# 2015-16

*Summer Term 2015/16: Fridays 2-3pm A1.01*

**Week 2: 6th May 2016**

Professor Richard Stockbridge, University of Wisconsin-Milwaukee

Title: One-directional Impulse Control of Diﬀusions: Long-term Average Criterion.

Abstract: This talk examines a control problem when, in the absence of impulses, the process has continuous sample paths. Examples of such processes include the amount of water behind a dam, the amount of natural gas in a storage facility or (approximately) the value of a businesss cash account. The talk will be framed in terms of management of inventory. The inventory process is modelled by a one-dimensional stochastic diﬀerential equation on some interval in which the left boundary is attracting, so as to capture the eﬀect that demand tends to decrease the inventory level, and the right boundary is non-attracting so that, on their own, returns of inventory do not increase the level to its upper limit. Orders instantaneously increase the inventory level and incur both positive ﬁxed and level-dependent costs. The manager’s inﬂuence on the inventory is limited solely to ordering policies that increase the current level; he is not allowed to take action so as to reduce the inventory level. Thus this talk examines an impulse control problem in which the impulses only aﬀect the process by jumps in a single direction. Minimal conditions are provided on the model which imply that an optimal ordering policy exists in the class of (s,S) policies. Examination of the steady state behavior of (s,S) policies leads to a two-dimensional nonlinear optimization problem for which an optimizing pair establishes the levels for an optimal (s,S) policy within a large class of ordering policies. The results will be illustrated using the classical model of a drifted Brownian motion for the underlying diﬀusion process as well as this process with reﬂection at 0. In addition, a class of non-Markovian policies will be examined and an intuitive condition on the parameters yields the somewhat surprising result that each (s,S) policy incurs a larger cost than a corresponding non-Markovian policy. Thus no (s,S) policy is optimal.

This is joint work with K. Helmes and C. Zhu.

**Week 3: 13th May 2016**

Professor Aleksandar Mijatovic, King's College London

Title: A weak multilevel Monte Carlo scheme for multidimensional L\'evy-type processes.

Abstract: In this talk we describe a novel weak multilevel approximation scheme for time-changed L\'evy processes and L\'evy driven SDEs. The scheme is based on the state space discretisation of the driving L\'evy process and is particularly well suited if the multidimensional driver is given by a L\'evy copula.The multilevel version of the scheme is genuinely weak as it does not require strong convergence to control the level variances. The analysis of the level variances rests upon a new coupling between the approximating Markov chains of the consecutive levels, which is defined via a coupling of the corresponding Poisson Point Processes and is easy to simulate. This is joint work with D. Belomestny.

**Week 4: 20th May 2016**

Professor Nizar Touzi, Ecole Polytechnique

Title: Branching diffusion representation of semilinear PDEs

Abstract: We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod, Watanabe , and McKean, by allowing for polynomial nonlinearity in the pair $(u, Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to ''small maturity'' or ''small nonlinearity'' of the PDE. Our main ingredient is the automatic differentiation technique, based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free Central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.

**Week 7: 10th June 2016
**

Dr Ania Aksamit, University of Oxford

Title: Quantification of additional information in a robust framework.

Abstract: In robust approach, instead of choosing one model, one considers superhedging simultaneously under a family of models, or pathwise on the set of feasible trajectories. Usually in the literature the focus is on the natural filtration $\mathbb F$ of the price process. Here we extend that to a general filtration $\mathbb G$ including the natural filtration of the price process $\mathbb F\subset \mathbb G$. Two filtrations can model asymmetry of information on the market.

We consider the price process as a canonical process on some restriction of space of $\mathbb R^d$-valued continuous functions on $[0, T]$. Price process represents underlying stocks and continuously traded options. Beside that we allow static position in options from a given set with given prices.

One may look at the superhedging prices for an informed agent or at the market model prices induced by appropriate sets of martingale measures. Our main result is showing that the pricing--hedging duality holds for the informed agent for some class of payoffs, in a number of interesting cases.

This is joint work with Zhaoxu Hou and Jan Ob\l\'{o}j.

**Week 8: 17th June 2016
**

Professor Martin Keller-Ressel, TU Dresden

Title: Implied Volatility for Strict Local Martingale Models

Abstract: Several authors have proposed to model price bubbles in stock markets by specifying a strict local martingale for the risk-neutral stock price process. Such models are consistent with absence of arbitrage (in the NFLVR sense) while allowing fundamental prices to diverge from actual prices and thus modeling investors’ exuberance during the appearance of a bubble. We show that the strict local martingale property as well as the “distance to a true martingale” can be detected from the asymptotic behavior of implied option volatilities for large strikes, thus providing a model-free asymptotic test for the strict local martingale property of the underlying. This talk is based on joint work with Antoine Jacquier.

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*Autumn Term 2015/16: Fridays 2-3pm C1.06 (except week 9, see below)*

### Week 1: Friday 9th October 2015

Alet Roux (York) - *American options with gradual exercise in models with proportional transaction costs*

The standard assumption in the literature is that an American option can only be exercised instantly at an ordinary stopping time. In this talk I will discuss the case when the holder of an American option is allowed to exercise it gradually at a so-called mixed (or randomised) stopping time in a model with proportional transaction costs. The introduction of gradual exercise leads to tighter bounds on option prices, in particular in models with large transaction costs, and it turns the hedging problem for the buyer into a convex problem, which makes for easier construction of bid prices, optimal stopping times and optimal hedging strategies. I will also present dual representations for bid and ask prices.

