London Mathematical Finance Group
London Mathematical Finance Group 

2016-17 London Mathematical Finance Seminar Series

 

The January-March 2017 programme is hosted by Cass Business School, City, University of London.

 

Date: Thursday 26 January 2017

 

Speaker #1: Ivar Ekeland, CEREMADE and Institut de Finance, Université Paris-Dauphine

 

Time: 18:15-19:00

 

Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Ivar Ekeland here


Storers, processors and speculators: an equilibrium model for commodity markets

We present an infinite-horizon model of a commodity market. In each period t, two markets are open: a spot market for the commodity, and a futures market for financial contracts. Contracts traded at time t are settled at time t+1. Market participants are either storers, who have a capacity to store physical quantities, processors, who need the commodity to produce consumer goods, and speculators, who trade futures in order to turn a profit or to hedge other portfolios. Speculators do not trade on the physical market, but storers and processors trade on the futures market. 

 

We seek an optimal Markov strategy for each participant, and we find it by solving a rational expectations equilibrium. We derive some insights concerning the impact of speculation on commodity markets, and we compare our findings with actual markets.

 

Joint work with Delphine Lautier, Bertrand Villeneuve and Edouard Jaeck.

 

Speaker #2: Mykhaylo Shkolnikov, ORFE, Princeton University


Time: 19:15-20:00

 

Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Mykhaylo Shkolnikov here

A random surface description of the capital distribution in large markets

 

We study the capital distribution in the context of the first-order models of Fernholz and Karatzas. We find that when the number of companies becomes large the capital distribution fluctuates around the solution of a porous medium PDE according to a linear parabolic SPDE with additive noise. Such a description opens the path to modelling the capital distribution surfaces in large markets by systems of a PDE and an SPDE and to understanding a variety of market characteristics and portfolio performances therein.

 

Joint work with Praveen Kolli.

 

 

Date: Thursday 9 February 2017


Speaker #1: Frank Riedel, Centre for Mathematical Economics, Bielefeld University

Time: 18:15-19:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

Find out more about Frank Riedel here


Incompleteness of Financial Markets under Knightian Uncertainty

 

In diffusion models, few suitably chosen financial securities allow to complete the market. As a consequence, the efficienta llocations of static Arrow-Debreu equilibria can be attained in Radner equilibria by dynamic trading. We show that this celebrated result generically fails if there is Knightian uncertainty about volatility. A Radner equilibrium with the same efficient allocation as in an Arrow-Debreu equilibrium exists if and only if the discounted net trades of the equilibrium allocation display no ambiguity in the mean. This property is violated generically in endowments, and thus Arrow-Debreu equilibrium allocations are generically unattainable by dynamically trading few long-lived assets.

 

 

Date: Thursday 23 February 2017


Speaker #1: Scott Robertson, Questrom School of Business, Boston University

Time: 18:15-19:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

Find out more about Scott Robertson here

Optimal Investment, Indifference Pricing, and Dynamic Default Insurance for Defaultable Assets

In this talk, we consider the optimal investment problem when the traded asset may default, causing a jump in its price.  Upon default, the investor will lose her dollar position in the stock. For an investor with constant absolute risk aversion, our goal is to explicitly compute both the indifference price for a defaultable bond, and a fair price for dynamic protection against default. For the latter problem, our work complements Sircar and Zariphopoulou (2007), where it is implicitly assumed the investor is protected against default. We consider a factor model where asset returns, variances, correlations and default intensities are driven by a time homogeneous diffusion X taking values in an arbitrary region E of R^d.  In addition to trading in the defaultable asset, the investor owns a non-traded asset whose terminal payoff depends upon the survival of the stock. Given X_t =x we identify the certainty equivalent with a semi-linear degenerate elliptic partial differential equation with quadratic growth in both the function and its gradient.  Under a minimal exponential integrability assumption on the market price of risk, we show the certainty equivalent is a classical solution.  In particular, our results cover when X is a one-dimensional affine diffusion, and when returns/variances are also affine. 

