2 Pricing and risk-neutral dynamics

2.1 SDF: Absence of Arbitrage Approach

Consider a period of interest \({\mathcal T} = \{0,1,2,...,T^*\}\). As in Section 1, vector \(w_t\) constitutes the new information in the economy at \(t\). The historical, or physical, dynamics of \(w_t\), \(f(\underline{w_t})\), is defined by \(f(w_{t+1}|\underline{w_t})\). The physical probability is denoted by \(\mathbb{P}\). \(L_{2t}, t \in {\mathcal T}\), is the (Hilbert) space of square integrate functions \(g(\underline{w_t})\), and we have \(L_{2t} \subset L_{2s}, t< s\).

2.1.1 Existence and unicity of the SDF

Hypothesis 2.1 (Price existence and uniqueness) For any \(\underline{w_t}\), there exists a unique \(p_t[g(\underline{w_s})]\), function of \(\underline{w_t}\), price at \(t\) of a payoff \(g(\underline{w_s})\) delivered at \(s, \forall t \le s\).

Hypothesis 2.2 (Linearity and continuity) For all \(t < s\), \(\underline{w_t}\), \(g_1\), \(g_2\), we have

• \(p_t[\lambda_1 g_1(\underline{w_s}) + \lambda_2g_2(\underline{w_s})] = \lambda_1p_t[g_1(\underline{w_s})]+\lambda_2 p_t[g_2(\underline{w_s})]\),
• If \(g_n(\underline{w_s}) \overset{L_{2s}}{\underset{n\rightarrow\infty}{\longrightarrow}} 0\), then \(p_t[g_n(\underline{w_s})] \underset{n\rightarrow\infty}{\longrightarrow} 0\).

Hypothesis 2.3 (Absence of Arbitrage Opportunity (AAO)) At any \(t\), it is impossible to constitute a portfolio of future payoffs, possibly modified at subsequent dates, such that:

• the price of the portfolio at \(t\) is zero,
• payoffs at subsequent dates are \(\ge 0\),
• there is at least one subsequent date \(s\) such that the net payoff at \(s\) is strictly positive with a non zero conditional probability at \(t\).

Theorem 2.1 (Riesz representation theorem) Under Assumptions 2.1 and 2.2, for all \(\underline{w_t}\), and \(s > t\), there exists a unique \(\mathcal{M}_{t,s}(\underline{w_s}) \in L_{2s}\) such that, \(\forall g(\underline{w_s}) \in L_{2s}\), \[ p_t[g(\underline{w_s})] = \mathbb{E}[\mathcal{M}_{t,s}(\underline{w_s})g(\underline{w_s})|\underline{w_t}]. \] In particular the price at \(t\) of a zero coupon bond maturing at \(s\) is \(\mathbb{E}(\mathcal{M}_{t,s}|\underline{w_t})\).

Proposition 2.1 (Positivity of M) If Assumption 2.3 is satisfied, then for all \(t\) and \(s\), \(\mathbb{P}(\mathcal{M}_{t,s}>0|\underline{w_t})=1\).

Proof. \(\Leftarrow\) is obvious. If \(\Rightarrow\) was not true, the payoff \(\textbf{1}_{\{\mathcal{M}_{t,s} \le 0\}}\), at \(s\), would be such that: \(\mathbb{P}[\textbf{1}_{\{\mathcal{M}_{t,s} \le 0\}}=1|\underline{w_t}] > 0\) and \(p_t[\textbf{1}_{\{\mathcal{M}_{t,s} \le 0\}}] = \mathbb{E}_t[\mathcal{M}_{t,s}\textbf{1}_{\{\mathcal{M}_{t,s} \le 0\}}] \le 0\).

Proposition 2.2 (Time consistency) For all \(t < r < s\), we have \(\mathcal{M}_{t,s} = \mathcal{M}_{t,r} \mathcal{M}_{r,s}\), which implies:

• \(\mathcal{M}_{t,s} = \mathcal{M}_{t,t+1} \mathcal{M}_{t+1,t+2}\dots\mathcal{M}_{s-1,s}\)
• \(\mathcal{M}_{0,t} = \Pi^{t-1}_{j=0} \mathcal{M}_{j,j+1}\) (\(\mathcal{M}_{0,t}\) is called pricing kernel).

