Continuous equilibrium in affine and information-based capital asset pricing models

Abstract

We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents’ risk aversion on the implied volatility of simultaneously-traded European-style options.

Appendices

Proofs

Proof of Theorem 2.1

Due to the time-consistency and strict monotonicity of the entropic preferences, it suffices to show that the strategies ${{\hat{\vartheta }}}^a$ are optimal for the utility maximization in $t=0$. Note first that (2.2) ensures that (2.3) and (2.4) are well-defined. In particular, the price process $S$ is a $Q$-martingale, and thus $Q\in \mathcal{P }$. Furthermore, the constant strategies ${{\hat{\vartheta }}}^a$ lie in $\varTheta$, since for any ${\tilde{Q}}\in \mathcal{P }$, the process $G_t({{\hat{\vartheta }}}^a)={{\hat{\vartheta }}}^a\cdot (S_t-S_0)$ is by assumption a ${\tilde{Q}}$-martingale, and hence in particular a ${\tilde{Q}}$-supermartingale.

We now show that the quantity $\gamma$ introduced in (2.1) can be seen as the risk aversion of some representative agent whose optimal utility is attained at the constant strategy ${\vartheta }^*\equiv n + \eta$. Indeed, since $S$ is a $Q$-martingale, and $(n+\eta )\cdot S_T\in L^1(Q)$, the utility maximization of the representative agent can be formulated as follows:¹⁴

$$\begin{aligned}&{\mathop {\text{ sup}}\limits _{{\vartheta }\in \varTheta , {E}_{Q}\left[{G_T({\vartheta })}\right]\le {E}_{Q}\left[{(n+\eta )\cdot S_T}\right]}} \left\{ { U_0^\gamma \big (G_T({\vartheta })\big ) }\right\} \nonumber \\&\quad \le {\mathop {\text{ sup}}\limits _{{\vartheta }\in \varTheta }} \left\{ { U_0^\gamma \big (G_T({\vartheta }) - {E}_{Q}\left[{G_T({\vartheta })}\right] + {E}_{Q}\left[{(n+\eta )\cdot S_T}\right]\big )}\right\} \nonumber \\&\quad = {\mathop {\text{ sup}}\limits _{{\vartheta }\in \varTheta }} \left\{ { U_0^\gamma \big (G_T({\vartheta })\big ) - {E}_{Q}\left[{G_T({\vartheta })}\right]}\right\} + {E}_{Q}\left[{(n+\eta )\cdot S_T}\right]\nonumber \\&\quad \le \frac{1}{\gamma }H(Q|P) + {E}_{Q}\left[{(n+\eta )\cdot S_T}\right]. \end{aligned}$$

(5.1)

The last inequality is derived from the dual representation of $U_0^\gamma$, where the relative entropy is given by $H(Q|P)=E[\frac{dQ}{dP}\log (\frac{dQ}{dP})]$. But $G_T({\vartheta }^*)$ with ${\vartheta }^*\equiv n+\eta$ plugged into the representative agent’s utility $U^\gamma _0(\cdot )$ yields

$$\begin{aligned} U^\gamma _0\big ((n+\eta )\cdot S_T\big ) = \frac{1}{\gamma }H(Q|P) + {E}_{Q}\left[{(n+\eta )\cdot S_T}\right]. \end{aligned}$$

Comparing this with (5.1) shows that ${\vartheta }^*\equiv n+\eta$ is indeed optimal for the representative agent when the price process $S$ is given by (2.4). Individual optimality of ${{\hat{\vartheta }}}^a$ for the single agents now follows by a scaling argument and the specific form of the aggregated endowment. Note that, for all $a\in \mathbb A$,

$$\begin{aligned} {\vartheta }^*={\mathop {\text{ arg} \text{ max}}\limits _{{\vartheta }\in \varTheta }}\left\{ {U_0^\gamma \big (G_T({\vartheta })\big )}\right\} \end{aligned}$$

is equivalent to

$$\begin{aligned} \frac{\gamma }{\gamma ^a} {\vartheta }^*={\mathop {\text{ arg} \text{ max}}\limits _{{\vartheta }\in \varTheta }}\left\{ {U_0^a\big (G_T({\vartheta })\big )}\right\} , \end{aligned}$$

which in turn is equivalent to

$$\begin{aligned} \frac{\gamma }{\gamma ^a} {\vartheta }^* - \eta ^a = {\mathop {\text{ arg} \text{ max}}\limits _{{\vartheta }\in \varTheta }}\left\{ {U_0^a\big (H^a + G_T({\vartheta })\big )}\right\} . \end{aligned}$$

