From Black-Scholes to Online Learning: Dynamic Hedging under Adversarial Environments

We consider a non-stochastic online learning approach to price financial options by modeling the market dynamic as a repeated game between the nature (adversary) and the investor. We demonstrate that such framework yields analogous structure as the Black-Scholes model, the widely popular option pricing model in stochastic finance, for both European and American options with convex payoffs. In the case of non-convex options, we construct approximate pricing algorithms, and demonstrate that their efficiency can be analyzed through the introduction of an artificial probability measure, in parallel to the so-called risk-neutral measure in the finance literature, even though our framework is completely adversarial. Continuous-time convergence results and extensions to incorporate price jumps are also presented

Variance-constrained multiobjective control and filtering for nonlinear stochastic systems: A survey

The multiobjective control and filtering problems for nonlinear stochastic systems with variance constraints are surveyed. First, the concepts of nonlinear stochastic systems are recalled along with the introduction of some recent advances. Then, the covariance control theory, which serves as a practical method for multi-objective control design as well as a foundation for linear system theory, is reviewed comprehensively. The multiple design requirements frequently applied in engineering practice for the use of evaluating system performances are introduced, including robustness, reliability, and dissipativity. Several design techniques suitable for the multi-objective variance-constrained control and filtering problems for nonlinear stochastic systems are discussed. In particular, as a special case for the multi-objective design problems, the mixed H 2 / H ∞ control and filtering problems are reviewed in great detail. Subsequently, some latest results on the variance-constrained multi-objective control and filtering problems for the nonlinear stochastic systems are summarized. Finally, conclusions are drawn, and several possible future research directions are pointed out

Strategically-Timed Actions in Stochastic Differential Games

Financial systems are rich in interactions amenable to description by stochastic control theory. Optimal stochastic control theory is an elegant mathematical framework in which a controller, profitably alters the dynamics of a stochastic system by exercising costly control inputs. If the system includes more than one agent, the appropriate modelling framework is stochastic differential game theory — a multiplayer generalisation of stochastic control theory. There are numerous environments in which financial agents incur fixed minimal costs when adjusting their investment positions; trading environments with transaction costs and real options pricing are important examples. The presence of fixed minimal adjustment costs produces adjustment stickiness as agents now enact their investment adjustments over a sequence of discrete points. Despite the fundamental relevance of adjustment stickiness within economic theory, in stochastic differential game theory, the set of players’ modifications to the system dynamics is mainly restricted to a continuous class of controls. Under this assumption, players modify their positions through infinitesimally fine adjustments over the problem horizon. This renders such models unsuitable for modelling systems with fixed minimal adjustment costs. To this end, we present a detailed study of strategic interactions with fixed minimal adjustment costs. We perform a comprehensive study of a new stochastic differential game of impulse control and stopping on a jump-diffusion process and, conduct a detailed investigation of two-player impulse control stochastic differential games. We establish the existence of a value of the games and show that the value is a unique (viscosity) solution to a double obstacle problem which is characterised in terms of a solution to a non-linear partial differential equation (PDE). The study is contextualised within two new models of investment that tackle a dynamic duopoly investment problem and an optimal liquidity control and lifetime ruin problem. It is then shown that each optimal investment strategy can be recovered from the equilibrium strategies of the corresponding stochastic differential game. Lastly, we introduce a dynamic principal-agent model with a self-interested agent that faces minimally bounded adjustment costs. For this setting, we show for the first time that the principal can sufficiently distort that agent’s preferences so that the agent finds it optimal to execute policies that maximise the principal’s payoff in the presence of fixed minimal costs

Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach

We propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Given the set of call and put dates, the callable and putable bond pricing function is the value function of a stochastic game with stopping times. Under some technical conditions, it is shown to have an eigenfunction expansion in eigenfunctions of the pricing operator with the expansion coefficients determined through a backward recursion. For popular short rate diffusion models, such as CIR, Vasicek, 3/2, the method is orders of magnitude faster than the alternative approaches in the literature. In contrast to the alternative approaches in the literature that have so far been limited to diffusions, the method is equally applicable to short rate jump-diffusion and pure jump models constructed from diffusion models by Bochner's subordination with a L\'{e}vy subordinator

Avoiding the Curse of Dimensionality in Dynamic Stochastic Games

Discrete-time stochastic games with a finite number of states have been widely ap- plied to study the strategic interactions among forward-looking players in dynamic en- vironments. However, these games suffer from a "curse of dimensionality" since the cost of computing players' expectations over all possible future states increases exponentially in the number of state variables. We explore the alternative of continuous-time stochas- tic games with a finite number of states, and show that continuous time has substantial computational and conceptual advantages. Most important, continuous time avoids the curse of dimensionality, thereby speeding up the computations by orders of magnitude in games with more than a few state variables. Overall, the continuous-time approach opens the way to analyze more complex and realistic stochastic games than currently feasible.