A multivariate stochastic volatility model with applications in the foreign exchange market

Abstract

The main objective of this paper is to study the behavior of a daily calibration of a multivariate stochastic volatility model, namely the principal component stochastic volatility (PCSV) model, to market data of plain vanilla options on foreign exchange rates. To this end, a general setting describing a foreign exchange market is introduced. Two adequate models—PCSV and a simpler multivariate Heston model—are adjusted to suit the foreign exchange setting. For both models, characteristic functions are found which allow for an almost instantaneous calculation of option prices using Fourier techniques. After presenting the general calibration procedure, both the multivariate Heston and the PCSV models are calibrated to a time series of option data on three exchange rates—USD-SEK, EUR-SEK, and EUR-USD—spanning more than 11 years. Finally, the benefits of the PCSV model which we find to be superior to the multivariate extension of the Heston model in replicating the dynamics of these options are highlighted.

Appendix: Proofs

Proof (Proposition 1)

The trivariate conditional characteristic function can be written as

$$\begin{aligned} \varPhi _{\mathbf {X}}(\mathbf {w},T\mid {\mathcal {F}}_t) = E_{{\mathbb {Q}}_{\textit{DOM}}}\left[ e^{i\mathbf {w}'\mathbf {X}(T)}| {\mathcal {F}}_t \right] , \end{aligned}$$

(15)

where $\mathbf {w} = \left( \begin{array}{c} w_1\\ w_2\\ w_3\\ \end{array} \right) \in {\mathbb {R}}^3$. With (D2) it holds for the log-exchange rates that

$$\begin{aligned} \mathbf {X}(t) = \left( \begin{array}{c} X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t)\\ X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t)\\ X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(t)\\ \end{array} \right) = \left( \begin{array}{c} X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t)\\ X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t)\\ X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t) - X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t)\\ \end{array} \right) . \end{aligned}$$

(16)

Using (15) and (16) we obtain the expression for the trivariate conditional characteristic function

$$\begin{aligned} \varPhi _{\mathbf {X}}(\mathbf {w},T\mid {\mathcal {F}}_t)&= E_{{\mathbb {Q}}_{\textit{DOM}}}\left[ e^{i\mathbf {w}'\mathbf {X}(T)} \bigg | {\mathcal {F}}_t \right] \\&= E_{{\mathbb {Q}}_{\textit{DOM}}}\left[ e^{i\left( (w_1 - w_3)\cdot X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T) + (w_2 + w_3)\cdot X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T) \right) } \bigg | {\mathcal {F}}_t \right] \\&= E_{{\mathbb {Q}}_{\textit{DOM}}}\left[ e^{i(w_1 - w_3)\cdot X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T)} \bigg | {\mathcal {F}}_t \right] \\&\quad \cdot E_{{\mathbb {Q}}_{\textit{DOM}}}\left[ e^{i(w_2 + w_3) \cdot X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T)} \bigg | {\mathcal {F}}_t \right] \\&= \varPhi _{X^{\textit{FOR}_1 \text {-} \textit{DOM}}}(w_1 - w_3,T\mid {\mathcal {F}}_t) \cdot \varPhi _{X^{\textit{FOR}_2 \text {-} \textit{DOM}}}(w_2 + w_3,T\mid {\mathcal {F}}_t), \end{aligned}$$

where $\varPhi _{X^{\textit{FOR}_1 \text {-} \textit{DOM}}}(w_1 - w_3,T\mid {\mathcal {F}}_t)$ and $\varPhi _{X^{\textit{FOR}_2 \text {-} \textit{DOM}}}(w_2 + w_3,T\mid {\mathcal {F}}_t)$ are the conditional characteristic functions of $X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T)$ and $X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T)$, respectively. The penultimate equation follows from the independence of $(X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t))_{t\ge 0 }$ and $(X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t))_{t\ge 0 }$.

Proof (Proposition 2)

For the conditional characteristic function of $\mathbf {X}(t)$ it holds that

$$\begin{aligned} \varPhi _{\mathbf {X}}\left( \mathbf {w},T \mid {\mathcal {F}}_t\right)&= E\left[ e^{i\mathbf {w}'{\mathbf {X}}(T)} \bigg | {\mathcal {F}}_t \right] \\&= E\left[ e^{i \left( w_1\cdot X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T) + w_2\cdot X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T) + w_3\cdot X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(T) \right) } \bigg | {\mathcal {F}}_t \right] \\&= e^{i\mathbf {w}'{\mathbf {X}}(t)} \cdot E\left[ e^{i \left( w_1\cdot \left( X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t)\right) + w_2 \cdot \left( X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t)\right) \right. } \right. \\&\quad \left. {}^{\left. + w_3 \cdot \left( X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(T) - X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(t) \right) \right) } \bigg | {\mathcal {F}}_t \right] \\&= e^{i\mathbf {w}'{\mathbf {X}}(t)} \cdot E\left[ e^{i\mathbf {w}'\left( \mathbf {X}(T) - \mathbf {X}(t)\right) } \bigg | {\mathcal {F}}_t \right] . \end{aligned}$$