Proof. Using Lemma 2.1 we have: \[\begin{eqnarray*} p_t(g_s) &=& \mathbb{E}(\mathcal{M}_{t,s}g_s|\underline{w_t}) = \mathbb{E}(\mathcal{M}_{t,r} p_r(g_s)|\underline{w_t}) \\ &=& \mathbb{E}[\mathcal{M}_{t,r}\mathbb{E}(\mathcal{M}_{r,s} g_s|\underline{w_r})|\underline{w_t}] = \mathbb{E}(\mathcal{M}_{t,r} \mathcal{M}_{r,s} g_s|\underline{w_t}), \forall g, \forall \underline{w}_{t} \end{eqnarray*}\] and, therefore, \(\mathcal{M}_{t,s} = \mathcal{M}_{t,r}\mathcal{M}_{r,s}\).

Lemma 2.1 For any payoff \(g_s\) at \(s\), \(p_t(g_s) = p_t[p_r(g_s)]\).

Proof. If this was not true, we could construct a sequence of portfolios with a strictly positive payoff at \(s\) with zero payoff at any other future date and with price zero at \(t\), contradicting Assumption 2.3. Indeed, assuming, for instance, \(p_t(g_s) > p_t[p_r(g_s)]\), the payoff at \(s\) is defined by the following strategy: (i) at \(t\): buy \(p_r(g_s)\), (short) sell \(g_s\), buy \(\frac{p_t(g_s)-p_t[p_r(g_s)]}{\mathbb{E}(\mathcal{M}_{t,s}|\underline{w_t})}\) zero-coupon bonds maturing at \(s\), at global price zero, (ii) at \(r\): buy \(g_s\) and sell \(p_r(g_s)\), generating a zero net payoff, (iii) at \(s\), the net payoff is: \(g_s-g_s+\frac{p_t(g_s)-p_t[p_r(g_s)]}{\mathbb{E}(\mathcal{M}_{t,s}|\underline{w_t})} > 0\).

Consider an asset whose payoff, on date \(s\), is \(g(\underline{w_s})\). We have, \(\forall t < s\): \[\begin{equation} \boxed{p_t[g(\underline{w_s})] = \mathbb{E}_t[\mathcal{M}_{t,t+1}...\mathcal{M}_{s-1,s}g(\underline{w_s})].}\tag{2.1} \end{equation}\] In particular, since \(L_{2,t+1}\) contains 1, the price at \(t\) of a zero-coupon with residual maturity one is given by: \[ B_{t,1} := \mathbb{E}_t [\mathcal{M}_{t,t+1}]. \] Denoting by \(i_t\) the continuously-compounded interest rate, defined through \(B_{t,1}=\exp(-i_{t})\), we get \[\begin{equation} \boxed{i_{t}=-\log \mathbb{E}_t [\mathcal{M}_{t,t+1}].}\tag{2.2} \end{equation}\] Denoting by \(B_{t,h}\) the price of a zero-coupon bond of maturity \(h\), we have: \[ B_{t,h} = \mathbb{E}_t(\mathcal{M}_{t,t+1}\times \dots \times \mathcal{M}_{t+h-1,t+h})=\mathbb{E}_t(\mathcal{M}_{t,t+h}), \] and the associated continuously-compounded yield is: \[\begin{equation} \boxed{i_{t,h}=-\frac{1}{h}\log \mathbb{E}_t [\mathcal{M}_{t,t+h}].}\tag{2.3} \end{equation}\]

Definition 2.1 (Bank account) The bank account process \(R_t\) is defined by \(R_{t} \equiv \exp(i_0+...+i_{t-1}) = \frac{1}{\mathbb{E}_0[ \mathcal{M}_{0,1}]\times ... \times \mathbb{E}_{t-1} [\mathcal{M}_{t-1,t}]}\).