See: http://maths.york.ac.uk/www/ar521

### Week 2: Friday 16th October 2015

Maria Claramunt (Barcelona) - Discrete Schur-constant models

This talk introduces a class of Schur-constant survival models, of dimension n, for arithmetic non-negative random variables. Such a model is defined through a univariate survival function that is shown to be n-monotone. Two general representations are obtained, by conditioning on the sum of the n variables or through a doubly mixed multinomial distribution. Several other properties including correlation measures are derived. Three processes in insurance theory are discussed for which the claim interarrival periods form a Schur-constant model. This talk is based on joint work with A. Castañer, C. Lefèvre and S. Loisel.

### Week 5: Friday 6th November 2015

Kris Glover (UTS Sydney, on sabbatical in UK) - The Optimal Time to Close an Open-Ended Mutual Fund

In this talk I will attempt to shed some light onto two intimately linked questions about empirically observed mutual fund behaviour: (1) Why do open-ended mutual funds decide to close their doors to new investors? and (2) Why do these funds underperform after closing? A theoretical model for the optimal closure of an open-ended mutual fund is developed in which the fund is subject to performance sensitive fund flows and a decreasing return-to-scale on its investment portfolio. Such funds are found to optimally close ‘too late’ from the perspective of the fund investors, since the optimal fund size that maximises the fund manager's expected fee income is larger than the size at which the fund’s ‘alpha’ is expected to becomes negative. In other words, the fund closes at a point where the decreasing returns-to-scale have already started to negatively affect fund performance; thus explaining the empirically observed underperformance after closure. Additional empirical predictions generated from the model will also be discussed.

See: https://sites.google.com/site/kristofferjglover/

### Week 6: Friday 13th November 2015

Sam Cohen (Oxford) - Control and Games with Ergodic BSDEs

When considering a variety of problems related to control, it is natural

to consider representing the value function of the control problem

through a backward stochastic differential equation (BSDE). In this

talk, we will consider how these equations can be adapted to consider

problems where the controller values the ergodic behaviour of a

controlled process, which may be in infinite dimensions, and may have

jumps. We shall also look at extensions of this theory, which allow us

to consider stochastic differential games, both zero-sum and

non-zero-sum, with a possibly large number of players.

### Week 9: **Wednesday** 2nd December 2015. C1.06.

Ilan Kramer (Warwick, Economics) - Robust Option Pricing: Hannan and Blackwell Meet Black and Scholes

Approachability and calibration are important concepts in a growing literature on robust optimization. While this literature focuses primarily on asymptotic performance, we provide a financial interpretation of these methods by demonstrating how the gradient strategies developed by Hannan and Blackwell to minimize asymptotic regret imply trading strategies that yield robust, arbitrage-based bounds for option prices. These bounds are both new and robust in that they do not depend on the continuity of the stock price process, complete markets, or an assumed pricing kernel. Rather, they depend only on the realized quadratic variation of the stock price process, which can be measured and, more importantly, hedged in financial markets using existing securities. We then argue that the Hannan–Blackwell strategy is path dependent and therefore suboptimal with a finite horizon. We solve for the optimal path-independent strategy, and compare the bounds achieved with Black-Scholes.

See: http://www2.warwick.ac.uk/fac/soc/economics/staff/ikremer/

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Spring Term 2015/16: Fridays 2-3pm C1.06 (except Week 1 in H0.58)

Week 2: 22nd January 2016

Chris Rogers (Statslab, Cambrdige)

Bermudan options by simulation

The aim of this study is to devise numerical methods for dealing with very

high-dimensional Bermudan-style derivatives. For such problems, we quickly

see that we can at best hope for price bounds, and we can only use a

simulation approach. We use the approach of Barraquand \& Martineau which

proposes that the reward process should be treated as if it were Markovian,

and then uses this to generate a stopping rule and hence a lower bound on

the price. Using the dual approach introduced by Rogers and Haugh \& Kogan,

this approximate Markov process leads us to hedging strategies, and upper

bounds on the price. The methodology is generic, and is illustrated on

eight examples of varying levels of difficulty. Run times are largely

insensitive to dimension.

Week 5: 12th February 2016

Winton Capital visit. No SF@W seminar.

Week 7: 26th February 2016

Sebastian Ebert (Tilburg)

Measuring Multivariate Risk Preferences

We measure risk preferences for decisions that involve more than a sin- gle, monetary attribute. According to theory, correlation aversion, cross- prudence and cross-temperance determine how risk preferences over two single attributes co-vary and interact. We obtain model-free measure- ments of these cross-risk attitudes in three economic domains, viz., time preferences, social preferences, and preferences over waiting time. This first systematic empirical exploration of multivariate risk preferences provides evidence for assumptions made in economic models on inequal- ity, labor, time preferences, saving, and insurance. We observe non-neutrality of cross-risk attitudes in all domains which questions the de- scriptive accuracy of economic models that assume that utility is addi- tively separable in its arguments.

See: https://sites.google.com/site/ebertecon/home