Here, the differences $X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t), X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t)$, and $X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(T) - X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(t)$ can be rewritten as

$$\begin{aligned} \begin{aligned} X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t)&= \sum \limits _{j=1}^2 \left( Z^{\textit{FOR}_1 \text {-} \textit{DOM}}_{j}(T) - Z^{\textit{FOR}_1 \text {-} \textit{DOM}}_{j}(t)\right) ,\\ X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t)&= \sum \limits _{j=1}^2 \left( Z^{\textit{FOR}_2 \text {-} \textit{DOM}}_{j}(T) - Z^{\textit{FOR}_2 \text {-} \textit{DOM}}_{j}(t)\right) ,\\ X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(T) - X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(t)&= \sum \limits _{j=1}^2 \left( Z^{\textit{FOR}_2 \text {-} \textit{FOR}_1}_{j}(T) - Z^{\textit{FOR}_2 \text {-} \textit{FOR}_1}_{j}(t)\right) , \end{aligned} \end{aligned}$$

with the dynamics of $Z^{\textit{FOR}_1 \text {-} \textit{DOM}}_{j}(t), Z^{\textit{FOR}_2 \text {-} \textit{DOM}}_{j}(t)$, and $Z^{\textit{FOR}_2 \text {-} \textit{FOR}_1}_{j}(t)$ defined by

$$\begin{aligned} dZ^{\textit{FOR}_1 \text {-} \textit{DOM}}_{j}(t)&= \left( \frac{r_{\textit{DOM}} - r_{\textit{FOR}_1}}{2}-\frac{1}{2}a_{1j}^{2}\lambda _{j}(t)\right) dt +a_{1j}\sqrt{\lambda _{j}(t)}dW_{j}(t),\\ dZ^{\textit{FOR}_2 \text {-} \textit{DOM}}_{j}(t)&= \left( \frac{r_{\textit{DOM}} - r_{\textit{FOR}_2}}{2}-\frac{1}{2}a_{2j}^{2}\lambda _{j}(t)\right) dt +a_{2j}\sqrt{\lambda _{j}(t)}dW_{j}(t),\\ dZ^{\textit{FOR}_2 \text {-} \textit{FOR}_1}_{j}(t)&= \left( \frac{r_{\textit{FOR}_1}-r_{\textit{FOR}_2}}{2}-\frac{1}{2}\left( a_{2j}^{2} - a_{1j}^{2}\right) \lambda _{j}(t)\right) dt \\&\quad +\left( a_{2j} - a_{1j}\right) \sqrt{\lambda _{j}(t)}dW_{j}(t). \end{aligned}$$

Defining $Z^*_{j}(t)$ as $Z^*_{j}(t) :={w_1 \cdot Z^{\textit{FOR}_1 \text {-} \textit{DOM}}_{j}(t) + w_2 \cdot Z^{\textit{FOR}_2 \text {-} \textit{DOM}}(t)} + w_3 \cdot Z^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(t)$, it follows that $Z^*_{j}(t)$ is a Heston-type stochastic process with dynamics

$$\begin{aligned} \begin{aligned} dZ^*_{j}(t)&= \left( r(\mathbf {w}) + \frac{b_{j}(\mathbf {w})}{c^2_{j}(\mathbf {w})}\lambda ^*_{j}(t)\right) dt + \sqrt{\lambda ^*_{j}(t)}dW_{j}(t),\\ d\lambda ^*_{j}(t)&= \kappa _{j}(c^2_{j}(\mathbf {w}) \cdot \theta _{j}-\lambda ^*_{j}(t))dt+c_{j}(\mathbf {w}) \cdot \sigma _{j} \cdot \sqrt{\lambda ^*_{j}(t)}dB_{j}(t), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} c_{j}(\mathbf {w})&:=(w_{1} - w_3)\cdot a_{1j} + (w_{2} + w_3)\cdot a_{2j},\\ b_{j}(\mathbf {w})&:=-\frac{1}{2}\left( (w_{1} - w_{3}) \cdot a_{1j}^{2} + (w_{2} + w_{3})\cdot a_{2j}^{2}\right) ,\\ r(\mathbf {w})&:=(w_1 - w_3) \cdot \frac{r_{\textit{DOM}}-r_{\textit{FOR}_1}}{2} + (w_2 + w_3) \cdot \frac{r_{\textit{DOM}}-r_{\textit{FOR}_2}}{2}, \end{aligned} \end{aligned}$$

(17)

and $\lambda ^*_{j}(t) :=c^2_{j}(\mathbf {w}) \cdot \lambda _{j}(t)$.

Due to the fact that $\left\langle dW_j(t),dB_{j'}(t)\right\rangle = \left\langle dW_{j}(t),dW_{j'}(t)\right\rangle = \left\langle dB_{j}(t),dB_{j'}(t)\right\rangle = 0$ for all $j\ne j'$ it can be shown that $Z^*_{j}(t)$ and $Z^*_{j'}(t)$ are independent.

