DH 1st floor SR

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.

This paper analyzes a generalized class of flat-top realized kernels for

estimation of the quadratic variation spectrum in the presence of a

market microstructure noise component that is allowed to exhibit both

endogenous and exogenous $\alpha$-mixing dependence with polynomially

decaying autocovariances. In the absence of jumps, the class of flat-top

estimators are shown to be consistent, asymptotically unbiased, and

mixed Gaussian with the optimal rate of convergence, $n^{1/4}$. Exact

bounds on lower order terms are obtained using maximal inequalities and

these are used to derive a conservative MSE-optimal flat-top shrinkage.

In a theoretical and/or a numerical comparison with alternative

estimators, including the realized kernel, the two-scale realized

kernel, and a proposed robust pre-averaging estimator, the flat-top

realized kernels are shown to have superior bias reduction properties

with little or no increase in finite sample variance.

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:

(i) The optimal terminal wealth X*(T) := Xφ* (T) of the classical problem to

maximise the expected U-utility of the terminal wealth Xφ(T) generated by admissible

portfolios φ(t); 0 ≤ t ≤ T in a market with the risky asset price process modeled as a semimartingale;

(ii) The optimal scenario dQ*/dP of the dual problem to minimise the expected

V -value of dQ/dP over a family of equivalent local martingale measures Q. Here V is

the convex dual function of the concave function U.

In this talk we consider markets modeled by Itô-Lėvy processes, and we present

in a first part a new proof of the above result in this setting, based on the maximum

principle in stochastic control theory. An advantage with our approach is that it also

gives an explicit relation between the optimal portfolio φ* and the optimal scenario

Q*, in terms of backward stochastic differential equations. In a second part we present

robust (model uncertainty) versions of the optimization problems in (i) and (ii), and

we prove a relation between them. We illustrate the results with explicit examples.

The presentation is based on recent joint work with Bernt ¬Oksendal, University of

Oslo, Norway.

The banking industry lost a trillion dollars during the global financial crisis. Some of these losses, if not most of them, were attributable to complex derivatives or securities being incorrectly priced and hedged. We introduce a new methodology which provides a better way of trying to hedge and mark-to-market complex derivatives and other illiquid securities which recognise the fundamental incompleteness of markets and the presence of model uncertainty. Our methodology combines elements of the No Good Deals methodology of Cochrane and Saa-Requejo with the Robustness methodology of Hansen and Sargent. We give some numerical examples for a range of both simple and complex problems encompassing not only financial derivatives but also “real options”occurring in commodity-related businesses.