 

Given the certainty equivalent we derive the indifference price for a defaultable bond as well as a fair price for dynamic protection against default. Numerical examples highlight the relationship between the factor process, and both the indifference price and default insurance.  In fact, we show the insurance protection does not coincide with the default intensity under the dual optimal measure, as one may expect.

 

This is joint work with Tetsuya Ishikawa of Morgan Stanley.
 

Speaker #2: Peter Tankov, ENSAE ParisTech, Université Paris-Saclay

Time: 19:15-20:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Peter Tankov here

 

Optimal forecast-based trading policies for a wind energy producer


We study the optimal trading policies for a wind energy producer who aims to sell the future production in the open electricity markets,  and who has access to imperfect dynamically updated forecasts of the future production. We construct a stochastic model for the evolution of probabilistic forecasts and determine the optimal trading policies which are updated dynamically as new forecast information becomes available. Our results allow to quantify the expected future gain of the wind producer and to determine the economic value of the forecasts.

 

 

Date: Thursday 9 March 2017


Speaker #1: Mattheus Grasselli, Mathematics and Statistics, McMaster University

Time: 18:15-19:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Matheus Grasselli here

 

Inequality in a monetary dynamic macroeconomic model

Thomas Piketty's influential book “Capital in the Twenty-First Century” documents the marked and unequivocal rise of income and wealth inequality observed across the developed world in the last three decades. His extrapolations into the distant future are much more controversial and have been subject to various criticisms from both mainstreams and heterodox economists. This motivates the search for an alternative standpoint incorporating heterodox insights such as endogenous money and the lessons from the Cambridge capital controversies. We argue that the Goodwin-Keen approach paves the road towards such an alternative.We first consider a modified Goodwin-Keen model driven by consumption by households, instead of investment by firms, leading to the same qualitative features of the original Keen 1995 model, namely the existence of an undesirable equilibrium characterized by infinite private debt ratio and zero employment, in addition to a desirable one with finite debt and non-zero employment. By further subdividing the household sector into workers and investors, we are able to investigate their relative income and wealth ratios for in the context of these two long-run equilibria, providing a testable link between asymptotic inequality and private debt accumulation

 

Speaker #2: Pretty Sagoo, Deutsche Bank

Time: 19:15-20:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Pretty Sagoo here

 

The Longevity Market according to Shrek

 

In 2009 the Life and Longevity Markets Association (‘The LLMA’) was set up, in anticipation of an explosion in derivatives linked to mortality rates, in particular to mortality improvement rates. The LLMA was formed by a collection of banks, insurers and reinsurers, all working towards standardization in mortality derivatives, especially those linked to aggregate and publicly available mortality data, such as E&W mortality data from the National Office of Statistics. In this talk we take a look at how that market developed and why it turned out very differently to that envisaged back in 2009.

 

 

Date: Thursday 23 March 2017


Speaker #1: Mathias Beiglböck, Mathematics, TU Vienna

Time: 18:15-19:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Mathias Beiglböck here

 

Brenier's Theorem, Martingale Optimal Transport and the Local Vol Model


A seminal result in optimal transport is Brenier's theorem on the structure of the optimal plan for squared distance costs. We briefly review related results on the martingale version of the transport problem and connections with robust finance. We then introduce a continuous time Brenier-type theorem for the martingale

transport problem which exhibits a particularly simple functional form. Finally, we explain a link of this result with the local vol model.

 

Speaker #2: Pierre Collin-Dufresne, École Polytechnique Fédérale de Lausanne

Time: 19:15-20:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Pierre Collin-Dufresne here

 

Activism, Strategic Trading and Liquidity

 

We analyze dynamic trading in an anonymous market by an activist investor who can expend costly effort to affect firm value. We obtain the equilibrium in closed form for a general activism technology, including both binary and continuous outcomes. The optimal continuous trading strategy is independent of the activism technology. Activism, prices, and liquidity are jointly determined in equilibrium. Variation in noise trading volatility can produce either positive or negative effects on both efficiency and liquidity, depending on the activism technology and model parameters, because future effort depends on the realized amount of noise trading. The ‘lock in’ effect emphasized in previous literature (e.g., Coffee (1991), Bhide (1993) and Maug (1998)) holds only for special forms of the activism technology. Reducing the uncertainty about the activist’s position improves market liquidity, but the effect on efficiency depends on the specification of the effort cost function. Variation in the activist’s productivity produces a negative cross-sectional relation between efficiency and liquidity as the possibility of more activism exacerbates the risk of adverse selection.

