This talk will discuss efficient numerical methods for estimating the
probability of a large portfolio loss, and associated risk measures such
as VaR and CVaR.  These involve nested expectations, and following
Bujok, Hambly & Reisinger (2015) we use the number of samples for the
inner conditional expectation as the key approximation parameter in the
Multilevel Monte Carlo formulation.  The main difference in this case is
the indicator function in the definition of the probability. Here we
build on previous work by Gordy & Juneja (2010) who analyse the use of a
fixed number of inner samples , and Broadie, Du & Moallemi (2011) who
develop and analyse an adaptive algorithm.  I will present the
algorithm, outline the main theoretical results and give the numerical
results for a representative model problem.  I will also discuss the
extension to real portfolios with a large number of options based on
multiple underlying assets.

Joint work with Abdul-Lateef Haji-Ali

In the 12 months from the middle of June 2016 to the middle of June 2017, a number of events occurred in a relatively short period of time, all of which either had, or had the potential to have,  a considerably volatile impact upon financial markets. The events referred to here are the Brexit  referendum (23 June 2016), the US election (8 November 2016), the 2017 French elections (23 April and 7 May 2017) and the surprise 2017 UK parliamentary election (8 June 2017). 
All of these events - the Brexit referendum and the Trump election in particular - were notable both for their impact upon financial markets after the event and the degree to which the markets failed to anticipate these events. A natural question to ask is whether these could have been predicted, given information freely available in the financial markets beforehand. In this talk, we focus on market expectations for price action around Brexit and the Trump election, based on information available in the traded foreign exchange options market. We also investigate the horizon date of 30 March 2019, when the two year time window that started with the Article 50 notification on 29 March 2017 will terminate.
Mathematically, we construct a mixture model corresponding to two scenarios for the GBPUSD exchange rate after the referendum vote, one scenario for “remain” and one for “leave”. Calibrating this model to four months of market data, from 24 February to 22 June 2016, we find that a “leave” vote was associated with a predicted devaluation of the British pound to approximately 1.37 USD per GBP, a 4.5% devaluation, and quite consistent with the observed post-referendum exchange rate move down from 1.4877 to 1.3622. We find similar predictive power for USDMXN in the case of the 2016 US presidential election. We argue that we can apply the same bimodal mixture model technique to construct two states of the world corresponding to soft Brexit (continued access to the single market) and hard Brexit (failure of negotiations in this regard).
 

Modular forms holomorphic functions on the upper half of the complex plane, H, invariant under certain matrix transformations of H. The have a very rich structure - they form a graded algebra over C and come equipped with a family of linear operators called Hecke operators. We can also view them as functions on a Riemann surface, which we refer to as a modular curve. It transpires that the integral homology of this curve is equipped with such a rich structure that we can use it to compute modular forms in an algorithmic way. The modular symbols are a finite presentation for this homology, and we will explore this a little and their connection to modular symbols.

Finding implementable descriptions of the possible configurations of a knotted DNA molecule has remarkable importance from a biological point of view, and it is a hard and well studied problem in mathematics.
Here we present two newly developed mathematical tools that describe the configuration space of knots and model the action of Type I and II Topoisomerases on a covalently closed circular DNA molecule: the Reidemeister graphs.
We determine some local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring.
Finally, we indicate how the Reidemeister graphs can be used to infer information about the proteins' action.