American options require numerical methods, namely lattice models, to provide accurate price estimates. The computations can become expensive when more than one state variable is involved. Analytical upper bounds can therefore provide a useful guideline for how high American values can reach. In this paper, we derive analytical (closed-form) upper bounds for American option prices under stochastic interest rates, stochastic volatility, and jumps where American option prices are difficult to compute with accuracy. In a stochastic volatility model (Heston (1993) and Scott (1997)) that has two random factors, we demonstrate that the upper bound only takes a very small fraction of the time that the American option needs to compute.

Liquidity plays an important role in financial markets, especially during a financial crisis. New Basel III regulatory framework highlights the importance of liquidity risk management implemented by financial institutions. Moreover, updated International Financial Reporting Standards (IFRS) require the improvements about fair value measurements and reinforce existing principles for disclosures about the liquidity risk associated with financial instruments. Using the liquidity discount model of Chen (2012), we are able to empirically classify Taiwan's financial institutions into three liquidity categories: safe, crisis contagious and vulnerable. Our findings can serve as an early warning signal for liquidity calamity. In addition, we investigate what factors affect firm-specific liquidity discounts for these institutions and conduct a sub-period analysis, which examines whether there is significant liquidity discounts changes before and after the 2008 financial crisis. We find that liquidity discounts change substantially during the financial crisis. Furthermore, we find that liquidity discounts can be attributed to some firm-specific performance.