Stochastic impulse control with discounted and ergodic optimization criteria: A comparative study for the control of risky holdings

We consider a single-asset investment fund that in the absence of transactions costs would hold a constant amount of wealth in the risky asset. In the presence of market frictions wealth is allowed to fluctuate within a control band: Its upper (lower) boundary is chosen so that gains (losses) from adjustments to the target minus (plus) fixed plus proportional transaction costs maximize (minimize) a power utility function. We compare stochastic impulse control policies derived via ergodic and discounted optimization criteria. For the solution of the ergodic problem we use basic tools from the theory of diffusions whereas the discounted problem is solved after being characterized as a system of quasi-variational inequalities. For both versions of the problem, derivation of the control bands pertains to the numerical solution of a system of nonlinear equations. We solve numerous such systems and present an extensive comparative sensitivity analysis with respect to the parameters that characterize investor’s preferences and market behavior.Transaction costs; stochastic impulse control; ergodic criteria

Controlling the risky fraction process with an ergodic criterion

This article examines a tracking problem, similar to the one presented in Pliska and Suzuki (Quantitative Finance, 2004): an investor would keep constant proportions of her wealth in different assets if markets were frictionless; however, in the presence of fixed and proportional transaction costs her implementation problem is to keep asset proportions close to the target levels whilst avoiding too much intervention costs. Instead of minimizing discounted tracking error plus transaction costs over an infinite horizon, the optimization objective here is minimization of long run tracking error plus intervention costs per unit time. This ergodic problem is treated via combining basic tools from diffusion theory and nonlinear optimization techniques. A comparative sensitivity analysis of the ergodic and discounted problems is undertaken.

Controlled diffusion processes

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Indifference Pricing and Hedging in a Multiple-Priors Model with Trading Constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.Comment: 28 pages in Science China Mathematics, 201

European Option Pricing and Hedging with both Fixed and Proportional Transaction Costs

In this paper we extend the utility based option pricing and hedging approach, pioneered by Hodges and Neuberger (1989) and further developed by Davis, Panas and Zariphopoulou (1993), for the market where each transaction has a fixed cost component. We present a model, where investors have a CARA utility, and derive some properties of reservation option prices. We suggest and implement discretization schemes for computing the reservation option prices. The numerical results of option pricing and hedging are presented for the case of European call options and the investors with different levels of ARA. We also try to reconcile our findings with such empirical pricing bias as the volatility smile.option pricing, transaction costs, stochastic control, Markov chain approximation

Applications of Stochastic Control in Energy Real Options and Market Illiquidity

We present three interesting applications of stochastic control in finance. The first is a real option model that considers the optimal entry into and subsequent operation of a biofuel production facility. We derive the associated Hamilton Jacobi Bellman (HJB) equation for the entry and operating decisions along with the econometric analysis of the stochastic price inputs. We follow with a Monte Carlo analysis of the risk profile for the facility. The second application expands on the analysis of the biofuel facility to account for the associated regulatory and taxation uncertainty experienced by players in the renewables and energy industries. A federal biofuel production subsidy per gallon has been available to producers for many years but the subsidy price level has changed repeatedly. We model this uncertain price as a jump process. We present and solve the HJB equations for the associated multidimensional jump diffusion problem which also addresses the model uncertainty pervasive in real option problems such as these. The novel real option framework we present has many applications for industry practitioners and policy makers dealing with country risk or regulatory uncertainty---which is a very real problem in our current global environment. Our final application (which, although apparently different from the first two applications, uses the same tools) addresses the problem of producing reliable bid-ask spreads for derivatives in illiquid markets. We focus on the hedging of over the counter (OTC) equity derivatives where the underlying assets realistically have transaction costs and possible illiquidity which standard finance models such as Black-Scholes neglect. We present a model for hedging under market impact (such as bid-ask spreads, order book depth, liquidity) using temporary and permanent equity price impact functions and derive the associated HJB equations for the problem. This model transitions from continuous to impulse trading (control) with the introduction of fixed trading costs. We then price and hedge via the economically sound framework of utility indifference pricing. The problem of hedging under liquidity impact is an on-going concern of market makers following the Global Financial Crisis

Game Theory for Security Investments in Cyber and Supply Chain Networks

In a constantly and intricately connected world that is going digital, cybersecurity is imperative to not just the success but also the survival of a business. The ubiquitous digital transformation is fueled by a convulsive growth of devices and data that are leading important innovations in the domain of cyber-physical systems. However, this growth has also enabled internal and external threats to skyrocket, depicting the inherent dichotomy. With an evolving threat landscape, a perpetrator has to be successful once, while the defenders have to continually succeed in fending-off attacks to protect critical infrastructure and digital assets. Businesses are facing a barrage of attacks, majority of which have a financial or an espionage motive. Against the hackers, businesses put forth an asymmetric and myopic struggle.Through this dissertation, I contribute to the modeling and analysis of security investments in cyber and supply chain networks considering network vulnerability also nonlinear budget constraints. The latter contributes significantly to the literature on variational inequality, game theory, and cybersecurity by being methodologically relevant to the application and solution of such problems. I also explore cooperation in terms of cybersecurity among firms in a network, providing a quantitative basis to information sharing to explore its financial and policy related benefits. This work extends the current literature in the cooperative game theory and cybersecurity domains