 

 

Date: Thursday 6 April 2017


Speaker #1: Peter Takáč, Mathematics, Universität Rostock

Time: 18:15-19:00


Place: Cass Business School, 106 Bunhill Row, Room 6001 (6th floor)

 

Find out more about Peter Takáč here

 

Stochastic Volatility Models, Complete Markets, and Analyticity of Solutions to Ito’s Parabolic Problems

 

The problem of constant volatility, σ > 0, in the Black-Scholes option pricing model has sparked a number of new research directions on the nature of volatility. We first briefly recall the “most popular” volatility models based on mean reversion (the Ornstein-Uhlenbeck process): (i) Heston’s stochastic volatility model (1993); (ii) an alternative, mathematically much “easier” model due to Fouque, Papanicolaou, and Sircar (2000); and (iii) a more sophisticated model due to Dupire (1992) based on local implied volatility. We will discuss Dupire’s model and a possible generalization to volatility depending also on the asset (stock) price (as suggested by Lewis (2000)). This generalization does not cause any new mathematical difficulty (from an analytic or probabilistic point of view; numerical implementation might be harder).

 

Although Heston’s model for option pricing (i) is the simpliest model with stochastic volatility (SV), its rigorous analytical treatment is quite involved due to the degeneracies in Ito’s parabolic problem for very low and very high volatility levels (2016). This analysis is motivated by the analytical treatment of the alternative SV model (ii) due to Fouque, Papanicolaou, and Sircar which is much “easier” to handle. Ito’s parabolic problem for this model turns out to be uniformly parabolic with bounded analytic coefficients. Hence, it is not surprising that the solution is analytic in both, space and time variables, even if the initial data are only continuous. The “space” variable stands for the pair of the asset price and the volatility. We will prove the analyticity result for the option price by standard L2-methods in the Hardy space H2 of holomorphic functions that extend the real analytic functions to a suitable complex parabolic domain (2012).

We finish our lecture by applying well-known results on complete markets by M. H. A. Davis and J. Obloj (2008) to model (ii): Analyticity result for the option price implies that this option completes the market.

 

 

The October-December 2016 programme was hosted by University College London (UCL).

The seminar was partially supported by INTECH.

Date: Thursday 6 October 2016

Speaker: Lukasz Szpruch, University of Edinburgh

Time: 17:00-18:00

Place: University College London - Physics Building A1/3, Gower Place

Find out more about Lukasz Szpruch here

Multilevel Monte Carlo for McKean-Vlasov SDEs

Stochastic Interacting Particle System (SIPS) and they limiting stochastic McKean-Vlasov equations offer a very rich and versatile modelling framework. On one hand, interactions allow to capture complex dependent structure, on the other provide a great challenge for Monte Carlo simulations. The non-linear dependence of the approximation bias on the statistical error renders classical variance reduction techniques not applicable. In this talk, we will propose a strategy to overcome this difficulty. In particular, we will establish Multilevel Monte Carlo estimator for SIPS and demonstrate its computational superiority over standard Monte Carlo techniques.

Speaker: Mikko Pakkanen, Imperial College London

Time: 18:00-19:00

Place: Univer/sity College London - Physics Building A1/3, Gower Place

Find out more about Mikko Pakkanen here

Modelling and forecasting rough volatility

In their recent, yet already seminal paper ("Volatility is rough", arXiv:1410.3394) Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum argued that financial market volatility should be modelled by stochastic processes that are rougher than Brownian motion. In my talk, I will first present some new corroborative empirical evidence of the roughness of volatility, drawn from high-frequency data on more than five thousand assets. I will then introduce a new stochastic volatility model, building on the so-called Brownian semistationary (BSS) process, which is able to conveniently capture both the roughness and highly persistent long-term behaviour of volatility. Finally, I will discuss two parsimonious parameterisations of the BSS model and demonstrate their remarkable performance in volatility forecasting. Joint work with Mikkel Bennedsen and Asger Lunde.


