A note on the pricing of multivariate contingent claims under a transformed-gamma distribution

Abstract

We develop a framework for pricing multivariate European-style contingent claims in a discrete-time economy based on a multivariate transformed-gamma distribution. In our model, each transformed-gamma distributed underlying asset depends on two terms: a idiosyncratic term and a systematic term, where the latter is the same for all underlying assets and has a direct impact on their correlation structure. Given our distributional assumptions and the existence of a representative agent with a standard utility function, we apply equilibrium arguments and provide sufficient conditions for obtaining preference-free contingent claim pricing equations. We illustrate the applicability of our framework by providing examples of preference-free contingent claim pricing models. Multivariate pricing models are of particular interest when payoffs depend on two or more underlying assets, such as crack and crush spread options, options to exchange one asset for another, and options with a stochastic strike price in general.

Appendix

Proof

(Proposition 1) In order to obtain a functional form for the asset specific pricing kernel in Eq. (2) Definitions 2, 3 and 4, and Eq. (5) are required.

The numerator in Eq. (2), $E\left[ U_{W}|\mathbf{X}\right]$, can be obtained directly from the definition of the conditional distribution, $f\left( W|\mathbf{X}\right) =f\left( x_{1},\ldots ,x_{M}\right) /f\left( x_{1},\ldots ,x_{M-1}\right)$, which yields

$$\begin{aligned} E\left[ U_{W}|\mathbf{X}\right]&=E\left[ \exp \left( \gamma h_{M}\left( W\right) \right) |x_{1},\ldots ,x_{M-1}\right] \nonumber \\&=\frac{1}{f\left( x_{1},\ldots ,x_{M-1}\right) }\int _{y_{0}}\frac{e^{\gamma \mu _{M}^{*}}\left( y_{0}^{*}\right) ^{p_{0}-1}e^{y_{0} ^{*}\left( \gamma \sigma _{M}-1\right) }}{{\varGamma }\left( p_{0}\right) \left( 1-\gamma \sigma _{M}\right) ^{p_{M}}}\nonumber \\&\prod _{i=1}^{M-1}\frac{\left| h_{i}^{\prime }\left( x_{i}\right) \right| }{{\varGamma }\left( p_{i}\right) \sigma _{i}^{p_{i}}}\left( h_{i}\left( x_{i}\right) -\sigma _{i}y_{0}^{*}-\mu _{i}^{*}\right) ^{p_{i}-1}\nonumber \\&\exp \left[ -\left( h_{i}\left( x_{i}\right) -\sigma _{i}y_{0}^{*} -\mu _{i}^{*}\right) /\sigma _{i}\right] dy_{0} , \end{aligned}$$

(19)

where $f\left( x_{1},\ldots ,x_{M-1}\right)$ is given by Eq. (8), $y_{0}^{*}=\left( y_{0}-\mu _{0}\right) /\sigma _{0}$, and $\mu _{i}^{*}=\mu _{i}+\mu _{0}\sigma _{i}/\sigma _{0}$ for $i=1,\ldots ,M$.

The denominator in Eq. (2) can be obtained by solving the expectation $E\left[ U_{W}\right]$, noting that from Definition (1) $x_{M}\sim G\left( p_{0}+p_{M},\sigma _{M},\mu _{M}+\mu _{0}\sigma _{M}/\sigma _{0}\right)$, which yields

$$\begin{aligned} E\left[ U_{W}\right] =E\left[ \exp \left( \gamma h_{M}\left( W\right) \right) \right] =\frac{\exp \left( \gamma \mu _{M}^{*}\right) }{\left( 1-\gamma \sigma _{M}\right) ^{p_{M}+p_{0}}}. \end{aligned}$$

(20)

Finally, substituting Eqs. (20) and (19) into Eq. (2) yields Eq. (9). $\square$