This shows that ${{\hat{\vartheta }}}^a$ is the optimal strategy for agent $a \in \mathbb{A }$. Since the strategies $({{\hat{\vartheta }}}^a)_{a\in \mathbb A }$ add up to $n$, the market clears at any time, and hence the pair $((S_t)_{t\in [0,T]},({{\hat{\vartheta }}}^a)_{a\in \mathbb{A }})$ forms an equilibrium. $\square$

Proof of Theorem 3.2

Part 1: Pricing Formula (3.7). From Sect. 3.1.1 it is known that $Y=(V,X)$ satisfies

$$\begin{aligned} {E}\left[{{\exp \left({u\cdot Y_T}\right)} \mid \mathcal{F }_{t}}\right] = \exp \left[{\phi (\tau ,u) + \psi (\tau ,u)\cdot Y_t}\right], \end{aligned}$$

(5.2)

for all $u=(u_v,u_x) \in \mathbb{C }^2$ such that $(T,u) \in \mathcal{E }_\mathbb{C }$, since the latter implies that (3.1) and thus (3.2) hold for all $t\in [0,T]$.

Let us assume for the moment that (2.2) holds. This will be verified later. We then know from (2.3) that the equilibrium pricing measure $Q$ is given by its Radon-Nikodym-density

$$\begin{aligned} \frac{dQ}{dP} = \frac{\exp \left({-{{\tilde{\gamma }}}\cdot S_T}\right)}{{E}\left[{\exp \left({-{{\tilde{\gamma }}}\cdot S_T}\right)}\right]} = \frac{\exp \left({-{{\tilde{\gamma }}}\cdot f(X_T)}\right)}{{E}\left[{\exp \left({-{{\tilde{\gamma }}}\cdot f(X_T)}\right)}\right]}. \end{aligned}$$

Hence, by applying Bayes formula and following (2.4), we obtain

$$\begin{aligned} S^k_t = {E}_{Q}\left[{ S^k_T} \mid \mathcal{F }_{t} \right] = \frac{{E}\left[{{f^k(X_T) \exp \left({-{{\tilde{\gamma }}}\cdot f(X_T)}\right)} \mid \mathcal{F }_{t}}\right]}{{E}\left[{{\exp \left({-{{\tilde{\gamma }}}\cdot f(X_T)}\right)} \mid \mathcal{F }_{t}}\right]} \end{aligned}$$

(5.3)

for the equilibrium price of the $k$th security. The Fourier transforms ${\hat{g}}^k$ and ${\hat{h}}$ defined in (3.4) and (3.5), respectively, exist and are integrable by assumption. Hence we apply the Fourier inversion formula¹⁵ to obtain

$$\begin{aligned} g^k(x) = \frac{1}{2\pi } \int \limits _\mathbb{R } \mathrm{e}^{isx} {\hat{g}}^k(s) ds\quad \text{ and}\quad h(x) = \frac{1}{2\pi } \int \limits _\mathbb{R } \mathrm{e}^{isx} {\hat{h}}(s) ds, \end{aligned}$$

