PhD Oral Exam - Shima Jalili, Mathematics

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

This thesis studies dynamic hedging of financial derivatives in incomplete markets, with a focus on how hedging decisions are shaped by risk criteria, jump risk, and model uncertainty. In such environments, exact replication is generally unattainable; hedging performance depends not only on the traded assets and market dynamics but also on the objective function used to measure risk and guide rebalancing decisions. The thesis develops and compares several frameworks for dynamic hedging, ranging from multistage risk-sensitive stochastic control to jump-risk hedging and data-driven deep hedging.

The first study considers discrete-time hedging under Conditional Value-at-Risk (CVaR) in a multiperiod setting. Since a global CVaR objective could lead to time inconsistency, the analysis enforces time consistency at the level of the hedging policy. This is achieved through a temporal decomposition of coherent risk functionals and the solution of a sequence of conditional subproblems by backward dynamic programming. Numerical experiments for European call options illustrate how imposing policy time consistency affects hedging behavior and performance.

The second study investigates dynamic hedging of American put options in jump-diffusion markets. Because closed-form continuous-time hedging solutions are generally unavailable in this setting, the continuous-time models are approximated by a discrete-time lattice framework. Within this framework, global quadratic hedging, local risk minimization, and Delta-Gamma hedging are compared under Kou and Merton jump-diffusion models calibrated to S\&P 500 and Bitcoin data. The analysis also examines an internal exercise specification, in which the exercise decision is linked to the current hedging portfolio value, and compares it with the standard externally imposed exercise rule.

The third study examines deep hedging of cryptocurrency derivatives under model misspecification. Cryptocurrency returns are modelled using both a non-causal AR--ARCH specification and a Gaussian GARCH(1,1) benchmark, and LSTM-based self-financing hedging strategies are trained under each environment. The empirical results do not support a systematic advantage of the noncausal specification: although there is some evidence of noncausal dependence, the estimated effect is small, the GARCH model often provides a better in-sample fit, and deep hedging performance remains sensitive to the assumed return dynamics. Misspecification experiments further show that adopting a more complex model without strong empirical support can lead to unstable or less reliable hedging conclusions.

Taken together, the thesis shows that effective hedging in incomplete markets requires careful alignment between the risk measure, the market model, and the optimization architecture. Its main contribution is to connect time-consistent risk control, jump-risk hedging, and deep hedging under model uncertainty within a unified perspective on dynamic risk management.