Mathematics (Oxford)

In this talk, We show that both reflected BSDE and its associated penalized BSDE admit both optimal stopping representation and optimal control

representation. We also show that both multidimensional reflected BSDE and its associated multidimensional penalized BSDE admit optimal switching representation. The corresponding optimal stopping problems for penalized BSDE have the feature that one is only allowed to stop at Poisson arrival times.

We consider the problem of optimizing a portfolio of medium to low frequency

quant strategies under heavy tailed distributions. Approaching this problem by modelling

returns through mixture distributions, we derive robust and relative robust methodologies

and discuss conic optimization approaches to solving these models.

The second order sensitivity of a trading position, the so

called gamma, has a very real and intuitive meaning to the traders.

People think that convex payoffs must generate convex prices. Being long

or short of gamma is a strategy used to balance risks in options books.

While the simples models, like Black Scholes, are consistent with this

intuition other popular models used in the industry are not. I will give

examples of simple and popular models which do not always convert a

convex payoff into a convex price. I will also give the necessary and

sufficient conditions under which the convexity is propagated.

We consider a problem of maximising lifetime utility of consumption subject to a drawdown constraint on undiscounted wealth

process. This problem was solved by Elie and Touzi in the case of zero interest rate. We apply methodology of Azema-Yor processes to connect

constrained and unconstrained wealth processes, which allows us to get the results for non-zero interest rate.

General Arrow-Debreu equilibrium can be determined for expected utility maximisers by explicit solutions for individual players. When the expected

utilities are distorted by probability weighting functions, players cannot find explicit optimal decisions. Zhou and Xia studied the existence of equilibrium when the probability weighting functions are the same for all individual players. In this paper, we investigate the same problem but with heterogeneous probability weighting function.