In recent years, borrowing with a tradable financial asset like gold and stock becomes popular in the financial service industry. In this paper, I study the valuation of infinite maturity stock loans and gold loans which can be formulated as the pricing problem of perpetual American call options with time-dependent strike prices. For the infinite maturity gold loans, a key condition under which the valuation of the gold loan is trivial is found under general exponential Lévy models and it is shown that the optimal stopping time is a first hitting time of an interval for a modified asset price when this condition fails. And the analytical solution to the infinity maturity gold loan price is derived under the double exponential jump-diffusion model. Besides, the gold loan price is obtai...[ Read more ]

In recent years, borrowing with a tradable financial asset like gold and stock becomes popular in the financial service industry. In this paper, I study the valuation of infinite maturity stock loans and gold loans which can be formulated as the pricing problem of perpetual American call options with time-dependent strike prices. For the infinite maturity gold loans, a key condition under which the valuation of the gold loan is trivial is found under general exponential Lévy models and it is shown that the optimal stopping time is a first hitting time of an interval for a modified asset price when this condition fails. And the analytical solution to the infinity maturity gold loan price is derived under the double exponential jump-diffusion model. Besides, the gold loan price is obtained in a constructive way under the hyper-exponential and phase-type jump-diffusion model. For the infinite maturity stock loans, a unified approach to its valuation is proposed under general regime switching exponential Lévy models with any finite numbers of regimes. More precisely, a key condition is found under which the stock loan price simply equals the initial stock price and the equivalent relationship is discovered between the stock loan and a conventional perpetual American put option with nonnegative interest rates in all regimes. In addition, a fixed point approach is developed for an auxiliary optimal stopping problem to determine all ”continuation” regimes within which waiting is always optimal, and a full characterization of the optimal stopping time is given for the auxiliary optimal stopping problem, including the valuation of the stock loan as a special case. Finally, the analytical solution to the infinity maturity stock loan price is obtained under a regime-switching phase-type jump-diffusion model.