Optimal portfolio choice of gold assets in the differential market and differential game structures

Abstract

Portfolio choices of gold-related assets for market investors and dealers may not only depend on price differences and the inflation rate, but may also react to the market participants’ strategic behavior and risk attitude. This study develops a two-agent stochastic differential game model to solve the portfolio choice problem of the asset allocations of gold spot, futures, and cash for market participators who are exposed to inflation risks. The equilibrium prices of spot and futures driven by the volatility rate and co-variances that reflect various risk sources are also determined. Specifically, regarding the choice of hedging tools, market participators may prefer gold spot to futures for the purpose of hedging inflation risk. By capturing the stylistic facts of differential market and multiple agent structures, the article can develop a more reasonable and practical model to usefully explain the gold portfolio choices and pricing in the gold markets.

Appendices

Appendix 1: Proof of Corollary 1

Through Bellman’s principle of optimality, we obtain the Hamilton–Jacobi-Bellman (henceforth HJB) equations as follows:

$$ \begin{aligned} \mu_{l} J & = \mathop {\hbox{max} }\limits_{C,w,m} U(C,t) + J_{W} \left[ {\mu_{l} W - SC + wW(\mu_{a} I + \mu_{b} (1 - I) - \mu_{l} ) + mW(\mu_{f} - \mu_{l} )} \right] \\ & \quad + \frac{1}{2}J_{WW} \left[ {w^{2} W^{2} (\sigma_{a}^{2} I^{2} + \sigma_{b}^{2} (1 - I)^{2} ) + m^{2} W^{2} \sigma_{f}^{2} + 2wW^{2} m\sigma_{f} (\sigma_{a} I\eta_{af} + \sigma_{b} (1 - I)\eta_{bf} )} \right] \\ & \quad + J_{S} \mu_{S} S + \frac{1}{2}J_{SS} S^{2} \sigma_{S}^{2} + J_{WS} \left[ {wW\sigma_{S} S(\sigma_{a} I\eta_{aS} + \sigma_{b} (1 - I)\eta_{bS} ) + mWS\sigma_{f} \sigma_{S} \eta_{Sf} } \right] \\ \end{aligned} $$

(26)

$$ \begin{aligned} \mu_{l} L & = \mathop {\hbox{max} }\limits_{D,n} V(D,t) + L_{K} \left[ {\mu_{l} K - SD - wW(\mu_{a} I + \mu_{b} (1 - I) - \mu_{l} ) + nK(\mu_{f} - \mu_{l} )} \right] \\ & \quad + \frac{1}{2}L_{KK} \left[ {w^{2} W^{2} (\sigma_{a}^{2} I^{2} + \sigma_{b}^{2} (1 - I)^{2} ) + n^{2} K^{2} \sigma_{f}^{2} - 2wnWK\sigma_{f} (\sigma_{a} I\eta_{af} + \sigma_{b} (1 - I)\eta_{bf} )} \right] \\ & \quad + L_{S} \mu_{S} S + \frac{1}{2}L_{SS} S^{2} \sigma_{S}^{2} + L_{KS} \left[ { - wW\sigma_{S} S(\sigma_{a}^{{}} I\eta_{aS} + \sigma_{b} (1 - I)\eta_{bS} ) + nKS\sigma_{f} \sigma_{S} \eta_{Sf} } \right] \\ \end{aligned} $$

(27)

where J _W , J _S , L _K , L _KS , J _WW and others denote the partial derivatives of J(W,S,t) and L(K,S,t) with respect to W and S. We differentiate Eqs. (26) and (27) with respect to control variables and let these results equal zero, and solve the algebraic system; the article then obtains the optimal decisions of Corollary 2.

Appendix 2: Proof of Corollary 2

Solving the first order condition of the HJB Eqs. (26) and (27) with respect to the holdings of the gold spot and futures, the article obtains two equations about w and m. Then the article sets w = 1 and m = 0 in the two equations above. Naturally, they yield the results of Eqs. (22) and (23).