\(R_t\) is the price of an investment initiated on date 0—when it was worth one dollar—and invested on each date at the risk-free rate (for one period).

For any price process \(p_t\), we have \(p_t = \mathbb{E}_t(\mathcal{M}_{t,s} p_s)\) (with \(s>t\)), or \(\mathcal{M}_{0,t} p_t = \mathbb{E}_t(\mathcal{M}_{0,s}p_s)\). That is, \(\mathcal{M}_{0,t} p_t\) is a martingale. In particular \(\mathcal{M}_{0,t} R_t\) is a martingale.

2.1.2 Exponential affine SDF

A specific (tractable) case is that of exponential affine SDF. Assume that \[ \mathcal{M}_{t,t+1}(\underline{w_{t+1}}) = \exp[\alpha_t(\underline{w_t})'w_{t+1}+\beta_t(\underline{w_t})], \] where \(\alpha_t\) defines the prices of risk or sensitivity vector. Using \(\mathbb{E}_t[\mathcal{M}_{t,t+1}]=\exp(-i_{t})=\exp[\psi_t(\alpha_t)+\beta_t]\), we get: \[\begin{equation} \boxed{\mathcal{M}_{t,t+1} = \exp[-i_{t}+\alpha'_tw_{t+1}-\psi_t(\alpha_t)].}\tag{2.4} \end{equation}\]

Example 2.1 (CCAPM/Power utility case) In the CCAPM-power-utility case (see Def. 3.4), we have (Eq. (3.10)): \[ \mathcal{M}_{t,t+1} = \exp(\log \delta + \log q_t + \gamma \log C_t - \log q_{t+1} - \gamma \log C_{t+1}), \] where \(q_t\) is the price of the consumption good, \(C_t\) is the quantity consumed at \(t\) and \(\delta\) is the intertemporal discount rate.

Hence, in that case, \(\mathcal{M}_{t,t+1}\) is exponential affine in \(w_{t+1} = (\pi_t, \Delta c_{t+1})'\), where \(\pi_{t+1}\) is the inflation between dates \(t\) and \(t+1\), i.e., \(\pi_t = \log q_{t+1} - \log q_{t}\) and \(\Delta c_{t+1}\) is the (log) consumption growth rate.

2.2 The risk-neutral (R.N.) dynamics

The historical Dynamics is characterized by \(f(\underline{w_{T^*}})\), or by the sequence of conditional p.d.f. \(f_{t+1}(w_{t+1}|\underline{w_t})\), or \(f_{t+1}(w_{t+1})\), with respect to (w.r.t.) some measure \(\mu\).

We define the conditional risk-neutral p.d.f. w.r.t. the conditional historical probability. For that, we employ the Radon-Nikodym derivative \(d^{\mathbb{Q}}_{t+1}(w_{t+1}|\underline{w_t})\):⁷ \[\begin{equation} d^{\mathbb{Q}}_{t+1}(w_{t+1}|\underline{w_t}) = \frac{\mathcal{M}_{t,t+1}(\underline{w_{t+1}})}{\mathbb{E}[\mathcal{M}_{t,t+1}(\underline{w_{t+1}})|\underline{w_t}]},\tag{2.5} \end{equation}\] or \[ d^{\mathbb{Q}}_{t+1}(w_{t+1})= \frac{\mathcal{M}_{t,t+1}}{\mathbb{E}_t(\mathcal{M}_{t,t+1})}=\exp(i_{t}) \mathcal{M}_{t,t+1}. \] In this context, the risk neutral conditional p.d.f. is: \[\begin{eqnarray} f^{\mathbb{Q}}_{t+1}(w_{t+1}) &=& f_{t+1}(w_{t+1})d^{\mathbb{Q}}_{t+1}(w_{t+1}) \nonumber \\ &=&f_{t+1} (w_{t+1}) \mathcal{M}_{t,t+1} (\underline{w_{t+1}}) \exp [i_{t} (\underline{w_t})].\tag{2.6} \end{eqnarray}\]