$dx$-almost surely. With this at hand, (5.3) transforms to

$$\begin{aligned} S^k_t&= \frac{{E}\left[{{\exp \left({-\alpha ^k X_T}\right) g^k(X_T)} \mid \mathcal{F }_{t}}\right]}{{E}\left[{{\exp \left({-\beta X_T}\right) h(X_T)} \mid \mathcal{F }_{t}}\right]}\nonumber \\&= \frac{{E}\left[{{ \int _\mathbb{R } \exp \left[{(-\alpha ^k + is)X_T}\right] {\hat{g}}^k(s) ds} \mid \mathcal{F }_{t}}\right]}{{E}\left[{{ \int _\mathbb{R } \exp \left[{(-\beta + is)X_T}\right] {\hat{h}}(s) ds} \mid \mathcal{F }_{t}}\right]}. \end{aligned}$$

(5.4)

Now we observe that

$$\begin{aligned}&{E}\left[{{ \,\,\bigg | \,\int \limits _\mathbb{R } \exp \left[{(-\alpha ^k + is)X_T}\right] {\hat{g}}^k(s) ds \,\,\bigg | \,\,} \mid \mathcal{F }_{t}}\right]\nonumber \\&\quad < {E}\left[{{ \exp \left({-\alpha ^k X_T}\right) \int \limits _\mathbb{R } \left|{{\hat{g}}^k(s)}\right| ds \,} \mid \mathcal{F }_{t}}\right] < \infty , \end{aligned}$$

(5.5)

since $\left({T,(0,-\alpha ^k)}\right)\in \mathcal{E }\subseteq \mathcal{E }_\mathbb{C }$ and ${\hat{g}}^k$ is integrable. The same holds analogously for the denominator in (5.4). In particular, we have

$$\begin{aligned} 0< {E}\left[{{\exp \left({-{{\tilde{\gamma }}}\cdot f(X_T)}\right)} \mid \mathcal{F }_{t}}\right] < \infty ,\text{ for} \text{ all} t\in [0,T]\,, \end{aligned}$$

(5.6)

since we required $Y^T$ to be conservative and $\left({T,(0,-\beta )}\right)$ to lie in $\mathcal{E }$. Thus, (5.5) and (5.6), in combination with (5.3), yield that (2.2) is indeed satisfied.  We may now apply Fubini’s Theorem to exchange the order of integration, and we get that

$$\begin{aligned}&{E}\left[{{ \int \limits _\mathbb{R } \exp \left[{(-\alpha ^k + is)X_T}\right] {\hat{g}}^k(s) ds} \mid \mathcal{F }_{t}}\right] \nonumber \\&\quad =\int \limits _\mathbb{R } {E}\left[{{ \exp \left[{(-\alpha ^k + is)X_T}\right]} \mid \mathcal{F }_{t}}\right] {\hat{g}}^k(s) ds\nonumber \\&\quad =\int \limits _\mathbb{R } \exp \left[{\phi \big (\tau ,(0,-\alpha ^k+is)\big )+\psi \big (\tau ,(0,-\alpha ^k+is)\big )\cdot Y_t}\right] {\hat{g}}^k(s) ds. \end{aligned}$$

(5.7)

The affine transformation formula (5.2) holds, since $\left({T,(0,-\alpha ^k)}\right) \in \mathcal{E }$. Applying the same arguments to the denominator in (5.4) combined with (5.7) yields the desired form of $S^k_t$ in (3.7).

Part 2: Pricing Formula (3.8). We outline the details for $K=1$, the rest follows by repeating the arguments for the partial derivative with respect to each $\zeta ^k$. So we assume we only have one security $S$ with corresponding ${{\tilde{\gamma }}} \in \mathbb{R }$ affecting the density of the pricing measure $Q$. It follows that

$$\begin{aligned} \frac{dQ}{dP} = \frac{\exp (-{\tilde{\gamma }} S_T)}{{E}\left[{\exp (-{\tilde{\gamma }} S_T)}\right]} = \frac{\exp (-{\tilde{\gamma }} f(X_T))}{{E}\left[{\exp (-{\tilde{\gamma }} f(X_T))}\right]} \end{aligned}$$

and the equilibrium price of $S$ at time $t$ can be obtained again by computing

$$\begin{aligned} S_t = \frac{{E}\left[{{f(X_T) \exp (-{\tilde{\gamma }} f(X_T))} \mid \mathcal{F }_{t}}\right]}{{E}\left[{{\exp (-{\tilde{\gamma }} f(X_T))} \mid \mathcal{F }_{t}}\right]}. \end{aligned}$$

