition of the multiplicative functionalM analyzed in Hansen and Scheinkman (2009).A shock price elasticity is given by the difference between an exposure elasticity and a5Borovička et al. (2011) consider responses over finite investment intervals and introduce a separateparameter that localizes the risk exposure. Similarly, Borovička and Hansen (2013) use a discrete timeeconomic environment and again introduce a parameter that localizes the risk. Here we avoid introducing anadditional parameter by letting the continuous-time approximation localize the risk exposure over arbitrarilyshort time intervals. In Section 8 we elaborate on the connection between calculation in this paper and ourprevious work Borovička et al. (2011).11

cost elasticity:η(x) ·E [Gt (D0 log Gt ) X0 x]E [St Gt (D0 log St D0 log Gt ) X0 x] η(x) ·E [Gt X0 x]E [St Gt X0 x](20)In a globally log-normal model D0 log Gt and D0 log St depend on t but are not random,and the weighting by either Gt or by St Gt is of no consequence. Moreover, in this casethe shock price elasticities can be computed directly from the impulse response functionfor log S to the underlying shocks since the expression in (20) is equal to D0 log St . Theresulting elasticities are the continuous time limits of the results from the log-normal exampleintroduced in Section 4.1.2.Instead of computing directly the Malliavin derivatives, there is a second approach thatsometimes gives a tractable alternative to computing the coefficients of the Haussmann–Clark–Ocone representation for M. This approach starts by computing E [Mt X0 x] andthen differentiating with respect to the state:σ (x)′ E [Mt X0 x] . xThe premultiplication by σ (x)′ acts as a measure of the local response of X to a shock. Asin Borovička et al. (2011) expression (19) can be written as6 ′ log E [Mt X0 x] .ε(t, x) η(x) · α (x) σ (x) x(21)This result separates the ‘instantaneous’ effect of the change in exposure, α (x), from theimpact that propagates through the nonlinear dynamics of the model, expressed by thesecond term in the bracket. In case of the shock-price elasticity, α (x) corresponds to thelocal price of risk.6Formal construction of shock elasticitiesTo construct the elasticities that interest us, we “perturb” the cash flows in alternative ways.Let N τ beNtτ Zτ t0η(Xu ) · dWu .6For example see the formulas (5) and (6) and the discussion in Section 4 provided in Borovička et al.(2011).12

The process N τ alters the exposure over the interval [0, τ ] and remains constant for t τ .We impose E η(X0 ) 2 1when X is stationary and X0 is initialized at the stationary distribution. While the finitesecond moment condition is a restriction, our choice of unity is made as a convenient normalization. We impose this second moment restriction to insure that N τ has a finite varianceτ for t τ . We use the vector η to select alternative risk exposures. Let E(N τ ) denote thestochastic exponential of N τ :Htτ 1 τ ττ Et (N ) exp Nt [N , N ]t 2 Z τ tZ1 τ t2 η(Xu ) du . expη(Xu ) · dWu 2 00τOur assumption that X is stationary with a finite second moment guarantees that the processH τ as constructed is a local martingale. We will in fact assume that the stochastic exponentials of the perturbations N τ are martingales. This normalization will be of no consequencefor our shock price elasticity calculations, but it is a natural scaling in any event.Form the perturbed payoff GH τ . This payoff changes the exposure of the payoff G byaltering the shock exposure of log Gt to beZ0tαg (Xu ) · dWu Z0τη(Xu ) · dWufor τ t. This exposure change is small for small τ . Construct the logarithm of the expectedreturn:εp (τ, t, x) log E [Gt Htτ X0 x] log E [St Gt Htτ X0 x] .log expected payofflog costNote thatεp (τ, t, x) εe (τ, t, x) εc (τ, t, x)whereεe (τ, t, x) log E [Gt Htτ X0 x]εc (τ, t, x) log E [Gt St Htτ X0 x] .We will calculate derivatives of εe and εc to compute a shock price elasticity. The first ofthese derivatives is a shock exposure elasticity.13

We use H τ to change the exposure to uncertainty. Since H τ is a positive martingale witha unit expectation, equivalently it can be used as a change in probability measure. Underthis interpretation, think of η(Xt ) as being a drift distortion of the Brownian motion W .ThendWt (ft 0 t τη(Xt )dt dWftdWt τf is a Brownian motion under the change of measure. This gives an alternativewhere Winterpretation to our calculations and a formal link to Malliavin differentiation. Bismut(1981) uses the change of measure to perform calculations typically associated with Malliavindifferentiation.It what follows we will shrink the interval [0, τ ] to focus on the instantaneous changein exposure, which we can think of equivalently as an instantaneous drift distortion in theBrownian motion.6.1Haussmann–Clark–Ocone formulaTo characterize the derivatives of interest, we apply the Haussmann–Clark–Ocone formula(17) to M G or M SG and represent Mt as a stochastic integral against the underlyingBrownian motion. The vector of state dependent impulse response functions for Mt for thedate zero Brownian increment is E [D0 Mt X0 x] when viewed as a function of t. Thisaverages over the random impacts in the future but still depends on X0 x.We will also use the following result that is a consequence of Proposition 5.6 in Øksendal(1997):Du E (Mt Fτ , X0 x) E (Du Mt Fτ , X0 x) 1[0,τ ] (u).6.2(22)Computing shock elasticitiesFix X0 x and τ t. The Haussmann–Clark–Ocone formula (17) implies that:E [E(Mt Fτ , X0 x)Htτ X0 Z t τ x] E HτE (Du E(Mt Fτ ) Fu , X0 x) · dWu X0 x E0τ[Hτ E(E(Mt Fτ ) X0 x) X0 x] .Hence, using equation (22),E(Mt Htτ X0 x) E HττZτ0 E (Du Mt Fu , X0 x) · dWu X0 x E (Mt X0 x)14

for t τ . Under the change of measure implied by H τ ,E(Mt Htτ X0e x) E Zτ0 Zu · η(Xu )du X0 x E (Mt X0 x)e is the expectation under the change in probability measure andwhere EZu E (Du Mt Fu , X0 x) .We compute the derivative with respect to τ at τ 0 by evaluating: Z τ Z τ 1e1τZu · η(Xu )du X0 xE (Du Mt Fu , X0 x) · dWu X0 x lim Elim E Hττ 0 ττ 0 τ00 Z0 · η(x) E (D0 Mt X0 x) · η(x) E [Mt (D0 log Mt ) X0 x] · η(x)where the last line follows from the formula:D0 Mt Mt D0 log Mt .For globally log normal models D0 log Mt depends only on t and not on random outcomes,but more generally D0 log Mt is random.Since our aim is to compute elasticities, the actual differentiation that interest us is thederivative of the logarithm of the conditional expectation:εm (t, x) η(x) · E [Mt (D0 log Mt ) X0 x]E [Mt X0 x] .Thus the elasticities are weighted averages of D0 log Mt weighted by Mt .Remark

tion, and by so doing introduces into the system the energy necessary to maintain the swings. Frisch (1933) 1 Introduction Impulse response function characterize the impact of "a stream of erratic shocks" on dy-namic economic models. They measure the consequences of alternative shocks on the future variables within the dynamical system.