The p.d.f. of \(\mathbb{Q}\) w.r.t. the historical dynamics \(\mathbb{P}\) is: \[ \xi_{T^*} = \frac{d\mathbb{Q}}{d\mathbb{P}} = \Pi^{T^{*}-1}_{t=0} d^{\mathbb{Q}}_{t+1}(w_{t+1}) = \Pi^{T^{*}-1}_{t=0} \exp(i_{t}) \mathcal{M}_{t,t+1}, \] and the p.d.f. of the R.N. distribution of \(\underline{w_t}\), w.r.t. the corresponding historical distribution is: \[ \xi_t= \Pi^{t-1}_{\tau=1} d^{\mathbb{Q}}_{\tau+1}(w_{\tau+1})=\mathbb{E}_t\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right) = \mathbb{E}_t\xi_{T^*}. \] Therefore, \(\xi_t\) is a \(\mathbb{P}\)-martingale.⁸

Consider the date-\(t\) price of a payoff \(g(\underline{w_s})\) at time \(s>t\). An equivalent form of the pricing formula (2.1) is: \[\begin{eqnarray*} p_t[g(\underline{w_s})] &=& \mathbb{E}_t[\mathcal{M}_{t,t+1}...\mathcal{M}_{s-1,s}g(\underline{w_s})] \\ &=& \mathbb{E}^{\mathbb{Q}}_t[\exp(-i_{t}-...-i_{s-1})g(\underline{w_s})], \end{eqnarray*}\] or, with simpler notations: \[ p_t = \mathbb{E}^{\mathbb{Q}}_t[\exp(-i_{t}-...-i_{s-1})p_s] = \mathbb{E}^{\mathbb{Q}}_t\left(\frac{R_t}{R_s} p_s\right), \] where \(R_t\) is the bank account (Def. 2.1).

In particular, considering a zero-coupon bond of maturity \(h\), we have \[ B_{t,h} = \mathbb{E}^{\mathbb{Q}}_t[\exp(-i_{t}-...-i_{s-1})], \] which gives: \[\begin{equation} \boxed{i_{t,h}=-\frac{1}{h}\log \mathbb{E}^{\mathbb{Q}}_t[\exp(-i_{t}-...-i_{s-1})].}\tag{2.7} \end{equation}\] (This is equivalent to (2.3).)

We also have \(p_t/R_t = \mathbb{E}^{\mathbb{Q}}_t\left( p_s/R_s\right)\), that is, \(p_t/R_t\) is a \(\mathbb{Q}\)-martingale. In particular \(p_t = \exp(-i_{t})\mathbb{E}^{\mathbb{Q}}_t(p_{t+1})\), or, using the arithmetic return of any payoff \((p_{t+1}-p_t)/p_t\), and the arithmetic return of the riskless asset \(r_{A,t+1}=\exp(i_{t})-1\), we get: \[ \mathbb{E}^{\mathbb{Q}}_t\left(\frac{p_{t+1}-p_t}{p_t}\right)=r_{A,t}. \] Moreover the excess arithmetic return process \((p_{t+1}-p_t)/p_t-r_{A,t}\) is a \(\mathbb{Q}\)-martingale difference and, therefore, \(\mathbb{Q}\)-serially uncorrelated.

Let us consider the case of an exponential affine SDF \(\mathcal{M}_{t,t+1}=\exp(\alpha'_t w_{t+1}+\beta_t)\): \[ d^{\mathbb{Q}}_{t+1}(w_{t+1}) = \frac{\mathcal{M}_{t,t+1}}{\mathbb{E}_t(\mathcal{M}_{t,t+1})} = \frac{\exp(\alpha'_t w_{t+1}+\beta_t)}{\exp[\psi_t(\alpha_t)+\beta_t]} = \exp[\alpha'_t w_{t+1}-\psi_t(\alpha_t)]. \] We then have that \(d^{\mathbb{Q}}_{t+1}(w_{t+1})\) is also exponential affine. Moreover: \[ f^{\mathbb{Q}}_{t+1} (w_{t+1}) = \frac{f_{t+1} (w_{t+1}) \exp (\alpha'_t w_{t+1})}{\varphi_t (\alpha_t)}. \] The previous equation shows that \(f^{\mathbb{Q}}_{t+1}\) is the Esscher transform of \(f_{t+1}\) evaluated at \(\alpha_t\).