(5.8)

Recall from Part 1 that

$$\begin{aligned} \exp \left({-{{\tilde{\gamma }}} f(X_T)}\right) \in L^1(P) \quad \text{ and}\quad f(X_T)\exp \left({-{{\tilde{\gamma }}} f(X_T)}\right) \in L^1(P)\,, \end{aligned}$$

(5.9)

due to the assumption of $(T,(0,-\alpha ))$ and $(T,(0,-\beta ))$ lying in $\mathcal{E }$. Since the set $\mathcal{E }_\mathbb{C }$ is open, compare (Keller-Ressel 2009, Lemmata 3.12 and 3.19), and due to the integrability assumptions on the functions $g^k_\zeta (s)$, the first integrability in (5.9) even holds in some neighbourhood of ${\tilde{\gamma }}$, allowing us to differentiate the function $\zeta \mapsto E[\exp (-\zeta f(X_T))|\mathcal{F }_t]$ at $\zeta ={{\tilde{\gamma }}}$. Indeed, by the smoothness of the mapping $\zeta \mapsto \exp (-\zeta f(X_T))$ and the integrability of the second term in (5.9), we obtain

$$\begin{aligned} {E}\left[{{ f(X_T) \exp \left({-{{\tilde{\gamma }}} f(X_T)}\right) } \mid \mathcal{F }_{t}}\right] \!=\! -\left.\frac{\partial }{\partial \zeta }{E}\left[{{ \exp \left({-\zeta f(X_T)}\right)} \mid \mathcal{F }_{t}}\right]\right|_{\zeta ={{\tilde{\gamma }}}},\qquad \end{aligned}$$

(5.10)

as an application of the triangular inequality and dominated convergence. On the other hand we know from an analogue of (5.4) and (5.7) that the denominator in (5.8) can be computed by

$$\begin{aligned} {E}\left[{{ \exp \left({-{{\tilde{\gamma }}} f(X_T)}\right) } \mid \mathcal{F }_{t}}\right]&= \frac{1}{2\pi }\int \limits _\mathbb{R } \exp \left[\phi \big (\tau ,(0,-\beta +is)\big ) \right.\nonumber \\&\left. + \,\psi \big (\tau , (0,-\beta + is)\big ) \cdot Y_t\right]\,{\hat{h}}_{{\tilde{\gamma }}}(s)\,\,ds, \end{aligned}$$

(5.11)

where we need the dependence of ${\hat{h}}(s)={\hat{h}}_{{\tilde{\gamma }}}(s)$ on ${\tilde{\gamma }}$. Combining (5.10) and (5.11) yields

$$\begin{aligned} {E}\left[{{ f(X_T) \exp \left({-{{\tilde{\gamma }}} f(X_T)}\right) } \mid \mathcal{F }_{t}}\right]&= - \frac{\partial }{\partial \zeta } \left( \frac{1}{2\pi }\int \limits _\mathbb{R } \exp \Big [\phi \big (\tau ,(0,-\beta +is)\big ) \right.\\&\left.+ \,\psi \big (\tau , (0,-\beta +is)\big ) \cdot Y_t\Big ]\,{\hat{h}}_{\zeta }(s)\,\,ds \Bigg ) \right|_{\zeta ={{\tilde{\gamma }}}}\,. \end{aligned}$$

$\square$

Proof of Corollary 3.3

Expression (3.9) is an immediate consequence of (3.8) in Theorem 3.2 with $f(x)=x$, and the fact that there is no need of Fourier methods to compute the denominator $H({{\tilde{\gamma }}})$ in the analogue to (5.8)

$$\begin{aligned} S^1_t = \frac{{E}\left[{{X_T \exp (-{{\tilde{\gamma }}}^1 X_T)} \mid \mathcal{F }_{t}}\right]}{{E}\left[{{\exp (-{{\tilde{\gamma }}}^1 X_T)} \mid \mathcal{F }_{t}}\right]} \,, \end{aligned}$$

