A Joint Stock and Bond Market based on the Hyperbolic Gaussian Model

Abstract

In this paper, we introduce a joint bond and stock market model based on the state price density approach as a mean to discount future payments—whether these are stochastic dividend payments or secure repayments of government zerobonds. Based upon a recipe of Rogers [20], we define a state price density model, the so-called Hyperbolic Gaussian model which allows for closed form zerobond prices and stock prices in an arbitrage-free way. It is particularly useful for insurance applications where large time horizons are considered. We estimate the joint factor model using the extended Kalman filter. The model we propose here is computationally much simpler than other models which have been considered in the literature.

Appendix

The proof of Theorem 2.7 is as follows:

Proof

Using the Itô-Doeblin formula it is easy to see that the state price density

$$ \varsigma_t = {1\over 2} e^{-\alpha t} ( e^{V_t+c}+e^{-V_t-c}) $$

satisfies the SDE

$$ d\varsigma_t = -\alpha \varsigma_t dt + \tanh(V_t+c) \varsigma_t d V_t + {1\over 2} \varsigma_t d \langle V \rangle_{t}. $$

Since $\langle V \rangle_{t}=\|\hat{C}\|^2 t$ we obtain

$$ d\varsigma_t = \varsigma_t \left(-\alpha + \gamma'\kappa (\tilde{\mu}-X_t) \tanh(V_t+c)+{1\over 2} \hat{C}'\hat{C}\right) dt + \varsigma_t\tanh(V_t+c) \hat{C} dZ_t $$

In view of Corollary 2.5 we obtain

$$ d\varsigma_t = -\varsigma_t r_t dt + \varsigma_t \tanh(V_t+c) \hat{C} dZ_t. $$

Let us denote by $\mathcal{E}(X)$ the stochastic exponential of the process X. Since $\int_0^\cdot r_sds$ is of bounded variation we obtain

$$ \begin{aligned} \varsigma_t &= {\mathcal{E}}\left(\int_0^\cdot r_sds\right)_t \cdot {\mathcal{E}}\left(\int_0^\cdot \tanh(V_s+c)\hat{C}dZ_s\right)_t\\ &= \exp\left(-\int_0^t r_s ds\right) \cdot {\mathcal{E}}\left(\int_0^\cdot \tanh(V_s+c)\hat{C}dZ_s\right)_t. \end{aligned} $$

(6.1)

If we define $B_t = \exp\Big(\int_0^t r_s ds\Big)$ and $L_t := \mathcal{E}(\int_0^\cdot \tanh(V_s+c)\hat{C}dZ_s)_t$ we obtain $\varsigma_t = B_t^{-1} L_t.$ Note that since $\tanh$ is bounded, (L _t ) is an $(\mathcal{F}_t)$-martingale with expectation 1. Hence we can define the probability measure ${\mathbb{Q}}$ by ${\frac{d\mathbb{Q}}{d\operatorname{\mathbb{P}}}\big|_{\mathcal{F}_t}=L_t. }$ Note that ${\mathbb{Q}}$ does not depend on T. From the Bayes formula we obtain

$$ P(t,T) = \frac{\operatorname{{\mathbb{E}}}[\varsigma_T|{\mathcal{F}}_t]}{\varsigma_t}= \frac{\operatorname{{\mathbb{E}}}[B_T^{-1}L_T|{\mathcal{F}}_t]}{B_t^{-1}L_t}= B_t \operatorname{{\mathbb{E}}}_{{\mathbb{Q}}}[B_T^{-1}|F_t]. $$

Hence the discounted bond price is a martingale under ${\mathbb{Q}}$ which shows that the market is free of arbitrage.□

The proof of Theorem 3.2 is as follows:

Proof

It is sufficient to show that ${\varsigma^{-1}B_t^{-1} \operatorname{\mathbb{E}}[\varsigma_{\tau_n} D_{\tau_n} | \mathcal{F}_t]}$ is a ${\mathbb{Q}}$-martingale for the same ${\mathbb{Q}}$ as in the proof of Theorem 2.7. However, the Bayes formula again implies

$$ \begin{aligned} \frac{\operatorname{{\mathbb{E}}}[\varsigma_{\tau_n} D_{\tau_n} | {\mathcal{F}}_t]}{ \varsigma_t } &= \frac{\operatorname{{\mathbb{E}}}[B_{\tau_n}^{-1}L_{\tau_n} D_{\tau_n} | {\mathcal{F}}_t]}{ B_t^{-1} L_t} \\ &= B_t \operatorname{{\mathbb{E}}}_{{\mathbb{Q}}}[B_{\tau_n}^{-1} D_{\tau_n} | {\mathcal{F}}_t] \end{aligned} $$

which implies the statement.□