We establish deterministic necessary and sufficient conditions for the no-arbitrage notions NA (“no arbitrage”), NUPBR (“no unbounded profit with bounded risk”) and NFLVR (“no free lunch with vanishing risk”) in one-dimensional general diffusion market models with finite and infinite time horizons. These are models whose (discounted) single risky asset price process S is a regular continuous strong Markov process that is also a semimartingale. We further characterise the existence of an equivalent martingale measure in such models. All deterministic criteria are provided in terms of the scale function and the speed measure of S .
We examine optimal portfolio and consumption choice with defined contribution plan (DCP) participation and taxation of labour and capital income. We rigorously establish the connection between the value function and the associated Hamilton–Jacobi–Bellman equation. We uncover the impact of tax sheltering, employer matching and potential diversification effects related to the DCP on the optimal choice. We calculate the value of participation and decompose it into four components corresponding to different types of utility gains from participation. We find that suboptimal asset location significantly reduces the participation value, while suboptimal capital gains tax-timing does not.
Vulnerable options of European and American style with a possible occurrence of an exogenous termination are studied under market incompleteness in a hazard-process setup. It is proved that the reduced upper price of a vulnerable European option coincides with the unique price of an American option with a properly defined payoff and holder’s exercise times constrained to the random set given by the right support of the hazard process. For a vulnerable American option, it is shown that the reduced upper price equals the price of a specific American option with unrestricted exercise times, whereas the reduced lower price coincides with the price of a particular game option in which the issuer’s exercise times are constrained to the above-mentioned random set.
We develop a nonparametric, semimartingale optimal transport, calibration methodology for local volatility models with stochastic interest rates. The method finds a fully calibrated model which is closest, in a way defined by a general cost function, to a given reference model. We establish a general duality result which allows to solve the problem by optimising over solutions to a second-order fully nonlinear Hamilton–Jacobi–Bellman equation. Our methodology is analogous to Guo et al. (SIAM J. Financ. Math. 13:1–31, 2022; Math. Finance 32:46–77, 2022), but features a novel element of solving for discounted densities, or sub-probability measures. As an example, we apply the method to a sequential calibration problem, where a Vašíček model is already given for the interest rates and we seek to calibrate a stock price’s local volatility model with a volatility coefficient depending on time, the underlying and the short rate process, and the two processes driven by possibly correlated Brownian motions. The equity model is calibrated to any number of European option prices.