(5.12)

since the affine transformation formula directly applies to the denominator in (5.12). We recall that $\left({T,(0,-{{\tilde{\gamma }}}^1)}\right) \in \mathcal{E }$. Now we only need to compute $\frac{\partial }{\partial \zeta }{E}\left[{{ \mathrm{e}^{-\zeta X_T} } \mid \mathcal{F }_{t}}\right]$, the actual derivative in formula (3.8). However, from (3.2) it follows that

$$\begin{aligned}&-\,\frac{\partial }{\partial \zeta }{E}\left[{{ \exp \left({-\zeta X_T}\right) } \mid \mathcal{F }_{t}}\right] \\&\quad \quad =\exp \left[{\phi \big (\tau ,u\big ) + \psi \big (\tau ,u\big )\cdot Y_t}\right] \left.\big [ \partial _{u_x} \phi (\tau ,u) + \partial _{u_x} \psi (\tau ,u)\cdot Y_t \big ] \right|_{u=(0,-\zeta )}. \end{aligned}$$

Combining the above with (5.12) yields

$$\begin{aligned} S^1_t = \left. \big [ \partial _{u_x} \phi (\tau ,u) + \partial _{u_x} \psi (\tau ,u)\cdot Y_t \big ] \right|_{u=(0,-{{\tilde{\gamma }}}^1)}. \end{aligned}$$

As to the remaining securities $S^2,\ldots ,S^K$, their price processes given in (3.10) directly follow from formula (3.7) in Theorem 3.2 and the discussion above. $\square$

Proof of Theorem 3.4

An application of Theorem 3.2 with $\alpha ^k = 0$, for all $k=1,\ldots ,N$, in addition to the observation that the Fourier transforms are all integrable functions yields the desired result. As to the second claim of integrability, straightforward calculations show that there exist constants ${\hat{M}}, {\hat{z}} > 0$, just depending on the model parameters, which give

$$\begin{aligned} {\mathop {\text{ max}}\limits _{f \in \{{\hat{g}}, {\hat{h}}, ({\hat{g}}^k)_{k=1}^N\}}}\int \limits _\mathbb{R } |f(s)| ds < {\hat{M}} \int \limits _\mathbb{R } \frac{1}{s^2 + {\hat{z}}} ds < \infty . \end{aligned}$$

$\square$

Proof of Theorem 3.5

The process $Y = (V,X)$ belongs to a subclass of affine processes, namely to the $\mathbb{R }^2$-valued affine diffusions.¹⁶ That is, $Y$ is a solution to the stochastic differential equation $dY_t = \mu (Y_t)dt + \rho (Y_t)dW_t$, with $Y_0 = y_0$, for a continuous function $b:D \rightarrow \mathbb{R }^2$ and a measurable function $\rho :D \rightarrow \mathbb{R }^{2\times 2}$ such that $y \mapsto \rho (y)\rho (y)^T$ is continuous. In particular, the set $\mathrm{int}\,\mathcal{D }_{0+}$ from Sect. 3.1.1 is non-empty and thus the affine transorm formula can be extended. See for instance the discussion on explosion times of the Heston model in Friz and Keller-Ressel (2010). Furthermore, the process $Y$ is conservative and, hence, so is the stopped process $Y^T$. Combining (6.3) with the fact that the generator of $(V,X)$ is determined by its diffusion matrix $\rho \rho ^T$ and its drift vector $b$, we identify the admissible parameters in (6.1), (6.2) and (6.3), where the parts connected with jumps do not play a role here. Hence we conclude that the conditional characteristic function of $Y$ allows a representation as follows

$$\begin{aligned} {E}\left[{{\exp \left({u\cdot Y_T}\right)} \mid \mathcal{F }_{t}}\right] = \exp \left[{\phi (\tau ,u) + \psi (\tau ,u)\cdot Y_t}\right], \end{aligned}$$

