This paper develops an economic framework to analyze optimal longevity risk transfers in both the reinsurance and capital markets, focusing on the differing risk aversions of buyers and sellers of longevity risk transfer contracts. Utilizing a Stackelberg game framework, we compare static longevity swap contracts, offering long-term protection with constant hedge ratios and predetermined hedging costs, against dynamic contracts, providing short-term coverage with variable contract terms. With real-life mortality data, our numerical analysis reveals that static contracts are preferred in the reinsurance market as they lead to larger welfare gains for both participating parties and more flexible conditions for market existence. Conversely, dynamic contracts are favored in the capital market due to sellers' higher risk aversion. Additionally, information asymmetry is incorporated in the form of ambiguity. While ambiguity reduces welfare gains for both parties and leads to more stringent conditions for market existence, it does not alter the contract preferences. Our analysis provides theoretical explanations for several key empirical observations in the current longevity risk transfer market and offers new insights into the development of the longevity-linked capital market.