Two problems in quantitative finance

We study two problems in quantitative finance. The first is the valuation of a path-dependent convertible bond with American-style options. The second is to determine optimal limit order submission strategies under a high-frequency setting.;A convertible bond has an embedded conditional American option to recall the bond. The constraint is that the underlying stock closing price is above trigger for at least 20 out of the previous 30 trading days (soft-call). This path-dependent constraint magnifies the computational cost significantly. We develop a computational method for this difficult valuation. The method introduces a new concept---convex path set and uses a new language for dealing with the complex path constraints. It can store the large quantity of intermediate values and makes computation under explicit finite difference PDEs more efficient.;A limit order is an order to buy or sell a security at a specific price. The quotes are assumed to be updated continuously without cost. We consider the arising optimal control problems. The underlying variable is the true value of a stock, the control variables are the bid and ask prices and the utility depends on the wealth at the terminal time. Under the full observation model, the optimal quotes depend on the true stock value and the current holding status. We develop a partial observation model, making the stock true value unobservable except at trading moments. Under this more realistic model, the optimal quotes depend on the last observed true stock value, last observation time and the current holding status, but not the unobserved current stock true value. Numerical solutions indicate it is still a profitable strategy that can handle the risk from the unobserved information. We also propose several interesting future problems in this area.