Date: Thursday 20 October 2016

Speaker: Christoph Reisinger

Time: 17:00-18:00

Place: University College London - Physics Building A1/3, Gower Place

Find out more about Christoph Reisinger here

High-order filtered schemes for time-dependent second order HJB equations

In this talk, we present and analyse a class of “filtered” numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations, with a focus on examples arising from stochastic control problems in financial engineering. We start by discussing more widely the difficulty in constructing compact and accurate approximations. The key obstacle is the requirement in the established convergence analysis of certain monotonicity properties of the schemes. We follow ideas in Oberman and Froese (2010) to introduce a suitable local modification of high order schemes, which are necessarily non-monotone, by "filtering" them with a monotone scheme. Thus, they can be proven to converge and still show an overall high order behaviour for smooth enough value functions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests.

This talk is based on joint work with Olivier Bokanowski and Athena Picarelli.

Speaker: Pietro Siorpaes, Mathematics, Imperial College London

Time: 18:00-19:00

Place: University College London - Physics Building A1/3, Gower Place

Find out more about Pietro Siorpaes here

 

The martingale polar sets

Martingale optimal transport (MOT) is a variant of the classical optimal transport problem where a martingale constraint is imposed on the coupling. In a recent paper, Beiglböck, Nutz and Touzi show that in dimension one there is no duality gap and that the dual problem admits an optimizer. A key step towards this achievement is the characterization of the polar sets of the family of all martingale couplings. Here we extend this characterization to arbitrary finite dimension through a deeper study of the convex order.


Date: Thursday 3 November 2016

Speaker: Aleš Černý, Cass Business School

Time: 17:00-18:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Aleš Černý here

 

Optimal trade execution under endogenous pressure to liquidate: theory and numerical solutions

We study optimal liquidation of a large trading position in a market with a temporary price impact. The novelty in our approach is that we endogenize the pressure to liquidate and hence allow the time horizon of liquidation to be determined endogenously, as part of the optimal strategy. The corresponding HJB equation leads to a severely singular initial value problem whose numerical solution we also study. In contrast to much of the existing literature spreading the liquidation strategy over a longer horizon is not necessarily beneficial to the trader.

 

Joint work with Pavol Brunovský and Ján Komadel.

Speaker: Antoine Jacquier, Mathematics, Imperial College London

Time: 18:00-19:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Antoine Jacquier here

Some recent results on (asymptotics of) fractional stochastic volatility models


Date: Thursday 17 November 2016

Speaker: Chris Rogers, Statistical Laboratory, University of Cambridge

Time: 17:00-18:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Chris Rogers here

 

High-frequency data: why are we looking at this?

High-frequency financial data is certainly a `big data' problem, with all of the associated issues: what are the stylized facts of the data? What are we trying to do with the data? What are appropriate models? Industry approaches get the first two of these questions, but do badly on the third. Most academic studies do badly on all three. For example, it is a fairy tale that we can propose a time-invariant model for the evolution of high-frequency data, estimate the parameters of this model, and then apply the conclusions of an analysis that assumes that the parameters were known with certainty. In this talk, I will try to identify what we might want to do with high-frequency data, critique some existing research agendas, and illustrate a possible way of trading in a high-frequency market.

Speaker: Umberto Cherubini, Statistical Sciences, Università di Bologna

Time: 18:00-19:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Umberto Cherubini here

No-Arbitrage Choquet Pricing with an Application to the Irrational Exercise Problem

 