Let us now consider the Laplace transform of the conditional R.N. probability, \(\varphi^{\mathbb{Q}}_t(u|\underline{w_t})\), also denoted by \(\varphi^{\mathbb{Q}}_t(u)\). We have: \[\begin{eqnarray*} \varphi^{\mathbb{Q}}_t(u) &=& \mathbb{E}^{\mathbb{Q}}_t \exp(u' w_{t+1}) \\ &=& \mathbb{E}_t \exp[(u+\alpha_t)'w_{t+1}-\psi_t(\alpha_t)] \\ &=& \exp[\psi_t(u+\alpha_t)-\psi_t(\alpha_t)] = \frac{\varphi_t(u+\alpha_t)}{\varphi_t(\alpha_t)}. \end{eqnarray*}\] Hence: \[\begin{equation} \boxed{\psi^{\mathbb{Q}}_t(u) = \psi_t(u+\alpha_t)-\psi_t(\alpha_t).}\tag{2.8} \end{equation}\] We check that, if \(\alpha_t=0\), \(\psi^{\mathbb{Q}}_t=\psi_t\) (since \(\psi_t(0)=0)\).

Moreover, putting \(u=-\alpha_t\) in the expression of \(\psi^{\mathbb{Q}}_t(u)\) we get \(\psi^{\mathbb{Q}}_t(-\alpha_t)=-\psi_t(\alpha_t)\), and, replacing \(u\) by \(u-\alpha_t\), we get: \[ \boxed{\psi_t(u) = \psi^{\mathbb{Q}}_t(u-\alpha_t)-\psi^{\mathbb{Q}}_t(-\alpha_t).} \] Also: \[\begin{equation*} \left\{ \begin{array}{ccl} d_{t+1}(w_{t+1}) &=& \exp[-\alpha'_t(w_{t+1})-\psi^{\mathbb{Q}}_t(-\alpha_t)] \\ d^{\mathbb{Q}}_{t+1}(w_{t+1}) &=& \exp[\alpha'_t(w_{t+1})+\psi^{\mathbb{Q}}_t(-\alpha_t)]. \end{array} \right. \end{equation*}\]

2.3 Typology of econometric asset-pricing models

Definition 2.2 (Econometric Asset Pricing Model (EAPM)) An Econometric Asset Pricing Model (EAPM) is defined by the following functions:

• \(i_{t}(\underline{w_t})\),
• \(f(w_{t+1}|\underline{w_t}))\) [or \(\psi_t(u)\)],
• \(\mathcal{M}_{t,t+1}(\underline{w_{t+1}})\),
• \(f^{\mathbb{Q}}(w_{t+1}|\underline{w_t})\) [or \(\psi^{\mathbb{Q}}_t(u)\)].

The previous functions have to to be specified and parameterized. They are linked by: \[ f^{\mathbb{Q}}(w_{t+1}|\underline{w_t}) = f(w_{t+1}|\underline{w_t}) \mathcal{M}_{t,t+1}(\underline{w_{t+1}}) \exp[i_{t}(\underline{w_t}))]. \]

In the following, we present three ways of specifying an EAPM:

1. the direct modelling,
2. the R.N.-constrained direct modelling (or mixed modelling),
3. the back modelling.

We focus on the case where \(\mathcal{M}_{t,t+1}\) is exponential affine, as in (2.4): \[ \mathcal{M}_{t,t+1} (\underline{w_{t+1}}) = \exp\left\{ -i_{t} (\underline{w_t}) + \alpha'_t(\underline{w_t})w_{t+1} - \psi_t [\alpha_t (w_t)]\right\}. \] Once the short-term rate function \(i_{t}(\underline{w_t})\) is specified, we have to specify \(\psi_t\), \(\alpha_t\), and \(\psi^{\mathbb{Q}}_t\), that are linked by (2.8).