(5.13)

whenever $(T,u)=(T,(u_v,u_x)) \in \mathcal{E }_\mathbb{C }$, so in particular for $(T,(u_v,u_x)) \in \mathcal{E }$. The functions $\phi$ and $\psi$ satisfy the following system of Riccati equations

$$\begin{aligned} \begin{array}{ll} \partial _t \phi (t,u)= \kappa \,\psi _1 (t,u) + \mu \, \psi _2 (t,u)&\quad \phi (0,u) = 0\\ \partial _t \psi _1 (t,u) = \frac{1}{2} \sigma ^2 {\psi _1 (t,u)}^2 - \lambda \psi _1 (t,u) + \frac{1}{2} {\psi _2 (t,u)}^2&\quad \psi _1 (0,u) = u_v\\ \partial _t \psi _2 (t,u) = 0&\quad \psi _2 (0,u) = u_x. \end{array} \end{aligned}$$

(R)

A solution to the above system (R), evaluated at the vector $u = (0,u_x)$, is given by¹⁷

$$\begin{aligned} \phi \big (t,(0,u_x)\big )&= \frac{2\kappa }{\sigma ^2}\log \left( \frac{2\theta (u_x) \exp \left({\frac{\theta (u_x)+\lambda }{2}t}\right)}{\theta (u_x)(\mathrm{e}^{\theta (u_x)t}+1)+ \lambda (\mathrm{e}^{\theta (u_x)t}-1)}\right) + \mu u_xt\\ \psi _1\big (t,(0,u_x)\big )&= \frac{u_x^2(\mathrm{e}^{\theta (u_x)t}-1)}{\theta (u_x)(\mathrm{e}^{\theta (u_x)t}+1) + \lambda (\mathrm{e}^{\theta (u_x)t}-1)}\nonumber \\ \psi _2\big (t,(0,u_x)\big )&= u_x \end{aligned}$$

where

$$\begin{aligned} \theta (u_x) = \left\{ \begin{array}{l@{\quad }l} \sqrt{\lambda ^2 - \sigma ^2\,u_x^2} \quad&\text{ if}\quad \left|{u_x}\right| < \frac{\lambda }{\sigma } \\ [2mm] i \sqrt{ \sigma ^2\,u_x^2 - \lambda ^2 } \quad&\text{ if}\quad \left|{u_x}\right| > \frac{\lambda }{\sigma } \end{array}\right.. \end{aligned}$$

Following Friz and Keller-Ressel (2010) and recalling that $\lambda >0$, we distinguish two different cases

$$\begin{aligned} t^+(u_x) = \left\{ \begin{array}{l@{\quad }l} +\infty \quad&\left|{u_x}\right| <\frac{\lambda }{\sigma } \\ [2mm] \frac{2}{|\theta (u_x)|}\left( \arctan \frac{|\theta (u_x)|}{-\lambda } + \pi \right)&\left|{u_x}\right| > \frac{\lambda }{\sigma } \end{array}\right. \end{aligned}$$

such that $(T,(0,u_x)) \in \mathcal{E } \subseteq \mathcal{E }_\mathbb{C }$, for all $T \le t^+(u_x)$.¹⁸ Hence, as long as $T < t^+(u_x)$, formula (5.13) holds for all $u=(0,u_x)$, where $u_x \in \mathbb{R }$. It now follows from (3.9) in Corollary 3.3 that, for all $t\in [0,T]$,

$$\begin{aligned} S_t = \left.\big [ \partial _{u_x} \phi (\tau ,u) + \partial _{u_x} \psi _1(\tau ,u) V_t + \partial _{u_x} \psi _2(\tau ,u)X_t \big ]\right|_{u=(0,-\gamma )}. \end{aligned}$$

(5.14)