In this paper we propose a pricing methodology that is based on the no-arbitrage assumption and that leads to option pricing formulas based on the Choquet integral, and interval based pricing. On one side, the Choquet pricing formula is typically derived from an approach based on non-additive expected utility theories, that is decision theory that take into account information ambiguity (such as the MMEU approach due to Gilboa and Schmeidler, 1989). On the other side, the no-arbitrage approach to interval based pricing is applied in the UVM pricing model based on interval-valued volatility due to Avellaneda, Levy and Paràs (1996).  In our model we impose no-arbitrage applying the standard martingale probability model to the “basic probability assignment” (BPA) instead of the probability measure itself. The BPA is a probability measure applied to the power set instead of the set of events. The approach produces a pair of capacities (that is non-additive probability measures), one sub-additive and the other super-additive, linked by a duality relationship (the sub-additive measure on a subset plus the dual-superadditive measure of the complement set must add to 1). This no-arbitrage condition collapses to the standard equivalent martingale measure approach when the BPA is zero on all the elements of the power set that are not singletons. We show that the pricing formula generated by this model is a Choquet integral. For illustration purposes, we show how to apply the model to the problem of irrational exercise of options. Finally, we explore specific parametric forms of the Choquet pricing formulas, and their link with a stream of literature, developed in physics and known as q-calculus (Tsallis).

Based on joint work with Sabrina Mulinacci



Date: Thursday 1 December 2016

Speaker: Michael Tehranchi (University of Cambridge)TBA

Time: 17:00-18:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Michael Tehranchi here

Black-Scholes inequalities: applications and generalisations

 

Various inequalities satisfied by the Black--Scholes call pricing formula and their applications to uniform bounds on implied volatility will be presented. To understand from where these inequalities arise, the family of call pricing functions is shown to be a noncommutative semigroup with involution with respect to a certain multiplication operator which is compatible with the convex order. A one-parameter sub-semigroup can be identified with a peacock, providing tractable call surface parametrisation in the spirit of the Gatheral-Jacquier SVI surface.

Speaker: Giorgia Callegaro, Università degli studi di Padova

Time: 18:00-19:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Giorgia Callegaro here

Optimal investment in markets with over and under-reaction to information

 

In this paper we introduce a jump-diffusion model of shot-noise type for stock prices, taking into account over and under-reaction of the market to incoming news. We work in a partial information setting, by supposing that standard investors do not have access to the market direction, the drift, (modelled via a random variable) after a jump. We focus on the expected (logarithmic) utility maximization problem by providing the optimal investment strategy in explicit form, both under full (i.e., from the insider point of view, aware of the right kind of market reaction at any time) and under partial information (i.e., from the standard investor viewpoint, who needs to infer the kind of market reaction from data). We test our results on market data relative to Enron and Ahold. The three main contributions of this paper are: the introduction of a new market model dealing with over and under-reaction to news, the explicit computation of the optimal filter dynamics using an original approach combining enlargement of filtrations with Innovation Theory and the application of the optimal portfolio allocation rule to market data.


Date: Thursday 15 December 2016

Speaker: Süleyman Başak, London Business School and CEPR

Time: 17:00-18:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Süleyman Başak here

Belief Dispersion in the Stock Market

 

We develop a dynamic model of belief dispersion which simultaneously explains the empirical regularities in a stock price, its mean return, volatility, and trading volume. Our model with a continuum of (possibly Bayesian) investors differing in beliefs is tractable and delivers exact closed-form solutions. Our model has the following implications. We find that the stock price is convex in cash-flow news, and it increases in belief dispersion while its mean return decreases when the view on the stock is optimistic, and vice versa when pessimistic. We also show that the presence of belief dispersion leads to a higher stock volatility, trading volume, and a positive relation between these two quantities. Furthermore, we demonstrate that the more familiar, otherwise identical, two-investor economies with heterogeneous beliefs do not necessarily generate our main results. Our quantitative analysis reveals that the effects of belief dispersion are economically significant and support the empirical evidence.

Speaker: Anes Dallagi, EDF France

Time: 18:00-19:00

Place: University College London - Physics Building A1/3, Gower Place


Find out more about Anes Dallagi here

UK energy market: the trilemma

 

The UK energy system is facing unprecedented changes due to changing market conditions and regulatory constraints. The operation of the market and the generation assets are strongly related to the technological characteristics of the generation mix and to the behaviours of the consumers that shapes the load. This lead to a delicate balance that need to be maintained between three main objectives:
- Security of supply
- Affordability
- Low carbon energy mix
How to achieve this balance? And what are the tools and the methodologies that are used in order to simulate the operation of the UK energy systems and determine at which condition of regulation and market framework this balance may be achieved?

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London Mathematical Finance Group: 2016-17