In all approaches, we have to specify the status of the short rate. The short rate \(i_{t}\) is a function of \(\underline{w_t}\), this function may be known or unknown by the econometrician. It is known in two cases: (a) \(i_{t}\) is exogenous (\(i_{t}(\underline{w_t})\) does not depend on \(\underline{w_t}\)) or (b) \(i_{t}\) is a component of \(w_t\). By contrast, if the function \(i_{t} (\underline{w_t})\) is unknown, it has to be specified parametrically: \[ \left\{ i_{t} (\underline{w_t}, \tilde{\theta}), \tilde{\theta}\in \tilde{\Theta} \right\}, \] where \(i_{t}(\bullet,\bullet)\) is a known function.

Let us now detail the three steps on which each of the three ways of defining an EAPM is based.

2.3.1 The direct modelling

• Step 1 – Specification of the historical dynamics. We choose a parametric family for the conditional historical Log-Laplace transform \(\psi_t(u|\underline{w_t})\): \(\left\{ \psi_t (u|\underline{w_t} ; \theta_1), \theta_1 \in \Theta_1 \right\}\).
• Step 2 – Specification of the SDF. Considering the affine specification of as (2.4), that is: \[ \mathcal{M}_{t,t+1} (\underline{w_{t+1}}) = \exp\left\{ -i_{t}(\underline{w_t}, \tilde{\theta}) + \alpha'_t(\underline{w_t})w_{t+1} - \psi_t [\alpha_t (w_t)|\underline{w_t} ; \theta_1]\right\}, \] we need to specifiy functions \(i_{t}(\underline{w_t}, \tilde{\theta})\) and \(\alpha_t(\underline{w_t})\). Assume that \(\alpha_t(\underline{w_t})\) belongs to a parametric family: \(\left\{ \alpha_t (\underline{w_t} ; \theta_2),\theta_2 \in \Theta_2 \right\}\). We then have: \[\begin{eqnarray*} &&\mathcal{M}_{t,t+1}(\underline{w_{t+1}}, \theta) \\ &=& \exp \left\{ - i_{t} (\underline{w_t}, \tilde{\theta}) + \alpha'_t (\underline{w_t},\theta_2) w_{t+1} - \psi_{t} \left[ \alpha_t (\underline{w_t}, \theta_2) | \underline{w_t} ; \theta_1 \right] \right\}, \end{eqnarray*}\] where \(\theta = (\tilde{\theta}', \theta'_1,\theta'_2)' \in \tilde{\Theta}\times \Theta_1 \times \Theta_2 = \Theta\).
• Step 3 – Internal consistency conditions (ICC). For any payoff \(g(\underline{w_s})\) settled at \(s>t\), with price \(p(\underline{w_t})\) at \(t\) which is a known function of \(\underline{w_t}\), we must have: \[\begin{equation*} p(\underline{w_t}) = \mathbb{E} \left\{\mathcal{M}_{t,t+1} (\theta) \dots \mathcal{M}_{s-1,s} (\theta) g(\underline{w_s}) | \underline{w_t}, \theta_1 \right\} \forall \; \underline{w_t}, \theta.\tag{2.9} \end{equation*}\] These ICC pricing conditions may imply strong constraints on \(\theta\). For instance, when components of \(w_t\) are returns of some assets: if \(w_{1,t} = \log(p_{1,t}/p_{1,t-1})\), then we must have \(\mathbb{E}_t [\mathcal{M}_{t,t+1} \exp (e'_1 w_{t+1})]= 1\) (Euler equation). Or, in the case of interest rates with various maturities: if \(w_{1,t} = -1/h\log B(t,h)\), then we must have \(e'_1 w_{t} = - 1/h \log \mathbb{E}_t (\mathcal{M}_{t,t+1}\times \dots \times \mathcal{M}_{t+h-1,t+h})\).