Next, we need to compute the derivatives of $\phi (t,u)$ and $\psi (t,u)$ with respect to $u_x$. Of course we have $\partial _{u_x} \psi _2(\tau ,u)\equiv 1$ and a straightforward calculation yields, with $\theta := \theta (-\gamma )$ and $\theta ^{\prime }:= [\partial _{u_x}\theta ](-\gamma )$,

$$\begin{aligned} \partial _{u_x} \phi (\tau ,(0,-\gamma ))= T(\tau ,\gamma )\quad \text{ and} \quad \partial _{u_x} \psi _1(\tau ,(0-\gamma ))=- \gamma \varGamma (\tau ,\gamma ). \end{aligned}$$

This, together with (5.14), is (3.14), the proof is complete. $\square$

Proof of Theorem 4.1

By assumption, the conditions of Theorem 2.1 are satisfied. Recall that the equilibrium price is obtained by the change of measure from $P$ to $Q$, that is:

$$\begin{aligned} S^k_t = {E}_{Q}\left[{S^k_T} \mid \mathcal{F }_{t} \right]&= \, {E}_{Q}\left[{f^k(X_1,\ldots ,X_N)} \mid \mathcal{F }_{t} \right] \\&= \, {E}\left[{{\frac{dQ}{dP}f^k(X_1,\ldots ,X_N)} \mid \mathcal{F }_{t}}\right] {E}\left[{{\frac{dQ}{dP}} \mid \mathcal{F }_{t}}\right]^{-1}\,. \end{aligned}$$

By (2.3), we know that $\frac{dQ}{dP}$ is a function of $S_T$ and hence of $X_1,\ldots ,X_N$, which is given in (4.4). Then we compute the regular conditional distribution of $\,(X_1,\ldots ,X_N)\,$ given $(\xi ^1_t,\ldots ,\xi ^N_t)$. Using the independence of the market factors, the Markov property of $\xi$, the Bayes formula, and observing that, given $(X_1,\ldots ,X_N)=(x_1,\ldots ,x_N),\,\xi ^i_t$ is Gaussian with mean $\sigma _ix_i t$ and variance $\frac{tT}{T-t}$, yields (4.5).$\square$

Proof of Proposition 4.2

The integrability assumptions on $X$ together with (Liptser and Shiryaev 2001, Theorem 7.17) yield that the innovation Brownian motion $W_t$ in (4.7) is well-defined for $t<T$. By the Fujisaki–Kallianpur–Kunita Theorem, see (Bain and Crisan 2009, Proposition 2.31), both expressions appearing in (4.6) allow for a representation with respect to $W$. Furthermore, we even know the structure of the integrands. Specifically, for every function $\varphi :\mathbb{R }\rightarrow \mathbb{R }$ such that $\varphi (X) \in L^2(P)$ and for $t<T$, we obtain that

$$\begin{aligned} {E}\left[{{\varphi (X)} \mid \mathcal{F }_{t}}\right] = {E}\left[{\varphi (X)}\right] + \int \limits _0^t\frac{\sigma T}{T-u}V^\varphi _udW_u\,, \end{aligned}$$

(5.15)

where $V^\varphi _t$, the conditional covariance of the market factor with the function $\varphi$, is given by

$$\begin{aligned} V^\varphi _t = {E}\left[{{\varphi (X)X} \mid \mathcal{F }_{t}}\right] - {E}\left[{{\varphi (X)} \mid \mathcal{F }_{t}}\right]{E}\left[{{X} \mid \mathcal{F }_{t}}\right]\,, \end{aligned}$$

(5.16)

as shown in (Brody et al. 2008, Section V). The dynamics (4.8) then follow by (5.15) in combination with (5.16) and an application of the Itô product rule to (4.6).

Proof of Corollary 4.3

The relation ${{\tilde{\gamma }}} > \kappa - 1$ ensures that the assumptions of Theorem 2.1 are met. It remains to apply Theorem 4.1 and explicitly work out the integrals in

$$\begin{aligned} \frac{\int _0^\infty x \,(1/\kappa )\exp \left({-x/\kappa }\right) \exp \left({-{{\tilde{\gamma }}} x}\right)\exp \left[{\frac{T}{T-t}\left({\sigma x \xi _t - \frac{1}{2}(\sigma x)^2 t}\right)}\right]dx}{\int _0^\infty (1/\kappa )\exp \left({-x/\kappa }\right) \exp \left({-{{\tilde{\gamma }}} x}\right)\exp \left[{\frac{T}{T-t}\left({\sigma x \xi _t - \frac{1}{2}(\sigma x)^2 t}\right)}\right]dx}\,, \end{aligned}$$

which is done by combining (Brody et al. 2008, Section VII) and (4.12), resulting in formulae (4.10) and (4.11). $\square$

