We would like to introduce a few different setups of martingale optimal transport problem in dimension greater than one. Optimal correlations of martingales become an interesting issue, and each problem has its own unique feature in this regard

# Past Mathematical Finance Internal Seminar

Joint work with Abdul-Lateef Haji-Ali

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.

The local volatility model is a celebrated model widely used for pricing and hedging financial derivatives. While the model’s main appeal is its capability of reproducing any given surface of observed option prices—it provides a perfect fit—the essential component of the model is a latent function which can only be unambiguously determined in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimates as result. We seek to look at the calibration problem in a probabilistic framework with a nonparametric approach based on Gaussian process priors. This immediately gives a way of encoding prior believes about the local volatility function, and a hypothesis model which is highly flexible whilst being prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate, we seek to draw posterior inference on the distribution over local volatility to better understand the uncertainty attached with the calibration. Further, we seek to understand dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this gives us means of inferring predictive distributions not only locally, but also for entire surfaces forward in time.

Leandro Sánchez Betancourt

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The Cost of Latency: Improving Fill Ratios in Foreign Exchange Markets

Latency is the time delay between an exchange streaming market data to a trader, the trader processing information and deciding to trade, and the exchange receiving the order from the trader. Liquidity takers face a moving target problem as a consequence of their latency in the marketplace -- they send marketable orders that aim at a price and quantity they observed in the LOB, but by the time their order was processed by the Exchange, prices (and/or quantities) may have worsened, so the order cannot be filled. If liquidity taking orders can walk the limit order book (LOB), then orders that arrive late may still be filled at worse prices. In this paper we show how to optimally choose the discretion of liquidity taking orders to walk the LOB. The optimal strategy balances the tradeoff between the costs of walking the LOB and targeting a desired percentage of filled orders over a period of time. We employ a proprietary data set of foreign exchange trades to analyze the performance of the strategy. Finally, we show the relationship between latency and the percentage of filled orders, and showcase the optimal strategy as an alternative investment to reduce latency.

Jasdeep Kalsi

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An SPDE model for the Limit Order Book

I will introduce a microscopic model for the Limit Order Book in a static setting i.e. in between price movements. Here, order flow at different price levels is given by Poisson processes which depend on the relative price and the depth of the book. I will discuss how reflected SPDEs can be obtained as scaling limits of such models. This motivates an SPDE with reflection and a moving boundary as a model for the dynamic Order Book. An outline for how to prove existence and uniqueness for the equation will be presented, as well as some simple simulations of the model.

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