The previous three steps imply the specification of the R.N. dynamics (according to Eq. (2.8)): \[\begin{equation*} \psi^{\mathbb{Q}} (u | \underline{w_t}, \theta_1, \theta_2) = \psi_t \left[ u + \alpha_t (\underline{w_t}, \theta_2) | \underline{w_t}, \theta_1 \right] - \psi_t \left[ \alpha_t (\underline{w_t}, \theta_2) | \underline{w_t}, \theta_1 \right]. \end{equation*}\]

2.3.2 The R.N.-constrained direct modelling (or mixed modelling)

• Step 1 – Specification of the physical dynamics. We select a family \(\{ \psi_t (u | \underline{w_t},\theta_1), \theta_1 \in \Theta_1 \}\).
• Step 2 – Specification of the risk-neutral dynamics. We select a family \(\{\psi^{\mathbb{Q}}_t (u | \underline{w_t}, \theta^*),\theta^* \in \Theta^* \}\) and, possibly, \(\{i_{t}(\underline{w_t},\tilde{\theta}),\tilde{\theta}\in\tilde{\Theta}\}\).
• Step 3 – Internal Consistency Conditions (ICC). Once the parameterization \((\tilde{\theta}, \theta_1, \theta^*) \in \tilde{\Theta} \times \Theta^*_1\) is defined, ICCs may be imposed. For instance, if \(w_{1,t} = \log(p_{1,t}/p_{1,t-1})\), then we must have \(\exp(-i_t)\mathbb{E}^{\mathbb{Q}}_t \exp (e_{1}' w_{t+1}) = 1\). Or if \(w_{1,t} = B(t,h)\), then \(e_{1}' w_{t} = \mathbb{E}_t^{\mathbb{Q}} \exp(-i_t - \dots - i_{t+h-1})\).

The SDF is a by-product. If we want an exponential affine SDF, for any pair \((\psi^{\mathbb{Q}}_t, \psi_t)\) belonging to these families, there must exist a unique function \(\alpha_t (\underline{w_t})\) denoted by \(\alpha_t (w_t ; \theta_1, \theta^*)\), and satisfying: \[\begin{equation*} \psi^{\mathbb{Q}}_t (u | \underline{w_t}) = \psi_t \left[ u + \alpha_t (w_t) | \underline{w_t} \right] - \psi_t \left[ \alpha_t (\underline{w_t}) | \underline{w_t} \right]. \end{equation*}\]

2.3.3 Back modelling (based on three steps)

• Step 1 – Specification of the R.N. dynamics, and possibly of \(i_{t}(\underline{w_t})\)]: \(\psi^{\mathbb{Q}}_t (u | \underline{w_t}; \theta^*_1)\).
• Step 2 – Internal consistency conditions (ICC), if relevant, are taken into account: \[\begin{equation*} \begin{array}{lll} && p(\underline{w_t}) = \mathbb{E}^{\mathbb{Q}}_t \left[ \exp (-i_{t} (\underline{w_t},\tilde{\theta}) - \dots - i_{s-1} (\underline{w_s}, \tilde{\theta}))g(\underline{w_s}) | \underline{w_t} , \theta^*_1\right] ,\\ && \forall \underline{w_t} , \tilde{\theta} , \theta^*_1. \end{array} \end{equation*}\]
• Step 3 – Choice of the specification of the prices of risk. One chooses function \(\alpha_t(\underline{w_t})\) without any constraint; this amounts to defining the family \(\{ \alpha_t (\underline{w_t}, \theta^*_2), \theta^*_2\in \Theta^*_2 \}\).

The historical dynamics is obtained as a by-product. Indeed: \[\begin{equation*} \psi_t(u | \underline{w_t} ; \theta^*_1, \theta^*_2) = \psi_t^{\mathbb{Q}}\left[ u -\alpha_t (\underline{w_t}, \theta^*_2)|\underline{w_t} ; \theta^*_1 \right] -\psi^{\mathbb{Q}}_t \left[- \alpha_t (\underline{w_t}, \theta^*_2) | \underline{w_t},\theta^*_1 \right]. \end{equation*}\]