As a Heston-type stochastic process, the characteristic function of $Z^*_{j}(t)$ at time T conditional on ${\mathcal {F}}_t$ is given by

$$\begin{aligned} \varPhi _{Z_j^*}\left( \phi ,T \mid {\mathcal {F}}_t\right)= & {} E\left[ e^{i\phi Z_j^*(T)} \bigg | {\mathcal {F}}_t \right] \nonumber \\= & {} e^{C_{j}(\phi , T-t ) + D_{j}(\phi , T-t )\cdot \lambda ^*_j(t) + i\phi Z_j^*(t)}, \end{aligned}$$

(18)

where

$$\begin{aligned} \begin{aligned} C_{j}(\phi , T-t)&= r(\mathbf {w})\phi i\cdot (T-t) +\frac{\kappa _j \theta _j }{\sigma _j ^{2}}\\&\quad \left\{ (\kappa _j -\rho _j c_{j}(\mathbf {w})\sigma _j \phi i - d_{j})\cdot (T-t) - 2\ln \left[ \frac{1-g_{j} \cdot e^{-d_{j}\cdot (T-t) }}{1-g_{j}}\right] \right\} ,\\ D_{j}(\phi , T-t)&= \frac{\kappa _j -\rho _j c_{j}(\mathbf {w})\sigma _j \phi i-d_{j}}{{c^{2}_{j}(\mathbf {w})}\sigma _j^{2}}\left[ \frac{1-e^{-d_{j}\cdot (T-t) }}{1-g_{j} \cdot e^{- d_{j}\cdot (T-t) }}\right] ,\\ g_{j}&= \frac{\kappa _j -\rho _j c_{j}(\mathbf {w})\sigma _j \phi i - d_{j}}{\kappa _j -\rho _j c_{j}(\mathbf {w})\sigma _j \phi i + d_{j}},\\ d_{j}&= \sqrt{(\rho _j c_{j}(\mathbf {w}) \sigma _j \phi i - \kappa _j )^{2} + \sigma _j ^{2}\left( -2 b_{j}(\mathbf {w}) \phi i + {c^{2}_{j}(\mathbf {w})}\phi ^{2}\right) }, \end{aligned} \end{aligned}$$

with $c_{j}(\mathbf {w}), b_{j}(\mathbf {w})$, and $r(\mathbf {w})$ as in (17). Hence, for the conditional characteristic function it follows that

$$\begin{aligned} \varPhi _{\mathbf {X}}\left( \mathbf {w},T \mid {\mathcal {F}}_t\right)&= e^{i\mathbf {w}'{\mathbf {X}}(t)} \cdot E\left[ e^{i\mathbf {w}'\left( \mathbf {X}(T) - \mathbf {X}(t)\right) } \bigg | {\mathcal {F}}_t \right] \\&= e^{i\mathbf {w}'{\mathbf {X}}(t)} \cdot E\left[ e^{i \left( w_1\cdot \left( X^{\textit{FOR}_1 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_1 \text {-} \textit{DOM}}(t)\right) + w_2\cdot \left( X^{\textit{FOR}_2 \text {-} \textit{DOM}}(T) - X^{\textit{FOR}_2 \text {-} \textit{DOM}}(t)\right) \right. } \right. \\&\quad \left. {}^{\left. + w_3 \cdot \left( X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(T) - X^{\textit{FOR}_2 \text {-} \textit{FOR}_1}(t) \right) \right) } \bigg | {\mathcal {F}}_t \right] \\&= e^{i\mathbf {w}'{\mathbf {X}}(t)} \cdot E\left[ e^{i \left( \sum \limits _{j=1}^2 Z^*_{j}(T) - Z^*_{j}(t)\right) } \bigg | {\mathcal {F}}_t \right] \\&= e^{i\mathbf {w}'\mathbf {X}(t)} \cdot \prod \limits ^2_{j=1} E\left[ e^{i \left( Z^*_{j}(T) - Z^*_{j}(t)\right) } \bigg | {\mathcal {F}}_t \right] \\&= e^{i\mathbf {w}'\mathbf {X}(t)} \cdot \prod \limits ^2_{j=1} e^{- i Z^*_{j}(t)} E\left[ e^{i Z^*_{j}(T)} \bigg | {\mathcal {F}}_t \right] \\&= e^{i\mathbf {w}'\mathbf {X}(t)} \cdot \prod \limits ^2_{j=1} e^{C_{j}(1, T-t ) + D_{j}(1, T-t )\cdot \lambda _j(t)\cdot c^2_{j}(\mathbf {w})}, \end{aligned}$$

where the third equation holds due to the independence of $Z^*_{1}(t)$ and $Z^*_{2}(t)$. Moreover, with $\phi = 1$ in Eq. (18) the desired form of the characteristic function is obtained.