Addendum to Section 3: regular affine processes

This proposition concerning the characterization of a regular affine process by its admissible parameters is stated without proof and we refer to (Duffie et al. 2003, Theorem 2.7) or (Keller-Ressel 2009, Theorem 2.6 and Equations (2.2a), (2.2b)) for two different approaches to prove it.

Proposition 6.1

Let $Y$ be a regular affine process with state space $D$. Let $F$ and $R$ be as in Definition 3.1. Then there exists a set of admissible parameters $(A,A^i,b,b^i,c,c^i,m,\mu ^i)_{i=1,\ldots ,d}$ such that $F$ and $R$ are of the Lévy–Khintchine form.

$$\begin{aligned}&F(u) = \frac{1}{2} \langle u,Au \rangle + \langle b, u\rangle - c + \int \limits _{\mathbb{R }^d\backslash \{0\}}\left(\mathrm{e}^{\langle \xi ,u \rangle } - 1 - \langle h(\xi ),u \rangle \right) m(d\xi )\end{aligned}$$

(6.1)

$$\begin{aligned}&R_i(u)=\frac{1}{2} \langle u,A^iu \rangle + \langle b^i, u\rangle - c^i \nonumber \\&\qquad \quad + \int \limits _{\mathbb{R }^d\backslash \{0\}}\left( \mathrm{e}^{\langle \xi ,u \rangle } - 1 - \langle \chi ^i(\xi ),u \rangle \right) \mu ^i(d\xi ) \end{aligned}$$

(6.2)

where $A,A^1,\ldots ,A^d$ are positive semi-definite real $d\times d$-matrices; $b,b^1,\ldots ,b^d$ are $\mathbb{R }^d$-valued vectors; $c,c^1,\ldots ,c^d$ are positive non-negative numbers; $m$ and $\mu ^1,\ldots ,\mu ^d$ are Lévy measures on $\mathbb{R }^d$, and finally $h$ and $\chi ^1,\ldots ,\chi ^d$ are suitably chosen truncation functions for the respective Lévy measures. Furthermore, the generator $\mathcal{A }$ of  $Y$ is given by

$$\begin{aligned} \mathcal{A }\varphi (x)&= \frac{1}{2} \sum _{k,l=1}^d\left({ A_{kl} + \sum _{i\in I} A^i_{kl} x_i }\right) \frac{\partial ^2 \varphi (x)}{\partial x_k\partial x_l} \nonumber \\&\quad + \,\,\langle b + \sum _{i=1}^d b^i x_i, \nabla \varphi (x) \rangle - \left({ c + \sum _{i\in I} c^ix_i }\right) \varphi (x) \nonumber \\&\quad + \,\int \limits _{D \backslash \{0\}} ( \varphi (c+\xi ) -\varphi (x) - \langle h(\xi ), \nabla \varphi (x) \rangle ) m(d\xi ) \nonumber \\&\quad +\,\sum _{i \in I} \int \limits _{D \backslash \{0\}} ( \varphi (c+\xi ) -\varphi (x) - \langle \chi ^i(\xi ), \nabla \varphi (x) \rangle ) x_i\mu ^i(d\xi )\,, \end{aligned}$$

(6.3)

and $\phi ,\,\psi$ satisfy the following system of ODEs

$$\begin{aligned} \partial _t \phi (t,u)&= F(\psi (t,u))\,,\quad \phi (0,u)=0 \end{aligned}$$

(6.4)

$$\begin{aligned} \partial _t \psi (t,u)&= R(\psi (t,u))\,,\quad \psi (0,u)=u\,. \end{aligned}$$

(6.5)